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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {419} \[ \frac {F\left (\sin ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[x], 1/2]/Sqrt[2]

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^2} \sqrt {2-x^2}} \, dx &=\frac {F\left (\sin ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ \frac {\operatorname {EllipticF}\left (\sin ^{-1}(x),\frac {1}{2}\right )}{\sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticF[ArcSin[x], 1/2]/Sqrt[2]

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} + 2} \sqrt {-x^{2} + 1}}{x^{4} - 3 \, x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.03, size = 13, normalized size = 1.08 \[ \frac {\sqrt {2}\, \EllipticF \left (x , \frac {\sqrt {2}}{2}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {1}{\sqrt {1-x^2}\,\sqrt {2-x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {2 - x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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