Example: Symbolic Integral Functions Using the Keyword fully

Symbolics > Working with Symbolics > Calculus > Example: Symbolic Integral Functions Using the Keyword fully

Use the keyword fully to receive a full detailed solution to a symbolic function. Using fully, you can receive solutions for different values of the function’s coefficients. In the following example: a, b, and c. To narrow the solution to only one value of the function’s coefficients, use the assume keyword.

1. Evaluate a symbolic integral.

2. Evaluate the full conditioned solution of the integral, using the keyword fully.

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                    <real>2</real>
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            <id xml:space="preserve">c</id>
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      <command>
        <id xml:space="preserve">fully</id>
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                <real>0</real>
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                    <div />
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                            <pow />
                            <id xml:space="preserve" labels="VARIABLE">b</id>
                            <real>2</real>
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                                  <real>2</real>
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                                    <nthRoot />
                                    <placeholder />
                                    <id xml:space="preserve" labels="VARIABLE">a</id>
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                                <apply>
                                  <nthRoot />
                                  <placeholder />
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                                    <plus />
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                                      <mult />
                                      <id xml:space="preserve" labels="VARIABLE">a</id>
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                                        <pow />
                                        <id xml:space="preserve" labels="VARIABLE">t</id>
                                        <real>2</real>
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                                    <real>2</real>
                                    <id xml:space="preserve" labels="VARIABLE">a</id>
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                                  <id xml:space="preserve" labels="VARIABLE">t</id>
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                                <id xml:space="preserve" labels="VARIABLE">b</id>
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                      <mult />
                      <real>8</real>
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                        <pow />
                        <id xml:space="preserve" labels="VARIABLE">a</id>
                        <apply>
                          <div />
                          <real>3</real>
                          <real>2</real>
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                    <plus />
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                      <div />
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                                <real>2</real>
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                              <id xml:space="preserve" labels="VARIABLE">t</id>
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                            <id xml:space="preserve" labels="VARIABLE">b</id>
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                              <mult />
                              <id xml:space="preserve" labels="VARIABLE">a</id>
                              <apply>
                                <pow />
                                <id xml:space="preserve" labels="VARIABLE">t</id>
                                <real>2</real>
                              </apply>
                            </apply>
                            <apply>
                              <mult />
                              <id xml:space="preserve" labels="VARIABLE">b</id>
                              <id xml:space="preserve" labels="VARIABLE">t</id>
                            </apply>
                          </apply>
                        </apply>
                      </apply>
                      <apply>
                        <mult />
                        <real>4</real>
                        <id xml:space="preserve" labels="VARIABLE">a</id>
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                    </apply>
                    <apply>
                      <mult />
                      <id xml:space="preserve" labels="VARIABLE">c</id>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
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                    <real>0</real>
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                    <notEqual />
                    <id xml:space="preserve" labels="VARIABLE">b</id>
                    <real>0</real>
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              </test>
              <then>
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                    <div />
                    <apply>
                      <plus />
                      <apply>
                        <mult />
                        <real>3</real>
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                          <mult />
                          <id xml:space="preserve" labels="VARIABLE">c</id>
                          <id xml:space="preserve" labels="VARIABLE">t</id>
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                          <real>2</real>
                          <id xml:space="preserve" labels="VARIABLE">t</id>
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                          <nthRoot />
                          <placeholder />
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                            <id xml:space="preserve" labels="VARIABLE">t</id>
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                        </apply>
                      </apply>
                    </apply>
                    <real>3</real>
                  </apply>
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              </then>
            </elseif>
            <elseif>
              <test>
                <apply>
                  <and />
                  <apply>
                    <equal />
                    <id xml:space="preserve" labels="VARIABLE">a</id>
                    <real>0</real>
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                  <apply>
                    <equal />
                    <id xml:space="preserve" labels="VARIABLE">b</id>
                    <real>0</real>
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                    <id xml:space="preserve" labels="VARIABLE">t</id>
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3. Evaluate the solution only for the cases when a≠0.

