Optimal. Leaf size=37 \[ \frac {2}{a c (1-a x)}+\frac {3 \log (1-a x)}{a c}+\frac {x}{c} \]
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Rubi [A] time = 0.12, antiderivative size = 37, 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} \[ \frac {2}{a c (1-a x)}+\frac {3 \log (1-a x)}{a c}+\frac {x}{c} \]
Antiderivative was successfully verified.
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Rule 77
Rule 6129
Rule 6131
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{c-\frac {c}{a x}} \, dx\\ &=\frac {a \int \frac {e^{2 \tanh ^{-1}(a x)} x}{1-a x} \, dx}{c}\\ &=\frac {a \int \frac {x (1+a x)}{(1-a x)^2} \, dx}{c}\\ &=\frac {a \int \left (\frac {1}{a}+\frac {2}{a (-1+a x)^2}+\frac {3}{a (-1+a x)}\right ) \, dx}{c}\\ &=\frac {x}{c}+\frac {2}{a c (1-a x)}+\frac {3 \log (1-a x)}{a c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 30, normalized size = 0.81 \[ \frac {a x+\frac {2}{1-a x}+3 \log (1-a x)}{a c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 40, normalized size = 1.08 \[ \frac {a^{2} x^{2} - a x + 3 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{a^{2} c x - a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 36, normalized size = 0.97 \[ \frac {x}{c} + \frac {3 \, \log \left ({\left | a x - 1 \right |}\right )}{a c} - \frac {2}{{\left (a x - 1\right )} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 36, normalized size = 0.97 \[ \frac {x}{c}+\frac {3 \ln \left (a x -1\right )}{c a}-\frac {2}{c a \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 35, normalized size = 0.95 \[ \frac {x}{c} - \frac {2}{a^{2} c x - a c} + \frac {3 \, \log \left (a x - 1\right )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 34, normalized size = 0.92 \[ \frac {x}{c}+\frac {2}{a\,\left (c-a\,c\,x\right )}+\frac {3\,\ln \left (a\,x-1\right )}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 26, normalized size = 0.70 \[ - \frac {2}{a^{2} c x - a c} + \frac {x}{c} + \frac {3 \log {\left (a x - 1 \right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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