Optimal. Leaf size=38 \[ -\frac {x}{c}-\frac {1}{a c (1-a x)}-\frac {2 \log (1-a x)}{a c} \]
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Rubi [A]
time = 0.09, antiderivative size = 38, 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 = {6292, 6285, 45}
\begin {gather*} -\frac {1}{a c (1-a x)}-\frac {2 \log (1-a x)}{a c}-\frac {x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6285
Rule 6292
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {a^2 \int \frac {e^{2 \tanh ^{-1}(a x)} x^2}{1-a^2 x^2} \, dx}{c}\\ &=-\frac {a^2 \int \frac {x^2}{(1-a x)^2} \, dx}{c}\\ &=-\frac {a^2 \int \left (\frac {1}{a^2}+\frac {1}{a^2 (-1+a x)^2}+\frac {2}{a^2 (-1+a x)}\right ) \, dx}{c}\\ &=-\frac {x}{c}-\frac {1}{a c (1-a x)}-\frac {2 \log (1-a x)}{a c}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 38, normalized size = 1.00 \begin {gather*} -\frac {x}{c}-\frac {1}{a c (1-a x)}-\frac {2 \log (1-a x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 37, normalized size = 0.97
method | result | size |
risch | \(-\frac {x}{c}+\frac {1}{a c \left (a x -1\right )}-\frac {2 \ln \left (a x -1\right )}{a c}\) | \(36\) |
default | \(\frac {a^{2} \left (-\frac {x}{a^{2}}+\frac {1}{a^{3} \left (a x -1\right )}-\frac {2 \ln \left (a x -1\right )}{a^{3}}\right )}{c}\) | \(37\) |
norman | \(\frac {\frac {2 x}{c}-\frac {a^{2} x^{3}}{c}+\frac {a \,x^{2}}{c}}{a^{2} x^{2}-1}-\frac {2 \ln \left (a x -1\right )}{a c}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 34, normalized size = 0.89 \begin {gather*} -\frac {x}{c} + \frac {1}{a^{2} c x - a c} - \frac {2 \, \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 = 41, normalized size = 1.08 \begin {gather*} -\frac {a^{2} x^{2} - a x + 2 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 1}{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 = 0.08, size = 37, normalized size = 0.97 \begin {gather*} - a^{2} \left (- \frac {1}{a^{4} c x - a^{3} c} + \frac {x}{a^{2} c} + \frac {2 \log {\left (a x - 1 \right )}}{a^{3} c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 36, normalized size = 0.95 \begin {gather*} -\frac {x}{c} - \frac {2 \, \log \left ({\left | a x - 1 \right |}\right )}{a c} + \frac {1}{{\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 = 0.05, size = 35, normalized size = 0.92 \begin {gather*} -\frac {x}{c}-\frac {1}{a\,\left (c-a\,c\,x\right )}-\frac {2\,\ln \left (a\,x-1\right )}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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