Optimal. Leaf size=23 \[ \frac {c \log (x)}{a}-\frac {4 c \log (a x+1)}{a}+c x \]
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Rubi [A] time = 0.08, 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 = {6167, 6131, 6129, 72} \[ \frac {c \log (x)}{a}-\frac {4 c \log (a x+1)}{a}+c x \]
Antiderivative was successfully verified.
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Rule 72
Rule 6129
Rule 6131
Rule 6167
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 = 0.04, size = 23, normalized size = 1.00 \[ \frac {c \log (x)}{a}-\frac {4 c \log (a x+1)}{a}+c x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 22, normalized size = 0.96 \[ \frac {a c x - 4 \, c \log \left (a x + 1\right ) + c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 25, normalized size = 1.09 \[ c x - \frac {4 \, c \log \left ({\left | a x + 1 \right |}\right )}{a} + \frac {c \log \left ({\left | x \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 24, normalized size = 1.04 \[ c x +\frac {c \ln \relax (x )}{a}-\frac {4 c \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 23, normalized size = 1.00 \[ c x - \frac {4 \, c \log \left (a x + 1\right )}{a} + \frac {c \log \relax (x)}{a} \]
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 \[ c\,x+\frac {c\,\ln \relax (x)}{a}-\frac {4\,c\,\ln \left (a\,x+1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 17, normalized size = 0.74 \[ c x + \frac {c \left (\log {\relax (x )} - 4 \log {\left (x + \frac {1}{a} \right )}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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