3.536 \(\int \frac{x}{1+2 x^2+x^4} \, dx\)

Optimal. Leaf size=11 \[ -\frac{1}{2 \left (x^2+1\right )} \]

[Out]

-1/(2*(1 + x^2))

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Rubi [A]  time = 0.0021301, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {28, 261} \[ -\frac{1}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + 2*x^2 + x^4),x]

[Out]

-1/(2*(1 + x^2))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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{x}{1+2 x^2+x^4} \, dx &=\int \frac{x}{\left (1+x^2\right )^2} \, dx\\ &=-\frac{1}{2 \left (1+x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0011637, size = 11, normalized size = 1. \[ -\frac{1}{2 \left (x^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(1 + 2*x^2 + x^4),x]

[Out]

-1/(2*(1 + x^2))

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Maple [A]  time = 0.043, size = 10, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}+2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4+2*x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.991789, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+2*x^2+1),x, algorithm="maxima")

[Out]

-1/2/(x^2 + 1)

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Fricas [A]  time = 1.45597, size = 22, normalized size = 2. \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+2*x^2+1),x, algorithm="fricas")

[Out]

-1/2/(x^2 + 1)

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Sympy [A]  time = 0.080482, size = 8, normalized size = 0.73 \begin{align*} - \frac{1}{2 x^{2} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**4+2*x**2+1),x)

[Out]

-1/(2*x**2 + 2)

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Giac [A]  time = 1.14512, size = 12, normalized size = 1.09 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4+2*x^2+1),x, algorithm="giac")

[Out]

-1/2/(x^2 + 1)