Optimal. Leaf size=53 \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^2}-\frac{x}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.245151, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6032, 6034, 5448, 3301, 5968, 3312} \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^2}-\frac{x}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 6032
Rule 6034
Rule 5448
Rule 3301
Rule 5968
Rule 3312
Rubi steps
\begin{align*} \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx &=-\frac{x}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\int \frac{1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx}{a}+(3 a) \int \frac{x^2}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{x}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh ^4(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (\frac{3}{8 x}+\frac{\cosh (2 x)}{2 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac{3 \operatorname{Subst}\left (\int \left (-\frac{1}{8 x}+\frac{\cosh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^2}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac{x}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}+\frac{\text{Chi}\left (4 \tanh ^{-1}(a x)\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0994367, size = 75, normalized size = 1.42 \[ \frac{\left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x) \text{Chi}\left (2 \tanh ^{-1}(a x)\right )+\left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x) \text{Chi}\left (4 \tanh ^{-1}(a x)\right )-2 a x}{2 a^2 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 54, normalized size = 1. \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{4\,{\it Artanh} \left ( ax \right ) }}+{\frac{{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{2}}-{\frac{\sinh \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{8\,{\it Artanh} \left ( ax \right ) }}+{\frac{{\it Chi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{2}} \right ) } \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*} -\frac{2 \, x}{{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (a x + 1\right ) -{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-a x + 1\right )} + \int -\frac{2 \,{\left (3 \, a^{2} x^{2} + 1\right )}}{{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) -{\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37035, size = 535, normalized size = 10.09 \begin{align*} -\frac{8 \, a x -{\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (\frac{a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) +{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (\frac{a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) +{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) +{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \logintegral \left (-\frac{a x - 1}{a x + 1}\right )\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{4 \,{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )} \log \left (-\frac{a x + 1}{a x - 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 \frac{x}{a^{6} x^{6} \operatorname{atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}^{2}{\left (a x \right )} - \operatorname{atanh}^{2}{\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}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname{artanh}\left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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