Optimal. Leaf size=30 \[ \frac {1}{a^2 c (1-a x)}+\frac {\log (1-a x)}{a^2 c} \]
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Rubi [A] time = 0.07, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6150, 43} \[ \frac {1}{a^2 c (1-a x)}+\frac {\log (1-a x)}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx &=\frac {\int \frac {x}{(1-a x)^2} \, dx}{c}\\ &=\frac {\int \left (\frac {1}{a (-1+a x)^2}+\frac {1}{a (-1+a x)}\right ) \, dx}{c}\\ &=\frac {1}{a^2 c (1-a x)}+\frac {\log (1-a x)}{a^2 c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 0.77 \[ \frac {\frac {1}{1-a x}+\log (1-a x)}{a^2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 30, normalized size = 1.00 \[ \frac {{\left (a x - 1\right )} \log \left (a x - 1\right ) - 1}{a^{3} c x - a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 30, normalized size = 1.00 \[ \frac {\log \left ({\left | a x - 1 \right |}\right )}{a^{2} c} - \frac {1}{{\left (a x - 1\right )} a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 30, normalized size = 1.00 \[ -\frac {1}{c \,a^{2} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{c \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 31, normalized size = 1.03 \[ -\frac {1}{a^{3} c x - a^{2} c} + \frac {\log \left (a x - 1\right )}{a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 27, normalized size = 0.90 \[ \frac {1}{a^2\,\left (c-a\,c\,x\right )}+\frac {\ln \left (a\,x-1\right )}{a^2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 24, normalized size = 0.80 \[ - \frac {1}{a^{3} c x - a^{2} c} + \frac {\log {\left (a x - 1 \right )}}{a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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