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            <real>0</real>
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                      <real>2</real>
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                              <id xml:space="preserve" labels="VARIABLE">a</id>
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                                  <real>2</real>
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                              <real>2</real>
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                            <id xml:space="preserve" labels="VARIABLE">t</id>
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                          <id xml:space="preserve" labels="VARIABLE">b</id>
                        </apply>
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                    </apply>
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                <mult />
                <real>8</real>
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                  <id xml:space="preserve" labels="VARIABLE">a</id>
                  <apply>
                    <div />
                    <real>3</real>
                    <real>2</real>
                  </apply>
                </apply>
              </apply>
            </apply>
            <apply>
              <div />
              <apply>
                <mult />
                <parens>
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                    <plus />
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                      <mult />
                      <apply>
                        <mult />
                        <real>2</real>
                        <id xml:space="preserve" labels="VARIABLE">a</id>
                      </apply>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
                    </apply>
                    <id xml:space="preserve" labels="VARIABLE">b</id>
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                  <nthRoot />
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                      <id xml:space="preserve" labels="VARIABLE">a</id>
                      <apply>
                        <pow />
                        <id xml:space="preserve" labels="VARIABLE">t</id>
                        <real>2</real>
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                      <mult />
                      <id xml:space="preserve" labels="VARIABLE">b</id>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
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              </apply>
              <apply>
                <mult />
                <real>4</real>
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              </apply>
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Linear Power Function

1. Evaluate a symbolic integral of a linear power function.

2. Evaluate the full conditioned solution of the integral, using the keyword fully.

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                <id xml:space="preserve">c</id>
              </apply>
            </parens>
            <id xml:space="preserve">n</id>
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        </lambda>
        <lowerBound>
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                <real>-1</real>
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                  <div />
                  <apply>
                    <pow />
                    <parens>
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                        <plus />
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                          <id xml:space="preserve" labels="VARIABLE">t</id>
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                        <id xml:space="preserve" labels="VARIABLE">c</id>
                      </apply>
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                    <apply>
                      <plus />
                      <id xml:space="preserve" labels="VARIABLE">n</id>
                      <real>1</real>
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                    <plus />
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                      <id xml:space="preserve" labels="VARIABLE">b</id>
                      <id xml:space="preserve" labels="VARIABLE">n</id>
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                    <id xml:space="preserve" labels="VARIABLE">b</id>
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                  <id xml:space="preserve" labels="VARIABLE">n</id>
                  <real>-1</real>
                </apply>
              </test>
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                    <div />
                    <apply>
                      <id xml:space="preserve" labels="FUNCTION">ln</id>
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                        <plus />
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                          <id xml:space="preserve" labels="VARIABLE">b</id>
                          <id xml:space="preserve" labels="VARIABLE">t</id>
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                    <id xml:space="preserve" labels="VARIABLE">b</id>
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Elliptic Integral Functions

The complete elliptic integral function of the first kind:

1. Evaluate a symbolic integral of the complete elliptic integral function of the first kind.

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          <div />
          <apply>
            <plus />
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                <real>2</real>
                <id xml:space="preserve" labels="VARIABLE">A</id>
              </apply>
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                <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
                <apply>
                  <plus />
                  <apply>
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                    <id xml:space="preserve" labels="VARIABLE">c</id>
                    <id xml:space="preserve" labels="VARIABLE">t</id>
                  </apply>
                  <id xml:space="preserve" labels="VARIABLE">d</id>
                </apply>
              </apply>
            </apply>
            <apply>
              <mult />
              <parens>
                <apply>
                  <plus />
                  <apply>
                    <mult />
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <real>2</real>
                        <id xml:space="preserve" labels="VARIABLE">A</id>
                      </apply>
                      <id xml:space="preserve" labels="VARIABLE">c</id>
                    </apply>
                    <id xml:space="preserve" labels="VARIABLE">t</id>
                  </apply>
                  <parens>
                    <apply>
                      <minus />
                      <apply>
                        <mult />
                        <apply>
                          <mult />
                          <real>2</real>
                          <id xml:space="preserve" labels="VARIABLE">A</id>
                        </apply>
                        <id xml:space="preserve" labels="VARIABLE">d</id>
                      </apply>
                      <apply>
                        <mult />
                        <real>2</real>
                        <id xml:space="preserve" labels="VARIABLE">A</id>
                      </apply>
                    </apply>
                  </parens>
                </apply>
              </parens>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
                <apply>
                  <plus />
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">c</id>
                    <id xml:space="preserve" labels="VARIABLE">t</id>
                  </apply>
                  <id xml:space="preserve" labels="VARIABLE">d</id>
                </apply>
              </apply>
            </apply>
          </apply>
          <id xml:space="preserve" labels="VARIABLE">c</id>
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2. Evaluate the full conditioned solution of the complete elliptic integral function of the first kind, using the keyword fully.

