Optimal. Leaf size=28 \[ \frac{\sin ^{-1}(a x)}{a}-\frac{\sqrt{1-a^2 x^2}}{a} \]
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Rubi [A] time = 0.0109539, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6123, 641, 216} \[ \frac{\sin ^{-1}(a x)}{a}-\frac{\sqrt{1-a^2 x^2}}{a} \]
Antiderivative was successfully verified.
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Rule 6123
Rule 641
Rule 216
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} \, dx &=\int \frac{1+a x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{a}+\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{a}+\frac{\sin ^{-1}(a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0103446, size = 25, normalized size = 0.89 \[ \frac{\sin ^{-1}(a x)-\sqrt{1-a^2 x^2}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 45, normalized size = 1.6 \begin{align*} -{\frac{1}{a}\sqrt{-{a}^{2}{x}^{2}+1}}+{\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 = 1.41809, size = 47, normalized size = 1.68 \begin{align*} \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09674, size = 92, normalized size = 3.29 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} + 2 \, \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.62957, size = 19, normalized size = 0.68 \begin{align*} \begin{cases} \frac{- \sqrt{- a^{2} x^{2} + 1} + \operatorname{asin}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17587, size = 39, normalized size = 1.39 \begin{align*} \frac{\arcsin \left (a x\right ) \mathrm{sgn}\left (a\right )}{{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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