Optimal. Leaf size=40 \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x^2-1}} \]
[Out]
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Rubi [A] time = 0.0522014, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 - x^2]),x]
[Out]
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Rubi in Sympy [A] time = 8.90966, size = 39, normalized size = 0.98 \[ \frac{\sqrt{5} \sqrt{x^{2} + 1} F\left (\operatorname{asin}{\left (\frac{\sqrt{10} x}{2} \right )}\middle | - \frac{2}{5}\right )}{5 \sqrt{- x^{2} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-5*x**2+2)**(1/2)/(-x**2-1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0377513, size = 40, normalized size = 1. \[ \frac{\sqrt{x^2+1} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{2}} x\right )|-\frac{2}{5}\right )}{\sqrt{5} \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[2 - 5*x^2]*Sqrt[-1 - x^2]),x]
[Out]
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Maple [A] time = 0.035, size = 37, normalized size = 0.9 \[{{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{2}\sqrt{5} \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-5*x^2+2)^(1/2)/(-x^2-1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} - 1} \sqrt{-5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-x^{2} - 1} \sqrt{-5 \, x^{2} + 2}}{5 \, x^{4} + 3 \, x^{2} - 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 5 x^{2} + 2} \sqrt{- x^{2} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-5*x**2+2)**(1/2)/(-x**2-1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-x^{2} - 1} \sqrt{-5 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-x^2 - 1)*sqrt(-5*x^2 + 2)),x, algorithm="giac")
[Out]