Optimal. Leaf size=48 \[ -\frac {1}{7} a^4 c^2 x^7-\frac {1}{3} a^3 c^2 x^6+\frac {1}{2} a c^2 x^4+\frac {c^2 x^3}{3} \]
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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} \[ -\frac {1}{7} a^4 c^2 x^7-\frac {1}{3} a^3 c^2 x^6+\frac {1}{2} a c^2 x^4+\frac {c^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 75
Rule 6150
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} x^2 \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int x^2 (1-a x) (1+a x)^3 \, dx\\ &=c^2 \int \left (x^2+2 a x^3-2 a^3 x^5-a^4 x^6\right ) \, dx\\ &=\frac {c^2 x^3}{3}+\frac {1}{2} a c^2 x^4-\frac {1}{3} a^3 c^2 x^6-\frac {1}{7} a^4 c^2 x^7\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.67 \[ -\frac {1}{42} c^2 x^3 \left (6 a^4 x^4+14 a^3 x^3-21 a x-14\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 40, normalized size = 0.83 \[ -\frac {1}{7} \, a^{4} c^{2} x^{7} - \frac {1}{3} \, a^{3} c^{2} x^{6} + \frac {1}{2} \, a c^{2} x^{4} + \frac {1}{3} \, c^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 40, normalized size = 0.83 \[ -\frac {1}{7} \, a^{4} c^{2} x^{7} - \frac {1}{3} \, a^{3} c^{2} x^{6} + \frac {1}{2} \, a c^{2} x^{4} + \frac {1}{3} \, c^{2} x^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 33, normalized size = 0.69 \[ c^{2} \left (-\frac {1}{7} x^{7} a^{4}-\frac {1}{3} x^{6} a^{3}+\frac {1}{2} x^{4} a +\frac {1}{3} x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 40, normalized size = 0.83 \[ -\frac {1}{7} \, a^{4} c^{2} x^{7} - \frac {1}{3} \, a^{3} c^{2} x^{6} + \frac {1}{2} \, a c^{2} x^{4} + \frac {1}{3} \, c^{2} x^{3} \]
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 \[ -\frac {a^4\,c^2\,x^7}{7}-\frac {a^3\,c^2\,x^6}{3}+\frac {a\,c^2\,x^4}{2}+\frac {c^2\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 41, normalized size = 0.85 \[ - \frac {a^{4} c^{2} x^{7}}{7} - \frac {a^{3} c^{2} x^{6}}{3} + \frac {a c^{2} x^{4}}{2} + \frac {c^{2} x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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