Optimal. Leaf size=12 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2}} \]
[Out]
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Rubi [A] time = 0.0228961, antiderivative size = 12, 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 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{2}\right )}{\sqrt{2}} \]
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 = 4.94708, size = 14, normalized size = 1.17 \[ \frac{\sqrt{2} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{2}\right )}{2} \]
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 = 0.0326594, size = 18, normalized size = 1.5 \[ -i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 - x^2]*Sqrt[2 + x^2]),x]
[Out]
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Maple [A] time = 0.037, size = 14, normalized size = 1.2 \[{\frac{{\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{2} \right ) \sqrt{2}}{2}} \]
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. \[{\rm integral}\left (\frac{1}{\sqrt{x^{2} + 2} \sqrt{-x^{2} + 1}}, x\right ) \]
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]
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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]
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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]