Optimal. Leaf size=89 \[ \frac {a^2}{1-a x}+\frac {a^2}{8 (a x+1)}+\frac {a^2}{8 (1-a x)^2}+3 a^2 \log (x)-\frac {39}{16} a^2 \log (1-a x)-\frac {9}{16} a^2 \log (a x+1)-\frac {a}{x}-\frac {1}{2 x^2} \]
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Rubi [A] time = 0.14, antiderivative size = 89, 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} \[ \frac {a^2}{1-a x}+\frac {a^2}{8 (a x+1)}+\frac {a^2}{8 (1-a x)^2}+3 a^2 \log (x)-\frac {39}{16} a^2 \log (1-a x)-\frac {9}{16} a^2 \log (a x+1)-\frac {a}{x}-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac {1}{x^3 (1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {3 a^2}{x}-\frac {a^3}{4 (-1+a x)^3}+\frac {a^3}{(-1+a x)^2}-\frac {39 a^3}{16 (-1+a x)}-\frac {a^3}{8 (1+a x)^2}-\frac {9 a^3}{16 (1+a x)}\right ) \, dx\\ &=-\frac {1}{2 x^2}-\frac {a}{x}+\frac {a^2}{8 (1-a x)^2}+\frac {a^2}{1-a x}+\frac {a^2}{8 (1+a x)}+3 a^2 \log (x)-\frac {39}{16} a^2 \log (1-a x)-\frac {9}{16} a^2 \log (1+a x)\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.93 \[ \frac {1}{16} \left (\frac {16 a^2}{1-a x}+\frac {2 a^2}{a x+1}+\frac {2 a^2}{(a x-1)^2}+48 a^2 \log (x)-39 a^2 \log (1-a x)-9 a^2 \log (a x+1)-\frac {16 a}{x}-\frac {8}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 172, normalized size = 1.93 \[ -\frac {30 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 44 \, a^{2} x^{2} + 8 \, a x + 9 \, {\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 39 \, {\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 48 \, {\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \relax (x) + 8}{16 \, {\left (a^{3} x^{5} - a^{2} x^{4} - a x^{3} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 82, normalized size = 0.92 \[ -\frac {9}{16} \, a^{2} \log \left ({\left | a x + 1 \right |}\right ) - \frac {39}{16} \, a^{2} \log \left ({\left | a x - 1 \right |}\right ) + 3 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac {15 \, a^{4} x^{4} - 3 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 4}{8 \, {\left (a x + 1\right )} {\left (a x - 1\right )}^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 78, normalized size = 0.88 \[ -\frac {1}{2 x^{2}}-\frac {a}{x}+3 a^{2} \ln \relax (x )-\frac {a^{2}}{a x -1}+\frac {a^{2}}{8 \left (a x -1\right )^{2}}-\frac {39 a^{2} \ln \left (a x -1\right )}{16}+\frac {a^{2}}{8 a x +8}-\frac {9 a^{2} \ln \left (a x +1\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 89, normalized size = 1.00 \[ -\frac {9}{16} \, a^{2} \log \left (a x + 1\right ) - \frac {39}{16} \, a^{2} \log \left (a x - 1\right ) + 3 \, a^{2} \log \relax (x) - \frac {15 \, a^{4} x^{4} - 3 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 4}{8 \, {\left (a^{3} x^{5} - a^{2} x^{4} - a x^{3} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 89, normalized size = 1.00 \[ 3\,a^2\,\ln \relax (x)-\frac {39\,a^2\,\ln \left (a\,x-1\right )}{16}-\frac {9\,a^2\,\ln \left (a\,x+1\right )}{16}+\frac {\frac {15\,a^4\,x^4}{8}-\frac {3\,a^3\,x^3}{8}-\frac {11\,a^2\,x^2}{4}+\frac {a\,x}{2}+\frac {1}{2}}{-a^3\,x^5+a^2\,x^4+a\,x^3-x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 95, normalized size = 1.07 \[ 3 a^{2} \log {\relax (x )} - \frac {39 a^{2} \log {\left (x - \frac {1}{a} \right )}}{16} - \frac {9 a^{2} \log {\left (x + \frac {1}{a} \right )}}{16} - \frac {15 a^{4} x^{4} - 3 a^{3} x^{3} - 22 a^{2} x^{2} + 4 a x + 4}{8 a^{3} x^{5} - 8 a^{2} x^{4} - 8 a x^{3} + 8 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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