Optimal. Leaf size=190 \[ \frac {1}{12} \tan ^{-1}\left (\frac {x}{2 \sqrt [3]{x^2+1}+1}\right )+\frac {i \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}-\frac {i x}{\sqrt {3}}-\frac {1}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right )}{8 \sqrt {3}}-\frac {i \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+1}}{\sqrt {3}}+\frac {i x}{\sqrt {3}}-\frac {1}{\sqrt {3}}}{\sqrt [3]{x^2+1}}\right )}{8 \sqrt {3}}-\frac {1}{24} i \tanh ^{-1}\left (\frac {2 i x-2 i x \sqrt [3]{x^2+1}}{x^2-4 \left (x^2+1\right )^{2/3}+2 \sqrt [3]{x^2+1}-1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 70, normalized size of antiderivative = 0.37, 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} \begin {gather*} \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 ) \end {gather*}
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 = 0.09, size = 124, normalized size = 0.65 \begin {gather*} -\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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1+x^2} \left (9+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.75, size = 1395, normalized size = 7.34
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 \begin {gather*} \int \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 10.57, size = 512, normalized size = 2.69 \begin {gather*} \frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {24 \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +576 \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}} x -48 \RootOf \left (\textit {\_Z}^{2}+1\right )^{2} \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x -1152 \RootOf \left (\textit {\_Z}^{2}+1\right ) \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right )^{2} x -12 \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+72 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )+36 \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -6 \left (x^{2}+1\right )^{\frac {2}{3}}+96 \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x +x^{2}-3}{x^{2}+9}\right )}{12}-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +48 \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -6 \left (x^{2}+1\right )^{\frac {2}{3}}+96 \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x -x^{2}-6 \left (x^{2}+1\right )^{\frac {1}{3}}+3}{x^{2}+9}\right ) \RootOf \left (\textit {\_Z}^{2}+1\right )}{12}-\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +48 \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -6 \left (x^{2}+1\right )^{\frac {2}{3}}+96 \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) x -x^{2}-6 \left (x^{2}+1\right )^{\frac {1}{3}}+3}{x^{2}+9}\right ) \RootOf \left (12 \textit {\_Z} \RootOf \left (\textit {\_Z}^{2}+1\right )+144 \textit {\_Z}^{2}-1\right ) \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (x^{2} + 9\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\left (x^2+1\right )}^{1/3}\,\left (x^2+9\right )} \,d x \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt [3]{x^{2} + 1} \left (x^{2} + 9\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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