Optimal. Leaf size=87 \[ \frac{1}{10} a^6 c^3 x^{10}+\frac{2}{9} a^5 c^3 x^9-\frac{1}{8} a^4 c^3 x^8-\frac{4}{7} a^3 c^3 x^7-\frac{1}{6} a^2 c^3 x^6+\frac{2}{5} a c^3 x^5+\frac{c^3 x^4}{4} \]
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Rubi [A] time = 0.0989541, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6150, 88} \[ \frac{1}{10} a^6 c^3 x^{10}+\frac{2}{9} a^5 c^3 x^9-\frac{1}{8} a^4 c^3 x^8-\frac{4}{7} a^3 c^3 x^7-\frac{1}{6} a^2 c^3 x^6+\frac{2}{5} a c^3 x^5+\frac{c^3 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^3 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^3+2 a x^4-a^2 x^5-4 a^3 x^6-a^4 x^7+2 a^5 x^8+a^6 x^9\right ) \, dx\\ &=\frac{c^3 x^4}{4}+\frac{2}{5} a c^3 x^5-\frac{1}{6} a^2 c^3 x^6-\frac{4}{7} a^3 c^3 x^7-\frac{1}{8} a^4 c^3 x^8+\frac{2}{9} a^5 c^3 x^9+\frac{1}{10} a^6 c^3 x^{10}\\ \end{align*}
Mathematica [A] time = 0.0334374, size = 70, normalized size = 0.8 \[ c^3 \left (\frac{a^6 x^{10}}{10}+\frac{2 a^5 x^9}{9}-\frac{a^4 x^8}{8}-\frac{4 a^3 x^7}{7}-\frac{a^2 x^6}{6}+\frac{2 a x^5}{5}+\frac{x^4}{4}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 57, normalized size = 0.7 \begin{align*}{c}^{3} \left ({\frac{{a}^{6}{x}^{10}}{10}}+{\frac{2\,{a}^{5}{x}^{9}}{9}}-{\frac{{a}^{4}{x}^{8}}{8}}-{\frac{4\,{x}^{7}{a}^{3}}{7}}-{\frac{{x}^{6}{a}^{2}}{6}}+{\frac{2\,a{x}^{5}}{5}}+{\frac{{x}^{4}}{4}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942435, size = 99, normalized size = 1.14 \begin{align*} \frac{1}{10} \, a^{6} c^{3} x^{10} + \frac{2}{9} \, a^{5} c^{3} x^{9} - \frac{1}{8} \, a^{4} c^{3} x^{8} - \frac{4}{7} \, a^{3} c^{3} x^{7} - \frac{1}{6} \, a^{2} c^{3} x^{6} + \frac{2}{5} \, a c^{3} x^{5} + \frac{1}{4} \, c^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63667, size = 163, normalized size = 1.87 \begin{align*} \frac{1}{10} \, a^{6} c^{3} x^{10} + \frac{2}{9} \, a^{5} c^{3} x^{9} - \frac{1}{8} \, a^{4} c^{3} x^{8} - \frac{4}{7} \, a^{3} c^{3} x^{7} - \frac{1}{6} \, a^{2} c^{3} x^{6} + \frac{2}{5} \, a c^{3} x^{5} + \frac{1}{4} \, c^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.096982, size = 82, normalized size = 0.94 \begin{align*} \frac{a^{6} c^{3} x^{10}}{10} + \frac{2 a^{5} c^{3} x^{9}}{9} - \frac{a^{4} c^{3} x^{8}}{8} - \frac{4 a^{3} c^{3} x^{7}}{7} - \frac{a^{2} c^{3} x^{6}}{6} + \frac{2 a c^{3} x^{5}}{5} + \frac{c^{3} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12946, size = 99, normalized size = 1.14 \begin{align*} \frac{1}{10} \, a^{6} c^{3} x^{10} + \frac{2}{9} \, a^{5} c^{3} x^{9} - \frac{1}{8} \, a^{4} c^{3} x^{8} - \frac{4}{7} \, a^{3} c^{3} x^{7} - \frac{1}{6} \, a^{2} c^{3} x^{6} + \frac{2}{5} \, a c^{3} x^{5} + \frac{1}{4} \, c^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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