Optimal. Leaf size=16 \[ \frac{c (1-a x)^3}{3 a} \]
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Rubi [A] time = 0.0340112, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6167, 6140, 32} \[ \frac{c (1-a x)^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6140
Rule 32
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx\\ &=-\left (c \int (1-a x)^2 \, dx\right )\\ &=\frac{c (1-a x)^3}{3 a}\\ \end{align*}
Mathematica [A] time = 0.013573, size = 21, normalized size = 1.31 \[ -c \left (\frac{a^2 x^3}{3}-a x^2+x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 14, normalized size = 0.9 \begin{align*} -{\frac{c \left ( ax-1 \right ) ^{3}}{3\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01997, size = 27, normalized size = 1.69 \begin{align*} -\frac{1}{3} \, a^{2} c x^{3} + a c x^{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52522, size = 43, normalized size = 2.69 \begin{align*} -\frac{1}{3} \, a^{2} c x^{3} + a c x^{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.071557, size = 19, normalized size = 1.19 \begin{align*} - \frac{a^{2} c x^{3}}{3} + a c x^{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14692, size = 27, normalized size = 1.69 \begin{align*} -\frac{1}{3} \, a^{2} c x^{3} + a c x^{2} - c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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