3.539 \(\int \frac{x}{81-18 x^2+x^4} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{2 \left (9-x^2\right )} \]

[Out]

1/(2*(9 - x^2))

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Rubi [A]  time = 0.0021261, antiderivative size = 13, 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} \[ \frac{1}{2 \left (9-x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x/(81 - 18*x^2 + x^4),x]

[Out]

1/(2*(9 - 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}{81-18 x^2+x^4} \, dx &=\int \frac{x}{\left (-9+x^2\right )^2} \, dx\\ &=\frac{1}{2 \left (9-x^2\right )}\\ \end{align*}

Mathematica [A]  time = 0.0018655, size = 11, normalized size = 0.85 \[ -\frac{1}{2 \left (x^2-9\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(81 - 18*x^2 + x^4),x]

[Out]

-1/(2*(-9 + x^2))

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Maple [A]  time = 0.045, size = 10, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,{x}^{2}-18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(x^4-18*x^2+81),x)

[Out]

-1/2/(x^2-9)

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Maxima [A]  time = 0.985302, size = 12, normalized size = 0.92 \begin{align*} -\frac{1}{2 \,{\left (x^{2} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4-18*x^2+81),x, algorithm="maxima")

[Out]

-1/2/(x^2 - 9)

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Fricas [A]  time = 1.68699, size = 22, normalized size = 1.69 \begin{align*} -\frac{1}{2 \,{\left (x^{2} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4-18*x^2+81),x, algorithm="fricas")

[Out]

-1/2/(x^2 - 9)

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Sympy [A]  time = 0.08119, size = 8, normalized size = 0.62 \begin{align*} - \frac{1}{2 x^{2} - 18} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x**4-18*x**2+81),x)

[Out]

-1/(2*x**2 - 18)

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Giac [A]  time = 1.12867, size = 12, normalized size = 0.92 \begin{align*} -\frac{1}{2 \,{\left (x^{2} - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(x^4-18*x^2+81),x, algorithm="giac")

[Out]

-1/2/(x^2 - 9)