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                  <div />
                  <apply>
                    <plus />
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <real>2</real>
                        <id xml:space="preserve" labels="VARIABLE">A</id>
                      </apply>
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                        <id xml:space="preserve" labels="FUNCTION">ellipticE</id>
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                            <id xml:space="preserve" labels="VARIABLE">c</id>
                            <id xml:space="preserve" labels="VARIABLE">t</id>
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                          <id xml:space="preserve" labels="VARIABLE">d</id>
                        </apply>
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                    </apply>
                    <apply>
                      <mult />
                      <parens>
                        <apply>
                          <plus />
                          <apply>
                            <mult />
                            <apply>
                              <mult />
                              <apply>
                                <mult />
                                <real>2</real>
                                <id xml:space="preserve" labels="VARIABLE">A</id>
                              </apply>
                              <id xml:space="preserve" labels="VARIABLE">c</id>
                            </apply>
                            <id xml:space="preserve" labels="VARIABLE">t</id>
                          </apply>
                          <parens>
                            <apply>
                              <minus />
                              <apply>
                                <mult />
                                <apply>
                                  <mult />
                                  <real>2</real>
                                  <id xml:space="preserve" labels="VARIABLE">A</id>
                                </apply>
                                <id xml:space="preserve" labels="VARIABLE">d</id>
                              </apply>
                              <apply>
                                <mult />
                                <real>2</real>
                                <id xml:space="preserve" labels="VARIABLE">A</id>
                              </apply>
                            </apply>
                          </parens>
                        </apply>
                      </parens>
                      <apply>
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                        <apply>
                          <plus />
                          <apply>
                            <mult />
                            <id xml:space="preserve" labels="VARIABLE">c</id>
                            <id xml:space="preserve" labels="VARIABLE">t</id>
                          </apply>
                          <id xml:space="preserve" labels="VARIABLE">d</id>
                        </apply>
                      </apply>
                    </apply>
                  </apply>
                  <id xml:space="preserve" labels="VARIABLE">c</id>
                </apply>
              </program>
            </then>
            <elseif>
              <test>
                <apply>
                  <equal />
                  <id xml:space="preserve" labels="VARIABLE">c</id>
                  <real>0</real>
                </apply>
              </test>
              <then>
                <program>
                  <apply>
                    <mult />
                    <apply>
                      <mult />
                      <id xml:space="preserve" labels="VARIABLE">A</id>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
                    </apply>
                    <apply>
                      <id xml:space="preserve" labels="FUNCTION">ellipticK</id>
                      <id xml:space="preserve" labels="VARIABLE">d</id>
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                  </apply>
                </program>
              </then>
            </elseif>
          </if>
        </program>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

The incomplete elliptic integral function of the first kind:

