Optimal. Leaf size=35 \[ \frac{c^2 (a x+1)^4}{2 a}-\frac{c^2 (a x+1)^5}{5 a} \]
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Rubi [A] time = 0.0370518, antiderivative size = 35, 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} \[ \frac{c^2 (a x+1)^4}{2 a}-\frac{c^2 (a x+1)^5}{5 a} \]
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) (1+a x)^3 \, 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 = 0.0130379, size = 23, normalized size = 0.66 \[ -\frac{c^2 (a x+1)^4 (2 a x-3)}{10 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 28, 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 = 0.950569, size = 49, normalized size = 1.4 \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 = 1.69535, size = 76, normalized size = 2.17 \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 = 0.085515, size = 36, normalized size = 1.03 \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 = 1.15739, size = 49, normalized size = 1.4 \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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