3.6.39 \(\int \frac {x}{81-18 x^2+x^4} \, dx\) [539]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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, 267} \begin {gather*} \frac {1}{2 \left (9-x^2\right )} \end {gather*}

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 267

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*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2 \left (-9+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(81 - 18*x^2 + x^4),x]

[Out]

-1/2*1/(-9 + x^2)

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Maple [A]
time = 0.03, size = 10, normalized size = 0.77

method	result	size
gosper	\(-\frac {1}{2 \left (x^{2}-9\right )}\)	\(10\)
default	\(-\frac {1}{2 \left (x^{2}-9\right )}\)	\(10\)
norman	\(-\frac {1}{2 \left (x^{2}-9\right )}\)	\(10\)
risch	\(-\frac {1}{2 \left (x^{2}-9\right )}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{2 \, {\left (x^{2} - 9\right )}} \end {gather*}

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 = 0.33, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{2 \, {\left (x^{2} - 9\right )}} \end {gather*}

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 = 0.02, size = 8, normalized size = 0.62 \begin {gather*} - \frac {1}{2 x^{2} - 18} \end {gather*}

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 = 4.10, size = 9, normalized size = 0.69 \begin {gather*} -\frac {1}{2 \, {\left (x^{2} - 9\right )}} \end {gather*}

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)

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Mupad [B]
time = 0.05, size = 11, normalized size = 0.85 \begin {gather*} -\frac {1}{2\,\left (x^2-9\right )} \end {gather*}

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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