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

Optimal. Leaf size=109 \[ \frac{x+1}{\sqrt{\frac{2 x}{x^2+1}+1}}-\frac{(x+1) \sinh ^{-1}(x)}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{\sqrt{2} (x+1) \tanh ^{-1}\left (\frac{1-x}{\sqrt{2} \sqrt{x^2+1}}\right )}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.164534, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{x+1}{\sqrt{\frac{2 x}{x^2+1}+1}}-\frac{(x+1) \sinh ^{-1}(x)}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{\sqrt{2} (x+1) \tanh ^{-1}\left (\frac{1-x}{\sqrt{2} \sqrt{x^2+1}}\right )}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RecursionError

_______________________________________________________________________________________

Mathematica [A]  time = 0.0651665, size = 82, normalized size = 0.75 \[ \frac{(x+1) \left (\sqrt{x^2+1}-\sqrt{2} \log \left (\sqrt{2} \sqrt{x^2+1}-x+1\right )+\sqrt{2} \log (x+1)-\sinh ^{-1}(x)\right )}{\sqrt{\frac{(x+1)^2}{x^2+1}} \sqrt{x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.047, size = 79, normalized size = 0.7 \[{(1+x){\frac{1}{\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{{x}^{2}+1}}}}}}+{(1+x) \left ( -{\it Arcsinh} \left ( x \right ) -\sqrt{2}{\it Artanh} \left ({\frac{ \left ( 2-2\,x \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-2\,x}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ \left ( 1+x \right ) ^{2}}{{x}^{2}+1}}}}}{\frac{1}{\sqrt{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+2*x/(x^2+1))^(1/2),x)

[Out]

1/((1+x)^2/(x^2+1))^(1/2)*(1+x)+(-arcsinh(x)-2^(1/2)*arctanh(1/4*(2-2*x)*2^(1/2)
/((1+x)^2-2*x)^(1/2)))/((1+x)^2/(x^2+1))^(1/2)/(x^2+1)^(1/2)*(1+x)

_______________________________________________________________________________________

Maxima [A]  time = 0.780275, size = 46, normalized size = 0.42 \[ \sqrt{2} \operatorname{arsinh}\left (\frac{x}{{\left | x + 1 \right |}} - \frac{1}{{\left | x + 1 \right |}}\right ) + \sqrt{x^{2} + 1} - \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.307585, size = 290, normalized size = 2.66 \[ \frac{\sqrt{2}{\left (x + 1\right )} \log \left (\frac{2 \, x^{3} + 4 \, x^{2} + \sqrt{2}{\left (2 \, x^{2} + 3 \, x + 1\right )} -{\left (2 \, x^{3} + 2 \, x^{2} + \sqrt{2}{\left (2 \, x^{2} + x + 1\right )} + 3 \, x + 1\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + 4 \, x + 2}{2 \, x^{3} + 4 \, x^{2} -{\left (2 \, x^{3} + 2 \, x^{2} + x + 1\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + 2 \, x}\right ) +{\left (x + 1\right )} \log \left (-\frac{x \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} - x - 1}{\sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}}}\right ) +{\left (x^{2} + 1\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}}}{x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(sqrt(2)*(x + 1)*log((2*x^3 + 4*x^2 + sqrt(2)*(2*x^2 + 3*x + 1) - (2*x^3 + 2*x^2
+ sqrt(2)*(2*x^2 + x + 1) + 3*x + 1)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + 4*x + 2)
/(2*x^3 + 4*x^2 - (2*x^3 + 2*x^2 + x + 1)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + 2*x)
) + (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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.307031, size = 119, normalized size = 1.09 \[ \frac{\sqrt{2}{\rm ln}\left (\frac{{\left | -2 \, x - 2 \, \sqrt{2} + 2 \, \sqrt{x^{2} + 1} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt{2} + 2 \, \sqrt{x^{2} + 1} - 2 \right |}}\right )}{{\rm sign}\left (x + 1\right )} + \frac{{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right )}{{\rm sign}\left (x + 1\right )} + \frac{\sqrt{x^{2} + 1}}{{\rm sign}\left (x + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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