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]

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

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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 1}}{\sqrt {2 \, x^{2} - 1}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {2 \, x^{2} - 1}}\,{d x} \]

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)

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maple [A]  time = 0.02, size = 32, normalized size = 0.80 \[ \frac {\left (\EllipticE \left (x , \sqrt {2}\right )+\EllipticF \left (x , \sqrt {2}\right )\right ) \sqrt {-2 x^{2}+1}}{2 \sqrt {2 x^{2}-1}} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{\sqrt {2 \, x^{2} - 1}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {2\,x^2-1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {2 x^{2} - 1}}\, dx \]

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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