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 = 0.0213207, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}
Mathematica [C] time = 0.23153, 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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Maple [B] time = 0.048, size = 83, normalized size = 1.8 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{1}{2}\ln \left ( -x\sqrt{{{\rm e}^{a}}}+1 \right ) \left ({{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2}\ln \left ( x\sqrt{{{\rm e}^{a}}}+1 \right ) \left ({{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{2}\ln \left ( -{{\rm e}^{2\,a}}x+ \left ( -{{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) \left ( -{{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}}-{\frac{1}{2}\ln \left ({{\rm e}^{2\,a}}x+ \left ( -{{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) \left ( -{{\rm e}^{a}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75688, size = 65, normalized size = 1.44 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.68469, size = 171, normalized size = 3.8 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \coth{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14067, size = 73, normalized size = 1.62 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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