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

Optimal. Leaf size=12 \[ \frac{\text{EllipticF}\left (\sin ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2}} \]

[Out]

EllipticF[ArcSin[x], -1/2]/Sqrt[2]

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Rubi [A]  time = 0.0065566, antiderivative size = 12, 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} \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 0.0194594, size = 18, normalized size = 1.5 \[ -i \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.022, size = 14, normalized size = 1.2 \begin{align*}{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{2} \right ) \sqrt{2}}{2}} \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*EllipticF(x,1/2*I*2^(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} + 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 [A]  time = 2.17779, size = 19, normalized size = 1.58 \begin{align*} \begin{cases} \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{2}\right )}{2} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \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]

Piecewise((sqrt(2)*elliptic_f(asin(x), -1/2)/2, (x > -1) & (x < 1)))

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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)