Optimal. Leaf size=76 \[ -\frac {1}{a^6 (1-a x)}+\frac {1}{8 a^6 (a x+1)}+\frac {1}{8 a^6 (1-a x)^2}-\frac {23 \log (1-a x)}{16 a^6}+\frac {7 \log (a x+1)}{16 a^6}-\frac {x}{a^5} \]
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Rubi [A] time = 0.13, antiderivative size = 76, 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} \[ -\frac {x}{a^5}-\frac {1}{a^6 (1-a x)}+\frac {1}{8 a^6 (a x+1)}+\frac {1}{8 a^6 (1-a x)^2}-\frac {23 \log (1-a x)}{16 a^6}+\frac {7 \log (a x+1)}{16 a^6} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
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*}
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Mathematica [A] time = 0.08, 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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fricas [A] time = 0.54, size = 117, normalized size = 1.54 \[ -\frac {16 \, a^{4} x^{4} - 16 \, a^{3} x^{3} - 34 \, a^{2} x^{2} + 18 \, a x - 7 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 23 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 12}{16 \, {\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 64, normalized size = 0.84 \[ -\frac {x}{a^{5}} + \frac {7 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{6}} - \frac {23 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{6}} + \frac {9 \, a^{2} x^{2} - a x - 6}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 65, normalized size = 0.86 \[ -\frac {x}{a^{5}}-\frac {23 \ln \left (a x -1\right )}{16 a^{6}}+\frac {1}{8 a^{6} \left (a x -1\right )^{2}}+\frac {1}{a^{6} \left (a x -1\right )}+\frac {1}{8 a^{6} \left (a x +1\right )}+\frac {7 \ln \left (a x +1\right )}{16 a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 72, normalized size = 0.95 \[ \frac {9 \, a^{2} x^{2} - a x - 6}{8 \, {\left (a^{9} x^{3} - a^{8} x^{2} - a^{7} x + a^{6}\right )}} - \frac {x}{a^{5}} + \frac {7 \, \log \left (a x + 1\right )}{16 \, a^{6}} - \frac {23 \, \log \left (a x - 1\right )}{16 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.93, size = 73, normalized size = 0.96 \[ \frac {7\,\ln \left (a\,x+1\right )}{16\,a^6}-\frac {23\,\ln \left (a\,x-1\right )}{16\,a^6}-\frac {x}{a^5}+\frac {\frac {x}{8}-\frac {9\,a\,x^2}{8}+\frac {3}{4\,a}}{-a^8\,x^3+a^7\,x^2+a^6\,x-a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.43, size = 71, normalized size = 0.93 \[ - \frac {- 9 a^{2} x^{2} + a x + 6}{8 a^{9} x^{3} - 8 a^{8} x^{2} - 8 a^{7} x + 8 a^{6}} - \frac {x}{a^{5}} - \frac {\frac {23 \log {\left (x - \frac {1}{a} \right )}}{16} - \frac {7 \log {\left (x + \frac {1}{a} \right )}}{16}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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