Optimal. Leaf size=27 \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a}+\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0662723, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {5968, 3312, 3301} \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a}+\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5968
Rule 3312
Rule 3301
Rubi steps
\begin{align*} \int \frac{1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\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}\\ &=\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a}+\frac{\operatorname{Subst}\left (\int \frac{\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=\frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a}+\frac{\log \left (\tanh ^{-1}(a x)\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0895055, size = 20, normalized size = 0.74 \[ \frac{\text{Chi}\left (2 \tanh ^{-1}(a x)\right )+\log \left (\tanh ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.057, size = 24, normalized size = 0.9 \begin{align*}{\frac{{\it Chi} \left ( 2\,{\it Artanh} \left ( ax \right ) \right ) }{2\,a}}+{\frac{\ln \left ({\it Artanh} \left ( ax \right ) \right ) }{2\,a}} \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{1}{{\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 = 2.02823, size = 157, normalized size = 5.81 \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} \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{1}{\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{1}{{\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]