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

Optimal. Leaf size=49 \[ \frac{\sqrt{x^2+2} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

[Out]

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

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Rubi [A]  time = 0.0087737, antiderivative size = 49, 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 = {418} \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[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{2+x^2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-1-x^2} \sqrt{\frac{2+x^2}{1+x^2}}}\\ \end{align*}

Mathematica [C]  time = 0.0351139, size = 53, normalized size = 1.08 \[ -\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2} \sqrt{-\left (x^2+1\right ) \left (x^2+2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.019, size = 33, normalized size = 0.7 \begin{align*}{{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ( ix,{\frac{\sqrt{2}}{2}} \right ) \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*2^(1/2))*2^(1/2)/(x^2+1)^(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} + 3 \, 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 + 3*x^2 + 2), x)

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