1. Evaluate a symbolic integral of the incomplete elliptic integral function of the first kind.

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                  <id xml:space="preserve">a</id>
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              <apply>
                <nthRoot />
                <placeholder />
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                  <real>1</real>
                  <apply>
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                        <id xml:space="preserve">c</id>
                        <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">t</id>
                      </apply>
                    </parens>
                    <real>2</real>
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                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
        <lowerBound>
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        <upperBound>
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        </upperBound>
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      <command>
        <placeholder />
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      <symResult>
        <apply>
          <div />
          <apply>
            <minus />
            <apply>
              <mult />
              <id xml:space="preserve" labels="VARIABLE">A</id>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">atan</id>
                <apply>
                  <div />
                  <apply>
                    <mult />
                    <apply>
                      <mult />
                      <id xml:space="preserve" labels="VARIABLE">c</id>
                      <id xml:space="preserve" labels="VARIABLE">t</id>
                    </apply>
                    <apply>
                      <nthRoot />
                      <placeholder />
                      <id xml:space="preserve" labels="VARIABLE">a</id>
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                  </apply>
                  <apply>
                    <nthRoot />
                    <placeholder />
                    <apply>
                      <plus />
                      <apply>
                        <neg />
                        <parens>
                          <apply>
                            <mult />
                            <apply>
                              <mult />
                              <id xml:space="preserve" labels="VARIABLE">a</id>
                              <apply>
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                                <id xml:space="preserve" labels="VARIABLE">c</id>
                                <real>2</real>
                              </apply>
                            </apply>
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="VARIABLE">t</id>
                              <real>2</real>
                            </apply>
                          </apply>
                        </parens>
                      </apply>
                      <real>1</real>
                    </apply>
                  </apply>
                </apply>
              </apply>
            </apply>
            <apply>
              <mult />
              <apply>
                <mult />
                <apply>
                  <mult />
                  <id xml:space="preserve" labels="VARIABLE">A</id>
                  <apply>
                    <nthRoot />
                    <placeholder />
                    <apply>
                      <plus />
                      <apply>
                        <neg />
                        <parens>
                          <apply>
                            <mult />
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="VARIABLE">c</id>
                              <real>2</real>
                            </apply>
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="VARIABLE">t</id>
                              <real>2</real>
                            </apply>
                          </apply>
                        </parens>
                      </apply>
                      <real>1</real>
                    </apply>
                  </apply>
                </apply>
                <apply>
                  <nthRoot />
                  <placeholder />
                  <id xml:space="preserve" labels="VARIABLE">a</id>
                </apply>
              </apply>
              <apply>
                <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
                <sequence>
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">c</id>
                    <id xml:space="preserve" labels="VARIABLE">t</id>
                  </apply>
                  <id xml:space="preserve" labels="VARIABLE">a</id>
                </sequence>
              </apply>
            </apply>
          </apply>
          <apply>
            <mult />
            <id xml:space="preserve" labels="VARIABLE">c</id>
            <apply>
              <nthRoot />
              <placeholder />
              <id xml:space="preserve" labels="VARIABLE">a</id>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
  </math>
</region>

2. Evaluate the full conditioned solution of the incomplete elliptic integral function of the first kind, using the keyword fully.

