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

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 = 0.0038201, 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, Rules used = {25, 261} \[ \frac{1}{\sqrt{\frac{1}{x^2}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1/Sqrt[1 + x^(-2)]

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[(u*(
a + b*x^n)^(m + p))/x^(n*p), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{1+\frac{1}{x^2}} x}{\left (1+x^2\right )^2} \, dx &=\int \frac{1}{\left (1+\frac{1}{x^2}\right )^{3/2} x^3} \, dx\\ &=\frac{1}{\sqrt{1+\frac{1}{x^2}}}\\ \end{align*}

Mathematica [B]  time = 0.0062454, 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)

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Maple [B]  time = 0.003, size = 23, normalized size = 2.6 \begin{align*}{\frac{{x}^{2}}{{x}^{2}+1}\sqrt{{\frac{{x}^{2}+1}{{x}^{2}}}}} \end{align*}

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)

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Maxima [A]  time = 1.54989, size = 15, normalized size = 1.67 \begin{align*} \frac{1}{\sqrt{\frac{x^{2} + 1}{x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+1/x^2)^(1/2)/(x^2+1)^2,x, algorithm="maxima")

[Out]

1/sqrt((x^2 + 1)/x^2)

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Fricas [B]  time = 1.44177, size = 63, normalized size = 7. \begin{align*} \frac{x^{2} \sqrt{\frac{x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+1/x^2)^(1/2)/(x^2+1)^2,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 = 2.40479, size = 8, normalized size = 0.89 \begin{align*} \frac{x}{\sqrt{x^{2} + 1}} \end{align*}

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)

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Giac [A]  time = 1.12942, size = 15, normalized size = 1.67 \begin{align*} \frac{x \mathrm{sgn}\left (x\right )}{\sqrt{x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1+1/x^2)^(1/2)/(x^2+1)^2,x, algorithm="giac")

[Out]

x*sgn(x)/sqrt(x^2 + 1)