Optimal. Leaf size=48 \[ \frac {2}{a c (1-a x)^2}-\frac {4}{a c (1-a x)}-\frac {\log (1-a x)}{a c} \]
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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6264, 45}
\begin {gather*} -\frac {4}{a c (1-a x)}+\frac {2}{a c (1-a x)^2}-\frac {\log (1-a x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6264
Rubi steps
\begin {align*} \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 = 0.01, size = 36, normalized size = 0.75 \begin {gather*} \frac {-2+4 a x-(-1+a x)^2 \log (1-a x)}{a c (-1+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.12, size = 41, normalized size = 0.85
method | result | size |
risch | \(\frac {4 x -\frac {2}{a}}{\left (a x -1\right )^{2} c}-\frac {\ln \left (a x -1\right )}{a c}\) | \(36\) |
default | \(\frac {\frac {4}{a \left (a x -1\right )}+\frac {2}{\left (a x -1\right )^{2} a}-\frac {\ln \left (a x -1\right )}{a}}{c}\) | \(41\) |
norman | \(\frac {\frac {2 a \,x^{2}}{c}+\frac {2 a^{2} x^{3}}{c}}{\left (a x +1\right ) \left (a x -1\right )^{2}}-\frac {\ln \left (a x -1\right )}{a c}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 44, normalized size = 0.92 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 49, normalized size = 1.02 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 37, normalized size = 0.77 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 37, normalized size = 0.77 \begin {gather*} -\frac {\log \left ({\left | a x - 1 \right |}\right )}{a c} + \frac {2 \, {\left (2 \, a x - 1\right )}}{{\left (a x - 1\right )}^{2} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 42, normalized size = 0.88 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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