3.1007 \(\int \frac{x^2}{\sqrt{1-x^2} \sqrt{-1+2 x^2}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{2} \text{EllipticF}\left (\cos ^{-1}(x),2\right )-\frac{1}{2} E\left (\left .\cos ^{-1}(x)\right |2\right ) \]

[Out]

-EllipticE[ArcCos[x], 2]/2 - EllipticF[ArcCos[x], 2]/2

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Rubi [A]  time = 0.0278453, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {493, 425, 420} \[ -\frac{1}{2} F\left (\left .\cos ^{-1}(x)\right |2\right )-\frac{1}{2} E\left (\left .\cos ^{-1}(x)\right |2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-EllipticE[ArcCos[x], 2]/2 - EllipticF[ArcCos[x], 2]/2

Rule 493

Int[(x_)^(n_)/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/b, Int[Sqrt[a +
b*x^n]/Sqrt[c + d*x^n], x], x] - Dist[a/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c,
d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) &&  !(EqQ[n, 2] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 425

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

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

Mathematica [B]  time = 0.0348798, size = 37, normalized size = 2.18 \[ \frac{\sqrt{1-2 x^2} \left (\text{EllipticF}\left (\sin ^{-1}(x),2\right )-E\left (\left .\sin ^{-1}(x)\right |2\right )\right )}{2 \sqrt{2 x^2-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.018, size = 34, normalized size = 2. \begin{align*}{\frac{{\it EllipticF} \left ( x,\sqrt{2} \right ) -{\it EllipticE} \left ( x,\sqrt{2} \right ) }{2}\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(x^2/(-x^2+1)^(1/2)/(2*x^2-1)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\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(x^2/(-x^2+1)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(2*x^2 - 1)*sqrt(-x^2 + 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^{2} + 1} x^{2}}{2 \, x^{4} - 3 \, x^{2} + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-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)*x^2/(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{x^{2}}{\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(x**2/(-x**2+1)**(1/2)/(2*x**2-1)**(1/2),x)

[Out]

Integral(x**2/(sqrt(-(x - 1)*(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{x^{2}}{\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(x^2/(-x^2+1)^(1/2)/(2*x^2-1)^(1/2),x, algorithm="giac")

[Out]

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