3.3.33 \(\int \frac {1}{\sqrt {2-x^2} \sqrt {1+x^2}} \, dx\) [233]

Optimal. Leaf size=10 \[ F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 10, 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*} F\left (\left .\text {ArcSin}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 10.04, size = 19, normalized size = 1.90 \begin {gather*} -\frac {i F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*EllipticF[I*ArcSinh[x], -1/2])/Sqrt[2]

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Maple [A]
time = 0.09, size = 14, normalized size = 1.40

method	result	size
default	\(\EllipticF \left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )\)	\(14\)
elliptic	\(\frac {\sqrt {-\left (x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {2}}{2}, i \sqrt {2}\right )}{2 \sqrt {-x^{2}+2}\, \sqrt {-x^{4}+x^{2}+2}}\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

EllipticF(1/2*x*2^(1/2),I*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+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 [A]
time = 0.18, size = 8, normalized size = 0.80 \begin {gather*} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x, -2\right ) \end {gather*}

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]

ellipticF(1/2*sqrt(2)*x, -2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 - x^{2}} \sqrt {x^{2} + 1}}\, dx \end {gather*}

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(2 - x**2)*sqrt(x**2 + 1)), x)

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.10 \begin {gather*} \int \frac {1}{\sqrt {x^2+1}\,\sqrt {2-x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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