Optimal. Leaf size=25 \[ -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \]
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Rubi [A] time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6131, 6129, 72} \[ -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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Rule 72
Rule 6129
Rule 6131
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\frac {c \int \frac {e^{4 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=-\frac {c \int \frac {(1+a x)^2}{x (1-a x)} \, dx}{a}\\ &=-\frac {c \int \left (-a+\frac {1}{x}-\frac {4 a}{-1+a x}\right ) \, dx}{a}\\ &=c x-\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 25, normalized size = 1.00 \[ -\frac {c \log (x)}{a}+\frac {4 c \log (1-a x)}{a}+c x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 23, normalized size = 0.92 \[ \frac {a c x + 4 \, c \log \left (a x - 1\right ) - c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 26, normalized size = 1.04 \[ c x + \frac {4 \, c \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac {c \log \left ({\left | x \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 1.00 \[ c x -\frac {c \ln \relax (x )}{a}+\frac {4 c \ln \left (a x -1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 24, normalized size = 0.96 \[ c x + \frac {4 \, c \log \left (a x - 1\right )}{a} - \frac {c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 24, normalized size = 0.96 \[ c\,x-\frac {c\,\ln \relax (x)}{a}+\frac {4\,c\,\ln \left (a\,x-1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 17, normalized size = 0.68 \[ c x + \frac {c \left (- \log {\relax (x )} + 4 \log {\left (x - \frac {1}{a} \right )}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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