Optimal. Leaf size=29 \[ \frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^2}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2} \]
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Rubi [A] time = 0.0941661, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6034, 5448, 3298} \[ \frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^2}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 6034
Rule 5448
Rule 3298
Rubi steps
\begin{align*} \int \frac{x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\sinh (2 x)}{4 x}+\frac{\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^2}+\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^2}\\ &=\frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^2}+\frac{\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2}\\ \end{align*}
Mathematica [A] time = 0.107331, size = 24, normalized size = 0.83 \[ \frac{2 \text{Shi}\left (2 \tanh ^{-1}(a x)\right )+\text{Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 24, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{\it Shi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{4}}+{\frac{{\it Shi} \left ( 4\,{\it Artanh} \left ( ax \right ) \right ) }{8}} \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*} -\int \frac{x}{{\left (a^{2} x^{2} - 1\right )}^{3} \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 = 2.56244, size = 281, normalized size = 9.69 \begin{align*} \frac{\logintegral \left (\frac{a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - \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 x + 1}{a x - 1}\right ) - 2 \, \logintegral \left (-\frac{a x - 1}{a x + 1}\right )}{16 \, a^{2}} \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}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname{atanh}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname{atanh}{\left (a x \right )} - \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}{{\left (a^{2} x^{2} - 1\right )}^{3} \operatorname{artanh}\left (a x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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