3.255 \(\int \frac{1}{\sqrt{2-3 x^2} \sqrt{-1-x^2}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-x^2-1}} \]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/(Sqrt[3]*Sqrt[-1 - x^2])

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Rubi [A]  time = 0.0156102, antiderivative size = 40, 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} \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 - x^2]),x]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/(Sqrt[3]*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{2-3 x^2} \sqrt{-1-x^2}} \, dx &=\frac{\sqrt{1+x^2} \int \frac{1}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx}{\sqrt{-1-x^2}}\\ &=\frac{\sqrt{1+x^2} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-1-x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0238062, size = 40, normalized size = 1. \[ \frac{\sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3} \sqrt{-x^2-1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[2 - 3*x^2]*Sqrt[-1 - x^2]),x]

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/(Sqrt[3]*Sqrt[-1 - x^2])

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Maple [A]  time = 0.019, size = 34, normalized size = 0.9 \begin{align*}{{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{6} \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/(-3*x^2+2)^(1/2)/(-x^2-1)^(1/2),x)

[Out]

1/2*I*EllipticF(I*x,1/2*I*6^(1/2))/(x^2+1)^(1/2)*(-x^2-1)^(1/2)*2^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} - 1} \sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+2)^(1/2)/(-x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^2 - 1)*sqrt(-3*x^2 + 2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{2} - 1} \sqrt{-3 \, x^{2} + 2}}{3 \, x^{4} + x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+2)^(1/2)/(-x^2-1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(-x^2 - 1)*sqrt(-3*x^2 + 2)/(3*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 - 3 x^{2}} \sqrt{- x^{2} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**2+2)**(1/2)/(-x**2-1)**(1/2),x)

[Out]

Integral(1/(sqrt(2 - 3*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} - 1} \sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2+2)^(1/2)/(-x^2-1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^2 - 1)*sqrt(-3*x^2 + 2)), x)