Optimal. Leaf size=27 \[ -\frac {1}{2} a c x^2-\frac {4 c \log (1-a x)}{a}-3 c x \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6129, 43} \[ -\frac {1}{2} a c x^2-\frac {4 c \log (1-a x)}{a}-3 c x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rubi steps
\begin {align*} \int e^{4 \tanh ^{-1}(a x)} (c-a c x) \, dx &=c \int \frac {(1+a x)^2}{1-a x} \, dx\\ &=c \int \left (-3-a x+\frac {4}{1-a x}\right ) \, dx\\ &=-3 c x-\frac {1}{2} a c x^2-\frac {4 c \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.96 \[ c \left (-\frac {a x^2}{2}-\frac {4 \log (1-a x)}{a}-3 x\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 28, normalized size = 1.04 \[ -\frac {a^{2} c x^{2} + 6 \, a c x + 8 \, c \log \left (a x - 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 35, normalized size = 1.30 \[ -\frac {4 \, c \log \left ({\left | a x - 1 \right |}\right )}{a} - \frac {a^{3} c x^{2} + 6 \, a^{2} c x}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 0.93 \[ -\frac {a c \,x^{2}}{2}-3 c x -\frac {4 c \ln \left (a x -1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 24, normalized size = 0.89 \[ -\frac {1}{2} \, a c x^{2} - 3 \, c x - \frac {4 \, c \log \left (a x - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 26, normalized size = 0.96 \[ -\frac {c\,\left (8\,\ln \left (a\,x-1\right )+6\,a\,x+a^2\,x^2\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 26, normalized size = 0.96 \[ - \frac {a c x^{2}}{2} - 3 c x - \frac {4 c \log {\left (a x - 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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