Optimal. Leaf size=43 \[ \frac{1}{2 a^3 (1-a x)}+\frac{3 \log (1-a x)}{4 a^3}+\frac{\log (a x+1)}{4 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.113338, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 88} \[ \frac{1}{2 a^3 (1-a x)}+\frac{3 \log (1-a x)}{4 a^3}+\frac{\log (a x+1)}{4 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac{x^2}{(1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac{1}{2 a^2 (-1+a x)^2}+\frac{3}{4 a^2 (-1+a x)}+\frac{1}{4 a^2 (1+a x)}\right ) \, dx\\ &=\frac{1}{2 a^3 (1-a x)}+\frac{3 \log (1-a x)}{4 a^3}+\frac{\log (1+a x)}{4 a^3}\\ \end{align*}
Mathematica [A] time = 0.0243129, size = 33, normalized size = 0.77 \[ \frac{\frac{2}{1-a x}+3 \log (1-a x)+\log (a x+1)}{4 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 36, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( ax+1 \right ) }{4\,{a}^{3}}}-{\frac{1}{2\,{a}^{3} \left ( ax-1 \right ) }}+{\frac{3\,\ln \left ( ax-1 \right ) }{4\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.969361, size = 51, normalized size = 1.19 \begin{align*} -\frac{1}{2 \,{\left (a^{4} x - a^{3}\right )}} + \frac{\log \left (a x + 1\right )}{4 \, a^{3}} + \frac{3 \, \log \left (a x - 1\right )}{4 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.74665, size = 101, normalized size = 2.35 \begin{align*} \frac{{\left (a x - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (a^{4} x - a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.385481, size = 34, normalized size = 0.79 \begin{align*} - \frac{1}{2 a^{4} x - 2 a^{3}} + \frac{\frac{3 \log{\left (x - \frac{1}{a} \right )}}{4} + \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19562, size = 50, normalized size = 1.16 \begin{align*} \frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a^{3}} + \frac{3 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a^{3}} - \frac{1}{2 \,{\left (a x - 1\right )} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]