Optimal. Leaf size=51 \[ \frac{\sqrt{2 x^2+1} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+1}{x^2+1}}} \]
[Out]
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Rubi [A] time = 0.0372169, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{\sqrt{2 x^2+1} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{2 x^2+1}{x^2+1}}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + 4*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 5.72887, size = 51, normalized size = 1. \[ \frac{\sqrt{2} \sqrt{4 x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | -1\right )}{2 \sqrt{- \frac{- 4 x^{2} - 2}{x^{2} + 1}} \sqrt{- x^{2} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**2-1)**(1/2)/(4*x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0432214, size = 37, normalized size = 0.73 \[ -\frac{i \sqrt{x^2+1} F\left (\left .i \sinh ^{-1}(x)\right |2\right )}{\sqrt{2} \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + 4*x^2]),x]
[Out]
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Maple [A] time = 0.06, size = 33, normalized size = 0.7 \[{{\frac{i}{2}}{\it EllipticF} \left ( ix\sqrt{2},{\frac{\sqrt{2}}{2}} \right ) \sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2-1)^(1/2)/(4*x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(4*x^2 + 2)*sqrt(-x^2 - 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{4 \, x^{2} + 2} \sqrt{-x^{2} - 1}}{2 \,{\left (2 \, x^{4} + 3 \, x^{2} + 1\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(4*x^2 + 2)*sqrt(-x^2 - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{2} \int \frac{1}{\sqrt{- x^{2} - 1} \sqrt{2 x^{2} + 1}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**2-1)**(1/2)/(4*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{4 \, x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(4*x^2 + 2)*sqrt(-x^2 - 1)),x, algorithm="giac")
[Out]