3.3.22 \(\int \frac {1}{\sqrt {1-x^2} \sqrt {2+x^2}} \, dx\) [222]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {430} \begin {gather*} \frac {F\left (\text {ArcSin}(x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2}} \end {gather*}

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 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[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] Result contains complex when optimal does not.
time = 10.05, size = 18, normalized size = 1.50 \begin {gather*} -i F\left (\left .i \sinh ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \end {gather*}

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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Maple [A]
time = 0.09, size = 14, normalized size = 1.17

method	result	size
default	\(\frac {\EllipticF \left (x , \frac {i \sqrt {2}}{2}\right ) \sqrt {2}}{2}\)	\(14\)
elliptic	\(\frac {\sqrt {-\left (x^{2}-1\right ) \left (x^{2}+2\right )}\, \sqrt {2 x^{2}+4}\, \EllipticF \left (x , \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {x^{2}+2}\, \sqrt {-x^{4}-x^{2}+2}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+1)^(1/2)/(x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.19, size = 8, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {2} {\rm ellipticF}\left (x, -\frac {1}{2}\right ) \end {gather*}

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]

1/2*sqrt(2)*ellipticF(x, -1/2)

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Sympy [A]
time = 1.22, size = 19, normalized size = 1.58 \begin {gather*} \begin {cases} \frac {\sqrt {2} F\left (\operatorname {asin}{\left (x \right )}\middle | - \frac {1}{2}\right )}{2} & \text {for}\: x > -1 \wedge x < 1 \end {cases} \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {1}{\sqrt {1-x^2}\,\sqrt {x^2+2}} \,d x \end {gather*}

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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