Optimal. Leaf size=26 \[ -\frac {1}{2} a c x^2-\frac {4 c \log (a x+1)}{a}+3 c x \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6167, 6129, 43} \[ -\frac {1}{2} a c x^2-\frac {4 c \log (a x+1)}{a}+3 c x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6129
Rule 6167
Rubi steps
\begin {align*} \int e^{-2 \coth ^{-1}(a x)} (c-a c x) \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} (c-a c x) \, dx\\ &=-\left (c \int \frac {(1-a x)^2}{1+a x} \, dx\right )\\ &=-\left (c \int \left (-3+a x+\frac {4}{1+a x}\right ) \, dx\right )\\ &=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 = 1.00 \[ -\frac {1}{2} a c x^2-\frac {4 c \log (a x+1)}{a}+3 c x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 28, normalized size = 1.08 \[ -\frac {a^{2} c x^{2} - 6 \, a c x + 8 \, c \log \left (a x + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 35, normalized size = 1.35 \[ -\frac {4 \, c \log \left ({\left | a x + 1 \right |}\right )}{a} - \frac {a^{3} c x^{2} - 6 \, a^{2} c x}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 25, normalized size = 0.96 \[ 3 c x -\frac {a c \,x^{2}}{2}-\frac {4 c \ln \left (a x +1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 24, normalized size = 0.92 \[ -\frac {1}{2} \, a c x^{2} + 3 \, c x - \frac {4 \, c \log \left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 26, normalized size = 1.00 \[ -\frac {c\,\left (8\,\ln \left (a\,x+1\right )-6\,a\,x+a^2\,x^2\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 24, normalized size = 0.92 \[ - \frac {a c x^{2}}{2} + 3 c x - \frac {4 c \log {\left (a x + 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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