Optimal. Leaf size=72 \[ \frac {1}{8 a^5 (1-a x)^2}-\frac {3}{4 a^5 (1-a x)}-\frac {1}{8 a^5 (1+a x)}-\frac {11 \log (1-a x)}{16 a^5}-\frac {5 \log (1+a x)}{16 a^5} \]
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Rubi [A]
time = 0.09, antiderivative size = 72, 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 = {6285, 90}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
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*}
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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.76 \begin {gather*} \frac {\frac {2 \left (-6+3 a x+5 a^2 x^2\right )}{(-1+a x)^2 (1+a x)}-11 \log (1-a x)-5 \log (1+a x)}{16 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 60, normalized size = 0.83
method | result | size |
default | \(-\frac {1}{8 a^{5} \left (a x +1\right )}-\frac {5 \ln \left (a x +1\right )}{16 a^{5}}+\frac {1}{8 a^{5} \left (a x -1\right )^{2}}+\frac {3}{4 a^{5} \left (a x -1\right )}-\frac {11 \ln \left (a x -1\right )}{16 a^{5}}\) | \(60\) |
norman | \(\frac {-\frac {3 x}{8 a^{4}}+\frac {5 x^{3}}{8 a^{2}}-\frac {1}{4 a^{5}}+\frac {x^{4}}{2 a}}{\left (a^{2} x^{2}-1\right )^{2}}-\frac {11 \ln \left (a x -1\right )}{16 a^{5}}-\frac {5 \ln \left (a x +1\right )}{16 a^{5}}\) | \(64\) |
risch | \(\frac {\frac {5 x^{2}}{8 a^{3}}+\frac {3 x}{8 a^{4}}-\frac {3}{4 a^{5}}}{\left (a x -1\right ) \left (a^{2} x^{2}-1\right )}-\frac {11 \ln \left (a x -1\right )}{16 a^{5}}-\frac {5 \ln \left (-a x -1\right )}{16 a^{5}}\) | \(64\) |
meijerg | \(\frac {-\frac {x \left (-a^{2}\right )^{\frac {5}{2}} \left (-25 a^{2} x^{2}+15\right )}{10 a^{4} \left (-a^{2} x^{2}+1\right )^{2}}+\frac {3 \left (-a^{2}\right )^{\frac {5}{2}} \arctanh \left (a x \right )}{2 a^{5}}}{4 a^{4} \sqrt {-a^{2}}}-\frac {\frac {x^{2} a^{2} \left (-9 a^{2} x^{2}+6\right )}{3 \left (-a^{2} x^{2}+1\right )^{2}}+2 \ln \left (-a^{2} x^{2}+1\right )}{4 a^{5}}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 66, normalized size = 0.92 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 101, normalized size = 1.40 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 68, normalized size = 0.94 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 58, normalized size = 0.81 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 67, normalized size = 0.93 \begin {gather*} -\frac {11\,\ln \left (a\,x-1\right )}{16\,a^5}-\frac {5\,\ln \left (a\,x+1\right )}{16\,a^5}-\frac {\frac {3\,x}{8\,a^4}-\frac {3}{4\,a^5}+\frac {5\,x^2}{8\,a^3}}{-a^3\,x^3+a^2\,x^2+a\,x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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