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]

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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.

[In]

Int[1/Sqrt[Pi - b*x^2],x]

[Out]

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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.

[In]

Integrate[1/Sqrt[Pi - b*x^2],x]

[Out]

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

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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.

[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))

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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.

[In]

integrate(1/(-b*x^2+pi)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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.

[In]

integrate(1/(-b*x^2+pi)^(1/2),x, algorithm="fricas")

[Out]

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

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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.

[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)
)

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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.

[In]

integrate(1/(-b*x^2+pi)^(1/2),x, algorithm="giac")

[Out]

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