Optimal. Leaf size=19 \[ c^2 x-\frac {1}{3} a^2 c^2 x^3 \]
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Rubi [A] time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 41} \[ c^2 x-\frac {1}{3} a^2 c^2 x^3 \]
Antiderivative was successfully verified.
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Rule 41
Rule 6129
Rubi steps
\begin {align*} \int e^{2 \tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c^2 \int (1-a x) (1+a x) \, dx\\ &=c^2 \int \left (1-a^2 x^2\right ) \, dx\\ &=c^2 x-\frac {1}{3} a^2 c^2 x^3\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.84 \[ c^2 \left (x-\frac {a^2 x^3}{3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 17, normalized size = 0.89 \[ -\frac {1}{3} \, a^{2} c^{2} x^{3} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 17, normalized size = 0.89 \[ -\frac {1}{3} \, a^{2} c^{2} x^{3} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 15, normalized size = 0.79 \[ c^{2} \left (-\frac {1}{3} x^{3} a^{2}+x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 17, normalized size = 0.89 \[ -\frac {1}{3} \, a^{2} c^{2} x^{3} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 15, normalized size = 0.79 \[ -\frac {c^2\,x\,\left (a^2\,x^2-3\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 15, normalized size = 0.79 \[ - \frac {a^{2} c^{2} x^{3}}{3} + c^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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