3.3.26 \(\int \frac {1}{\sqrt {2-4 x^2} \sqrt {1-x^2}} \, dx\) [226]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {430} \begin {gather*} \frac {F(\text {ArcSin}(x)|2)}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[x], 2]/Sqrt[2]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[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 {F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 10, normalized size = 1.00 \begin {gather*} \frac {F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[x], 2]/Sqrt[2]

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Maple [A]
time = 0.10, size = 11, normalized size = 1.10

method	result	size
default	\(\frac {\EllipticF \left (x , \sqrt {2}\right ) \sqrt {2}}{2}\)	\(11\)
elliptic	\(\frac {\sqrt {\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, \EllipticF \left (x , \sqrt {2}\right )}{\sqrt {4 x^{4}-6 x^{2}+2}}\)	\(36\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.18, size = 8, normalized size = 0.80 \begin {gather*} \frac {1}{2} \, \sqrt {2} {\rm ellipticF}\left (x, 2\right ) \end {gather*}

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]

1/2*sqrt(2)*ellipticF(x, 2)

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Sympy [A]
time = 1.86, size = 39, normalized size = 3.90 \begin {gather*} \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 {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.10 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-4\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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