3.73.69 \(\int \frac {e^{e^x} (-54 x+e^x (-2+27 x^2))}{16-432 x^2+2916 x^4} \, dx\)

Optimal. Leaf size=22 \[ 3-\frac {e^{e^x}}{-4+3 \left (4-36 x^2\right )} \]

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Rubi [A]  time = 0.07, antiderivative size = 18, normalized size of antiderivative = 0.82, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {28, 2288} \begin {gather*} -\frac {e^{e^x}}{4 \left (2-27 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^E^x*(-54*x + E^x*(-2 + 27*x^2)))/(16 - 432*x^2 + 2916*x^4),x]

[Out]

-1/4*E^E^x/(2 - 27*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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2916 \int \frac {e^{e^x} \left (-54 x+e^x \left (-2+27 x^2\right )\right )}{\left (-216+2916 x^2\right )^2} \, dx\\ &=-\frac {e^{e^x}}{4 \left (2-27 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 0.82 \begin {gather*} \frac {e^{e^x}}{4 \left (-2+27 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^E^x*(-54*x + E^x*(-2 + 27*x^2)))/(16 - 432*x^2 + 2916*x^4),x]

[Out]

E^E^x/(4*(-2 + 27*x^2))

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fricas [A]  time = 0.88, size = 14, normalized size = 0.64 \begin {gather*} \frac {e^{\left (e^{x}\right )}}{4 \, {\left (27 \, x^{2} - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2-2)*exp(x)-54*x)*exp(exp(x))/(2916*x^4-432*x^2+16),x, algorithm="fricas")

[Out]

1/4*e^(e^x)/(27*x^2 - 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (27 \, x^{2} - 2\right )} e^{x} - 54 \, x\right )} e^{\left (e^{x}\right )}}{4 \, {\left (729 \, x^{4} - 108 \, x^{2} + 4\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2-2)*exp(x)-54*x)*exp(exp(x))/(2916*x^4-432*x^2+16),x, algorithm="giac")

[Out]

integrate(1/4*((27*x^2 - 2)*e^x - 54*x)*e^(e^x)/(729*x^4 - 108*x^2 + 4), x)

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maple [A]  time = 0.04, size = 15, normalized size = 0.68

method	result	size
norman	\(\frac {{\mathrm e}^{{\mathrm e}^{x}}}{108 x^{2}-8}\)	\(15\)
risch	\(\frac {{\mathrm e}^{{\mathrm e}^{x}}}{108 x^{2}-8}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^2-2)*exp(x)-54*x)*exp(exp(x))/(2916*x^4-432*x^2+16),x,method=_RETURNVERBOSE)

[Out]

1/4*exp(exp(x))/(27*x^2-2)

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maxima [A]  time = 0.39, size = 14, normalized size = 0.64 \begin {gather*} \frac {e^{\left (e^{x}\right )}}{4 \, {\left (27 \, x^{2} - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^2-2)*exp(x)-54*x)*exp(exp(x))/(2916*x^4-432*x^2+16),x, algorithm="maxima")

[Out]

1/4*e^(e^x)/(27*x^2 - 2)

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mupad [B]  time = 4.40, size = 13, normalized size = 0.59 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{108\,\left (x^2-\frac {2}{27}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x))*(54*x - exp(x)*(27*x^2 - 2)))/(2916*x^4 - 432*x^2 + 16),x)

[Out]

exp(exp(x))/(108*(x^2 - 2/27))

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sympy [A]  time = 0.15, size = 10, normalized size = 0.45 \begin {gather*} \frac {e^{e^{x}}}{108 x^{2} - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x**2-2)*exp(x)-54*x)*exp(exp(x))/(2916*x**4-432*x**2+16),x)

[Out]

exp(exp(x))/(108*x**2 - 8)

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