Optimal. Leaf size=173 \[ -\frac {3 e^{-3 a/2} \log \left (e^a x^2-\sqrt {2} e^{a/2} x+1\right )}{4 \sqrt {2}}+\frac {3 e^{-3 a/2} \log \left (e^a x^2+\sqrt {2} e^{a/2} x+1\right )}{4 \sqrt {2}}+\frac {x^3}{e^{2 a} x^4+1}+\frac {3 e^{-3 a/2} \tan ^{-1}\left (1-\sqrt {2} e^{a/2} x\right )}{2 \sqrt {2}}-\frac {3 e^{-3 a/2} \tan ^{-1}\left (\sqrt {2} e^{a/2} x+1\right )}{2 \sqrt {2}}+\frac {x^3}{3} \]
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Rubi [F] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^2 \tanh ^2(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \tanh ^2(a+2 \log (x)) \, dx &=\int x^2 \tanh ^2(a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.71, size = 174, normalized size = 1.01 \[ \frac {1}{12} \left (\frac {12 x^3}{e^{2 a} x^4+1}+9 (-1)^{3/4} e^{-3 a/2} \log \left (\sqrt [4]{-1} e^{-3 a/2}-e^{-a} x\right )+9 \sqrt [4]{-1} e^{-3 a/2} \log \left ((-1)^{3/4} e^{-3 a/2}-e^{-a} x\right )-9 (-1)^{3/4} e^{-3 a/2} \log \left (e^{-a} x+\sqrt [4]{-1} e^{-3 a/2}\right )-9 \sqrt [4]{-1} e^{-3 a/2} \log \left (e^{-a} x+(-1)^{3/4} e^{-3 a/2}\right )+4 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 231, normalized size = 1.34 \[ \frac {8 \, x^{7} e^{\left (2 \, a\right )} + 32 \, x^{3} + 36 \, {\left (\sqrt {2} x^{4} e^{\left (2 \, a\right )} + \sqrt {2}\right )} \arctan \left (-\sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} + \sqrt {2} \sqrt {\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}} e^{\left (\frac {1}{2} \, a\right )} - 1\right ) e^{\left (-\frac {3}{2} \, a\right )} + 36 \, {\left (\sqrt {2} x^{4} e^{\left (2 \, a\right )} + \sqrt {2}\right )} \arctan \left (-\sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} + \sqrt {2} \sqrt {-\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}} e^{\left (\frac {1}{2} \, a\right )} + 1\right ) e^{\left (-\frac {3}{2} \, a\right )} + 9 \, {\left (\sqrt {2} x^{4} e^{\left (2 \, a\right )} + \sqrt {2}\right )} e^{\left (-\frac {3}{2} \, a\right )} \log \left (\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) - 9 \, {\left (\sqrt {2} x^{4} e^{\left (2 \, a\right )} + \sqrt {2}\right )} e^{\left (-\frac {3}{2} \, a\right )} \log \left (-\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right )}{24 \, {\left (x^{4} e^{\left (2 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 139, normalized size = 0.80 \[ \frac {1}{3} \, x^{3} - \frac {3}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} + 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} - \frac {3}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} - 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{8} \, \sqrt {2} e^{\left (-\frac {3}{2} \, a\right )} \log \left (\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) - \frac {3}{8} \, \sqrt {2} e^{\left (-\frac {3}{2} \, a\right )} \log \left (-\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.08, size = 53, normalized size = 0.31 \[ \frac {x^{3}}{3}+\frac {x^{3}}{1+{\mathrm e}^{2 a} x^{4}}-\frac {3 \,{\mathrm e}^{-2 a} \left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{2 a} \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 144, normalized size = 0.83 \[ \frac {1}{3} \, x^{3} - \frac {3}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x e^{a} + \sqrt {2} e^{\left (\frac {1}{2} \, a\right )}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} - \frac {3}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x e^{a} - \sqrt {2} e^{\left (\frac {1}{2} \, a\right )}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {3}{8} \, \sqrt {2} e^{\left (-\frac {3}{2} \, a\right )} \log \left (x^{2} e^{a} + \sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} + 1\right ) - \frac {3}{8} \, \sqrt {2} e^{\left (-\frac {3}{2} \, a\right )} \log \left (x^{2} e^{a} - \sqrt {2} x e^{\left (\frac {1}{2} \, a\right )} + 1\right ) + \frac {x^{3}}{x^{4} e^{\left (2 \, a\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 67, normalized size = 0.39 \[ \frac {x^3}{{\mathrm {e}}^{2\,a}\,x^4+1}+\frac {3\,\mathrm {atan}\left (x\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{3/4}}+\frac {x^3}{3}+\frac {\mathrm {atan}\left (x\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{3/4}} \]
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 \[ \int x^{2} \tanh ^{2}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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