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

Optimal. Leaf size=40 \[ \frac{\sqrt{1-2 x^2} E\left (\sin ^{-1}\left (\sqrt{2} x\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{2 x^2-1}} \]

[Out]

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

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Rubi [A]  time = 0.014525, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {427, 424} \[ \frac{\sqrt{1-2 x^2} E\left (\sin ^{-1}\left (\sqrt{2} x\right )|\frac{1}{2}\right )}{\sqrt{2} \sqrt{2 x^2-1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[-1 + 2*x^2],x]

[Out]

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

Rule 427

Int[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[Sqrt[a + b*x^2]/Sqrt[1 + (d*x^2)/c], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] &&  !GtQ[c, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

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

Mathematica [A]  time = 0.0240141, size = 35, normalized size = 0.88 \[ \frac{\sqrt{1-2 x^2} E\left (\sin ^{-1}\left (\sqrt{2} x\right )|\frac{1}{2}\right )}{\sqrt{4 x^2-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 32, normalized size = 0.8 \begin{align*}{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) +{\it EllipticE} \left ( x,\sqrt{2} \right ) }{2}\sqrt{-2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(EllipticF(x,2^(1/2))+EllipticE(x,2^(1/2)))*(-2*x^2+1)^(1/2)/(2*x^2-1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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