Optimal. Leaf size=63 \[ -\frac {1}{2 x^2}-\frac {a}{x}+\frac {a^2}{2 (1-a x)}+2 a^2 \log (x)-\frac {7}{4} a^2 \log (1-a x)-\frac {1}{4} a^2 \log (1+a x) \]
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Rubi [A]
time = 0.08, antiderivative size = 63, 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 {a^2}{2 (1-a x)}+2 a^2 \log (x)-\frac {7}{4} a^2 \log (1-a x)-\frac {1}{4} a^2 \log (a x+1)-\frac {a}{x}-\frac {1}{2 x^2} \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^3 \left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{x^3 (1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {2 a^2}{x}+\frac {a^3}{2 (-1+a x)^2}-\frac {7 a^3}{4 (-1+a x)}-\frac {a^3}{4 (1+a x)}\right ) \, dx\\ &=-\frac {1}{2 x^2}-\frac {a}{x}+\frac {a^2}{2 (1-a x)}+2 a^2 \log (x)-\frac {7}{4} a^2 \log (1-a x)-\frac {1}{4} a^2 \log (1+a x)\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 59, normalized size = 0.94 \begin {gather*} \frac {1}{4} \left (-\frac {2}{x^2}-\frac {4 a}{x}+\frac {2 a^2}{1-a x}+8 a^2 \log (x)-7 a^2 \log (1-a x)-a^2 \log (1+a x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 54, normalized size = 0.86
method | result | size |
default | \(-\frac {a^{2} \ln \left (a x +1\right )}{4}-\frac {1}{2 x^{2}}-\frac {a}{x}+2 a^{2} \ln \left (x \right )-\frac {a^{2}}{2 \left (a x -1\right )}-\frac {7 a^{2} \ln \left (a x -1\right )}{4}\) | \(54\) |
risch | \(\frac {-\frac {3}{2} a^{2} x^{2}+\frac {1}{2} a x +\frac {1}{2}}{\left (a x -1\right ) x^{2}}-\frac {a^{2} \ln \left (a x +1\right )}{4}+2 a^{2} \ln \left (-x \right )-\frac {7 a^{2} \ln \left (-a x +1\right )}{4}\) | \(59\) |
norman | \(\frac {\frac {1}{2}+a x -a^{4} x^{4}-\frac {3}{2} a^{3} x^{3}}{\left (a^{2} x^{2}-1\right ) x^{2}}+2 a^{2} \ln \left (x \right )-\frac {7 a^{2} \ln \left (a x -1\right )}{4}-\frac {a^{2} \ln \left (a x +1\right )}{4}\) | \(67\) |
meijerg | \(-\frac {a^{3} \left (-\frac {2 \left (-3 a^{2} x^{2}+2\right )}{x \sqrt {-a^{2}}\, \left (-2 a^{2} x^{2}+2\right )}+\frac {3 a \arctanh \left (a x \right )}{\sqrt {-a^{2}}}\right )}{2 \sqrt {-a^{2}}}-\frac {a^{2} \left (-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{2}\) | \(122\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 59, normalized size = 0.94 \begin {gather*} -\frac {1}{4} \, a^{2} \log \left (a x + 1\right ) - \frac {7}{4} \, a^{2} \log \left (a x - 1\right ) + 2 \, a^{2} \log \left (x\right ) - \frac {3 \, a^{2} x^{2} - a x - 1}{2 \, {\left (a x^{3} - x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 96, normalized size = 1.52 \begin {gather*} -\frac {6 \, a^{2} x^{2} - 2 \, a x + {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 7 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 8 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (x\right ) - 2}{4 \, {\left (a x^{3} - x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 58, normalized size = 0.92 \begin {gather*} 2 a^{2} \log {\left (x \right )} - \frac {7 a^{2} \log {\left (x - \frac {1}{a} \right )}}{4} - \frac {a^{2} \log {\left (x + \frac {1}{a} \right )}}{4} + \frac {- 3 a^{2} x^{2} + a x + 1}{2 a x^{3} - 2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 59, normalized size = 0.94 \begin {gather*} -\frac {1}{4} \, a^{2} \log \left ({\left | a x + 1 \right |}\right ) - \frac {7}{4} \, a^{2} \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac {3 \, a^{2} x^{2} - a x - 1}{2 \, {\left (a x - 1\right )} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 58, normalized size = 0.92 \begin {gather*} 2\,a^2\,\ln \left (x\right )-\frac {7\,a^2\,\ln \left (a\,x-1\right )}{4}-\frac {a^2\,\ln \left (a\,x+1\right )}{4}+\frac {-\frac {3\,a^2\,x^2}{2}+\frac {a\,x}{2}+\frac {1}{2}}{a\,x^3-x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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