Optimal. Leaf size=28 \[ \frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}} \]
[Out]
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Rubi [A] time = 0.0113364, antiderivative size = 28, 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 \[ \frac{\sqrt{x^2+1} \tan ^{-1}(x)}{\sqrt{2} \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[-1 - x^2]*Sqrt[2 + 2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 3.12783, size = 24, normalized size = 0.86 \[ \frac{\sqrt{2 x^{2} + 2} \operatorname{atan}{\left (x \right )}}{2 \sqrt{- x^{2} - 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-x**2-1)**(1/2)/(2*x**2+2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0168138, size = 26, normalized size = 0.93 \[ \frac{\left (x^2+1\right ) \tan ^{-1}(x)}{\sqrt{2} \sqrt{-\left (x^2+1\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[-1 - x^2]*Sqrt[2 + 2*x^2]),x]
[Out]
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Maple [A] time = 0.01, size = 24, normalized size = 0.9 \[ -{\frac{\arctan \left ( x \right ) \sqrt{2}}{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2-1)^(1/2)/(2*x^2+2)^(1/2),x)
[Out]
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Maxima [A] time = 1.52956, size = 9, normalized size = 0.32 \[ -\frac{1}{2} i \, \sqrt{2} \arctan \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*x^2 + 2)*sqrt(-x^2 - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233575, size = 136, normalized size = 4.86 \[ \frac{1}{8} \, \sqrt{2}{\left (\log \left (\frac{2 \, \sqrt{2}{\left (x^{4} + \sqrt{2} \sqrt{2 \, x^{2} + 2} \sqrt{-x^{2} - 1} x - 1\right )}}{x^{4} + 2 \, x^{2} + 1}\right ) - \log \left (-\frac{2 \, \sqrt{2}{\left (x^{4} - \sqrt{2} \sqrt{2 \, x^{2} + 2} \sqrt{-x^{2} - 1} x - 1\right )}}{x^{4} + 2 \, x^{2} + 1}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*x^2 + 2)*sqrt(-x^2 - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt{2} \int \frac{1}{\sqrt{- x^{2} - 1} \sqrt{x^{2} + 1}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-x**2-1)**(1/2)/(2*x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(2*x^2 + 2)*sqrt(-x^2 - 1)),x, algorithm="giac")
[Out]