Optimal. Leaf size=173 \[ \frac{x^3}{e^{2 a} x^4+1}-\frac{3 e^{-3 a/2} \log \left (e^a x^2-\sqrt{2} e^{a/2} x+1\right )}{4 \sqrt{2}}+\frac{3 e^{-3 a/2} \log \left (e^a x^2+\sqrt{2} e^{a/2} x+1\right )}{4 \sqrt{2}}+\frac{3 e^{-3 a/2} \tan ^{-1}\left (1-\sqrt{2} e^{a/2} x\right )}{2 \sqrt{2}}-\frac{3 e^{-3 a/2} \tan ^{-1}\left (\sqrt{2} e^{a/2} x+1\right )}{2 \sqrt{2}}+\frac{x^3}{3} \]
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Rubi [F] time = 0.0479177, 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 = {} \[ \int x^2 \tanh ^2(a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^2 \tanh ^2(a+2 \log (x)) \, dx &=\int x^2 \tanh ^2(a+2 \log (x)) \, dx\\ \end{align*}
Mathematica [A] time = 0.649469, size = 174, normalized size = 1.01 \[ \frac{1}{12} \left (\frac{12 x^3}{e^{2 a} x^4+1}+9 (-1)^{3/4} e^{-3 a/2} \log \left (\sqrt [4]{-1} e^{-3 a/2}-e^{-a} x\right )+9 \sqrt [4]{-1} e^{-3 a/2} \log \left ((-1)^{3/4} e^{-3 a/2}-e^{-a} x\right )-9 (-1)^{3/4} e^{-3 a/2} \log \left (e^{-a} x+\sqrt [4]{-1} e^{-3 a/2}\right )-9 \sqrt [4]{-1} e^{-3 a/2} \log \left (e^{-a} x+(-1)^{3/4} e^{-3 a/2}\right )+4 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.022, size = 53, normalized size = 0.3 \begin{align*}{\frac{{x}^{3}}{3}}+{\frac{{x}^{3}}{1+{{\rm e}^{2\,a}}{x}^{4}}}-{\frac{3\,{{\rm e}^{-2\,a}}}{4}\sum _{{\it \_R}={\it RootOf} \left ({{\rm e}^{2\,a}}{{\it \_Z}}^{4}+1 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{\it \_R}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77848, size = 194, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} - \frac{3}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x e^{a} + \sqrt{2} e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} - \frac{3}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x e^{a} - \sqrt{2} e^{\left (\frac{1}{2} \, a\right )}\right )} e^{\left (-\frac{1}{2} \, a\right )}\right ) e^{\left (-\frac{3}{2} \, a\right )} + \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (x^{2} e^{a} + \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + 1\right ) - \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (x^{2} e^{a} - \sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + 1\right ) + \frac{x^{3}}{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 = 2.1458, size = 693, normalized size = 4.01 \begin{align*} \frac{8 \, x^{7} e^{\left (2 \, a\right )} + 32 \, x^{3} + 36 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} \arctan \left (-\sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + \sqrt{2} \sqrt{\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}} e^{\left (\frac{1}{2} \, a\right )} - 1\right ) e^{\left (-\frac{3}{2} \, a\right )} + 36 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} \arctan \left (-\sqrt{2} x e^{\left (\frac{1}{2} \, a\right )} + \sqrt{2} \sqrt{-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}} e^{\left (\frac{1}{2} \, a\right )} + 1\right ) e^{\left (-\frac{3}{2} \, a\right )} + 9 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} e^{\left (-\frac{3}{2} \, a\right )} \log \left (\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) - 9 \,{\left (\sqrt{2} x^{4} e^{\left (2 \, a\right )} + \sqrt{2}\right )} e^{\left (-\frac{3}{2} \, a\right )} \log \left (-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right )}{24 \,{\left (x^{4} e^{\left (2 \, a\right )} + 1\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^{2} \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 = 1.18396, size = 188, normalized size = 1.09 \begin{align*} \frac{1}{3} \, x^{3} - \frac{3}{4} \, \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{3}{2} \, a\right )} - \frac{3}{4} \, \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{3}{2} \, a\right )} + \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) - \frac{3}{8} \, \sqrt{2} e^{\left (-\frac{3}{2} \, a\right )} \log \left (-\sqrt{2} x e^{\left (-\frac{1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac{x^{3}}{x^{4} e^{\left (2 \, a\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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