Optimal. Leaf size=11 \[ \frac{\sin ^{-1}\left (\frac{b x}{2}\right )}{b} \]
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Rubi [A] time = 0.0033367, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {41, 216} \[ \frac{\sin ^{-1}\left (\frac{b x}{2}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{2-b x} \sqrt{2+b x}} \, dx &=\int \frac{1}{\sqrt{4-b^2 x^2}} \, dx\\ &=\frac{\sin ^{-1}\left (\frac{b x}{2}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0076968, size = 11, normalized size = 1. \[ \frac{\sin ^{-1}\left (\frac{b x}{2}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 56, normalized size = 5.1 \begin{align*}{\sqrt{ \left ( -bx+2 \right ) \left ( bx+2 \right ) }\arctan \left ({x\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}+4}}}} \right ){\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.429, size = 24, normalized size = 2.18 \begin{align*} \frac{\arcsin \left (\frac{b^{2} x}{2 \, \sqrt{b^{2}}}\right )}{\sqrt{b^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02964, size = 74, normalized size = 6.73 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b x + 2} - 2}{b x}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.18199, size = 76, normalized size = 6.91 \begin{align*} - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{4}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07294, size = 20, normalized size = 1.82 \begin{align*} \frac{2 \, \arcsin \left (\frac{1}{2} \, \sqrt{b x + 2}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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