Optimal. Leaf size=76 \[ -\frac{x}{a^5}-\frac{1}{a^6 (1-a x)}+\frac{1}{8 a^6 (a x+1)}+\frac{1}{8 a^6 (1-a x)^2}-\frac{23 \log (1-a x)}{16 a^6}+\frac{7 \log (a x+1)}{16 a^6} \]
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Rubi [A] time = 0.126824, antiderivative size = 76, 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} \[ -\frac{x}{a^5}-\frac{1}{a^6 (1-a x)}+\frac{1}{8 a^6 (a x+1)}+\frac{1}{8 a^6 (1-a x)^2}-\frac{23 \log (1-a x)}{16 a^6}+\frac{7 \log (a x+1)}{16 a^6} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{x^5}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac{1}{a^5}-\frac{1}{4 a^5 (-1+a x)^3}-\frac{1}{a^5 (-1+a x)^2}-\frac{23}{16 a^5 (-1+a x)}-\frac{1}{8 a^5 (1+a x)^2}+\frac{7}{16 a^5 (1+a x)}\right ) \, dx\\ &=-\frac{x}{a^5}+\frac{1}{8 a^6 (1-a x)^2}-\frac{1}{a^6 (1-a x)}+\frac{1}{8 a^6 (1+a x)}-\frac{23 \log (1-a x)}{16 a^6}+\frac{7 \log (1+a x)}{16 a^6}\\ \end{align*}
Mathematica [A] time = 0.0721872, size = 55, normalized size = 0.72 \[ \frac{2 \left (-8 a x+\frac{8}{a x-1}+\frac{1}{a x+1}+\frac{1}{(a x-1)^2}\right )-23 \log (1-a x)+7 \log (a x+1)}{16 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 65, normalized size = 0.9 \begin{align*} -{\frac{x}{{a}^{5}}}+{\frac{1}{8\,{a}^{6} \left ( ax+1 \right ) }}+{\frac{7\,\ln \left ( ax+1 \right ) }{16\,{a}^{6}}}-{\frac{23\,\ln \left ( ax-1 \right ) }{16\,{a}^{6}}}+{\frac{1}{8\,{a}^{6} \left ( ax-1 \right ) ^{2}}}+{\frac{1}{{a}^{6} \left ( ax-1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.952949, size = 97, normalized size = 1.28 \begin{align*} \frac{9 \, a^{2} x^{2} - a x - 6}{8 \,{\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right )}} - \frac{x}{a^{5}} + \frac{7 \, \log \left (a x + 1\right )}{16 \, a^{6}} - \frac{23 \, \log \left (a x - 1\right )}{16 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78357, size = 255, normalized size = 3.36 \begin{align*} -\frac{16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 23 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 12}{16 \,{\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.615176, size = 70, normalized size = 0.92 \begin{align*} \frac{9 a^{2} x^{2} - a x - 6}{8 a^{9} x^{3} - 8 a^{8} x^{2} - 8 a^{7} x + 8 a^{6}} - \frac{x}{a^{5}} - \frac{\frac{23 \log{\left (x - \frac{1}{a} \right )}}{16} - \frac{7 \log{\left (x + \frac{1}{a} \right )}}{16}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2163, size = 86, normalized size = 1.13 \begin{align*} -\frac{x}{a^{5}} + \frac{7 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{6}} - \frac{23 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{6}} + \frac{9 \, a^{2} x^{2} - a x - 6}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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