Optimal. Leaf size=58 \[ \frac {1}{2 a^5 (1-a x)}+\frac {7 \log (1-a x)}{4 a^5}+\frac {\log (a x+1)}{4 a^5}+\frac {x}{a^4}+\frac {x^2}{2 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \[ \frac {x^2}{2 a^3}+\frac {x}{a^4}+\frac {1}{2 a^5 (1-a x)}+\frac {7 \log (1-a x)}{4 a^5}+\frac {\log (a x+1)}{4 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 88
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac {x^4}{(1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac {1}{a^4}+\frac {x}{a^3}+\frac {1}{2 a^4 (-1+a x)^2}+\frac {7}{4 a^4 (-1+a x)}+\frac {1}{4 a^4 (1+a x)}\right ) \, dx\\ &=\frac {x}{a^4}+\frac {x^2}{2 a^3}+\frac {1}{2 a^5 (1-a x)}+\frac {7 \log (1-a x)}{4 a^5}+\frac {\log (1+a x)}{4 a^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 45, normalized size = 0.78 \[ \frac {2 \left (a^2 x^2+2 a x+\frac {1}{1-a x}\right )+7 \log (1-a x)+\log (a x+1)}{4 a^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 62, normalized size = 1.07 \[ \frac {2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 4 \, a x + {\left (a x - 1\right )} \log \left (a x + 1\right ) + 7 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \, {\left (a^{6} x - a^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 56, normalized size = 0.97 \[ \frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a^{5}} + \frac {7 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a^{5}} + \frac {a^{3} x^{2} + 2 \, a^{2} x}{2 \, a^{6}} - \frac {1}{2 \, {\left (a x - 1\right )} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 49, normalized size = 0.84 \[ \frac {x^{2}}{2 a^{3}}+\frac {x}{a^{4}}-\frac {1}{2 a^{5} \left (a x -1\right )}+\frac {7 \ln \left (a x -1\right )}{4 a^{5}}+\frac {\ln \left (a x +1\right )}{4 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 52, normalized size = 0.90 \[ -\frac {1}{2 \, {\left (a^{6} x - a^{5}\right )}} + \frac {a x^{2} + 2 \, x}{2 \, a^{4}} + \frac {\log \left (a x + 1\right )}{4 \, a^{5}} + \frac {7 \, \log \left (a x - 1\right )}{4 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.91, size = 54, normalized size = 0.93 \[ \frac {7\,\ln \left (a\,x-1\right )}{4\,a^5}+\frac {\ln \left (a\,x+1\right )}{4\,a^5}-\frac {1}{2\,a\,\left (a^5\,x-a^4\right )}+\frac {x}{a^4}+\frac {x^2}{2\,a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.26, size = 48, normalized size = 0.83 \[ - \frac {1}{2 a^{6} x - 2 a^{5}} + \frac {x^{2}}{2 a^{3}} + \frac {x}{a^{4}} + \frac {\frac {7 \log {\left (x - \frac {1}{a} \right )}}{4} + \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________