Optimal. Leaf size=48 \[ -\frac {1}{8} a^4 c^2 x^8-\frac {2}{7} a^3 c^2 x^7+\frac {2}{5} a c^2 x^5+\frac {c^2 x^4}{4} \]
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Rubi [A] time = 0.09, antiderivative size = 48, 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, 75} \[ -\frac {1}{8} a^4 c^2 x^8-\frac {2}{7} a^3 c^2 x^7+\frac {2}{5} a c^2 x^5+\frac {c^2 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x^3 (1-a x) (1+a x)^3 \, dx\\ &=c^2 \int \left (x^3+2 a x^4-2 a^3 x^6-a^4 x^7\right ) \, dx\\ &=\frac {c^2 x^4}{4}+\frac {2}{5} a c^2 x^5-\frac {2}{7} a^3 c^2 x^7-\frac {1}{8} a^4 c^2 x^8\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.83 \[ c^2 \left (-\frac {1}{8} a^4 x^8-\frac {2 a^3 x^7}{7}+\frac {2 a x^5}{5}+\frac {x^4}{4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 40, normalized size = 0.83 \[ -\frac {1}{8} \, a^{4} c^{2} x^{8} - \frac {2}{7} \, a^{3} c^{2} x^{7} + \frac {2}{5} \, a c^{2} x^{5} + \frac {1}{4} \, c^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 40, normalized size = 0.83 \[ -\frac {1}{8} \, a^{4} c^{2} x^{8} - \frac {2}{7} \, a^{3} c^{2} x^{7} + \frac {2}{5} \, a c^{2} x^{5} + \frac {1}{4} \, c^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 33, normalized size = 0.69 \[ c^{2} \left (-\frac {1}{8} a^{4} x^{8}-\frac {2}{7} a^{3} x^{7}+\frac {2}{5} a \,x^{5}+\frac {1}{4} x^{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 40, normalized size = 0.83 \[ -\frac {1}{8} \, a^{4} c^{2} x^{8} - \frac {2}{7} \, a^{3} c^{2} x^{7} + \frac {2}{5} \, a c^{2} x^{5} + \frac {1}{4} \, c^{2} x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 40, normalized size = 0.83 \[ -\frac {a^4\,c^2\,x^8}{8}-\frac {2\,a^3\,c^2\,x^7}{7}+\frac {2\,a\,c^2\,x^5}{5}+\frac {c^2\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 44, normalized size = 0.92 \[ - \frac {a^{4} c^{2} x^{8}}{8} - \frac {2 a^{3} c^{2} x^{7}}{7} + \frac {2 a c^{2} x^{5}}{5} + \frac {c^{2} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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