3.253 \(\int \frac{1}{\sqrt{-1-x^2} \sqrt{2-x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )}{\sqrt{-x^2-1}} \]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[x/Sqrt[2]], -2])/Sqrt[-1 - x^2]

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Rubi [A]  time = 0.0156464, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {421, 419} \[ \frac{\sqrt{x^2+1} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{\sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 - x^2]*Sqrt[2 - x^2]),x]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[x/Sqrt[2]], -2])/Sqrt[-1 - x^2]

Rule 421

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

Rule 419

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1-x^2} \sqrt{2-x^2}} \, dx &=\frac{\sqrt{1+x^2} \int \frac{1}{\sqrt{2-x^2} \sqrt{1+x^2}} \, dx}{\sqrt{-1-x^2}}\\ &=\frac{\sqrt{1+x^2} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{\sqrt{-1-x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0248291, size = 39, normalized size = 1.26 \[ -\frac{i \sqrt{x^2+1} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 - x^2]),x]

[Out]

((-I)*Sqrt[1 + x^2]*EllipticF[I*ArcSinh[x], -1/2])/(Sqrt[2]*Sqrt[-1 - x^2])

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Maple [A]  time = 0.02, size = 34, normalized size = 1.1 \begin{align*}{{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{2} \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2-1)^(1/2)/(-x^2+2)^(1/2),x)

[Out]

1/2*I*EllipticF(I*x,1/2*I*2^(1/2))/(x^2+1)^(1/2)*2^(1/2)*(-x^2-1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-x^2 + 2)*sqrt(-x^2 - 1)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{2} + 2} \sqrt{-x^{2} - 1}}{x^{4} - x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(-x^2 + 2)*sqrt(-x^2 - 1)/(x^4 - x^2 - 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 - x^{2}} \sqrt{- x^{2} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2-1)**(1/2)/(-x**2+2)**(1/2),x)

[Out]

Integral(1/(sqrt(2 - x**2)*sqrt(-x**2 - 1)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-x^2 + 2)*sqrt(-x^2 - 1)), x)