3.746 \(\int \frac{\sqrt{1+\frac{2 x}{1+x^2}}}{1+x^2} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(1-x) \sqrt{\frac{2 x}{x^2+1}+1}}{x+1} \]

[Out]

-(((1 - x)*Sqrt[1 + (2*x)/(1 + x^2)])/(1 + x))

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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]

-(((1 - x)*Sqrt[1 + (2*x)/(1 + x^2)])/(1 + x))

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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]

-(-2*x + 2)*sqrt(2*x/(x**2 + 1) + 1)/(2*(x + 1))

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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]

((-1 + x)*Sqrt[1 + (2*x)/(1 + x^2)])/(1 + x)

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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]

(-1+x)/(1+x)*((x^2+2*x+1)/(x^2+1))^(1/2)

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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]

x/sqrt(x^2 + 1) - 1/sqrt(x^2 + 1)

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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]

((x - 1)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + x + 1)/(x + 1)

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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]

Integral(sqrt((x + 1)**2/(x**2 + 1))/(x**2 + 1), x)

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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]

sqrt(2)*sign(x + 1) + (x*sign(x + 1) - sign(x + 1))/sqrt(x^2 + 1)