Optimal. Leaf size=28 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-\pi }}\right )}{\sqrt{b}} \]
[Out]
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Rubi [A] time = 0.0192495, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-\pi }}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[-Pi - b*x^2],x]
[Out]
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Rubi in Sympy [A] time = 2.34624, size = 24, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{- b x^{2} - \pi }} \right )}}{\sqrt{b}} \]
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 = 0.0110877, size = 28, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-\pi }}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[-Pi - b*x^2],x]
[Out]
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Maple [A] time = 0.006, size = 23, normalized size = 0.8 \[{1\arctan \left ({x\sqrt{b}{\frac{1}{\sqrt{-b{x}^{2}-\pi }}}} \right ){\frac{1}{\sqrt{b}}}} \]
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 = 0.252196, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (-2 \, \sqrt{-\pi - b x^{2}} b x -{\left (\pi + 2 \, b x^{2}\right )} \sqrt{-b}\right )}{2 \, \sqrt{-b}}, \frac{\arctan \left (\frac{\sqrt{b} x}{\sqrt{-\pi - b x^{2}}}\right )}{\sqrt{b}}\right ] \]
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 = 3.47881, size = 20, normalized size = 0.71 \[ - \frac{i \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \right )}}{\sqrt{b}} \]
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 = 0.214004, size = 41, normalized size = 1.46 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{-b} x + \sqrt{-\pi - b x^{2}} \right |}\right )}{\sqrt{-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-pi - b*x^2),x, algorithm="giac")
[Out]