Optimal. Leaf size=37 \[ \frac{c^5 (1-a x)^6}{6 a}-\frac{2 c^5 (1-a x)^5}{5 a} \]
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Rubi [A] time = 0.0339982, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6129, 43} \[ \frac{c^5 (1-a x)^6}{6 a}-\frac{2 c^5 (1-a x)^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 6129
Rule 43
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^5 \, dx &=c^5 \int (1-a x)^4 (1+a x) \, dx\\ &=c^5 \int \left (2 (1-a x)^4-(1-a x)^5\right ) \, dx\\ &=-\frac{2 c^5 (1-a x)^5}{5 a}+\frac{c^5 (1-a x)^6}{6 a}\\ \end{align*}
Mathematica [A] time = 0.018634, size = 23, normalized size = 0.62 \[ \frac{c^5 (a x-1)^5 (5 a x+7)}{30 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 45, normalized size = 1.2 \begin{align*}{c}^{5} \left ({\frac{{x}^{6}{a}^{5}}{6}}-{\frac{3\,{x}^{5}{a}^{4}}{5}}+{\frac{{x}^{4}{a}^{3}}{2}}+{\frac{2\,{x}^{3}{a}^{2}}{3}}-{\frac{3\,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 = 0.946944, size = 80, normalized size = 2.16 \begin{align*} \frac{1}{6} \, a^{5} c^{5} x^{6} - \frac{3}{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{3}{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 = 1.49136, size = 128, normalized size = 3.46 \begin{align*} \frac{1}{6} \, a^{5} c^{5} x^{6} - \frac{3}{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{3}{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 [B] time = 0.104863, size = 66, normalized size = 1.78 \begin{align*} \frac{a^{5} c^{5} x^{6}}{6} - \frac{3 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{3 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 = 1.24932, size = 80, normalized size = 2.16 \begin{align*} \frac{1}{6} \, a^{5} c^{5} x^{6} - \frac{3}{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{3}{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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