Optimal. Leaf size=32 \[ -\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \]
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Rubi [A]
time = 0.04, antiderivative size = 32, 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 = {6302, 6264, 45}
\begin {gather*} -\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6264
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{c-a c x} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{c-a c x} \, dx\\ &=-\frac {\int \frac {1+a x}{(1-a x)^2} \, dx}{c}\\ &=-\frac {\int \left (\frac {2}{(-1+a x)^2}+\frac {1}{-1+a x}\right ) \, dx}{c}\\ &=-\frac {2}{a c (1-a x)}-\frac {\log (1-a x)}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.94 \begin {gather*} -\frac {\frac {2}{a (1-a x)}+\frac {\log (1-a x)}{a}}{c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 29, normalized size = 0.91
method | result | size |
default | \(\frac {\frac {2}{a \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{a}}{c}\) | \(29\) |
norman | \(\frac {2 x}{c \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{a c}\) | \(29\) |
risch | \(\frac {2}{a c \left (a x -1\right )}-\frac {\ln \left (a x -1\right )}{a c}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 30, normalized size = 0.94 \begin {gather*} \frac {2}{a^{2} c x - a c} - \frac {\log \left (a x - 1\right )}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 29, normalized size = 0.91 \begin {gather*} -\frac {{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{a^{2} c x - a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.62 \begin {gather*} \frac {2}{a^{2} c x - a c} - \frac {\log {\left (a x - 1 \right )}}{a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 31, normalized size = 0.97 \begin {gather*} -\frac {\log \left ({\left | a x - 1 \right |}\right )}{a c} + \frac {2}{{\left (a x - 1\right )} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 29, normalized size = 0.91 \begin {gather*} -\frac {2}{a\,\left (c-a\,c\,x\right )}-\frac {\ln \left (a\,x-1\right )}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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