Optimal. Leaf size=147 \[ \frac{e^{a/2} \log \left (e^a x^2-\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}-\frac{e^{a/2} \log \left (e^a x^2+\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}-\frac{e^{a/2} \tan ^{-1}\left (1-\sqrt{2} e^{a/2} x\right )}{\sqrt{2}}+\frac{e^{a/2} \tan ^{-1}\left (\sqrt{2} e^{a/2} x+1\right )}{\sqrt{2}}+\frac{1}{x} \]
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Rubi [F] time = 0.0237645, 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 = {} \[ \int \frac{\tanh (a+2 \log (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\tanh (a+2 \log (x))}{x^2} \, dx &=\int \frac{\tanh (a+2 \log (x))}{x^2} \, dx\\ \end{align*}
Mathematica [C] time = 0.159208, size = 59, normalized size = 0.4 \[ \frac{2-x (\sinh (a)+\cosh (a))^2 \text{RootSum}\left [-\text{$\#$1}^4 \sinh (a)+\text{$\#$1}^4 \cosh (a)+\sinh (a)+\cosh (a)\& ,\frac{\log \left (\frac{1}{x}-\text{$\#$1}\right )+\log (x)}{\text{$\#$1}^3}\& \right ]}{2 x} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 42, normalized size = 0.3 \begin{align*}{x}^{-1}+{\frac{\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}+{{\rm e}^{2\,a}} \right ) }{\it \_R}\,\ln \left ( \left ( 5\,{{\it \_R}}^{4}+4\,{{\rm e}^{2\,a}} \right ) x-{{\it \_R}}^{3} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.90063, size = 169, normalized size = 1.15 \begin{align*} -\frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (\frac{1}{2} \, a\right )} + \frac{2}{x}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} - \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (\frac{1}{2} \, a\right )} - \frac{2}{x}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} - \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (\frac{\sqrt{2} e^{\left (\frac{1}{2} \, a\right )}}{x} + \frac{1}{x^{2}} + e^{a}\right ) + \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (-\frac{\sqrt{2} e^{\left (\frac{1}{2} \, a\right )}}{x} + \frac{1}{x^{2}} + e^{a}\right ) + \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19939, size = 597, normalized size = 4.06 \begin{align*} -\frac{4 \, \sqrt{2} x \arctan \left (-{\left (\sqrt{2} x e^{\left (\frac{5}{2} \, a\right )} - \sqrt{2} \sqrt{x^{2} e^{\left (4 \, a\right )} + \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac{1}{2} \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + 4 \, \sqrt{2} x \arctan \left (-{\left (\sqrt{2} x e^{\left (\frac{5}{2} \, a\right )} - \sqrt{2} \sqrt{x^{2} e^{\left (4 \, a\right )} - \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}} e^{\left (\frac{1}{2} \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (-2 \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} + \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} \log \left (x^{2} e^{\left (4 \, a\right )} - \sqrt{2} x e^{\left (\frac{7}{2} \, a\right )} + e^{\left (3 \, a\right )}\right ) - 4}{4 \, x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18468, size = 163, normalized size = 1.11 \begin{align*} \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (-\frac{1}{2} \, a\right )} + 2 \, x\right )} e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + \frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} e^{\left (-\frac{1}{2} \, a\right )} - 2 \, x\right )} e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} - \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac{1}{4} \, \sqrt{2} e^{\left (\frac{1}{2} \, a\right )} \log \left (-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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