Optimal. Leaf size=48 \[ -\frac {1}{6} a^4 c^2 x^6-\frac {2}{5} a^3 c^2 x^5+\frac {2}{3} a c^2 x^3+\frac {c^2 x^2}{2} \]
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Rubi [A] time = 0.07, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6150, 75} \[ -\frac {1}{6} a^4 c^2 x^6-\frac {2}{5} a^3 c^2 x^5+\frac {2}{3} a c^2 x^3+\frac {c^2 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x (1-a x) (1+a x)^3 \, dx\\ &=c^2 \int \left (x+2 a x^2-2 a^3 x^4-a^4 x^5\right ) \, dx\\ &=\frac {c^2 x^2}{2}+\frac {2}{3} a c^2 x^3-\frac {2}{5} a^3 c^2 x^5-\frac {1}{6} a^4 c^2 x^6\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.67 \[ -\frac {1}{30} c^2 x^2 \left (5 a^4 x^4+12 a^3 x^3-20 a x-15\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 40, normalized size = 0.83 \[ -\frac {1}{6} \, a^{4} c^{2} x^{6} - \frac {2}{5} \, a^{3} c^{2} x^{5} + \frac {2}{3} \, a c^{2} x^{3} + \frac {1}{2} \, c^{2} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 40, normalized size = 0.83 \[ -\frac {1}{6} \, a^{4} c^{2} x^{6} - \frac {2}{5} \, a^{3} c^{2} x^{5} + \frac {2}{3} \, a c^{2} x^{3} + \frac {1}{2} \, c^{2} x^{2} \]
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 \[ c^{2} \left (-\frac {1}{6} x^{6} a^{4}-\frac {2}{5} a^{3} x^{5}+\frac {2}{3} x^{3} a +\frac {1}{2} x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 40, normalized size = 0.83 \[ -\frac {1}{6} \, a^{4} c^{2} x^{6} - \frac {2}{5} \, a^{3} c^{2} x^{5} + \frac {2}{3} \, a c^{2} x^{3} + \frac {1}{2} \, c^{2} x^{2} \]
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 \[ -\frac {a^4\,c^2\,x^6}{6}-\frac {2\,a^3\,c^2\,x^5}{5}+\frac {2\,a\,c^2\,x^3}{3}+\frac {c^2\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 44, normalized size = 0.92 \[ - \frac {a^{4} c^{2} x^{6}}{6} - \frac {2 a^{3} c^{2} x^{5}}{5} + \frac {2 a c^{2} x^{3}}{3} + \frac {c^{2} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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