Optimal. Leaf size=73 \[ -\frac {1}{3 x^3}-\frac {a}{2 x^2}-\frac {2 a^2}{x}+\frac {a^3}{2 (1-a x)}+2 a^3 \log (x)-\frac {9}{4} a^3 \log (1-a x)+\frac {1}{4} a^3 \log (1+a x) \]
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Rubi [A]
time = 0.09, antiderivative size = 73, 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^3}{2 (1-a x)}+2 a^3 \log (x)-\frac {9}{4} a^3 \log (1-a x)+\frac {1}{4} a^3 \log (a x+1)-\frac {2 a^2}{x}-\frac {a}{2 x^2}-\frac {1}{3 x^3} \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 )^{3/2}} \, dx &=\int \frac {1}{x^4 (1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac {1}{x^4}+\frac {a}{x^3}+\frac {2 a^2}{x^2}+\frac {2 a^3}{x}+\frac {a^4}{2 (-1+a x)^2}-\frac {9 a^4}{4 (-1+a x)}+\frac {a^4}{4 (1+a x)}\right ) \, dx\\ &=-\frac {1}{3 x^3}-\frac {a}{2 x^2}-\frac {2 a^2}{x}+\frac {a^3}{2 (1-a x)}+2 a^3 \log (x)-\frac {9}{4} a^3 \log (1-a x)+\frac {1}{4} a^3 \log (1+a x)\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 67, normalized size = 0.92 \begin {gather*} \frac {1}{12} \left (-\frac {4}{x^3}-\frac {6 a}{x^2}-\frac {24 a^2}{x}+\frac {6 a^3}{1-a x}+24 a^3 \log (x)-27 a^3 \log (1-a x)+3 a^3 \log (1+a x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 62, normalized size = 0.85
method | result | size |
default | \(\frac {a^{3} \ln \left (a x +1\right )}{4}-\frac {1}{3 x^{3}}-\frac {a}{2 x^{2}}-\frac {2 a^{2}}{x}+2 a^{3} \ln \left (x \right )-\frac {a^{3}}{2 \left (a x -1\right )}-\frac {9 a^{3} \ln \left (a x -1\right )}{4}\) | \(62\) |
risch | \(\frac {-\frac {5}{2} a^{3} x^{3}+\frac {3}{2} a^{2} x^{2}+\frac {1}{6} a x +\frac {1}{3}}{\left (a x -1\right ) x^{3}}+2 a^{3} \ln \left (-x \right )+\frac {a^{3} \ln \left (a x +1\right )}{4}-\frac {9 a^{3} \ln \left (-a x +1\right )}{4}\) | \(67\) |
norman | \(\frac {\frac {1}{3}-x^{5} a^{5}+\frac {1}{2} a x +\frac {5}{3} a^{2} x^{2}-\frac {5}{2} a^{4} x^{4}}{\left (a^{2} x^{2}-1\right ) x^{3}}+2 a^{3} \ln \left (x \right )-\frac {9 a^{3} \ln \left (a x -1\right )}{4}+\frac {a^{3} \ln \left (a x +1\right )}{4}\) | \(76\) |
meijerg | \(-\frac {a^{3} \left (-\frac {3 a^{2} x^{2}}{-3 a^{2} x^{2}+3}+2 \ln \left (-a^{2} x^{2}+1\right )-1-4 \ln \left (x \right )-2 \ln \left (-a^{2}\right )+\frac {1}{a^{2} x^{2}}\right )}{2}+\frac {a^{4} \left (-\frac {2 \left (-15 a^{4} x^{4}+10 a^{2} x^{2}+2\right )}{3 x^{3} \left (-a^{2}\right )^{\frac {3}{2}} \left (-2 a^{2} x^{2}+2\right )}+\frac {5 a^{3} \arctanh \left (a x \right )}{\left (-a^{2}\right )^{\frac {3}{2}}}\right )}{2 \sqrt {-a^{2}}}\) | \(132\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 67, normalized size = 0.92 \begin {gather*} \frac {1}{4} \, a^{3} \log \left (a x + 1\right ) - \frac {9}{4} \, a^{3} \log \left (a x - 1\right ) + 2 \, a^{3} \log \left (x\right ) - \frac {15 \, a^{3} x^{3} - 9 \, a^{2} x^{2} - a x - 2}{6 \, {\left (a x^{4} - x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 105, normalized size = 1.44 \begin {gather*} -\frac {30 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 2 \, a x - 3 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x + 1\right ) + 27 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 24 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 4}{12 \, {\left (a x^{4} - x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 66, normalized size = 0.90 \begin {gather*} 2 a^{3} \log {\left (x \right )} - \frac {9 a^{3} \log {\left (x - \frac {1}{a} \right )}}{4} + \frac {a^{3} \log {\left (x + \frac {1}{a} \right )}}{4} + \frac {- 15 a^{3} x^{3} + 9 a^{2} x^{2} + a x + 2}{6 a x^{4} - 6 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 67, normalized size = 0.92 \begin {gather*} \frac {1}{4} \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) - \frac {9}{4} \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a^{3} \log \left ({\left | x \right |}\right ) - \frac {15 \, a^{3} x^{3} - 9 \, a^{2} x^{2} - a x - 2}{6 \, {\left (a x - 1\right )} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.91, size = 66, normalized size = 0.90 \begin {gather*} 2\,a^3\,\ln \left (x\right )-\frac {9\,a^3\,\ln \left (a\,x-1\right )}{4}+\frac {a^3\,\ln \left (a\,x+1\right )}{4}+\frac {-\frac {5\,a^3\,x^3}{2}+\frac {3\,a^2\,x^2}{2}+\frac {a\,x}{6}+\frac {1}{3}}{a\,x^4-x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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