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

Optimal. Leaf size=30 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt{2} \sqrt{x^2-1}} \]

[Out]

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

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Rubi [A]  time = 0.0150831, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {421, 419} \[ \frac{\sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt{2} \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.029037, size = 36, normalized size = 1.2 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),\frac{1}{2}\right )}{2 \sqrt{x^2-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 27, normalized size = 0.9 \begin{align*}{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) \sqrt{2}}{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/(-4*x^2+2)^(1/2)/(x^2-1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(x^2 - 1)*sqrt(-4*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{-4 \, x^{2} + 2}}{2 \,{\left (2 \, x^{4} - 3 \, x^{2} + 1\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/2*sqrt(x^2 - 1)*sqrt(-4*x^2 + 2)/(2*x^4 - 3*x^2 + 1), x)

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Sympy [A]  time = 8.98279, size = 42, normalized size = 1.4 \begin{align*} \frac{\sqrt{2} \left (\begin{cases} - \frac{\sqrt{2} i F\left (\operatorname{asin}{\left (\sqrt{2} x \right )}\middle | \frac{1}{2}\right )}{2} & \text{for}\: x > - \frac{\sqrt{2}}{2} \wedge x < \frac{\sqrt{2}}{2} \end{cases}\right )}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} - 1} \sqrt{-4 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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