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 = 0.0156127, 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 = {} \[ \int x \coth (a+2 \log (x)) \, dx \]
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*}
Mathematica [A] time = 0.181986, 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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Maple [A] time = 0.024, size = 37, normalized size = 1.6 \begin{align*}{\frac{{x}^{2}}{2}}-{\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 = 1.04383, size = 49, normalized size = 2.13 \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.59376, size = 81, normalized size = 3.52 \begin{align*} \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 )} \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{\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 = 1.1168, size = 50, normalized size = 2.17 \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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