3.101.93 \(\int \frac {36 e^{2 e^x} x^2+e^{e^x} (-3 e^{2 x} x^2+e^x (-6 x+3 x^2))}{4 e^{2 x}+e^{e^x+x} (-96 x+4 x^2)+e^{2 e^x} (576 x^2-48 x^3+x^4)} \, dx\)

Optimal. Leaf size=29 \[ \frac {x}{16-x+\frac {1}{3} \left (-\frac {4 e^{-e^x+x}}{x}+x\right )} \]

________________________________________________________________________________________

Rubi [F]  time = 11.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(36*E^(2*E^x)*x^2 + E^E^x*(-3*E^(2*x)*x^2 + E^x*(-6*x + 3*x^2)))/(4*E^(2*x) + E^(E^x + x)*(-96*x + 4*x^2)
+ E^(2*E^x)*(576*x^2 - 48*x^3 + x^4)),x]

[Out]

(-3*Defer[Int][E^E^x*x^2, x])/4 - 36*Defer[Int][(E^(2*E^x)*x^2)/(2*E^x + E^E^x*(-24 + x)*x)^2, x] + 39*Defer[I
nt][(E^(2*E^x)*x^3)/(2*E^x + E^E^x*(-24 + x)*x)^2, x] - (3*Defer[Int][(E^(2*E^x)*x^4)/(2*E^x + E^E^x*(-24 + x)
*x)^2, x])/2 - 432*Defer[Int][(E^(3*E^x)*x^4)/(2*E^x + E^E^x*(-24 + x)*x)^2, x] + 36*Defer[Int][(E^(3*E^x)*x^5
)/(2*E^x + E^E^x*(-24 + x)*x)^2, x] - (3*Defer[Int][(E^(3*E^x)*x^6)/(2*E^x + E^E^x*(-24 + x)*x)^2, x])/4 - 3*D
efer[Int][(E^E^x*x)/(2*E^x + E^E^x*(-24 + x)*x), x] + (3*Defer[Int][(E^E^x*x^2)/(2*E^x + E^E^x*(-24 + x)*x), x
])/2 - 36*Defer[Int][(E^(2*E^x)*x^3)/(2*E^x + E^E^x*(-24 + x)*x), x] + (3*Defer[Int][(E^(2*E^x)*x^4)/(2*E^x +
E^E^x*(-24 + x)*x), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{e^x} x \left (e^x (-2+x)+12 e^{e^x} x-e^{2 x} x\right )}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ &=3 \int \frac {e^{e^x} x \left (e^x (-2+x)+12 e^{e^x} x-e^{2 x} x\right )}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ &=3 \int \left (-\frac {1}{4} e^{e^x} x^2+\frac {e^{e^x} x \left (-2+x-24 e^{e^x} x^2+e^{e^x} x^3\right )}{2 \left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )}-\frac {e^{2 e^x} x^2 \left (48-52 x+2 x^2+576 e^{e^x} x^2-48 e^{e^x} x^3+e^{e^x} x^4\right )}{4 \left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{2 e^x} x^2 \left (48-52 x+2 x^2+576 e^{e^x} x^2-48 e^{e^x} x^3+e^{e^x} x^4\right )}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+\frac {3}{2} \int \frac {e^{e^x} x \left (-2+x-24 e^{e^x} x^2+e^{e^x} x^3\right )}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \left (\frac {48 e^{2 e^x} x^2}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}-\frac {52 e^{2 e^x} x^3}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}+\frac {2 e^{2 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}+\frac {576 e^{3 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}-\frac {48 e^{3 e^x} x^5}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}+\frac {e^{3 e^x} x^6}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}\right ) \, dx+\frac {3}{2} \int \frac {e^{e^x} x \left (-2+x-24 e^{e^x} x^2+e^{e^x} x^3\right )}{2 e^x+e^{e^x} (-24+x) x} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{3 e^x} x^6}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx-\frac {3}{2} \int \frac {e^{2 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+\frac {3}{2} \int \left (-\frac {2 e^{e^x} x}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}+\frac {e^{e^x} x^2}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}-\frac {24 e^{2 e^x} x^3}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}+\frac {e^{2 e^x} x^4}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}\right ) \, dx-36 \int \frac {e^{2 e^x} x^2}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+36 \int \frac {e^{3 e^x} x^5}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+39 \int \frac {e^{2 e^x} x^3}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx-432 \int \frac {e^{3 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{3 e^x} x^6}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-\frac {3}{2} \int \frac {e^{2 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+\frac {3}{2} \int \frac {e^{e^x} x^2}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx+\frac {3}{2} \int \frac {e^{2 e^x} x^4}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx-3 \int \frac {e^{e^x} x}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx-36 \int \frac {e^{2 e^x} x^2}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+36 \int \frac {e^{3 e^x} x^5}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-36 \int \frac {e^{2 e^x} x^3}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx+39 \int \frac {e^{2 e^x} x^3}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-432 \int \frac {e^{3 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{3 e^x} x^6}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-\frac {3}{2} \int \frac {e^{2 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+\frac {3}{2} \int \frac {e^{e^x} x^2}{2 e^x+e^{e^x} (-24+x) x} \, dx+\frac {3}{2} \int \frac {e^{2 e^x} x^4}{2 e^x+e^{e^x} (-24+x) x} \, dx-3 \int \frac {e^{e^x} x}{2 e^x+e^{e^x} (-24+x) x} \, dx-36 \int \frac {e^{2 e^x} x^2}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+36 \int \frac {e^{3 e^x} x^5}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-36 \int \frac {e^{2 e^x} x^3}{2 e^x+e^{e^x} (-24+x) x} \, dx+39 \int \frac {e^{2 e^x} x^3}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-432 \int \frac {e^{3 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 4.39, size = 32, normalized size = 1.10 \begin {gather*} \frac {3 \left (e^x-12 e^{e^x} x\right )}{2 e^x+e^{e^x} (-24+x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36*E^(2*E^x)*x^2 + E^E^x*(-3*E^(2*x)*x^2 + E^x*(-6*x + 3*x^2)))/(4*E^(2*x) + E^(E^x + x)*(-96*x + 4
*x^2) + E^(2*E^x)*(576*x^2 - 48*x^3 + x^4)),x]

