Optimal. Leaf size=20 \[ e^a \tan ^{-1}\left (e^a x^2\right )+\frac{1}{2 x^2} \]
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Rubi [F] time = 0.0244611, 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 = {} \[ \int \frac{\tanh (a+2 \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tanh (a+2 \log (x))}{x^3} \, dx &=\int \frac{\tanh (a+2 \log (x))}{x^3} \, dx\\ \end{align*}
Mathematica [A] time = 0.153145, size = 40, normalized size = 2. \[ \cosh (a) \left (-\tan ^{-1}\left (\frac{\cosh (a)-\sinh (a)}{x^2}\right )\right )-\sinh (a) \tan ^{-1}\left (\frac{\cosh (a)-\sinh (a)}{x^2}\right )+\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 44, normalized size = 2.2 \begin{align*}{\frac{1}{2\,{x}^{2}}}+{\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 = 1.6208, size = 26, normalized size = 1.3 \begin{align*} -\arctan \left (\frac{e^{\left (-a\right )}}{x^{2}}\right ) e^{a} + \frac{1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09518, size = 55, normalized size = 2.75 \begin{align*} \frac{2 \, x^{2} \arctan \left (x^{2} e^{a}\right ) e^{a} + 1}{2 \, x^{2}} \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{\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 = 1.20641, size = 22, normalized size = 1.1 \begin{align*} \arctan \left (x^{2} e^{a}\right ) e^{a} + \frac{1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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