Optimal. Leaf size=41 \[ -e^{-a} \tanh ^{-1}\left (e^a x^2\right )+\frac {x^2}{1-e^{2 a} x^4}+\frac {x^2}{2} \]
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Rubi [F] time = 0.03, 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 = {} \[ \int x \coth ^2(a+2 \log (x)) \, dx \]
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*}
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Mathematica [C] time = 3.07, size = 163, normalized size = 3.98 \[ \frac {2}{105} e^{2 a} x^6 \left (e^{2 a} x^4+1\right )^2 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};e^{2 a} x^4\right )+\frac {e^{-4 a} \left (61 e^{6 a} x^{12}-181 e^{4 a} x^8-713 e^{2 a} x^4+\frac {3 \left (e^{8 a} x^{16}-52 e^{6 a} x^{12}-14 e^{4 a} x^8+196 e^{2 a} x^4+125\right ) \tanh ^{-1}\left (\sqrt {e^{2 a} x^4}\right )}{\sqrt {e^{2 a} x^4}}-375\right )}{96 x^6} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.40, size = 74, normalized size = 1.80 \[ \frac {x^{6} e^{\left (3 \, a\right )} - 3 \, x^{2} e^{a} - {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} + 1\right ) + {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} - 1\right )}{2 \, {\left (x^{4} e^{\left (3 \, a\right )} - e^{a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 54, normalized size = 1.32 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) - \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 54, normalized size = 1.32 \[ \frac {x^{2}}{2}-\frac {x^{2}}{-1+{\mathrm e}^{2 a} x^{4}}+\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}-1\right )}{2}-\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 53, normalized size = 1.29 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 42, normalized size = 1.02 \[ \frac {x^2}{2}-\frac {x^2}{x^4\,{\mathrm {e}}^{2\,a}-1}-\frac {\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )}{\sqrt {{\mathrm {e}}^{2\,a}}} \]
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 \[ \int x \coth ^{2}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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