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

Optimal. Leaf size=16 \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right ) \]

[Out]

EllipticF[ArcSin[Sqrt[2]*x], -1/2]/2

________________________________________________________________________________________

Rubi [A]  time = 0.0066735, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {419} \[ \frac{1}{2} F\left (\sin ^{-1}\left (\sqrt{2} x\right )|-\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0062921, size = 16, normalized size = 1. \[ \frac{1}{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-\frac{1}{2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[Sqrt[2]*x], -1/2]/2

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 15, normalized size = 0.9 \begin{align*}{\frac{{\it EllipticF} \left ( x\sqrt{2},{\frac{i}{2}}\sqrt{2} \right ) }{2}} \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),1/2*I*2^(1/2))

________________________________________________________________________________________

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)

________________________________________________________________________________________

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} + 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 + x^2 - 1), x)

________________________________________________________________________________________

Sympy [A]  time = 8.5156, size = 41, normalized size = 2.56 \begin{align*} \frac{\sqrt{2} \left (\begin{cases} \frac{\sqrt{2} 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)*elliptic_f(asin(sqrt(2)*x), -1/2)/2, (x > -sqrt(2)/2) & (x < sqrt(2)/2)))/2

________________________________________________________________________________________

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)