Optimal. Leaf size=40 \[ 2 a^3 \log (x)-2 a^3 \log (a x+1)+\frac {2 a^2}{x}-\frac {a}{x^2}+\frac {1}{3 x^3} \]
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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} \[ \frac {2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (a x+1)-\frac {a}{x^2}+\frac {1}{3 x^3} \]
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 \[ 2 a^3 \log (x)-2 a^3 \log (a x+1)+\frac {2 a^2}{x}-\frac {a}{x^2}+\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, 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 = 0.15, 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 = 39, normalized size = 0.98 \[ \frac {1}{3 x^{3}}-\frac {a}{x^{2}}+\frac {2 a^{2}}{x}+2 a^{3} \ln \relax (x )-2 a^{3} \ln \left (a x +1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, 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 = 1.21, size = 31, normalized size = 0.78 \[ \frac {2\,a^2\,x^2-a\,x+\frac {1}{3}}{x^3}-4\,a^3\,\mathrm {atanh}\left (2\,a\,x+1\right ) \]
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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