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

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

[Out]

EllipticF[ArcSin[x], -2]/Sqrt[2]

_______________________________________________________________________________________

Rubi [A]  time = 0.0273416, 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 \[ \frac{F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

EllipticF[ArcSin[x], -2]/Sqrt[2]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.24884, size = 12, normalized size = 1.2 \[ \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | -2\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2+1)**(1/2)/(4*x**2+2)**(1/2),x)

[Out]

sqrt(2)*elliptic_f(asin(x), -2)/2

_______________________________________________________________________________________

Mathematica [C]  time = 0.0512027, size = 58, normalized size = 5.8 \[ -\frac{i \sqrt{1-x^2} \sqrt{2 x^2+1} F\left (i \sinh ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right )}{2 \sqrt{-2 x^4+x^2+1}} \]

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])/Sqr
t[1 + x^2 - 2*x^4]

_______________________________________________________________________________________

Maple [A]  time = 0.038, size = 14, normalized size = 1.4 \[{\frac{{\it EllipticF} \left ( x,i\sqrt{2} \right ) \sqrt{2}}{2}} \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{4 \, x^{2} + 2} \sqrt{-x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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(-x**2 + 1)*sqrt(2*x**2 + 1)), x)/2

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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