Optimal. Leaf size=72 \[ -\frac{3}{4 a^5 (1-a x)}-\frac{1}{8 a^5 (a x+1)}+\frac{1}{8 a^5 (1-a x)^2}-\frac{11 \log (1-a x)}{16 a^5}-\frac{5 \log (a x+1)}{16 a^5} \]
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Rubi [A] time = 0.124757, antiderivative size = 72, 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} \[ -\frac{3}{4 a^5 (1-a x)}-\frac{1}{8 a^5 (a x+1)}+\frac{1}{8 a^5 (1-a x)^2}-\frac{11 \log (1-a x)}{16 a^5}-\frac{5 \log (a x+1)}{16 a^5} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^4}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac{1}{4 a^4 (-1+a x)^3}-\frac{3}{4 a^4 (-1+a x)^2}-\frac{11}{16 a^4 (-1+a x)}+\frac{1}{8 a^4 (1+a x)^2}-\frac{5}{16 a^4 (1+a x)}\right ) \, dx\\ &=\frac{1}{8 a^5 (1-a x)^2}-\frac{3}{4 a^5 (1-a x)}-\frac{1}{8 a^5 (1+a x)}-\frac{11 \log (1-a x)}{16 a^5}-\frac{5 \log (1+a x)}{16 a^5}\\ \end{align*}
Mathematica [A] time = 0.057497, size = 55, normalized size = 0.76 \[ \frac{\frac{2 \left (5 a^2 x^2+3 a x-6\right )}{(a x-1)^2 (a x+1)}-11 \log (1-a x)-5 \log (a x+1)}{16 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 60, normalized size = 0.8 \begin{align*} -{\frac{1}{8\,{a}^{5} \left ( ax+1 \right ) }}-{\frac{5\,\ln \left ( ax+1 \right ) }{16\,{a}^{5}}}+{\frac{1}{8\,{a}^{5} \left ( ax-1 \right ) ^{2}}}+{\frac{3}{4\,{a}^{5} \left ( ax-1 \right ) }}-{\frac{11\,\ln \left ( ax-1 \right ) }{16\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949897, size = 89, normalized size = 1.24 \begin{align*} \frac{5 \, a^{2} x^{2} + 3 \, a x - 6}{8 \,{\left (a^{8} x^{3} - a^{7} x^{2} - a^{6} x + a^{5}\right )}} - \frac{5 \, \log \left (a x + 1\right )}{16 \, a^{5}} - \frac{11 \, \log \left (a x - 1\right )}{16 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06296, size = 217, normalized size = 3.01 \begin{align*} \frac{10 \, a^{2} x^{2} + 6 \, a x - 5 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) - 11 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) - 12}{16 \,{\left (a^{8} x^{3} - a^{7} x^{2} - a^{6} x + a^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.548471, size = 66, normalized size = 0.92 \begin{align*} \frac{5 a^{2} x^{2} + 3 a x - 6}{8 a^{8} x^{3} - 8 a^{7} x^{2} - 8 a^{6} x + 8 a^{5}} - \frac{\frac{11 \log{\left (x - \frac{1}{a} \right )}}{16} + \frac{5 \log{\left (x + \frac{1}{a} \right )}}{16}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20655, size = 78, normalized size = 1.08 \begin{align*} -\frac{5 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{5}} - \frac{11 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{5}} + \frac{5 \, a^{2} x^{2} + 3 \, a x - 6}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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