Optimal. Leaf size=42 \[ -\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.090742, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 44} \[ -\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 44
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^4 \sqrt{1-a^2 x^2}} \, dx &=\int \frac{1}{x^4 (1-a x)} \, dx\\ &=\int \left (\frac{1}{x^4}+\frac{a}{x^3}+\frac{a^2}{x^2}+\frac{a^3}{x}-\frac{a^4}{-1+a x}\right ) \, dx\\ &=-\frac{1}{3 x^3}-\frac{a}{2 x^2}-\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)\\ \end{align*}
Mathematica [A] time = 0.015382, size = 42, normalized size = 1. \[ -\frac{a^2}{x}+a^3 \log (x)-a^3 \log (1-a x)-\frac{a}{2 x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 38, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-{\frac{a}{2\,{x}^{2}}}-{\frac{{a}^{2}}{x}}+{a}^{3}\ln \left ( x \right ) -{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 = 0.957407, size = 50, normalized size = 1.19 \begin{align*} -a^{3} \log \left (a x - 1\right ) + a^{3} \log \left (x\right ) - \frac{6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85445, size = 105, normalized size = 2.5 \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 + 2}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.340384, size = 34, normalized size = 0.81 \begin{align*} - a^{3} \left (- \log{\left (x \right )} + \log{\left (x - \frac{1}{a} \right )}\right ) - \frac{6 a^{2} x^{2} + 3 a x + 2}{6 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13409, size = 53, normalized size = 1.26 \begin{align*} -a^{3} \log \left ({\left | a x - 1 \right |}\right ) + a^{3} \log \left ({\left | x \right |}\right ) - \frac{6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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