Optimal. Leaf size=33 \[ \frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \sin ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.0198065, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {6127, 195, 216} \[ \frac{1}{2} c x \sqrt{1-a^2 x^2}+\frac{c \sin ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 6127
Rule 195
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} (c-a c x) \, dx &=c \int \sqrt{1-a^2 x^2} \, dx\\ &=\frac{1}{2} c x \sqrt{1-a^2 x^2}+\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 \sin ^{-1}(a x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0093686, size = 30, normalized size = 0.91 \[ \frac{c \left (a x \sqrt{1-a^2 x^2}+\sin ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 46, normalized size = 1.4 \begin{align*}{\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 = 1.41016, size = 49, normalized size = 1.48 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} c x + \frac{c \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{2 \, \sqrt{a^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75386, size = 107, normalized size = 3.24 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1} a c x - 2 \, c \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.39108, size = 46, normalized size = 1.39 \begin{align*} \begin{cases} - \frac{c \left (\begin{cases} - \frac{a x \sqrt{- a^{2} x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (a x \right )}}{2} & \text{for}\: a x > -1 \wedge a x < 1 \end{cases}\right ) - c \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\c x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20023, size = 41, normalized size = 1.24 \begin{align*} \frac{1}{2} \, \sqrt{-a^{2} x^{2} + 1} c x + \frac{c \arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{2 \,{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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