Optimal. Leaf size=53 \[ \frac {8}{a c (1-a x)}-\frac {2}{a c (1-a x)^2}+\frac {5 \log (1-a x)}{a c}+\frac {x}{c} \]
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Rubi [A] time = 0.13, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6167, 6131, 6129, 77} \[ \frac {8}{a c (1-a x)}-\frac {2}{a c (1-a x)^2}+\frac {5 \log (1-a x)}{a c}+\frac {x}{c} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{4 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=\int \frac {e^{4 \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\\ &=-\frac {a \int \frac {e^{4 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=-\frac {a \int \frac {x (1+a x)^2}{(1-a x)^3} \, dx}{c}\\ &=-\frac {a \int \left (-\frac {1}{a}-\frac {4}{a (-1+a x)^3}-\frac {8}{a (-1+a x)^2}-\frac {5}{a (-1+a x)}\right ) \, dx}{c}\\ &=\frac {x}{c}-\frac {2}{a c (1-a x)^2}+\frac {8}{a c (1-a x)}+\frac {5 \log (1-a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.96 \[ -\frac {a \left (-\frac {8}{a^2 (1-a x)}+\frac {2}{a^2 (1-a x)^2}-\frac {5 \log (1-a x)}{a^2}-\frac {x}{a}\right )}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 64, normalized size = 1.21 \[ \frac {a^{3} x^{3} - 2 \, a^{2} x^{2} - 7 \, a x + 5 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 6}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 74, normalized size = 1.40 \[ \frac {a x - 1}{a c} - \frac {5 \, \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a c} - \frac {2 \, {\left (\frac {4 \, a^{3} c}{a x - 1} + \frac {a^{3} c}{{\left (a x - 1\right )}^{2}}\right )}}{a^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 51, normalized size = 0.96 \[ \frac {x}{c}+\frac {5 \ln \left (a x -1\right )}{c a}-\frac {8}{c a \left (a x -1\right )}-\frac {2}{c a \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 49, normalized size = 0.92 \[ -\frac {2 \, {\left (4 \, a x - 3\right )}}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} + \frac {x}{c} + \frac {5 \, \log \left (a x - 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 48, normalized size = 0.91 \[ \frac {x}{c}-\frac {8\,x-\frac {6}{a}}{c\,a^2\,x^2-2\,c\,a\,x+c}+\frac {5\,\ln \left (a\,x-1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 41, normalized size = 0.77 \[ \frac {- 8 a x + 6}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac {x}{c} + \frac {5 \log {\left (a x - 1 \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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