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

Optimal. Leaf size=47 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0285857, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{x^2+2}{x^2+1}}} \]

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)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.87794, size = 42, normalized size = 0.89 \[ \frac{\sqrt{2} \sqrt{x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{\frac{x^{2} + 2}{x^{2} + 1}} \sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2+1)**(1/2)/(x**2+2)**(1/2),x)

[Out]

sqrt(2)*sqrt(x**2 + 2)*elliptic_f(atan(x), 1/2)/(2*sqrt((x**2 + 2)/(x**2 + 1))*s
qrt(x**2 + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0288724, size = 19, normalized size = 0.4 \[ -\frac{i F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [C]  time = 0.034, size = 15, normalized size = 0.3 \[ -i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \]

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]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 1} \sqrt{x^{2} + 2}}\, dx \]

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**2 + 1)*sqrt(x**2 + 2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} + 2} \sqrt{x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 + 2)*sqrt(x^2 + 1)),x, algorithm="giac")

[Out]

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