Optimal. Leaf size=37 \[ \frac{c^2 (1-a x)^5}{5 a}-\frac{c^2 (1-a x)^4}{2 a} \]
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Rubi [A] time = 0.0367321, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6140, 43} \[ \frac{c^2 (1-a x)^5}{5 a}-\frac{c^2 (1-a x)^4}{2 a} \]
Antiderivative was successfully verified.
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Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=c^2 \int (1-a x)^3 (1+a x) \, dx\\ &=c^2 \int \left (2 (1-a x)^3-(1-a x)^4\right ) \, dx\\ &=-\frac{c^2 (1-a x)^4}{2 a}+\frac{c^2 (1-a x)^5}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0160266, size = 32, normalized size = 0.86 \[ c^2 \left (-\frac{1}{5} a^4 x^5+\frac{a^3 x^4}{2}-a x^2+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 29, normalized size = 0.8 \begin{align*}{c}^{2} \left ( -{\frac{{x}^{5}{a}^{4}}{5}}+{\frac{{x}^{4}{a}^{3}}{2}}-a{x}^{2}+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967953, size = 50, normalized size = 1.35 \begin{align*} -\frac{1}{5} \, a^{4} c^{2} x^{5} + \frac{1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19162, size = 76, normalized size = 2.05 \begin{align*} -\frac{1}{5} \, a^{4} c^{2} x^{5} + \frac{1}{2} \, a^{3} c^{2} x^{4} - a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.097322, size = 36, normalized size = 0.97 \begin{align*} - \frac{a^{4} c^{2} x^{5}}{5} + \frac{a^{3} c^{2} x^{4}}{2} - a c^{2} x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1257, size = 73, normalized size = 1.97 \begin{align*} -\frac{{\left (2 \, c^{2} - \frac{15 \, c^{2}}{a x + 1} + \frac{40 \, c^{2}}{{\left (a x + 1\right )}^{2}} - \frac{40 \, c^{2}}{{\left (a x + 1\right )}^{3}}\right )}{\left (a x + 1\right )}^{5}}{10 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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