Optimal. Leaf size=42 \[ a^3 \log (x)-a^3 \log (1-a x)-\frac {a^2}{x}-\frac {a}{2 x^2}-\frac {1}{3 x^3} \]
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Rubi [A] time = 0.09, antiderivative size = 42, normalized size of antiderivative = 1.00, 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 44
Rule 6150
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*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 1.00 \[ a^3 \log (x)-a^3 \log (1-a x)-\frac {a^2}{x}-\frac {a}{2 x^2}-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 43, normalized size = 1.02 \[ -\frac {6 \, a^{3} x^{3} \log \left (a x - 1\right ) - 6 \, a^{3} x^{3} \log \relax (x) + 6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 39, normalized size = 0.93 \[ -a^{3} \log \left ({\left | a x - 1 \right |}\right ) + a^{3} \log \left ({\left | x \right |}\right ) - \frac {6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 38, normalized size = 0.90 \[ -\frac {1}{3 x^{3}}-\frac {a}{2 x^{2}}-\frac {a^{2}}{x}+a^{3} \ln \relax (x )-a^{3} \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 37, normalized size = 0.88 \[ -a^{3} \log \left (a x - 1\right ) + a^{3} \log \relax (x) - \frac {6 \, a^{2} x^{2} + 3 \, a x + 2}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 31, normalized size = 0.74 \[ 2\,a^3\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {a^2\,x^2+\frac {a\,x}{2}+\frac {1}{3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 34, normalized size = 0.81 \[ - a^{3} \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {6 a^{2} x^{2} + 3 a x + 2}{6 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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