Optimal. Leaf size=53 \[ \frac {5}{a c (1-a x)}-\frac {1}{a c (1-a x)^2}+\frac {4 \log (1-a x)}{a c}+\frac {x}{c} \]
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Rubi [A] time = 0.13, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6157, 6150, 77} \[ \frac {5}{a c (1-a x)}-\frac {1}{a c (1-a x)^2}+\frac {4 \log (1-a x)}{a c}+\frac {x}{c} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6150
Rule 6157
Rubi steps
\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {a^2 \int \frac {e^{4 \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \int \frac {x^2 (1+a x)}{(1-a x)^3} \, dx}{c}\\ &=-\frac {a^2 \int \left (-\frac {1}{a^2}-\frac {2}{a^2 (-1+a x)^3}-\frac {5}{a^2 (-1+a x)^2}-\frac {4}{a^2 (-1+a x)}\right ) \, dx}{c}\\ &=\frac {x}{c}-\frac {1}{a c (1-a x)^2}+\frac {5}{a c (1-a x)}+\frac {4 \log (1-a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.00 \[ \frac {5}{a c (1-a x)}-\frac {1}{a c (1-a x)^2}+\frac {4 \log (1-a x)}{a c}+\frac {x}{c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 64, normalized size = 1.21 \[ \frac {a^{3} x^{3} - 2 \, a^{2} x^{2} - 4 \, a x + 4 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (a x - 1\right ) + 4}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 42, normalized size = 0.79 \[ \frac {x}{c} + \frac {4 \, \log \left ({\left | a x - 1 \right |}\right )}{a c} - \frac {5 \, a x - 4}{{\left (a x - 1\right )}^{2} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 51, normalized size = 0.96 \[ \frac {x}{c}-\frac {5}{c a \left (a x -1\right )}+\frac {4 \ln \left (a x -1\right )}{a c}-\frac {1}{a c \left (a x -1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 49, normalized size = 0.92 \[ -\frac {5 \, a x - 4}{a^{3} c x^{2} - 2 \, a^{2} c x + a c} + \frac {x}{c} + \frac {4 \, \log \left (a x - 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 48, normalized size = 0.91 \[ \frac {x}{c}-\frac {5\,x-\frac {4}{a}}{c\,a^2\,x^2-2\,c\,a\,x+c}+\frac {4\,\ln \left (a\,x-1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 41, normalized size = 0.77 \[ \frac {- 5 a x + 4}{a^{3} c x^{2} - 2 a^{2} c x + a c} + \frac {x}{c} + \frac {4 \log {\left (a x - 1 \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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