Optimal. Leaf size=48 \[ -\frac {4}{a c (1-a x)}+\frac {2}{a c (1-a x)^2}-\frac {\log (1-a x)}{a c} \]
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Rubi [A] time = 0.07, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ -\frac {4}{a c (1-a x)}+\frac {2}{a c (1-a x)^2}-\frac {\log (1-a x)}{a c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{c-a c x} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{c-a c x} \, dx\\ &=\frac {\int \frac {(1+a x)^2}{(1-a x)^3} \, dx}{c}\\ &=\frac {\int \left (\frac {1}{1-a x}-\frac {4}{(-1+a x)^3}-\frac {4}{(-1+a x)^2}\right ) \, dx}{c}\\ &=\frac {2}{a c (1-a x)^2}-\frac {4}{a c (1-a x)}-\frac {\log (1-a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 0.75 \[ \frac {4 a x+(a x-1)^2 (-\log (1-a x))-2}{a c (a x-1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 49, normalized size = 1.02 \[ \frac {4 \, a x - {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) - 2}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 57, normalized size = 1.19 \[ \frac {\log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c} + \frac {2 \, {\left (\frac {2 \, a c}{a x - 1} + \frac {a c}{{\left (a x - 1\right )}^{2}}\right )}}{a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 46, normalized size = 0.96 \[ \frac {2}{c a \left (a x -1\right )^{2}}-\frac {\ln \left (a x -1\right )}{c a}+\frac {4}{c a \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 44, normalized size = 0.92 \[ \frac {2 \, {\left (2 \, a x - 1\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} - \frac {\log \left (a x - 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 42, normalized size = 0.88 \[ \frac {4\,x-\frac {2}{a}}{c\,a^2\,x^2-2\,c\,a\,x+c}-\frac {\ln \left (a\,x-1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 37, normalized size = 0.77 \[ - \frac {- 4 a x + 2}{a^{3} c x^{2} - 2 a^{2} c x + a c} - \frac {\log {\left (a x - 1 \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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