3.3.52 \(\int \frac {1}{\sqrt {-1-x^2} \sqrt {2+x^2}} \, dx\) [252]

Optimal. Leaf size=49 \[ \frac {\sqrt {2+x^2} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1-x^2} \sqrt {\frac {2+x^2}{1+x^2}}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 49, 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 = {429} \begin {gather*} \frac {\sqrt {x^2+2} F\left (\text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{\sqrt {2} \sqrt {-x^2-1} \sqrt {\frac {x^2+2}{x^2+1}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.03, size = 53, normalized size = 1.08 \begin {gather*} -\frac {i \sqrt {1+x^2} \sqrt {2+x^2} F\left (i \sinh ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2} \sqrt {-\left (\left (1+x^2\right ) \left (2+x^2\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains complex when optimal does not.
time = 0.07, size = 33, normalized size = 0.67

method	result	size
default	\(\frac {i \EllipticF \left (i x , \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-x^{2}-1}}{2 \sqrt {x^{2}+1}}\)	\(33\)
elliptic	\(-\frac {i \sqrt {-\left (x^{2}+1\right ) \left (x^{2}+2\right )}\, \sqrt {x^{2}+1}\, \sqrt {2 x^{2}+4}\, \EllipticF \left (i x , \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-x^{2}-1}\, \sqrt {x^{2}+2}\, \sqrt {-x^{4}-3 x^{2}-2}}\)	\(74\)

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*I*EllipticF(I*x,1/2*2^(1/2))*2^(1/2)/(x^2+1)^(1/2)*(-x^2-1)^(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 [C] Result contains complex when optimal does not.
time = 0.18, size = 16, normalized size = 0.33 \begin {gather*} \frac {1}{2} i \, \sqrt {2} \sqrt {-2} {\rm ellipticF}\left (\frac {1}{2} i \, \sqrt {2} x, 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*I*sqrt(2)*sqrt(-2)*ellipticF(1/2*I*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 {- x^{2} - 1} \sqrt {x^{2} + 2}}\, dx \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]

Integral(1/(sqrt(-x**2 - 1)*sqrt(x**2 + 2)), 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^2-1)^(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 = -0.02 \begin {gather*} \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {x^2+2}} \,d x \end {gather*}

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]

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

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