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              <apply>
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                  <real>1</real>
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                        <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">t</id>
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                    <real>2</real>
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                </apply>
              </apply>
            </apply>
          </apply>
        </lambda>
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          <placeholder />
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        <upperBound>
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      </apply>
      <command>
        <id xml:space="preserve">fully</id>
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      <symResult>
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                  <real>0</real>
                </apply>
                <apply>
                  <notEqual />
                  <id xml:space="preserve" labels="VARIABLE">a</id>
                  <real>0</real>
                </apply>
              </apply>
            </test>
            <then>
              <program>
                <apply>
                  <div />
                  <apply>
                    <minus />
                    <apply>
                      <mult />
                      <id xml:space="preserve" labels="VARIABLE">A</id>
                      <apply>
                        <id xml:space="preserve" labels="FUNCTION">atan</id>
                        <apply>
                          <div />
                          <apply>
                            <mult />
                            <apply>
                              <mult />
                              <id xml:space="preserve" labels="VARIABLE">c</id>
                              <id xml:space="preserve" labels="VARIABLE">t</id>
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                            <apply>
                              <nthRoot />
                              <placeholder />
                              <id xml:space="preserve" labels="VARIABLE">a</id>
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                          <apply>
                            <nthRoot />
                            <placeholder />
                            <apply>
                              <plus />
                              <apply>
                                <neg />
                                <parens>
                                  <apply>
                                    <mult />
                                    <apply>
                                      <mult />
                                      <id xml:space="preserve" labels="VARIABLE">a</id>
                                      <apply>
                                        <pow />
                                        <id xml:space="preserve" labels="VARIABLE">c</id>
                                        <real>2</real>
                                      </apply>
                                    </apply>
                                    <apply>
                                      <pow />
                                      <id xml:space="preserve" labels="VARIABLE">t</id>
                                      <real>2</real>
                                    </apply>
                                  </apply>
                                </parens>
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                              <real>1</real>
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                      </apply>
                    </apply>
                    <apply>
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                      <apply>
                        <mult />
                        <apply>
                          <mult />
                          <id xml:space="preserve" labels="VARIABLE">A</id>
                          <apply>
                            <nthRoot />
                            <placeholder />
                            <apply>
                              <plus />
                              <apply>
                                <neg />
                                <parens>
                                  <apply>
                                    <mult />
                                    <apply>
                                      <pow />
                                      <id xml:space="preserve" labels="VARIABLE">c</id>
                                      <real>2</real>
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                                    <apply>
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                                      <id xml:space="preserve" labels="VARIABLE">t</id>
                                      <real>2</real>
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                              <real>1</real>
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                          </apply>
                        </apply>
                        <apply>
                          <nthRoot />
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                          <id xml:space="preserve" labels="VARIABLE">a</id>
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                  </apply>
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">c</id>
                    <apply>
                      <nthRoot />
                      <placeholder />
                      <id xml:space="preserve" labels="VARIABLE">a</id>
                    </apply>
                  </apply>
                </apply>
              </program>
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            <elseif>
              <test>
                <apply>
                  <and />
                  <apply>
                    <notEqual />
                    <id xml:space="preserve" labels="VARIABLE">c</id>
                    <real>0</real>
                  </apply>
                  <apply>
                    <equal />
                    <id xml:space="preserve" labels="VARIABLE">a</id>
                    <real>0</real>
                  </apply>
                </apply>
              </test>
              <then>
                <program>
                  <apply>
                    <div />
                    <apply>
                      <plus />
                      <apply>
                        <neg />
                        <parens>
                          <apply>
                            <mult />
                            <apply>
                              <mult />
                              <id xml:space="preserve" labels="VARIABLE">A</id>
                              <apply>
                                <nthRoot />
                                <placeholder />
                                <apply>
                                  <plus />
                                  <apply>
                                    <neg />
                                    <parens>
                                      <apply>
                                        <mult />
                                        <apply>
                                          <pow />
                                          <id xml:space="preserve" labels="VARIABLE">c</id>
                                          <real>2</real>
                                        </apply>
                                        <apply>
                                          <pow />
                                          <id xml:space="preserve" labels="VARIABLE">t</id>
                                          <real>2</real>
                                        </apply>
                                      </apply>
                                    </parens>
                                  </apply>
                                  <real>1</real>
                                </apply>
                              </apply>
                            </apply>
                            <apply>
                              <id xml:space="preserve" labels="FUNCTION">ellipticF</id>
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                                <apply>
                                  <mult />
                                  <id xml:space="preserve" labels="VARIABLE">c</id>
                                  <id xml:space="preserve" labels="VARIABLE">t</id>
                                </apply>
                                <real>0</real>
                              </sequence>
                            </apply>
                          </apply>
                        </parens>
                      </apply>
                      <apply>
                        <mult />
                        <apply>
                          <mult />
                          <id xml:space="preserve" labels="VARIABLE">A</id>
                          <id xml:space="preserve" labels="VARIABLE">c</id>
                        </apply>
                        <id xml:space="preserve" labels="VARIABLE">t</id>
                      </apply>
                    </apply>
                    <id xml:space="preserve" labels="VARIABLE">c</id>
                  </apply>
                </program>
              </then>
            </elseif>
            <elseif>
              <test>
                <apply>
                  <equal />
                  <id xml:space="preserve" labels="VARIABLE">c</id>
                  <real>0</real>
                </apply>
              </test>
              <then>
                <program>
                  <real>0</real>
                </program>
              </then>
            </elseif>
          </if>
        </program>
      </symResult>
    </symEval>
    <resultFormat>
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    </resultFormat>
  </math>
</region>

Trigonometric Functions

The cos function:

