Optimal. Leaf size=28 \[ -\frac{(1-x) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1} \]
[Out]
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Rubi [A] time = 0.211871, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{(1-x) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + (2*x)/(1 + x^2)]/(1 + x^2),x]
[Out]
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Rubi in Sympy [A] time = 9.25262, size = 24, normalized size = 0.86 \[ - \frac{\left (- 2 x + 2\right ) \sqrt{\frac{2 x}{x^{2} + 1} + 1}}{2 \left (x + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+2*x/(x**2+1))**(1/2)/(x**2+1),x)
[Out]
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Mathematica [A] time = 0.0213153, size = 25, normalized size = 0.89 \[ \frac{(x-1) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + (2*x)/(1 + x^2)]/(1 + x^2),x]
[Out]
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Maple [A] time = 0.006, size = 28, normalized size = 1. \[{\frac{-1+x}{1+x}\sqrt{{\frac{{x}^{2}+2\,x+1}{{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x/(x^2+1))^(1/2)/(x^2+1),x)
[Out]
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Maxima [A] time = 0.78526, size = 26, normalized size = 0.93 \[ \frac{x}{\sqrt{x^{2} + 1}} - \frac{1}{\sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x/(x^2 + 1) + 1)/(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.263292, size = 42, normalized size = 1.5 \[ \frac{{\left (x - 1\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + x + 1}{x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x/(x^2 + 1) + 1)/(x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{\left (x + 1\right )^{2}}{x^{2} + 1}}}{x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+2*x/(x**2+1))**(1/2)/(x**2+1),x)
[Out]
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GIAC/XCAS [A] time = 0.266658, size = 41, normalized size = 1.46 \[ \sqrt{2}{\rm sign}\left (x + 1\right ) + \frac{x{\rm sign}\left (x + 1\right ) -{\rm sign}\left (x + 1\right )}{\sqrt{x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x/(x^2 + 1) + 1)/(x^2 + 1),x, algorithm="giac")
[Out]