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

Optimal. Leaf size=12 \[ -\text{EllipticF}\left (\cos ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right ) \]

[Out]

-EllipticF[ArcCos[x/Sqrt[2]], 2]

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Rubi [A]  time = 0.0065863, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {420} \[ -F\left (\left .\cos ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-EllipticF[ArcCos[x/Sqrt[2]], 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{2-x^2} \sqrt{-1+x^2}} \, dx &=-F\left (\left .\cos ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )\\ \end{align*}

Mathematica [B]  time = 0.0221592, size = 47, normalized size = 3.92 \[ \frac{\sqrt{1-x^2} \sqrt{1-\frac{x^2}{2}} \text{EllipticF}\left (\sin ^{-1}(x),\frac{1}{2}\right )}{\sqrt{-x^4+3 x^2-2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.02, size = 28, normalized size = 2.3 \begin{align*}{{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},\sqrt{2} \right ) \sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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