Optimal. Leaf size=55 \[ \frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.0299767, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6139, 641, 195, 216} \[ \frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 6139
Rule 641
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x) \sqrt{1-a^2 x^2} \, dx\\ &=\frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a}+c \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{1}{2} c \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac{c \sin ^{-1}(a x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0807683, size = 57, normalized size = 1.04 \[ \frac{c \left (\left (-2 a^2 x^2+3 a x+2\right ) \sqrt{1-a^2 x^2}-6 \sin ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 64, normalized size = 1.2 \begin{align*}{\frac{c}{3\,a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{cx}{2}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{c}{2}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45104, size = 61, normalized size = 1.11 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} c x + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} c}{3 \, a} + \frac{c \arcsin \left (a x\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16619, size = 140, normalized size = 2.55 \begin{align*} -\frac{6 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (2 \, a^{2} c x^{2} - 3 \, a c x - 2 \, c\right )} \sqrt{-a^{2} x^{2} + 1}}{6 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.65976, size = 109, normalized size = 1.98 \begin{align*} - a c \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: a^{2} = 0 \\- \frac{\left (- a^{2} x^{2} + 1\right )^{\frac{3}{2}}}{3 a^{2}} & \text{otherwise} \end{cases}\right ) + c \left (\begin{cases} \frac{i a^{2} x^{3}}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i x}{2 \sqrt{a^{2} x^{2} - 1}} - \frac{i \operatorname{acosh}{\left (a x \right )}}{2 a} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac{x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2 a} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26892, size = 62, normalized size = 1.13 \begin{align*} \frac{c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} - \frac{1}{6} \, \sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, a c x - 3 \, c\right )} x - \frac{2 \, c}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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