1. Evaluate a symbolic integral of the cos function.

2. Evaluate the full conditioned solution of the cos function, using the keyword fully.

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                  <id xml:space="preserve">c</id>
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">t</id>
                </apply>
                <id xml:space="preserve">d</id>
              </apply>
            </apply>
            <real>3</real>
          </apply>
        </lambda>
        <lowerBound>
          <placeholder />
        </lowerBound>
        <upperBound>
          <placeholder />
        </upperBound>
      </apply>
      <command>
        <id xml:space="preserve">fully</id>
      </command>
      <symResult>
        <program>
          <if>
            <test>
              <apply>
                <notEqual />
                <id xml:space="preserve" labels="VARIABLE">c</id>
                <real>0</real>
              </apply>
            </test>
            <then>
              <program>
                <apply>
                  <div />
                  <apply>
                    <mult />
                    <apply>
                      <id xml:space="preserve" labels="FUNCTION">sin</id>
                      <apply>
                        <plus />
                        <apply>
                          <mult />
                          <id xml:space="preserve" labels="VARIABLE">c</id>
                          <id xml:space="preserve" labels="VARIABLE">t</id>
                        </apply>
                        <id xml:space="preserve" labels="VARIABLE">d</id>
                      </apply>
                    </apply>
                    <parens>
                      <apply>
                        <plus />
                        <apply>
                          <pow />
                          <apply>
                            <id xml:space="preserve" labels="FUNCTION">cos</id>
                            <apply>
                              <plus />
                              <apply>
                                <mult />
                                <id xml:space="preserve" labels="VARIABLE">c</id>
                                <id xml:space="preserve" labels="VARIABLE">t</id>
                              </apply>
                              <id xml:space="preserve" labels="VARIABLE">d</id>
                            </apply>
                          </apply>
                          <real>2</real>
                        </apply>
                        <real>2</real>
                      </apply>
                    </parens>
                  </apply>
                  <apply>
                    <mult />
                    <real>3</real>
                    <id xml:space="preserve" labels="VARIABLE">c</id>
                  </apply>
                </apply>
              </program>
            </then>
            <elseif>
              <test>
                <apply>
                  <equal />
                  <id xml:space="preserve" labels="VARIABLE">c</id>
                  <real>0</real>
                </apply>
              </test>
              <then>
                <program>
                  <apply>
                    <mult />
                    <id xml:space="preserve" labels="VARIABLE">t</id>
                    <apply>
                      <pow />
                      <apply>
                        <id xml:space="preserve" labels="FUNCTION">cos</id>
                        <id xml:space="preserve" labels="VARIABLE">d</id>
                      </apply>
                      <real>3</real>
                    </apply>
                  </apply>
                </program>
              </then>
            </elseif>
          </if>
        </program>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
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  </math>
</region>

The sin function:

