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

Optimal. Leaf size=20 \[ \frac{\sqrt{x^2} \sqrt{x^2+1}}{x} \]

[Out]

(Sqrt[x^2]*Sqrt[1 + x^2])/x

_______________________________________________________________________________________

Rubi [A]  time = 0.0108797, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{x^2} \sqrt{x^2+1}}{x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x^2/(1 + x^2)],x]

[Out]

(Sqrt[x^2]*Sqrt[1 + x^2])/x

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.40483, size = 15, normalized size = 0.75 \[ \frac{\sqrt{x^{2} + 1} \sqrt{x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x**2 + 1)*sqrt(x**2)/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.00926511, size = 17, normalized size = 0.85 \[ \frac{x}{\sqrt{\frac{x^2}{x^2+1}}} \]

Antiderivative was successfully verified.

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

[Out]

x/Sqrt[x^2/(1 + x^2)]

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 23, normalized size = 1.2 \[{\frac{{x}^{2}+1}{x}\sqrt{{\frac{{x}^{2}}{{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2/(x^2+1))^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.757597, size = 9, normalized size = 0.45 \[ \sqrt{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.262364, size = 30, normalized size = 1.5 \[ \frac{{\left (x^{2} + 1\right )} \sqrt{\frac{x^{2}}{x^{2} + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x^2 + 1)*sqrt(x^2/(x^2 + 1))/x

_______________________________________________________________________________________

Sympy [A]  time = 1.8012, size = 36, normalized size = 1.8 \[ x \sqrt{x^{2}} \sqrt{\frac{1}{x^{2} + 1}} + \frac{\sqrt{x^{2}} \sqrt{\frac{1}{x^{2} + 1}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.258509, size = 20, normalized size = 1. \[ \sqrt{x^{2} + 1}{\rm sign}\left (x\right ) -{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x^2 + 1)*sign(x) - sign(x)