Optimal. Leaf size=25 \[ -\frac {c \log (x)}{a}+\frac {4 c \log (a x+1)}{a}-c x \]
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Rubi [A] time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {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
Rubi steps
\begin {align*} \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 = 25, 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 = 0.51, size = 23, normalized size = 0.92 \[ -\frac {a c x - 4 \, c \log \left (a x + 1\right ) + c \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 56, normalized size = 2.24 \[ -\frac {{\left (a x + 1\right )} c}{a} - \frac {3 \, c \log \left (\frac {{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {c \log \left ({\left | -\frac {1}{a x + 1} + 1 \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 26, 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 = 25, 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 = 0.07, size = 25, normalized size = 1.00 \[ \frac {4\,c\,\ln \left (a\,x+1\right )}{a}-\frac {c\,\ln \relax (x)}{a}-c\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 19, normalized size = 0.76 \[ - c x - \frac {c \left (\log {\relax (x )} - 4 \log {\left (x + \frac {1}{a} \right )}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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