Optimal. Leaf size=49 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]
[Out]
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Rubi [A] time = 0.0313948, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + x^2]),x]
[Out]
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Rubi in Sympy [A] time = 5.4532, size = 48, normalized size = 0.98 \[ \frac{\sqrt{2} \sqrt{x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{- \frac{- x^{2} - 2}{x^{2} + 1}} \sqrt{- x^{2} - 1}} \]
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]
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Mathematica [C] time = 0.0537498, size = 53, normalized size = 1.08 \[ -\frac{i \sqrt{x^2+1} \sqrt{x^2+2} F\left (i \sinh ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-\left (x^2+1\right ) \left (x^2+2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + x^2]),x]
[Out]
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Maple [C] time = 0.028, size = 33, normalized size = 0.7 \[{{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ( ix,{\frac{\sqrt{2}}{2}} \right ) \sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]
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]
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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]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{x^{2} + 2} \sqrt{-x^{2} - 1}}{x^{4} + 3 \, x^{2} + 2}, x\right ) \]
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]
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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]
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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]