Optimal. Leaf size=46 \[ -\frac {c (a x+1)^3}{3 a}-\frac {c (a x+1)^2}{a}-\frac {8 c \log (1-a x)}{a}-4 c x \]
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Rubi [A] time = 0.05, antiderivative size = 46, 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 = {6167, 6140, 43} \[ -\frac {c (a x+1)^3}{3 a}-\frac {c (a x+1)^2}{a}-\frac {8 c \log (1-a x)}{a}-4 c x \]
Antiderivative was successfully verified.
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Rule 43
Rule 6140
Rule 6167
Rubi steps
\begin {align*} \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=\int e^{4 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx\\ &=c \int \frac {(1+a x)^3}{1-a x} \, dx\\ &=c \int \left (-4+\frac {8}{1-a x}-2 (1+a x)-(1+a x)^2\right ) \, dx\\ &=-4 c x-\frac {c (1+a x)^2}{a}-\frac {c (1+a x)^3}{3 a}-\frac {8 c \log (1-a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 0.78 \[ -\frac {1}{3} a^2 c x^3-2 a c x^2-\frac {8 c \log (1-a x)}{a}-7 c x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 37, normalized size = 0.80 \[ -\frac {a^{3} c x^{3} + 6 \, a^{2} c x^{2} + 21 \, a c x + 24 \, c \log \left (a x - 1\right )}{3 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 60, normalized size = 1.30 \[ -\frac {{\left (a x - 1\right )}^{3} {\left (c + \frac {9 \, c}{a x - 1} + \frac {36 \, c}{{\left (a x - 1\right )}^{2}}\right )}}{3 \, a} + \frac {8 \, c \log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 34, normalized size = 0.74 \[ -\frac {a^{2} c \,x^{3}}{3}-2 a c \,x^{2}-7 c x -\frac {8 c \ln \left (a x -1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 33, normalized size = 0.72 \[ -\frac {1}{3} \, a^{2} c x^{3} - 2 \, a c x^{2} - 7 \, c x - \frac {8 \, c \log \left (a x - 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 33, normalized size = 0.72 \[ -7\,c\,x-\frac {a^2\,c\,x^3}{3}-\frac {8\,c\,\ln \left (a\,x-1\right )}{a}-2\,a\,c\,x^2 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 36, normalized size = 0.78 \[ - \frac {a^{2} c x^{3}}{3} - 2 a c x^{2} - 7 c x - \frac {8 c \log {\left (a x - 1 \right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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