Optimal. Leaf size=14 \[ \frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0696786, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6034, 5448, 12, 3298} \[ \frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6034
Rule 5448
Rule 12
Rule 3298
Rubi steps
\begin{align*} \int \frac{x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^2}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 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 )}{2 a^2}\\ &=\frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0637272, size = 14, normalized size = 1. \[ \frac{\text{Shi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.058, size = 13, normalized size = 0.9 \begin{align*}{\frac{{\it Shi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{2\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.96954, size = 112, normalized size = 8. \begin{align*} \frac{\logintegral \left (-\frac{a x + 1}{a x - 1}\right ) - \logintegral \left (-\frac{a x - 1}{a x + 1}\right )}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]