Optimal. Leaf size=59 \[ -\frac{3 e^{2 a} x^2}{2 \left (e^{2 a} x^4+1\right )}-\frac{1}{2 x^2 \left (e^{2 a} x^4+1\right )}-e^a \tan ^{-1}\left (e^a x^2\right ) \]
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Rubi [F] time = 0.0534242, 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 = {} \[ \int \frac{\tanh ^2(a+2 \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tanh ^2(a+2 \log (x))}{x^3} \, dx &=\int \frac{\tanh ^2(a+2 \log (x))}{x^3} \, dx\\ \end{align*}
Mathematica [A] time = 0.355313, size = 40, normalized size = 0.68 \[ \frac{-\frac{2}{e^{-2 (a+2 \log (x))}+1}-1}{2 x^2}+e^a \tan ^{-1}\left (\frac{e^{-a}}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.029, size = 66, normalized size = 1.1 \begin{align*}{\frac{1}{{x}^{2} \left ( 1+{{\rm e}^{2\,a}}{x}^{4} \right ) } \left ( -{\frac{3\,{{\rm e}^{2\,a}}{x}^{4}}{2}}-{\frac{1}{2}} \right ) }+{\frac{\sum _{{\it \_R}={\it RootOf} \left ({{\rm e}^{2\,a}}+{{\it \_Z}}^{2} \right ) }{\it \_R}\,\ln \left ( \left ( -4\,{{\rm e}^{2\,a}}-5\,{{\it \_R}}^{2} \right ){x}^{2}-{\it \_R} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.81886, size = 50, normalized size = 0.85 \begin{align*} \arctan \left (\frac{e^{\left (-a\right )}}{x^{2}}\right ) e^{a} - \frac{1}{2 \, x^{2}} - \frac{e^{\left (2 \, a\right )}}{x^{2}{\left (\frac{1}{x^{4}} + e^{\left (2 \, a\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88606, size = 122, normalized size = 2.07 \begin{align*} -\frac{3 \, x^{4} e^{\left (2 \, a\right )} + 2 \,{\left (x^{6} e^{\left (3 \, a\right )} + x^{2} e^{a}\right )} \arctan \left (x^{2} e^{a}\right ) + 1}{2 \,{\left (x^{6} e^{\left (2 \, a\right )} + x^{2}\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 \frac{\tanh ^{2}{\left (a + 2 \log{\left (x \right )} \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40139, size = 53, normalized size = 0.9 \begin{align*} -\arctan \left (x^{2} e^{a}\right ) e^{a} - \frac{3 \, x^{4} e^{\left (2 \, a\right )} + 1}{2 \,{\left (x^{6} e^{\left (2 \, a\right )} + x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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