Optimal. Leaf size=27 \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10304, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6034, 3312, 3301} \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6034
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\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^3}\\ &=-\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a^3}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}\\ &=\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.112589, size = 27, normalized size = 1. \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^3}-\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.064, size = 24, normalized size = 0.9 \begin{align*}{\frac{{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{2\,{a}^{3}}}-{\frac{\ln \left ({\it Artanh} \left ( ax \right ) \right ) }{2\,{a}^{3}}} \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^{2}}{{\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 = 2.01163, size = 161, normalized size = 5.96 \begin{align*} -\frac{2 \, \log \left (\log \left (-\frac{a x + 1}{a x - 1}\right )\right ) - \logintegral \left (-\frac{a x + 1}{a x - 1}\right ) - \logintegral \left (-\frac{a x - 1}{a x + 1}\right )}{4 \, a^{3}} \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^{2}}{\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^{2}}{{\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]