Optimal. Leaf size=55 \[ -\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^3}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a^3}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{16 a^3} \]
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Rubi [A] time = 0.142112, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6034, 5448, 3301} \[ -\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^3}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a^3}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 6034
Rule 5448
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2}{\left (1-a^2 x^2\right )^4 \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^4(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{16 x}-\frac{\cosh (2 x)}{32 x}+\frac{\cosh (4 x)}{16 x}+\frac{\cosh (6 x)}{32 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\log \left (\tanh ^{-1}(a x)\right )}{16 a^3}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (6 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{32 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{16 a^3}\\ &=-\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^3}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a^3}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{16 a^3}\\ \end{align*}
Mathematica [A] time = 0.159147, size = 55, normalized size = 1. \[ -\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{32 a^3}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{16 a^3}+\frac{\text{Chi}\left (6 \tanh ^{-1}(a x)\right )}{32 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{16 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 48, normalized size = 0.9 \begin{align*} -{\frac{{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{32\,{a}^{3}}}+{\frac{{\it Chi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{16\,{a}^{3}}}+{\frac{{\it Chi} \left ( 6\,{\it Artanh} \left ( ax \right ) \right ) }{32\,{a}^{3}}}-{\frac{\ln \left ({\it Artanh} \left ( ax \right ) \right ) }{16\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.96815, size = 559, normalized size = 10.16 \begin{align*} -\frac{4 \, \log \left (\log \left (-\frac{a x + 1}{a x - 1}\right )\right ) - \logintegral \left (-\frac{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) - \logintegral \left (-\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}{a^{3} x^{3} + 3 \, a^{2} x^{2} + 3 \, a x + 1}\right ) - 2 \, \logintegral \left (\frac{a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - 2 \, \logintegral \left (\frac{a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) + \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) + \logintegral \left (-\frac{a x - 1}{a x + 1}\right )}{64 \, a^{3}} \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{x^{2}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4} \operatorname{atanh}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (a^{2} x^{2} - 1\right )}^{4} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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