Optimal. Leaf size=19 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0138777, 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 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[Pi - b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 2.23522, size = 17, normalized size = 0.89 \[ \frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-b*x**2+pi)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0132435, size = 19, normalized size = 1. \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[Pi - b*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 21, normalized size = 1.1 \[{1\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}+\pi }}}} \right ){\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-b*x^2+Pi)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(pi - b*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225467, size = 1, normalized size = 0.05 \[ \left [\frac{\log \left (2 \, \sqrt{\pi - b x^{2}} b x -{\left (\pi - 2 \, b x^{2}\right )} \sqrt{-b}\right )}{2 \, \sqrt{-b}}, \frac{\arctan \left (\frac{\sqrt{b} x}{\sqrt{\pi - b x^{2}}}\right )}{\sqrt{b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(pi - b*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.60544, size = 46, normalized size = 2.42 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x^{2}}\right |}{\pi } > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-b*x**2+pi)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.223298, size = 38, normalized size = 2. \[ -\frac{{\rm ln}\left ({\left | -\sqrt{-b} x + \sqrt{\pi - b x^{2}} \right |}\right )}{\sqrt{-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(pi - b*x^2),x, algorithm="giac")
[Out]