3.6.81 \(\int \frac {1}{\sqrt {\pi -b x^2}} \, dx\) [581]

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 = {222} \begin {gather*} \frac {\text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {\pi }}\right )}{\sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Pi - b*x^2],x]

[Out]

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

Rule 222

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 = 0.02, size = 33, normalized size = 1.74 \begin {gather*} \frac {b \log \left (-\sqrt {-b} x+\sqrt {\pi -b x^2}\right )}{(-b)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Pi - b*x^2],x]

[Out]

(b*Log[-(Sqrt[-b]*x) + Sqrt[Pi - b*x^2]])/(-b)^(3/2)

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Maple [A]
time = 0.04, size = 21, normalized size = 1.11

method	result	size
meijerg	\(\frac {\arcsin \left (\frac {x \sqrt {b}}{\sqrt {\pi }}\right )}{\sqrt {b}}\)	\(14\)
default	\(\frac {\arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+\pi }}\right )}{\sqrt {b}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^2+Pi)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 13, normalized size = 0.68 \begin {gather*} \frac {\arcsin \left (\frac {b x}{\sqrt {\pi b}}\right )}{\sqrt {b}} \end {gather*}

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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Fricas [A]
time = 1.62, size = 62, normalized size = 3.26 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.44, size = 46, normalized size = 2.42 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (13) = 26\).
time = 1.19, size = 43, normalized size = 2.26 \begin {gather*} \frac {1}{2} \, \sqrt {\pi - b x^{2}} x - \frac {\pi \log \left ({\left | -\sqrt {-b} x + \sqrt {\pi - b x^{2}} \right |}\right )}{2 \, \sqrt {-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(pi - b*x^2)*x - 1/2*pi*log(abs(-sqrt(-b)*x + sqrt(pi - b*x^2)))/sqrt(-b)

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

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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