Optimal. Leaf size=23 \[ \frac {x^2}{2}-e^{-a} \tanh ^{-1}\left (e^a x^2\right ) \]
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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 = {} \[ \int x \coth (a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int x \coth (a+2 \log (x)) \, dx &=\int x \coth (a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [A] time = 0.19, size = 26, normalized size = 1.13 \[ (\sinh (a)-\cosh (a)) \tanh ^{-1}\left (x^2 (\sinh (a)+\cosh (a))\right )+\frac {x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 33, normalized size = 1.43 \[ \frac {1}{2} \, {\left (x^{2} e^{a} - \log \left (x^{2} e^{a} + 1\right ) + \log \left (x^{2} e^{a} - 1\right )\right )} e^{\left (-a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 37, normalized size = 1.61 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 37, normalized size = 1.61 \[ \frac {x^{2}}{2}-\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}+1\right )}{2}+\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}-1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 36, normalized size = 1.57 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 25, normalized size = 1.09 \[ \frac {x^2}{2}-\frac {\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )}{\sqrt {{\mathrm {e}}^{2\,a}}} \]
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 \[ \int x \coth {\left (a + 2 \log {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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