Optimal. Leaf size=37 \[ \frac{c^2 (1-a x)^4}{2 a}-\frac{c^2 (1-a x)^5}{5 a} \]
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Rubi [A] time = 0.063194, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6167, 6140, 43} \[ \frac{c^2 (1-a x)^4}{2 a}-\frac{c^2 (1-a x)^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 43
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx\\ &=-\left (c^2 \int (1-a x)^3 (1+a x) \, dx\right )\\ &=-\left (c^2 \int \left (2 (1-a x)^3-(1-a x)^4\right ) \, dx\right )\\ &=\frac{c^2 (1-a x)^4}{2 a}-\frac{c^2 (1-a x)^5}{5 a}\\ \end{align*}
Mathematica [A] time = 0.0165207, size = 30, normalized size = 0.81 \[ \frac{1}{10} c^2 x \left (2 a^4 x^4-5 a^3 x^3+10 a x-10\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 30, 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 = 1.13044, 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 = 1.5016, size = 74, normalized size = 2. \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 = 0.083979, 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 = 1.13428, 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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