Optimal. Leaf size=102 \[ \frac {5 a^3}{4 (1-a x)}-\frac {a^3}{8 (a x+1)}+\frac {a^3}{8 (1-a x)^2}+3 a^3 \log (x)-\frac {59}{16} a^3 \log (1-a x)+\frac {11}{16} a^3 \log (a x+1)-\frac {3 a^2}{x}-\frac {a}{2 x^2}-\frac {1}{3 x^3} \]
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Rubi [A] time = 0.15, antiderivative size = 102, 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} \[ \frac {5 a^3}{4 (1-a x)}-\frac {a^3}{8 (a x+1)}+\frac {a^3}{8 (1-a x)^2}-\frac {3 a^2}{x}+3 a^3 \log (x)-\frac {59}{16} a^3 \log (1-a x)+\frac {11}{16} a^3 \log (a x+1)-\frac {a}{2 x^2}-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
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 {1}{x^4 (1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (\frac {1}{x^4}+\frac {a}{x^3}+\frac {3 a^2}{x^2}+\frac {3 a^3}{x}-\frac {a^4}{4 (-1+a x)^3}+\frac {5 a^4}{4 (-1+a x)^2}-\frac {59 a^4}{16 (-1+a x)}+\frac {a^4}{8 (1+a x)^2}+\frac {11 a^4}{16 (1+a x)}\right ) \, dx\\ &=-\frac {1}{3 x^3}-\frac {a}{2 x^2}-\frac {3 a^2}{x}+\frac {a^3}{8 (1-a x)^2}+\frac {5 a^3}{4 (1-a x)}-\frac {a^3}{8 (1+a x)}+3 a^3 \log (x)-\frac {59}{16} a^3 \log (1-a x)+\frac {11}{16} a^3 \log (1+a x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 91, normalized size = 0.89 \[ \frac {1}{48} \left (\frac {60 a^3}{1-a x}-\frac {6 a^3}{a x+1}+\frac {6 a^3}{(a x-1)^2}+144 a^3 \log (x)-177 a^3 \log (1-a x)+33 a^3 \log (a x+1)-\frac {144 a^2}{x}-\frac {24 a}{x^2}-\frac {16}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 180, normalized size = 1.76 \[ -\frac {210 \, a^{5} x^{5} - 138 \, a^{4} x^{4} - 212 \, a^{3} x^{3} + 104 \, a^{2} x^{2} + 8 \, a x - 33 \, {\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (a x + 1\right ) + 177 \, {\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 144 \, {\left (a^{6} x^{6} - a^{5} x^{5} - a^{4} x^{4} + a^{3} x^{3}\right )} \log \relax (x) + 16}{48 \, {\left (a^{3} x^{6} - a^{2} x^{5} - a x^{4} + x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 90, normalized size = 0.88 \[ \frac {11}{16} \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) - \frac {59}{16} \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 3 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac {105 \, a^{5} x^{5} - 69 \, a^{4} x^{4} - 106 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 4 \, a x + 8}{24 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 86, normalized size = 0.84 \[ -\frac {1}{3 x^{3}}-\frac {a}{2 x^{2}}-\frac {3 a^{2}}{x}+3 a^{3} \ln \relax (x )+\frac {a^{3}}{8 \left (a x -1\right )^{2}}-\frac {5 a^{3}}{4 \left (a x -1\right )}-\frac {59 a^{3} \ln \left (a x -1\right )}{16}-\frac {a^{3}}{8 \left (a x +1\right )}+\frac {11 a^{3} \ln \left (a x +1\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 97, normalized size = 0.95 \[ \frac {11}{16} \, a^{3} \log \left (a x + 1\right ) - \frac {59}{16} \, a^{3} \log \left (a x - 1\right ) + 3 \, a^{3} \log \relax (x) - \frac {105 \, a^{5} x^{5} - 69 \, a^{4} x^{4} - 106 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 4 \, a x + 8}{24 \, {\left (a^{3} x^{6} - a^{2} x^{5} - a x^{4} + x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 97, normalized size = 0.95 \[ 3\,a^3\,\ln \relax (x)-\frac {59\,a^3\,\ln \left (a\,x-1\right )}{16}+\frac {11\,a^3\,\ln \left (a\,x+1\right )}{16}+\frac {\frac {35\,a^5\,x^5}{8}-\frac {23\,a^4\,x^4}{8}-\frac {53\,a^3\,x^3}{12}+\frac {13\,a^2\,x^2}{6}+\frac {a\,x}{6}+\frac {1}{3}}{-a^3\,x^6+a^2\,x^5+a\,x^4-x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 104, normalized size = 1.02 \[ 3 a^{3} \log {\relax (x )} - \frac {59 a^{3} \log {\left (x - \frac {1}{a} \right )}}{16} + \frac {11 a^{3} \log {\left (x + \frac {1}{a} \right )}}{16} - \frac {105 a^{5} x^{5} - 69 a^{4} x^{4} - 106 a^{3} x^{3} + 52 a^{2} x^{2} + 4 a x + 8}{24 a^{3} x^{6} - 24 a^{2} x^{5} - 24 a x^{4} + 24 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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