Optimal. Leaf size=41 \[ \frac{x^2}{1-e^{2 a} x^4}-e^{-a} \tanh ^{-1}\left (e^a x^2\right )+\frac{x^2}{2} \]
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Rubi [F] time = 0.0298197, 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 = {} \[ \int x \coth ^2(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x \coth ^2(a+2 \log (x)) \, dx &=\int x \coth ^2(a+2 \log (x)) \, dx\\ \end{align*}
Mathematica [C] time = 2.93267, size = 163, normalized size = 3.98 \[ \frac{2}{105} e^{2 a} x^6 \left (e^{2 a} x^4+1\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2\right \},\left \{1,1,\frac{9}{2}\right \},e^{2 a} x^4\right )+\frac{e^{-4 a} \left (61 e^{6 a} x^{12}-181 e^{4 a} x^8-713 e^{2 a} x^4+\frac{3 \left (e^{8 a} x^{16}-52 e^{6 a} x^{12}-14 e^{4 a} x^8+196 e^{2 a} x^4+125\right ) \tanh ^{-1}\left (\sqrt{e^{2 a} x^4}\right )}{\sqrt{e^{2 a} x^4}}-375\right )}{96 x^6} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.02, size = 54, normalized size = 1.3 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{{x}^{2}}{{{\rm e}^{2\,a}}{x}^{4}-1}}+{\frac{{{\rm e}^{-a}}\ln \left ({{\rm e}^{a}}{x}^{2}-1 \right ) }{2}}-{\frac{{{\rm e}^{-a}}\ln \left ({{\rm e}^{a}}{x}^{2}+1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06055, size = 72, normalized size = 1.76 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac{x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.57195, size = 169, normalized size = 4.12 \begin{align*} \frac{x^{6} e^{\left (3 \, a\right )} - 3 \, x^{2} e^{a} -{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} + 1\right ) +{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} - 1\right )}{2 \,{\left (x^{4} e^{\left (3 \, a\right )} - e^{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 \coth ^{2}{\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 = 1.11119, size = 73, normalized size = 1.78 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-a\right )} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) - \frac{x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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