Optimal. Leaf size=70 \[ \frac {1}{12} \tan ^{-1}\left (\frac {\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \tan ^{-1}\left (\frac {x}{3}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {394} \[ \frac {1}{12} \tan ^{-1}\left (\frac {\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt {3}}+\frac {1}{12} \tan ^{-1}\left (\frac {x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 394
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx &=\frac {1}{12} \tan ^{-1}\left (\frac {x}{3}\right )+\frac {1}{12} \tan ^{-1}\left (\frac {\left (1-\sqrt [3]{1+x^2}\right )^2}{3 x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+x^2}\right )}{x}\right )}{4 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 124, normalized size = 1.77 \[ -\frac {27 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,-\frac {x^2}{9}\right )}{\sqrt [3]{x^2+1} \left (x^2+9\right ) \left (2 x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-x^2,-\frac {x^2}{9}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-x^2,-\frac {x^2}{9}\right )\right )-27 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,-\frac {x^2}{9}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 5.41, size = 1395, normalized size = 19.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 10.59, size = 512, normalized size = 7.31 \[ -\RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right ) \ln \left (\frac {-x^{2}+48 \left (x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right )+96 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right )+2 \left (x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 x \RootOf \left (\textit {\_Z}^{2}+1\right )-6 \left (x^{2}+1\right )^{\frac {2}{3}}-6 \left (x^{2}+1\right )^{\frac {1}{3}}+3}{x^{2}+9}\right )-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-x^{2}+48 \left (x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right )+96 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right )+2 \left (x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 x \RootOf \left (\textit {\_Z}^{2}+1\right )-6 \left (x^{2}+1\right )^{\frac {2}{3}}-6 \left (x^{2}+1\right )^{\frac {1}{3}}+3}{x^{2}+9}\right )}{12}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-12 x^{2} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )+576 \left (x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+1\right )-1152 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+1\right )+24 \left (x^{2}+1\right )^{\frac {1}{3}} x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )^{2}-48 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )^{2}+x^{2}+96 x \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right )+4 x \RootOf \left (\textit {\_Z}^{2}+1\right )+72 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )+36 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )-6 \left (x^{2}+1\right )^{\frac {2}{3}}-3}{x^{2}+9}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x^2+1\right )}^{1/3}\,\left (x^2+9\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [3]{x^{2} + 1} \left (x^{2} + 9\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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