Optimal. Leaf size=40 \[ -2 a^3 \log (x)+2 a^3 \log (1-a x)+\frac {2 a^2}{x}+\frac {a}{x^2}+\frac {1}{3 x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 40, 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, 77} \[ \frac {2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (1-a x)+\frac {a}{x^2}+\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6126
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{x^4} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{x^4} \, dx\\ &=-\int \frac {1+a x}{x^4 (1-a x)} \, dx\\ &=-\int \left (\frac {1}{x^4}+\frac {2 a}{x^3}+\frac {2 a^2}{x^2}+\frac {2 a^3}{x}-\frac {2 a^4}{-1+a x}\right ) \, dx\\ &=\frac {1}{3 x^3}+\frac {a}{x^2}+\frac {2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (1-a x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.00 \[ -2 a^3 \log (x)+2 a^3 \log (1-a x)+\frac {2 a^2}{x}+\frac {a}{x^2}+\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 43, normalized size = 1.08 \[ \frac {6 \, a^{3} x^{3} \log \left (a x - 1\right ) - 6 \, a^{3} x^{3} \log \relax (x) + 6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 40, normalized size = 1.00 \[ 2 \, a^{3} \log \left ({\left | a x - 1 \right |}\right ) - 2 \, a^{3} \log \left ({\left | x \right |}\right ) + \frac {6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 0.95 \[ \frac {1}{3 x^{3}}+\frac {a}{x^{2}}+\frac {2 a^{2}}{x}-2 a^{3} \ln \relax (x )+2 a^{3} \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 38, normalized size = 0.95 \[ 2 \, a^{3} \log \left (a x - 1\right ) - 2 \, a^{3} \log \relax (x) + \frac {6 \, a^{2} x^{2} + 3 \, a x + 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 30, normalized size = 0.75 \[ \frac {2\,a^2\,x^2+a\,x+\frac {1}{3}}{x^3}-4\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 34, normalized size = 0.85 \[ 2 a^{3} \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{a} \right )}\right ) + \frac {6 a^{2} x^{2} + 3 a x + 1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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