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

Optimal. Leaf size=6 \[ -\text{EllipticF}\left (\cos ^{-1}(x),2\right ) \]

[Out]

-EllipticF[ArcCos[x], 2]

________________________________________________________________________________________

Rubi [A]  time = 0.0060251, antiderivative size = 6, normalized size of antiderivative = 1., 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 = {420} \[ -F\left (\left .\cos ^{-1}(x)\right |2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-EllipticF[ArcCos[x], 2]

Rule 420

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

Rubi steps

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

Mathematica [B]  time = 0.0245561, size = 27, normalized size = 4.5 \[ \frac{\sqrt{1-2 x^2} \text{EllipticF}\left (\sin ^{-1}(x),2\right )}{\sqrt{2 x^2-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 25, normalized size = 4.2 \begin{align*}{{\it EllipticF} \left ( x,\sqrt{2} \right ) \sqrt{-2\,{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{2 \, x^{2} - 1} \sqrt{-x^{2} + 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/(-x^2+1)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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