Optimal. Leaf size=32 \[ \frac{1}{5} a^4 c^4 x^5-\frac{2}{3} a^2 c^4 x^3+c^4 x \]
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Rubi [A] time = 0.0550847, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {6167, 6129, 41, 194} \[ \frac{1}{5} a^4 c^4 x^5-\frac{2}{3} a^2 c^4 x^3+c^4 x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6129
Rule 41
Rule 194
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} (c-a c x)^4 \, dx &=\int e^{4 \tanh ^{-1}(a x)} (c-a c x)^4 \, dx\\ &=c^4 \int (1-a x)^2 (1+a x)^2 \, dx\\ &=c^4 \int \left (1-a^2 x^2\right )^2 \, dx\\ &=c^4 \int \left (1-2 a^2 x^2+a^4 x^4\right ) \, dx\\ &=c^4 x-\frac{2}{3} a^2 c^4 x^3+\frac{1}{5} a^4 c^4 x^5\\ \end{align*}
Mathematica [A] time = 0.0161992, size = 26, normalized size = 0.81 \[ c^4 \left (\frac{a^4 x^5}{5}-\frac{2 a^2 x^3}{3}+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 23, normalized size = 0.7 \begin{align*}{c}^{4} \left ({\frac{{x}^{5}{a}^{4}}{5}}-{\frac{2\,{x}^{3}{a}^{2}}{3}}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03821, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{2}{3} \, a^{2} c^{4} x^{3} + c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35828, size = 58, normalized size = 1.81 \begin{align*} \frac{1}{5} \, a^{4} c^{4} x^{5} - \frac{2}{3} \, a^{2} c^{4} x^{3} + c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.172421, size = 29, normalized size = 0.91 \begin{align*} \frac{a^{4} c^{4} x^{5}}{5} - \frac{2 a^{2} c^{4} x^{3}}{3} + c^{4} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16814, size = 57, normalized size = 1.78 \begin{align*} \frac{{\left (3 \, c^{4} + \frac{15 \, c^{4}}{a x - 1} + \frac{20 \, c^{4}}{{\left (a x - 1\right )}^{2}}\right )}{\left (a x - 1\right )}^{5}}{15 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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