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 = 0.0052011, 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, Rules used = {217, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-b x^2-\pi }}\right )}{\sqrt{b}} \]

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]

Rule 217

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-\pi -b x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+b x^2} \, dx,x,\frac{x}{\sqrt{-\pi -b x^2}}\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{-\pi -b x^2}}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0056439, 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 = 0.004, size = 23, normalized size = 0.8 \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]

arctan(x*b^(1/2)/(-b*x^2-Pi)^(1/2))/b^(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 = 1.43177, size = 178, normalized size = 6.36 \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{-\pi - b x^{2}} \sqrt{b} x}{\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(-pi - b*x^2)*sqrt(b)*x/(pi
+ b*x^2))/sqrt(b)]

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Sympy [C]  time = 0.947158, size = 20, normalized size = 0.71 \begin{align*} - \frac{i \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{\pi }} \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)

[Out]

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

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Giac [A]  time = 1.50849, size = 41, normalized size = 1.46 \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)