Optimal. Leaf size=68 \[ \frac{2}{a^2 \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0983558, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {5994, 5958} \[ \frac{2}{a^2 \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5994
Rule 5958
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac{\tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a}\\ &=\frac{2}{a^2 \sqrt{1-a^2 x^2}}-\frac{2 x \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}}+\frac{\tanh ^{-1}(a x)^2}{a^2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.055104, size = 34, normalized size = 0.5 \[ \frac{\tanh ^{-1}(a x)^2-2 a x \tanh ^{-1}(a x)+2}{a^2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.211, size = 82, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}-2\,{\it Artanh} \left ( ax \right ) +2}{2\,{a}^{2} \left ( ax-1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }}+{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}+2\,{\it Artanh} \left ( ax \right ) +2}{2\,{a}^{2} \left ( ax+1 \right ) }\sqrt{- \left ( ax-1 \right ) \left ( ax+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.952745, size = 84, normalized size = 1.24 \begin{align*} -\frac{2 \, x \operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} a} + \frac{\operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{2}{\sqrt{-a^{2} x^{2} + 1} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.00201, size = 146, normalized size = 2.15 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}{\left (4 \, a x \log \left (-\frac{a x + 1}{a x - 1}\right ) - \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 8\right )}}{4 \,{\left (a^{4} x^{2} - a^{2}\right )}} \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 \operatorname{atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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 \operatorname{artanh}\left (a x\right )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]