\(\int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx\) [256]

Optimal result

Integrand size = 23, antiderivative size = 36 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-\frac {1}{2}\right )}{2 \sqrt {-1-x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {432, 430} \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-\frac {1}{2}\right )}{2 \sqrt {-x^2-1}} \]

[In]

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

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1/2])/(2*Sqrt[-1 - x^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])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*
x^2], Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + (d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^2} \int \frac {1}{\sqrt {2-4 x^2} \sqrt {1+x^2}} \, dx}{\sqrt {-1-x^2}} \\ & = \frac {\sqrt {1+x^2} F\left (\sin ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{2}\right )}{2 \sqrt {-1-x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {2} x\right ),-\frac {1}{2}\right )}{2 \sqrt {-1-x^2}} \]

[In]

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

[Out]

(Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[2]*x], -1/2])/(2*Sqrt[-1 - x^2])

Maple [A] (verified)

Time = 2.47 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {i F\left (i x , i \sqrt {2}\right ) \sqrt {2}\, \sqrt {-x^{2}-1}}{2 \sqrt {x^{2}+1}}\)	\(34\)
elliptic	\(-\frac {i \sqrt {\left (2 x^{2}-1\right ) \left (x^{2}+1\right )}\, \sqrt {x^{2}+1}\, F\left (i x , i \sqrt {2}\right )}{\sqrt {-x^{2}-1}\, \sqrt {4 x^{4}+2 x^{2}-2}}\)	\(60\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=-\frac {1}{4} \, \sqrt {2} \sqrt {-2} F(\arcsin \left (\sqrt {2} x\right )\,|\,-\frac {1}{2}) \]

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(-2)*elliptic_f(arcsin(sqrt(2)*x), -1/2)

Sympy [A] (verification not implemented)

Time = 1.98 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=\frac {\sqrt {2} \left (\begin {cases} - \frac {\sqrt {2} i F\left (\operatorname {asin}{\left (\sqrt {2} x \right )}\middle | - \frac {1}{2}\right )}{2} & \text {for}\: x > - \frac {\sqrt {2}}{2} \wedge x < \frac {\sqrt {2}}{2} \end {cases}\right )}{2} \]

[In]

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

[Out]

sqrt(2)*Piecewise((-sqrt(2)*I*elliptic_f(asin(sqrt(2)*x), -1/2)/2, (x > -sqrt(2)/2) & (x < sqrt(2)/2)))/2

Maxima [F]

\[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-4 \, x^{2} + 2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-4 \, x^{2} + 2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {2-4 x^2} \sqrt {-1-x^2}} \, dx=\int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-4\,x^2}} \,d x \]

[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)