Optimal. Leaf size=52 \[ \frac{\sqrt{-\frac{x^2}{1-x^2}} \sqrt{x^2-1} \tan ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]
[Out]
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Rubi [A] time = 0.177504, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{\sqrt{-\frac{x^2}{1-x^2}} \sqrt{x^2-1} \tan ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x^2/(-1 + x^2)]/(1 + x^2),x]
[Out]
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Rubi in Sympy [A] time = 15.4958, size = 44, normalized size = 0.85 \[ \frac{\sqrt{2} \sqrt{\frac{x^{2}}{x^{2} - 1}} \sqrt{x^{2} - 1} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{x^{2} - 1}}{2} \right )}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2/(x**2-1))**(1/2)/(x**2+1),x)
[Out]
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Mathematica [A] time = 0.0298416, size = 49, normalized size = 0.94 \[ \frac{\sqrt{\frac{x^2}{x^2-1}} \sqrt{x^2-1} \tan ^{-1}\left (\frac{\sqrt{x^2-1}}{\sqrt{2}}\right )}{\sqrt{2} x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x^2/(-1 + x^2)]/(1 + x^2),x]
[Out]
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Maple [A] time = 0.016, size = 42, normalized size = 0.8 \[{\frac{\sqrt{2}}{2\,x}\sqrt{{\frac{{x}^{2}}{{x}^{2}-1}}}\sqrt{{x}^{2}-1}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{{x}^{2}-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2/(x^2-1))^(1/2)/(x^2+1),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{x^{2}}{x^{2} - 1}}}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2/(x^2 - 1))/(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272557, size = 81, normalized size = 1.56 \[ -\frac{1}{2} \, \sqrt{2} \arctan \left (-\frac{\sqrt{2} x^{2} - \sqrt{2}{\left (x^{2} - 1\right )} \sqrt{\frac{x^{2}}{x^{2} - 1}}}{2 \,{\left (x \sqrt{\frac{x^{2}}{x^{2} - 1}} - x\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2/(x^2 - 1))/(x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{x^{2}}{x^{2} - 1}}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2/(x**2-1))**(1/2)/(x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.281761, size = 55, normalized size = 1.06 \[ \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x^{2} - 1}\right ){\rm sign}\left (x^{2} - 1\right ){\rm sign}\left (x\right ) + \frac{1}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} i\right ){\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^2/(x^2 - 1))/(x^2 + 1),x, algorithm="giac")
[Out]