\(\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx\) [656]

Optimal result

Integrand size = 26, antiderivative size = 52 \[ \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx=\frac {\sqrt {1-2 x^2} \sqrt {1-x^2} \operatorname {EllipticF}(\arcsin (x),2)}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {533, 432, 430} \[ \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx=\frac {\sqrt {1-2 x^2} \sqrt {1-x^2} \operatorname {EllipticF}(\arcsin (x),2)}{\sqrt {x-1} \sqrt {x+1} \sqrt {2 x^2-1}} \]

[In]

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

[Out]

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

Rule 533

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPa
rt[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(52)=104\).

Time = 34.62 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.06 \[ \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx=-\frac {2 (-1+x)^{3/2} \sqrt {\frac {1+x}{1-x}} \sqrt {\frac {1-2 x^2}{(-1+x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2+\sqrt {2}+\frac {1}{-1+x}}}{2^{3/4}}\right ),4 \left (-4+3 \sqrt {2}\right )\right )}{\sqrt {3+2 \sqrt {2}} \sqrt {1+x} \sqrt {-1+2 x^2}} \]

[In]

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

[Out]

(-2*(-1 + x)^(3/2)*Sqrt[(1 + x)/(1 - x)]*Sqrt[(1 - 2*x^2)/(-1 + x)^2]*EllipticF[ArcSin[Sqrt[2 + Sqrt[2] + (-1
+ x)^(-1)]/2^(3/4)], 4*(-4 + 3*Sqrt[2])])/(Sqrt[3 + 2*Sqrt[2]]*Sqrt[1 + x]*Sqrt[-1 + 2*x^2])

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.12

method	result	size
default	\(\frac {\sqrt {-1+x}\, \sqrt {1+x}\, \sqrt {2 x^{2}-1}\, \sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, F\left (x , \sqrt {2}\right )}{2 x^{4}-3 x^{2}+1}\)	\(58\)
elliptic	\(\frac {\sqrt {\left (2 x^{2}-1\right ) \left (x^{2}-1\right )}\, \sqrt {-x^{2}+1}\, \sqrt {-2 x^{2}+1}\, F\left (x , \sqrt {2}\right )}{\sqrt {-1+x}\, \sqrt {1+x}\, \sqrt {2 x^{2}-1}\, \sqrt {2 x^{4}-3 x^{2}+1}}\)	\(73\)

[In]

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

[Out]

(-1+x)^(1/2)*(1+x)^(1/2)*(2*x^2-1)^(1/2)/(2*x^4-3*x^2+1)*(-x^2+1)^(1/2)*(-2*x^2+1)^(1/2)*EllipticF(x,2^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.08 \[ \int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \sqrt {-1+2 x^2}} \, dx=F(\arcsin \left (x\right )\,|\,2) \]

[In]

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

[Out]

elliptic_f(arcsin(x), 2)

Sympy [F]

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

[In]

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

[Out]

Integral(1/(sqrt(x - 1)*sqrt(x + 1)*sqrt(2*x**2 - 1)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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