1. Evaluate a symbolic integral of the sin function.

2. Evaluate the full conditioned solution of the sin function, using the keyword fully.

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              <id xml:space="preserve">n</id>
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      <command>
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      </command>
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                <id xml:space="preserve" labels="VARIABLE">a</id>
                <real>0</real>
              </apply>
            </test>
            <then>
              <program>
                <apply>
                  <div />
                  <apply>
                    <minus />
                    <apply>
                      <neg />
                      <parens>
                        <apply>
                          <mult />
                          <apply>
                            <mult />
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="CONSTANT">e</id>
                              <apply>
                                <neg />
                                <parens>
                                  <apply>
                                    <mult />
                                    <id xml:space="preserve" labels="VARIABLE">b</id>
                                    <imag symbol="i">1</imag>
                                  </apply>
                                </parens>
                              </apply>
                            </apply>
                            <apply>
                              <pow />
                              <parens>
                                <apply>
                                  <mult />
                                  <id xml:space="preserve" labels="VARIABLE">a</id>
                                  <imag symbol="i">1</imag>
                                </apply>
                              </parens>
                              <id xml:space="preserve" labels="VARIABLE">n</id>
                            </apply>
                          </apply>
                          <apply>
                            <id xml:space="preserve" labels="FUNCTION">Γ</id>
                            <sequence>
                              <apply>
                                <plus />
                                <apply>
                                  <neg />
                                  <id xml:space="preserve" labels="VARIABLE">n</id>
                                </apply>
                                <real>1</real>
                              </apply>
                              <apply>
                                <mult />
                                <apply>
                                  <mult />
                                  <id xml:space="preserve" labels="VARIABLE">a</id>
                                  <id xml:space="preserve" labels="VARIABLE">x</id>
                                </apply>
                                <imag symbol="i">1</imag>
                              </apply>
                            </sequence>
                          </apply>
                        </apply>
                      </parens>
                    </apply>
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <apply>
                          <pow />
                          <id xml:space="preserve" labels="CONSTANT">e</id>
                          <apply>
                            <mult />
                            <id xml:space="preserve" labels="VARIABLE">b</id>
                            <imag symbol="i">1</imag>
                          </apply>
                        </apply>
                        <apply>
                          <pow />
                          <parens>
                            <apply>
                              <neg />
                              <parens>
                                <apply>
                                  <mult />
                                  <id xml:space="preserve" labels="VARIABLE">a</id>
                                  <imag symbol="i">1</imag>
                                </apply>
                              </parens>
                            </apply>
                          </parens>
                          <id xml:space="preserve" labels="VARIABLE">n</id>
                        </apply>
                      </apply>
                      <apply>
                        <id xml:space="preserve" labels="FUNCTION">Γ</id>
                        <sequence>
                          <apply>
                            <plus />
                            <apply>
                              <neg />
                              <id xml:space="preserve" labels="VARIABLE">n</id>
                            </apply>
                            <real>1</real>
                          </apply>
                          <apply>
                            <neg />
                            <parens>
                              <apply>
                                <mult />
                                <apply>
                                  <mult />
                                  <id xml:space="preserve" labels="VARIABLE">a</id>
                                  <id xml:space="preserve" labels="VARIABLE">x</id>
                                </apply>
                                <imag symbol="i">1</imag>
                              </apply>
                            </parens>
                          </apply>
                        </sequence>
                      </apply>
                    </apply>
                  </apply>
                  <apply>
                    <mult />
                    <real>2</real>
                    <id xml:space="preserve" labels="VARIABLE">a</id>
                  </apply>
                </apply>
              </program>
            </then>
            <elseif>
              <test>
                <apply>
                  <equal />
                  <id xml:space="preserve" labels="VARIABLE">a</id>
                  <real>0</real>
                </apply>
              </test>
              <then>
                <program>
                  <apply>
                    <div />
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <apply>
                          <mult />
                          <id xml:space="preserve" labels="VARIABLE">x</id>
                          <imag symbol="i">1</imag>
                        </apply>
                        <parens>
                          <apply>
                            <minus />
                            <apply>
                              <pow />
                              <id xml:space="preserve" labels="CONSTANT">e</id>
                              <apply>
                                <mult />
                                <apply>
                                  <mult />
                                  <real>2</real>
                                  <id xml:space="preserve" labels="VARIABLE">b</id>
                                </apply>
                                <imag symbol="i">1</imag>
                              </apply>
                            </apply>
                            <real>1</real>
                          </apply>
                        </parens>
                      </apply>
                      <apply>
                        <pow />
                        <id xml:space="preserve" labels="CONSTANT">e</id>
                        <apply>
                          <neg />
                          <parens>
                            <apply>
                              <mult />
                              <id xml:space="preserve" labels="VARIABLE">b</id>
                              <imag symbol="i">1</imag>
                            </apply>
                          </parens>
                        </apply>
                      </apply>
                    </apply>
                    <apply>
                      <mult />
                      <apply>
                        <mult />
                        <real>2</real>
                        <parens>
                          <apply>
                            <minus />
                            <id xml:space="preserve" labels="VARIABLE">n</id>
                            <real>1</real>
                          </apply>
                        </parens>
                      </apply>
                      <apply>
                        <pow />
                        <id xml:space="preserve" labels="VARIABLE">x</id>
                        <id xml:space="preserve" labels="VARIABLE">n</id>
                      </apply>
                    </apply>
                  </apply>
                </program>
              </then>
            </elseif>
          </if>
        </program>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
    <formattingOverride BackgroundColor="#00FFFFFF" />
  </math>
</region>