Optimal. Leaf size=53 \[ -\frac{c^5 (1-a x)^6}{6 a}+\frac{4 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^4}{a} \]
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Rubi [A] time = 0.0680539, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6167, 6129, 43} \[ -\frac{c^5 (1-a x)^6}{6 a}+\frac{4 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^4}{a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^5 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^5 \, dx\\ &=c^5 \int (1-a x)^3 (1+a x)^2 \, dx\\ &=c^5 \int \left (4 (1-a x)^3-4 (1-a x)^4+(1-a x)^5\right ) \, dx\\ &=-\frac{c^5 (1-a x)^4}{a}+\frac{4 c^5 (1-a x)^5}{5 a}-\frac{c^5 (1-a x)^6}{6 a}\\ \end{align*}
Mathematica [A] time = 0.0208371, size = 31, normalized size = 0.58 \[ -\frac{c^5 (a x-1)^4 \left (5 a^2 x^2+14 a x+11\right )}{30 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 45, normalized size = 0.9 \begin{align*}{c}^{5} \left ( -{\frac{{x}^{6}{a}^{5}}{6}}+{\frac{{x}^{5}{a}^{4}}{5}}+{\frac{{x}^{4}{a}^{3}}{2}}-{\frac{2\,{x}^{3}{a}^{2}}{3}}-{\frac{a{x}^{2}}{2}}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992122, size = 80, normalized size = 1.51 \begin{align*} -\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{1}{5} \, a^{4} c^{5} x^{5} + \frac{1}{2} \, a^{3} c^{5} x^{4} - \frac{2}{3} \, a^{2} c^{5} x^{3} - \frac{1}{2} \, a c^{5} x^{2} + c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47756, size = 130, normalized size = 2.45 \begin{align*} -\frac{1}{6} \, a^{5} c^{5} x^{6} + \frac{1}{5} \, a^{4} c^{5} x^{5} + \frac{1}{2} \, a^{3} c^{5} x^{4} - \frac{2}{3} \, a^{2} c^{5} x^{3} - \frac{1}{2} \, a c^{5} x^{2} + c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.19778, size = 63, normalized size = 1.19 \begin{align*} - \frac{a^{5} c^{5} x^{6}}{6} + \frac{a^{4} c^{5} x^{5}}{5} + \frac{a^{3} c^{5} x^{4}}{2} - \frac{2 a^{2} c^{5} x^{3}}{3} - \frac{a c^{5} x^{2}}{2} + c^{5} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15922, size = 57, normalized size = 1.08 \begin{align*} -\frac{{\left (5 \, c^{5} + \frac{24 \, c^{5}}{a x - 1} + \frac{30 \, c^{5}}{{\left (a x - 1\right )}^{2}}\right )}{\left (a x - 1\right )}^{6}}{30 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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