Optimal. Leaf size=39 \[ -\frac{2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (a x+1)+\frac{a}{x^2}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.0317947, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {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 6126
Rule 77
Rubi steps
\begin{align*} \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*}
Mathematica [A] time = 0.0117499, size = 39, normalized size = 1. \[ -\frac{2 a^2}{x}-2 a^3 \log (x)+2 a^3 \log (a x+1)+\frac{a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 38, normalized size = 1. \begin{align*} -{\frac{1}{3\,{x}^{3}}}+{\frac{a}{{x}^{2}}}-2\,{\frac{{a}^{2}}{x}}-2\,{a}^{3}\ln \left ( x \right ) +2\,{a}^{3}\ln \left ( ax+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949299, size = 51, normalized size = 1.31 \begin{align*} 2 \, a^{3} \log \left (a x + 1\right ) - 2 \, a^{3} \log \left (x\right ) - \frac{6 \, a^{2} x^{2} - 3 \, a x + 1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82843, size = 104, normalized size = 2.67 \begin{align*} \frac{6 \, a^{3} x^{3} \log \left (a x + 1\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) - 6 \, a^{2} x^{2} + 3 \, a x - 1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.603, size = 36, normalized size = 0.92 \begin{align*} - 2 a^{3} \left (\log{\left (x \right )} - \log{\left (x + \frac{1}{a} \right )}\right ) - \frac{6 a^{2} x^{2} - 3 a x + 1}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19352, size = 84, normalized size = 2.15 \begin{align*} -2 \, a^{3} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right ) + \frac{10 \, a^{3} - \frac{24 \, a^{3}}{a x + 1} + \frac{15 \, a^{3}}{{\left (a x + 1\right )}^{2}}}{3 \,{\left (\frac{1}{a x + 1} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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