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

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.01, 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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fricas [B]  time = 0.46, size = 28, normalized size = 3.11 \[ \frac {x^{2} \sqrt {\frac {x^{2} + 1}{x^{2}}} + x^{2} + 1}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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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giac [A]  time = 0.34, size = 11, normalized size = 1.22 \[ \frac {x \mathrm {sgn}\relax (x)}{\sqrt {x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sgn(x)/sqrt(x^2 + 1)

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maple [B]  time = 0.01, size = 23, normalized size = 2.56 \[ \frac {\sqrt {\frac {x^{2}+1}{x^{2}}}\, x^{2}}{x^{2}+1} \]

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 = 1.40, size = 11, normalized size = 1.22 \[ \frac {1}{\sqrt {\frac {x^{2} + 1}{x^{2}}}} \]

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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mupad [B]  time = 3.11, size = 18, normalized size = 2.00 \[ \frac {x^2\,\sqrt {\frac {1}{x^2}+1}}{x^2+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*(1/x^2 + 1)^(1/2))/(x^2 + 1)

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sympy [A]  time = 2.83, 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)

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