Optimal. Leaf size=32 \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0438645, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-1 + x^2]*Sqrt[2 + x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.31563, size = 31, normalized size = 0.97 \[ \frac{\sqrt{2} \sqrt{- x^{2} + 1} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{2}\right )}{2 \sqrt{x^{2} - 1}} \]
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]
_______________________________________________________________________________________
Mathematica [A] time = 0.0295661, size = 32, normalized size = 1. \[ \frac{\sqrt{1-x^2} F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-1 + x^2]*Sqrt[2 + x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.035, size = 34, normalized size = 1.1 \[{-i{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},i\sqrt{2} \right ) \sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{{x}^{2}-1}}}} \]
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
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]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \sqrt{x^{2} + 2}}\, dx \]
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]
_______________________________________________________________________________________
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]