Optimal. Leaf size=190 \[ \frac {1}{12} \text {ArcTan}\left (\frac {x}{1+2 \sqrt [3]{1+x^2}}\right )+\frac {i \text {ArcTan}\left (\frac {-\frac {1}{\sqrt {3}}-\frac {i x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt {3}}-\frac {i \text {ArcTan}\left (\frac {-\frac {1}{\sqrt {3}}+\frac {i x}{\sqrt {3}}+\frac {\sqrt [3]{1+x^2}}{\sqrt {3}}}{\sqrt [3]{1+x^2}}\right )}{8 \sqrt {3}}-\frac {1}{24} i \tanh ^{-1}\left (\frac {2 i x-2 i x \sqrt [3]{1+x^2}}{-1+x^2+2 \sqrt [3]{1+x^2}-4 \left (1+x^2\right )^{2/3}}\right ) \]
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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 = {403}
\begin {gather*} \frac {1}{12} \text {ArcTan}\left (\frac {\left (1-\sqrt [3]{x^2+1}\right )^2}{3 x}\right )+\frac {1}{12} \text {ArcTan}\left (\frac {x}{3}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{x^2+1}\right )}{x}\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 403
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] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 4.27, 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]{1+x^2} \left (9+x^2\right ) \left (-27 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-x^2,-\frac {x^2}{9}\right )+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 )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 6.17, size = 624, normalized size = 3.28
method | result | size |
trager | \(144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} \ln \left (-\frac {497664 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -995328 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -6912 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +20736 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}-864 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+6 \left (x^{2}+1\right )^{\frac {2}{3}}-432 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-96 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x -x^{2}+3}{x^{2}+9}\right )-\RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {497664 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -995328 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x -6912 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +20736 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x +144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}-864 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}+6 \left (x^{2}+1\right )^{\frac {2}{3}}-432 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-96 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x -x^{2}+3}{x^{2}+9}\right )-\RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-82944 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x +165888 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{5} x +1728 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x -2304 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{3} x -24 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} x^{2}+144 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2} \left (x^{2}+1\right )^{\frac {1}{3}}-8 \left (x^{2}+1\right )^{\frac {1}{3}} \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right ) x +\left (x^{2}+1\right )^{\frac {2}{3}}+72 \RootOf \left (20736 \textit {\_Z}^{4}-144 \textit {\_Z}^{2}+1\right )^{2}-\left (x^{2}+1\right )^{\frac {1}{3}}}{x^{2}+9}\right )\) | \(624\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1395 vs.
\(2 (137) = 274\).
time = 1.32, size = 1395, normalized size = 7.34 \begin {gather*} \text {Too large to display} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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