Optimal. Leaf size=40 \[ \frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (a x+1)-\frac{a}{x^2}+\frac{1}{3 x^3} \]
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Rubi [A] time = 0.0496812, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 6167
Rule 6126
Rule 77
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*}
Mathematica [A] time = 0.0132844, size = 40, 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 = 0.046, size = 39, 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 = 0.985085, size = 51, normalized size = 1.27 \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 = 1.47225, size = 105, normalized size = 2.62 \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 = 0.360706, size = 34, normalized size = 0.85 \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 = 1.15059, size = 54, normalized size = 1.35 \begin{align*} -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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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