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

Optimal. Leaf size=19 \[ \frac{\sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{\pi }}\right )}{\sqrt{b}} \]

[Out]

ArcSin[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]

_______________________________________________________________________________________

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]

ArcSin[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]

_______________________________________________________________________________________

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]

asin(sqrt(b)*x/sqrt(pi))/sqrt(b)

_______________________________________________________________________________________

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]

ArcSin[(Sqrt[b]*x)/Sqrt[Pi]]/Sqrt[b]

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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]

Piecewise((-I*acosh(sqrt(b)*x/sqrt(pi))/sqrt(b), Abs(b*x**2)/pi > 1), (asin(sqrt
(b)*x/sqrt(pi))/sqrt(b), True))

_______________________________________________________________________________________

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]

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