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 = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 195
Rule 216
Rule 6127
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.73, size = 47, normalized size = 1.42 \[ \frac {\sqrt {-a^{2} x^{2} + 1} a c x - 2 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 30, normalized size = 0.91 \[ \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c x + \frac {c \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{2 \, {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 46, normalized size = 1.39 \[ \frac {c x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 27, normalized size = 0.82 \[ \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c x + \frac {c \arcsin \left (a x\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.81, size = 37, normalized size = 1.12 \[ \frac {c\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.64, size = 37, normalized size = 1.12 \[ \begin {cases} \frac {c \left (\begin {cases} \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\c x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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