Optimal. Leaf size=40 \[ \frac{x^2}{e^{2 a} x^4+1}-e^{-a} \tan ^{-1}\left (e^a x^2\right )+\frac{x^2}{2} \]
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Rubi [F] time = 0.0302817, 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 = {} \[ \int x \tanh ^2(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x \tanh ^2(a+2 \log (x)) \, dx &=\int x \tanh ^2(a+2 \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.377913, size = 41, normalized size = 1.02 \[ \frac{x^2}{e^{2 (a+2 \log (x))}+1}-e^{-a} \tan ^{-1}\left (e^a x^2\right )+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.033, size = 57, normalized size = 1.4 \begin{align*}{\frac{{x}^{2}}{2}}+{\frac{{x}^{2}}{1+{{\rm e}^{2\,a}}{x}^{4}}}+{\frac{i}{2}}{{\rm e}^{-a}}\ln \left ({{\rm e}^{a}}{x}^{2}-i \right ) -{\frac{i}{2}}{{\rm e}^{-a}}\ln \left ({{\rm e}^{a}}{x}^{2}+i \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.86376, size = 47, normalized size = 1.18 \begin{align*} \frac{1}{2} \, x^{2} - \arctan \left (x^{2} e^{a}\right ) e^{\left (-a\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 [A] time = 2.02991, size = 120, normalized size = 3. \begin{align*} \frac{x^{6} e^{\left (3 \, a\right )} + 3 \, x^{2} e^{a} - 2 \,{\left (x^{4} e^{\left (2 \, a\right )} + 1\right )} \arctan \left (x^{2} e^{a}\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 \tanh ^{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 = 1.23874, size = 47, normalized size = 1.18 \begin{align*} \frac{1}{2} \, x^{2} - \arctan \left (x^{2} e^{a}\right ) e^{\left (-a\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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