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

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

[Out]

1/2*EllipticF(x,1/2*I*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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, 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 [C]  time = 0.02, size = 18, normalized size = 1.50 \[ -i \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-I)*EllipticF[I*ArcSinh[x/Sqrt[2]], -2]

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{2} + 2} \sqrt {-x^{2} + 1}}{x^{4} + x^{2} - 2}, x\right ) \]

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 + 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 = 14, normalized size = 1.17 \[ \frac {\sqrt {2}\, \EllipticF \left (x , \frac {i \sqrt {2}}{2}\right )}{2} \]

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*I*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 \[ \int \frac {1}{\sqrt {1-x^2}\,\sqrt {x^2+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.50, size = 19, normalized size = 1.58 \[ \begin {cases} \frac {\sqrt {2} F\left (\operatorname {asin}{\relax (x )}\middle | - \frac {1}{2}\right )}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \]

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]

Piecewise((sqrt(2)*elliptic_f(asin(x), -1/2)/2, (x > -1) & (x < 1)))

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