Optimal. Leaf size=41 \[ -\frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac{a}{x^2}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.0356681, antiderivative size = 41, 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} \[ -\frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac{a}{x^2}-\frac{1}{3 x^3} \]
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 = 0.0131947, size = 41, normalized size = 1. \[ -\frac{2 a^2}{x}+2 a^3 \log (x)-2 a^3 \log (1-a x)-\frac{a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, 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 = 0.951531, size = 51, normalized size = 1.24 \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 = 1.69405, size = 105, normalized size = 2.56 \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 = 0.485669, size = 36, normalized size = 0.88 \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 = 1.14824, size = 54, normalized size = 1.32 \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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