Optimal. Leaf size=88 \[ -\frac {5}{4 a^7 (1-a x)}-\frac {1}{8 a^7 (a x+1)}+\frac {1}{8 a^7 (1-a x)^2}-\frac {39 \log (1-a x)}{16 a^7}-\frac {9 \log (a x+1)}{16 a^7}-\frac {x}{a^6}-\frac {x^2}{2 a^5} \]
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Rubi [A] time = 0.14, antiderivative size = 88, 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} \[ -\frac {x^2}{2 a^5}-\frac {x}{a^6}-\frac {5}{4 a^7 (1-a x)}-\frac {1}{8 a^7 (a x+1)}+\frac {1}{8 a^7 (1-a x)^2}-\frac {39 \log (1-a x)}{16 a^7}-\frac {9 \log (a x+1)}{16 a^7} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac {x^6}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac {1}{a^6}-\frac {x}{a^5}-\frac {1}{4 a^6 (-1+a x)^3}-\frac {5}{4 a^6 (-1+a x)^2}-\frac {39}{16 a^6 (-1+a x)}+\frac {1}{8 a^6 (1+a x)^2}-\frac {9}{16 a^6 (1+a x)}\right ) \, dx\\ &=-\frac {x}{a^6}-\frac {x^2}{2 a^5}+\frac {1}{8 a^7 (1-a x)^2}-\frac {5}{4 a^7 (1-a x)}-\frac {1}{8 a^7 (1+a x)}-\frac {39 \log (1-a x)}{16 a^7}-\frac {9 \log (1+a x)}{16 a^7}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 65, normalized size = 0.74 \[ \frac {2 \left (-4 a^2 x^2-8 a x+\frac {10}{a x-1}-\frac {1}{a x+1}+\frac {1}{(a x-1)^2}\right )-39 \log (1-a x)-9 \log (a x+1)}{16 a^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 125, normalized size = 1.42 \[ -\frac {8 \, a^{5} x^{5} + 8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} - 26 \, a^{2} x^{2} + 10 \, a x + 9 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 39 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 20}{16 \, {\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 77, normalized size = 0.88 \[ -\frac {9 \, \log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{7}} - \frac {39 \, \log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{7}} - \frac {a^{5} x^{2} + 2 \, a^{4} x}{2 \, a^{10}} + \frac {9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 74, normalized size = 0.84 \[ -\frac {x^{2}}{2 a^{5}}-\frac {x}{a^{6}}+\frac {1}{8 a^{7} \left (a x -1\right )^{2}}+\frac {5}{4 a^{7} \left (a x -1\right )}-\frac {39 \ln \left (a x -1\right )}{16 a^{7}}-\frac {1}{8 a^{7} \left (a x +1\right )}-\frac {9 \ln \left (a x +1\right )}{16 a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 80, normalized size = 0.91 \[ \frac {9 \, a^{2} x^{2} + 3 \, a x - 10}{8 \, {\left (a^{10} x^{3} - a^{9} x^{2} - a^{8} x + a^{7}\right )}} - \frac {a x^{2} + 2 \, x}{2 \, a^{6}} - \frac {9 \, \log \left (a x + 1\right )}{16 \, a^{7}} - \frac {39 \, \log \left (a x - 1\right )}{16 \, a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 82, normalized size = 0.93 \[ -\frac {39\,\ln \left (a\,x-1\right )}{16\,a^7}-\frac {9\,\ln \left (a\,x+1\right )}{16\,a^7}-\frac {x}{a^6}-\frac {\frac {3\,x}{8}+\frac {9\,a\,x^2}{8}-\frac {5}{4\,a}}{-a^9\,x^3+a^8\,x^2+a^7\,x-a^6}-\frac {x^2}{2\,a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 82, normalized size = 0.93 \[ - \frac {- 9 a^{2} x^{2} - 3 a x + 10}{8 a^{10} x^{3} - 8 a^{9} x^{2} - 8 a^{8} x + 8 a^{7}} - \frac {x^{2}}{2 a^{5}} - \frac {x}{a^{6}} - \frac {3 \left (\frac {13 \log {\left (x - \frac {1}{a} \right )}}{16} + \frac {3 \log {\left (x + \frac {1}{a} \right )}}{16}\right )}{a^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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