3.520 \(\int \frac{\sqrt{1+\frac{1}{x^2}} x}{\left (1+x^2\right )^2} \, dx\)

Optimal. Leaf size=9 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]

[Out]

1/Sqrt[1 + x^(-2)]

_______________________________________________________________________________________

Rubi [A]  time = 0.0140857, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]

Antiderivative was successfully verified.

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

[Out]

1/Sqrt[1 + x^(-2)]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.41876, size = 10, normalized size = 1.11 \[ \frac{1}{\sqrt{1 + \frac{1}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/sqrt(1 + x**(-2))

_______________________________________________________________________________________

Mathematica [B]  time = 0.0128918, size = 20, normalized size = 2.22 \[ \frac{\sqrt{\frac{1}{x^2}+1} x^2}{x^2+1} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + x^(-2)]*x^2)/(1 + x^2)

_______________________________________________________________________________________

Maple [B]  time = 0.005, size = 23, normalized size = 2.6 \[{\frac{{x}^{2}}{{x}^{2}+1}\sqrt{{\frac{{x}^{2}+1}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.804031, size = 15, normalized size = 1.67 \[ \frac{1}{\sqrt{\frac{x^{2} + 1}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/sqrt((x^2 + 1)/x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.261212, size = 38, normalized size = 4.22 \[ \frac{x^{2} \sqrt{\frac{x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 7.11112, size = 8, normalized size = 0.89 \[ \frac{x}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/sqrt(x**2 + 1)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.264072, size = 15, normalized size = 1.67 \[ \frac{x{\rm sign}\left (x\right )}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sign(x)/sqrt(x^2 + 1)