Optimal. Leaf size=19 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0030614, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {216} \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\pi -b x^2}} \, dx &=\frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0071897, size = 19, normalized size = 1. \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 21, normalized size = 1.1 \begin{align*}{\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+\pi }}}} \right ){\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28524, size = 159, normalized size = 8.37 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-\pi + 2 \, b x^{2} - 2 \, \sqrt{\pi - b x^{2}} \sqrt{-b} x\right )}{2 \, b}, -\frac{\arctan \left (-\frac{\sqrt{b} x}{\sqrt{\pi - b x^{2}}}\right )}{\sqrt{b}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01061, size = 46, normalized size = 2.42 \begin{align*} \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x^{2}}\right |}{\pi } > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.15331, size = 38, normalized size = 2. \begin{align*} -\frac{\log \left ({\left | -\sqrt{-b} x + \sqrt{\pi - b x^{2}} \right |}\right )}{\sqrt{-b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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