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

Optimal. Leaf size=61 \[ \frac{\sqrt{\frac{2 x}{x^2+1}+1} \left (x^2+1\right )}{x+1}+\frac{\sqrt{\frac{2 x}{x^2+1}+1} \sqrt{x^2+1} \sinh ^{-1}(x)}{x+1} \]

[Out]

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

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Rubi [A]  time = 0.0727453, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{\frac{2 x}{x^2+1}+1} \left (x^2+1\right )}{x+1}+\frac{\sqrt{\frac{2 x}{x^2+1}+1} \sqrt{x^2+1} \sinh ^{-1}(x)}{x+1} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + (2*x)/(1 + x^2)],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.69614, size = 49, normalized size = 0.8 \[ \frac{\sqrt{x^{2} + 1} \sqrt{\frac{2 x}{x^{2} + 1} + 1} \operatorname{asinh}{\left (x \right )}}{x + 1} + \frac{\left (x^{2} + 1\right ) \sqrt{\frac{2 x}{x^{2} + 1} + 1}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+2*x/(x**2+1))**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0264008, size = 40, normalized size = 0.66 \[ \frac{\sqrt{\frac{(x+1)^2}{x^2+1}} \left (x^2+\sqrt{x^2+1} \sinh ^{-1}(x)+1\right )}{x+1} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + (2*x)/(1 + x^2)],x]

[Out]

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

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Maple [A]  time = 0.008, size = 42, normalized size = 0.7 \[{\frac{1}{1+x}\sqrt{{\frac{{x}^{2}+2\,x+1}{{x}^{2}+1}}}\sqrt{{x}^{2}+1} \left ({\it Arcsinh} \left ( x \right ) +\sqrt{{x}^{2}+1} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+2*x/(x^2+1))^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.785498, size = 14, normalized size = 0.23 \[ \sqrt{x^{2} + 1} + \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*x/(x^2 + 1) + 1),x, algorithm="maxima")

[Out]

sqrt(x^2 + 1) + arcsinh(x)

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Fricas [A]  time = 0.268413, size = 112, normalized size = 1.84 \[ -\frac{{\left (x + 1\right )} \log \left (-\frac{x \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} - x - 1}{\sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}}}\right ) -{\left (x^{2} + 1\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*x/(x^2 + 1) + 1),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{2 x}{x^{2} + 1} + 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+2*x/(x**2+1))**(1/2),x)

[Out]

Integral(sqrt(2*x/(x**2 + 1) + 1), x)

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GIAC/XCAS [A]  time = 0.263335, size = 66, normalized size = 1.08 \[ -{\left (\sqrt{2} -{\rm ln}\left (\sqrt{2} + 1\right )\right )}{\rm sign}\left (x + 1\right ) -{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ){\rm sign}\left (x + 1\right ) + \sqrt{x^{2} + 1}{\rm sign}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(2*x/(x^2 + 1) + 1),x, algorithm="giac")

[Out]

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