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

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

[Out]

1/Sqrt[1 + x^(-2)]

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Rubi [A]  time = 0.0139321, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]

Antiderivative was successfully verified.

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

[Out]

1/Sqrt[1 + x^(-2)]

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Rubi in Sympy [A]  time = 1.44591, 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(1/x/(x**2+1)/(1+1/x**2)**(1/2),x)

[Out]

1/sqrt(1 + x**(-2))

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Mathematica [B]  time = 0.00927791, size = 20, normalized size = 2.22 \[ \frac{\sqrt{\frac{1}{x^2}+1} x^2}{x^2+1} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[1 + x^(-2)]*x^2)/(1 + x^2)

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Maple [A]  time = 0.005, size = 12, normalized size = 1.3 \[{\frac{1}{\sqrt{{\frac{{x}^{2}+1}{{x}^{2}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.813992, size = 12, normalized size = 1.33 \[ \frac{x}{\sqrt{x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/sqrt(x^2 + 1)

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Fricas [A]  time = 0.260374, 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(1/((x^2 + 1)*x*sqrt(1/x^2 + 1)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.91692, 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]  integrate(1/x/(x**2+1)/(1+1/x**2)**(1/2),x)

[Out]

1/sqrt(1 + x**(-2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} + 1\right )} x \sqrt{\frac{1}{x^{2}} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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