3.219 \(\int \frac{1}{\sqrt{1-x^2} \sqrt{2+4 x^2}} \, dx\)

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

[Out]

EllipticF[ArcSin[x], -2]/Sqrt[2]

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Rubi [A]  time = 0.007433, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {419} \[ \frac{F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - x^2]*Sqrt[2 + 4*x^2]),x]

[Out]

EllipticF[ArcSin[x], -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+4 x^2}} \, dx &=\frac{F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.030298, size = 58, normalized size = 5.8 \[ -\frac{i \sqrt{1-x^2} \sqrt{2 x^2+1} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right )}{2 \sqrt{-2 x^4+x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + 4*x^2]),x]

[Out]

((-I/2)*Sqrt[1 - x^2]*Sqrt[1 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2]*x], -1/2])/Sqrt[1 + x^2 - 2*x^4]

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Maple [A]  time = 0.023, size = 14, normalized size = 1.4 \begin{align*}{\frac{{\it EllipticF} \left ( x,i\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)/(4*x^2+2)^(1/2),x)

[Out]

1/2*EllipticF(x,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{4 \, 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)/(4*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/2)/(4*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+1)**(1/2)/(4*x**2+2)**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{4 \, 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)/(4*x^2+2)^(1/2),x, algorithm="giac")

[Out]

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