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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {419} \[ \frac {F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[x], -2]/Sqrt[2]

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 {1-x^2} \sqrt {2+4 x^2}} \, dx &=\frac {F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 58, normalized size = 5.80 \[ -\frac {i \sqrt {1-x^2} \sqrt {2 x^2+1} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {2} x\right ),-\frac {1}{2}\right )}{2 \sqrt {-2 x^4+x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/2*I)*Sqrt[1 - x^2]*Sqrt[1 + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2]*x], -1/2])/Sqrt[1 + x^2 - 2*x^4]

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {4 \, x^{2} + 2} \sqrt {-x^{2} + 1}}{2 \, {\left (2 \, x^{4} - x^{2} - 1\right )}}, x\right ) \]

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, algorithm="fricas")

[Out]

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

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

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, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 14, normalized size = 1.40 \[ \frac {\sqrt {2}\, \EllipticF \left (x , i \sqrt {2}\right )}{2} \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {1}{\sqrt {1-x^2}\,\sqrt {4\,x^2+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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(1 - x**2)*sqrt(2*x**2 + 1)), x)/2

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