3.6.58 \(\int \frac {4 e^2 x}{9-6 x^2+x^4} \, dx\)

Optimal. Leaf size=21 \[ 4-2 \left (-\frac {e^2}{3-x^2}+\log (4)\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 0.67, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {12, 28, 261} \begin {gather*} \frac {2 e^2}{3-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^2*x)/(9 - 6*x^2 + x^4),x]

[Out]

(2*E^2)/(3 - x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 {gather*} \begin {aligned} \text {integral} &=\left (4 e^2\right ) \int \frac {x}{9-6 x^2+x^4} \, dx\\ &=\left (4 e^2\right ) \int \frac {x}{\left (-3+x^2\right )^2} \, dx\\ &=\frac {2 e^2}{3-x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.57 \begin {gather*} -\frac {2 e^2}{-3+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^2*x)/(9 - 6*x^2 + x^4),x]

[Out]

(-2*E^2)/(-3 + x^2)

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fricas [A]  time = 0.55, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2 \, e^{2}}{x^{2} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(2)/(x^4-6*x^2+9),x, algorithm="fricas")

[Out]

-2*e^2/(x^2 - 3)

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giac [A]  time = 0.65, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2 \, e^{2}}{x^{2} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(2)/(x^4-6*x^2+9),x, algorithm="giac")

[Out]

-2*e^2/(x^2 - 3)

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maple [A]  time = 0.02, size = 12, normalized size = 0.57

method	result	size
gosper	\(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\)	\(12\)
default	\(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\)	\(12\)
norman	\(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\)	\(12\)
risch	\(-\frac {2 \,{\mathrm e}^{2}}{x^{2}-3}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x*exp(2)/(x^4-6*x^2+9),x,method=_RETURNVERBOSE)

[Out]

-2*exp(2)/(x^2-3)

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maxima [A]  time = 0.50, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2 \, e^{2}}{x^{2} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(2)/(x^4-6*x^2+9),x, algorithm="maxima")

[Out]

-2*e^2/(x^2 - 3)

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mupad [B]  time = 0.08, size = 11, normalized size = 0.52 \begin {gather*} -\frac {2\,{\mathrm {e}}^2}{x^2-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x*exp(2))/(x^4 - 6*x^2 + 9),x)

[Out]

-(2*exp(2))/(x^2 - 3)

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sympy [A]  time = 0.08, size = 12, normalized size = 0.57 \begin {gather*} - \frac {4 e^{2}}{2 x^{2} - 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x*exp(2)/(x**4-6*x**2+9),x)

[Out]

-4*exp(2)/(2*x**2 - 6)

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