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

Optimal. Leaf size=10 \[ \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right ) \]

[Out]

EllipticF[ArcSin[x/Sqrt[2]], -2]

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Rubi [A]  time = 0.0061151, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {419} \[ F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[x/Sqrt[2]], -2]

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

Mathematica [C]  time = 0.0183213, size = 19, normalized size = 1.9 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.022, size = 14, normalized size = 1.4 \begin{align*}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

EllipticF(1/2*x*2^(1/2),I*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{-x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^2 + 1)*sqrt(-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{-x^{2} + 2}}{x^{4} - x^{2} - 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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