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

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

[Out]

1/2*EllipticF(x*2^(1/2),1/2*I*2^(1/2))

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \frac {1}{2} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {2} x\right ),-\frac {1}{2}\right ) \]

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

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fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{2} + 1} \sqrt {-4 \, x^{2} + 2}}{2 \, {\left (2 \, x^{4} + x^{2} - 1\right )}}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {-4 \, x^{2} + 2}}\,{d x} \]

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)

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maple [A]  time = 0.03, size = 15, normalized size = 0.94 \[ \frac {\EllipticF \left (\sqrt {2}\, x , \frac {i \sqrt {2}}{2}\right )}{2} \]

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(2^(1/2)*x,1/2*I*2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {-4 \, x^{2} + 2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {x^2+1}\,\sqrt {2-4\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 6.02, size = 41, normalized size = 2.56 \[ \frac {\sqrt {2} \left (\begin {cases} \frac {\sqrt {2} F\left (\operatorname {asin}{\left (\sqrt {2} x \right )}\middle | - \frac {1}{2}\right )}{2} & \text {for}\: x > - \frac {\sqrt {2}}{2} \wedge x < \frac {\sqrt {2}}{2} \end {cases}\right )}{2} \]

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

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