3.583 \(\int \frac{1}{\sqrt{-\pi -b x^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-\pi }}\right )}{\sqrt{b}} \]

[Out]

ArcTan[(Sqrt[b]*x)/Sqrt[-Pi - b*x^2]]/Sqrt[b]

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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]

ArcTan[(Sqrt[b]*x)/Sqrt[-Pi - b*x^2]]/Sqrt[b]

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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]

atan(sqrt(b)*x/sqrt(-b*x**2 - pi))/sqrt(b)

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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]

ArcTan[(Sqrt[b]*x)/Sqrt[-Pi - b*x^2]]/Sqrt[b]

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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]

arctan(x*b^(1/2)/(-b*x^2-Pi)^(1/2))/b^(1/2)

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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]

Exception raised: ValueError

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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]

[1/2*log(-2*sqrt(-pi - b*x^2)*b*x - (pi + 2*b*x^2)*sqrt(-b))/sqrt(-b), arctan(sq
rt(b)*x/sqrt(-pi - b*x^2))/sqrt(b)]

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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]

-I*asinh(sqrt(b)*x/sqrt(pi))/sqrt(b)

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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]

-ln(abs(-sqrt(-b)*x + sqrt(-pi - b*x^2)))/sqrt(-b)