Optimal. Leaf size=87 \[ \frac {1}{11} a^6 c^3 x^{11}+\frac {1}{5} a^5 c^3 x^{10}-\frac {1}{9} a^4 c^3 x^9-\frac {1}{2} a^3 c^3 x^8-\frac {1}{7} a^2 c^3 x^7+\frac {1}{3} a c^3 x^6+\frac {c^3 x^5}{5} \]
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Rubi [A] time = 0.10, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6150, 88} \[ \frac {1}{11} a^6 c^3 x^{11}+\frac {1}{5} a^5 c^3 x^{10}-\frac {1}{9} a^4 c^3 x^9-\frac {1}{2} a^3 c^3 x^8-\frac {1}{7} a^2 c^3 x^7+\frac {1}{3} a c^3 x^6+\frac {c^3 x^5}{5} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^4 \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int x^4 (1-a x)^2 (1+a x)^4 \, dx\\ &=c^3 \int \left (x^4+2 a x^5-a^2 x^6-4 a^3 x^7-a^4 x^8+2 a^5 x^9+a^6 x^{10}\right ) \, dx\\ &=\frac {c^3 x^5}{5}+\frac {1}{3} a c^3 x^6-\frac {1}{7} a^2 c^3 x^7-\frac {1}{2} a^3 c^3 x^8-\frac {1}{9} a^4 c^3 x^9+\frac {1}{5} a^5 c^3 x^{10}+\frac {1}{11} a^6 c^3 x^{11}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 0.80 \[ c^3 \left (\frac {a^6 x^{11}}{11}+\frac {a^5 x^{10}}{5}-\frac {a^4 x^9}{9}-\frac {a^3 x^8}{2}-\frac {a^2 x^7}{7}+\frac {a x^6}{3}+\frac {x^5}{5}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 73, normalized size = 0.84 \[ \frac {1}{11} \, a^{6} c^{3} x^{11} + \frac {1}{5} \, a^{5} c^{3} x^{10} - \frac {1}{9} \, a^{4} c^{3} x^{9} - \frac {1}{2} \, a^{3} c^{3} x^{8} - \frac {1}{7} \, a^{2} c^{3} x^{7} + \frac {1}{3} \, a c^{3} x^{6} + \frac {1}{5} \, c^{3} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 73, normalized size = 0.84 \[ \frac {1}{11} \, a^{6} c^{3} x^{11} + \frac {1}{5} \, a^{5} c^{3} x^{10} - \frac {1}{9} \, a^{4} c^{3} x^{9} - \frac {1}{2} \, a^{3} c^{3} x^{8} - \frac {1}{7} \, a^{2} c^{3} x^{7} + \frac {1}{3} \, a c^{3} x^{6} + \frac {1}{5} \, c^{3} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 57, normalized size = 0.66 \[ c^{3} \left (\frac {1}{11} a^{6} x^{11}+\frac {1}{5} a^{5} x^{10}-\frac {1}{9} a^{4} x^{9}-\frac {1}{2} a^{3} x^{8}-\frac {1}{7} a^{2} x^{7}+\frac {1}{3} x^{6} a +\frac {1}{5} x^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 73, normalized size = 0.84 \[ \frac {1}{11} \, a^{6} c^{3} x^{11} + \frac {1}{5} \, a^{5} c^{3} x^{10} - \frac {1}{9} \, a^{4} c^{3} x^{9} - \frac {1}{2} \, a^{3} c^{3} x^{8} - \frac {1}{7} \, a^{2} c^{3} x^{7} + \frac {1}{3} \, a c^{3} x^{6} + \frac {1}{5} \, c^{3} x^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 73, normalized size = 0.84 \[ \frac {a^6\,c^3\,x^{11}}{11}+\frac {a^5\,c^3\,x^{10}}{5}-\frac {a^4\,c^3\,x^9}{9}-\frac {a^3\,c^3\,x^8}{2}-\frac {a^2\,c^3\,x^7}{7}+\frac {a\,c^3\,x^6}{3}+\frac {c^3\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 76, normalized size = 0.87 \[ \frac {a^{6} c^{3} x^{11}}{11} + \frac {a^{5} c^{3} x^{10}}{5} - \frac {a^{4} c^{3} x^{9}}{9} - \frac {a^{3} c^{3} x^{8}}{2} - \frac {a^{2} c^{3} x^{7}}{7} + \frac {a c^{3} x^{6}}{3} + \frac {c^{3} x^{5}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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