Optimal. Leaf size=27 \[ \frac{\sqrt{1-a^2 x^2}}{a}+\frac{\sin ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.0108152, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6123, 641, 216} \[ \frac{\sqrt{1-a^2 x^2}}{a}+\frac{\sin ^{-1}(a x)}{a} \]
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 = 0.0123919, size = 23, normalized size = 0.85 \[ \frac{\sqrt{1-a^2 x^2}+\sin ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 66, normalized size = 2.4 \begin{align*}{\frac{1}{a}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \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 = 1.44107, size = 34, normalized size = 1.26 \begin{align*} \frac{\arcsin \left (a x\right )}{a} + \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 = 1.9401, size = 90, normalized size = 3.33 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17369, size = 38, normalized size = 1.41 \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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