Optimal. Leaf size=23 \[ c x+\frac {c \log (x)}{a}-\frac {4 c \log (1+a x)}{a} \]
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Rubi [A]
time = 0.06, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6302, 6266,
6264, 84} \begin {gather*} \frac {c \log (x)}{a}-\frac {4 c \log (a x+1)}{a}+c x \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 6264
Rule 6266
Rule 6302
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right ) \, dx\\ &=\frac {c \int \frac {e^{-2 \tanh ^{-1}(a x)} (1-a x)}{x} \, dx}{a}\\ &=\frac {c \int \frac {(1-a x)^2}{x (1+a x)} \, dx}{a}\\ &=\frac {c \int \left (a+\frac {1}{x}-\frac {4 a}{1+a x}\right ) \, dx}{a}\\ &=c x+\frac {c \log (x)}{a}-\frac {4 c \log (1+a x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 1.00 \begin {gather*} c x+\frac {c \log (x)}{a}-\frac {4 c \log (1+a x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 20, normalized size = 0.87
method | result | size |
default | \(\frac {c \left (a x -4 \ln \left (a x +1\right )+\ln \left (x \right )\right )}{a}\) | \(20\) |
norman | \(c x +\frac {c \ln \left (x \right )}{a}-\frac {4 c \ln \left (a x +1\right )}{a}\) | \(24\) |
risch | \(c x -\frac {4 c \ln \left (a x +1\right )}{a}+\frac {c \ln \left (-x \right )}{a}\) | \(26\) |
meijerg | \(\frac {c \left (a x -\ln \left (a x +1\right )\right )}{a}-\frac {2 c \ln \left (a x +1\right )}{a}+\frac {c \left (-\ln \left (a x +1\right )+\ln \left (x \right )+\ln \left (a \right )\right )}{a}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 23, normalized size = 1.00 \begin {gather*} c x - \frac {4 \, c \log \left (a x + 1\right )}{a} + \frac {c \log \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 22, normalized size = 0.96 \begin {gather*} \frac {a c x - 4 \, c \log \left (a x + 1\right ) + c \log \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 17, normalized size = 0.74 \begin {gather*} c x + \frac {c \left (\log {\left (x \right )} - 4 \log {\left (x + \frac {1}{a} \right )}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 25, normalized size = 1.09 \begin {gather*} c x - \frac {4 \, c \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {c \log \left ({\left | x \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 23, normalized size = 1.00 \begin {gather*} c\,x+\frac {c\,\ln \left (x\right )}{a}-\frac {4\,c\,\ln \left (a\,x+1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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