Optimal. Leaf size=54 \[ \frac {4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac {8 a^2}{x}-\frac {2 a}{x^2}-\frac {1}{3 x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6167, 6126, 88} \[ \frac {4 a^3}{1-a x}-\frac {8 a^2}{x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac {2 a}{x^2}-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{x^4} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{x^4} \, dx\\ &=\int \frac {(1+a x)^2}{x^4 (1-a x)^2} \, dx\\ &=\int \left (\frac {1}{x^4}+\frac {4 a}{x^3}+\frac {8 a^2}{x^2}+\frac {12 a^3}{x}+\frac {4 a^4}{(-1+a x)^2}-\frac {12 a^4}{-1+a x}\right ) \, dx\\ &=-\frac {1}{3 x^3}-\frac {2 a}{x^2}-\frac {8 a^2}{x}+\frac {4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 54, normalized size = 1.00 \[ \frac {4 a^3}{1-a x}+12 a^3 \log (x)-12 a^3 \log (1-a x)-\frac {8 a^2}{x}-\frac {2 a}{x^2}-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 81, normalized size = 1.50 \[ -\frac {36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x + 36 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (a x - 1\right ) - 36 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \relax (x) - 1}{3 \, {\left (a x^{4} - x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 74, normalized size = 1.37 \[ 12 \, a^{3} \log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right ) - \frac {4 \, a^{3}}{a x - 1} + \frac {31 \, a^{3} + \frac {69 \, a^{3}}{a x - 1} + \frac {39 \, a^{3}}{{\left (a x - 1\right )}^{2}}}{3 \, {\left (\frac {1}{a x - 1} + 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 0.94 \[ -\frac {1}{3 x^{3}}-\frac {2 a}{x^{2}}-\frac {8 a^{2}}{x}+12 a^{3} \ln \relax (x )-\frac {4 a^{3}}{a x -1}-12 a^{3} \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 56, normalized size = 1.04 \[ -12 \, a^{3} \log \left (a x - 1\right ) + 12 \, a^{3} \log \relax (x) - \frac {36 \, a^{3} x^{3} - 18 \, a^{2} x^{2} - 5 \, a x - 1}{3 \, {\left (a x^{4} - x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 49, normalized size = 0.91 \[ 24\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right )+\frac {-12\,a^3\,x^3+6\,a^2\,x^2+\frac {5\,a\,x}{3}+\frac {1}{3}}{a\,x^4-x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 49, normalized size = 0.91 \[ 12 a^{3} \left (\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {- 36 a^{3} x^{3} + 18 a^{2} x^{2} + 5 a x + 1}{3 a x^{4} - 3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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