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

Optimal. Leaf size=52 \[ \frac{\sqrt{1-2 x^2} \sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt{x-1} \sqrt{x+1} \sqrt{2 x^2-1}} \]

[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])

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Rubi [A]  time = 0.0463509, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {519, 421, 419} \[ \frac{\sqrt{1-2 x^2} \sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |2\right )}{\sqrt{x-1} \sqrt{x+1} \sqrt{2 x^2-1}} \]

Antiderivative was successfully verified.

[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 519

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)^FracP
art[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])

Rule 421

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

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} \sqrt{1+x} \sqrt{-1+2 x^2}} \, dx &=\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]  time = 0.238413, size = 107, normalized size = 2.06 \[ -\frac{2 (x-1)^{3/2} \sqrt{\frac{x+1}{1-x}} \sqrt{\frac{1-2 x^2}{(x-1)^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{1}{x-1}+\sqrt{2}+2}}{2^{3/4}}\right ),4 \left (3 \sqrt{2}-4\right )\right )}{\sqrt{3+2 \sqrt{2}} \sqrt{x+1} \sqrt{2 x^2-1}} \]

Warning: Unable to verify antiderivative.

[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])

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Maple [A]  time = 0.115, size = 58, normalized size = 1.1 \begin{align*}{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) }{2\,{x}^{4}-3\,{x}^{2}+1}\sqrt{-1+x}\sqrt{1+x}\sqrt{2\,{x}^{2}-1}\sqrt{-{x}^{2}+1}\sqrt{-2\,{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{2} - 1} \sqrt{x + 1} \sqrt{x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{2 \, x^{2} - 1} \sqrt{x + 1} \sqrt{x - 1}}{2 \, x^{4} - 3 \, x^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x - 1} \sqrt{x + 1} \sqrt{2 x^{2} - 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{2} - 1} \sqrt{x + 1} \sqrt{x - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)