Optimal. Leaf size=36 \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^2}-\frac{x}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.207256, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6032, 6034, 3312, 3301, 5968} \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^2}-\frac{x}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 6032
Rule 6034
Rule 3312
Rule 3301
Rule 5968
Rubi steps
\begin{align*} \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx &=-\frac{x}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a}+a \int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac{x}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}-\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}+\frac{\operatorname{Subst}\left (\int \left (\frac{1}{2 x}+\frac{\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=-\frac{x}{a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+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 ) \tanh ^{-1}(a x)}+\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0715305, size = 32, normalized size = 0.89 \[ \frac{\frac{a x}{\left (a^2 x^2-1\right ) \tanh ^{-1}(a x)}+\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 28, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sinh \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{2\,{\it Artanh} \left ( ax \right ) }}+{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) \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^{3} x^{2} - a\right )} \log \left (a x + 1\right ) -{\left (a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )} - \int -\frac{2 \,{\left (a^{2} x^{2} + 1\right )}}{{\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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0095, size = 252, normalized size = 7. \begin{align*} \frac{4 \, a x +{\left ({\left (a^{2} x^{2} - 1\right )} \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) +{\left (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 )}{2 \,{\left (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}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \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 )}^{2} \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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