Optimal. Leaf size=45 \[ e^{-3 a/2} \tan ^{-1}\left (e^{a/2} x\right )-e^{-3 a/2} \tanh ^{-1}\left (e^{a/2} x\right )+\frac {x^3}{3} \]
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Rubi [F] time = 0.02, 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 = {} \[ \int x^2 \coth (a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x^2 \coth (a+2 \log (x)) \, dx &=\int x^2 \coth (a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [C] time = 0.24, size = 64, normalized size = 1.42 \[ \frac {1}{6} \left (3 (\sinh (2 a)-\cosh (2 a)) \text {RootSum}\left [\text {$\#$1}^4 \sinh (a)+\text {$\#$1}^4 \cosh (a)+\sinh (a)-\cosh (a)\& ,\frac {\log (x)-\log (x-\text {$\#$1})}{\text {$\#$1}}\& \right ]+2 x^3\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 62, normalized size = 1.38 \[ \frac {1}{6} \, {\left (2 \, x^{3} e^{\left (2 \, a\right )} + 6 \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + 3 \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )\right )} e^{\left (-2 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 54, normalized size = 1.20 \[ \frac {1}{3} \, x^{3} + \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 83, normalized size = 1.84 \[ \frac {x^{3}}{3}+\frac {\ln \left (-{\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {\ln \left ({\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{2 \left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}}+\frac {\ln \left (-\sqrt {{\mathrm e}^{a}}\, x +1\right )}{2 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}}-\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{2 \left ({\mathrm e}^{a}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 48, normalized size = 1.07 \[ \frac {1}{3} \, x^{3} + \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {3}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {3}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 39, normalized size = 0.87 \[ \frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}}-\frac {\mathrm {atanh}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{3/4}}+\frac {x^3}{3} \]
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 \[ \int x^{2} \coth {\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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