Optimal. Leaf size=60 \[ \frac {a^2}{c (1-a x)}+\frac {3 a^2 \log (x)}{c}-\frac {3 a^2 \log (1-a x)}{c}-\frac {2 a}{c x}-\frac {1}{2 c x^2} \]
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Rubi [A] time = 0.10, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 44} \[ \frac {a^2}{c (1-a x)}+\frac {3 a^2 \log (x)}{c}-\frac {3 a^2 \log (1-a x)}{c}-\frac {2 a}{c x}-\frac {1}{2 c x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx &=\frac {\int \frac {1}{x^3 (1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {1}{x^3}+\frac {2 a}{x^2}+\frac {3 a^2}{x}+\frac {a^3}{(-1+a x)^2}-\frac {3 a^3}{-1+a x}\right ) \, dx}{c}\\ &=-\frac {1}{2 c x^2}-\frac {2 a}{c x}+\frac {a^2}{c (1-a x)}+\frac {3 a^2 \log (x)}{c}-\frac {3 a^2 \log (1-a x)}{c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 52, normalized size = 0.87 \[ \frac {\frac {-6 a^2 x^2+3 a x+1}{x^2 (a x-1)}+6 a^2 \log (x)-6 a^2 \log (1-a x)}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 75, normalized size = 1.25 \[ -\frac {6 \, a^{2} x^{2} - 3 \, a x + 6 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 6 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \relax (x) - 1}{2 \, {\left (a c x^{3} - c x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 56, normalized size = 0.93 \[ -\frac {3 \, a^{2} \log \left ({\left | a x - 1 \right |}\right )}{c} + \frac {3 \, a^{2} \log \left ({\left | x \right |}\right )}{c} - \frac {6 \, a^{2} x^{2} - 3 \, a x - 1}{2 \, {\left (a x - 1\right )} c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 58, normalized size = 0.97 \[ -\frac {1}{2 c \,x^{2}}-\frac {2 a}{c x}+\frac {3 a^{2} \ln \relax (x )}{c}-\frac {a^{2}}{c \left (a x -1\right )}-\frac {3 a^{2} \ln \left (a x -1\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 56, normalized size = 0.93 \[ -\frac {3 \, a^{2} \log \left (a x - 1\right )}{c} + \frac {3 \, a^{2} \log \relax (x)}{c} - \frac {6 \, a^{2} x^{2} - 3 \, a x - 1}{2 \, {\left (a c x^{3} - c x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 47, normalized size = 0.78 \[ \frac {6\,a^2\,\mathrm {atanh}\left (2\,a\,x-1\right )}{c}-\frac {-3\,a^2\,x^2+\frac {3\,a\,x}{2}+\frac {1}{2}}{c\,x^2-a\,c\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 46, normalized size = 0.77 \[ \frac {3 a^{2} \left (\log {\relax (x )} - \log {\left (x - \frac {1}{a} \right )}\right )}{c} + \frac {- 6 a^{2} x^{2} + 3 a x + 1}{2 a c x^{3} - 2 c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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