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

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}} \]

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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fricas [A]  time = 0.96, size = 62, normalized size = 3.26 \[ \left [-\frac {\sqrt {-b} \log \left (-\pi + 2 \, b x^{2} - 2 \, \sqrt {\pi - b x^{2}} \sqrt {-b} x\right )}{2 \, b}, -\frac {\arctan \left (-\frac {\sqrt {b} x}{\sqrt {\pi - b x^{2}}}\right )}{\sqrt {b}}\right ] \]

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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giac [B]  time = 1.05, size = 28, normalized size = 1.47 \[ -\frac {\log \left ({\left | -\sqrt {-b} x + \sqrt {\pi - b x^{2}} \right |}\right )}{\sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 21, normalized size = 1.11 \[ \frac {\arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+\pi }}\right )}{\sqrt {b}} \]

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))

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maxima [A]  time = 2.86, size = 13, normalized size = 0.68 \[ \frac {\arcsin \left (\frac {b x}{\sqrt {\pi b}}\right )}{\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.10, size = 25, normalized size = 1.32 \[ \frac {\ln \left (\sqrt {\Pi -b\,x^2}+\sqrt {-b}\,x\right )}{\sqrt {-b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(Pi - b*x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 1.04, 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)
)

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