[Out]

(3*(E^x - 12*E^E^x*x))/(2*E^x + E^E^x*(-24 + x)*x)

________________________________________________________________________________________

fricas [A]  time = 0.78, size = 39, normalized size = 1.34 \begin {gather*} -\frac {3 \, {\left (12 \, x e^{\left (x + e^{x}\right )} - e^{\left (2 \, x\right )}\right )}}{{\left (x^{2} - 24 \, x\right )} e^{\left (x + e^{x}\right )} + 2 \, e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(exp(x)))/((x^4-48*x^3+576*x^2)*exp(ex
p(x))^2+(4*x^2-96*x)*exp(x)*exp(exp(x))+4*exp(x)^2),x, algorithm="fricas")

[Out]

-3*(12*x*e^(x + e^x) - e^(2*x))/((x^2 - 24*x)*e^(x + e^x) + 2*e^(2*x))

________________________________________________________________________________________

giac [A]  time = 0.19, size = 33, normalized size = 1.14 \begin {gather*} -\frac {3 \, {\left (12 \, x e^{\left (e^{x}\right )} - e^{x}\right )}}{x^{2} e^{\left (e^{x}\right )} - 24 \, x e^{\left (e^{x}\right )} + 2 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(exp(x)))/((x^4-48*x^3+576*x^2)*exp(ex
p(x))^2+(4*x^2-96*x)*exp(x)*exp(exp(x))+4*exp(x)^2),x, algorithm="giac")

[Out]

-3*(12*x*e^(e^x) - e^x)/(x^2*e^(e^x) - 24*x*e^(e^x) + 2*e^x)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 39, normalized size = 1.34

method	result	size
risch	\(-\frac {36}{x -24}+\frac {3 x \,{\mathrm e}^{x}}{\left (x -24\right ) \left ({\mathrm e}^{{\mathrm e}^{x}} x^{2}-24 x \,{\mathrm e}^{{\mathrm e}^{x}}+2 \,{\mathrm e}^{x}\right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(exp(x)))/((x^4-48*x^3+576*x^2)*exp(exp(x))^
2+(4*x^2-96*x)*exp(x)*exp(exp(x))+4*exp(x)^2),x,method=_RETURNVERBOSE)

[Out]

-36/(x-24)+3*x*exp(x)/(x-24)/(exp(exp(x))*x^2-24*x*exp(exp(x))+2*exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 31, normalized size = 1.07 \begin {gather*} -\frac {3 \, {\left (12 \, x e^{\left (e^{x}\right )} - e^{x}\right )}}{{\left (x^{2} - 24 \, x\right )} e^{\left (e^{x}\right )} + 2 \, e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x^2*exp(exp(x))^2+(-3*exp(x)^2*x^2+(3*x^2-6*x)*exp(x))*exp(exp(x)))/((x^4-48*x^3+576*x^2)*exp(ex
p(x))^2+(4*x^2-96*x)*exp(x)*exp(exp(x))+4*exp(x)^2),x, algorithm="maxima")

[Out]

-3*(12*x*e^(e^x) - e^x)/((x^2 - 24*x)*e^(e^x) + 2*e^x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (3\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (6\,x-3\,x^2\right )\right )-36\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{4\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (x^4-48\,x^3+576\,x^2\right )-{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\left (96\,x-4\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x))*(3*x^2*exp(2*x) + exp(x)*(6*x - 3*x^2)) - 36*x^2*exp(2*exp(x)))/(4*exp(2*x) + exp(2*exp(x))*
(576*x^2 - 48*x^3 + x^4) - exp(exp(x))*exp(x)*(96*x - 4*x^2)),x)

[Out]

-int((exp(exp(x))*(3*x^2*exp(2*x) + exp(x)*(6*x - 3*x^2)) - 36*x^2*exp(2*exp(x)))/(4*exp(2*x) + exp(2*exp(x))*
(576*x^2 - 48*x^3 + x^4) - exp(x + exp(x))*(96*x - 4*x^2)), x)

________________________________________________________________________________________

sympy [A]  time = 0.27, size = 39, normalized size = 1.34 \begin {gather*} \frac {3 x e^{x}}{2 x e^{x} + \left (x^{3} - 48 x^{2} + 576 x\right ) e^{e^{x}} - 48 e^{x}} - \frac {36}{x - 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((36*x**2*exp(exp(x))**2+(-3*exp(x)**2*x**2+(3*x**2-6*x)*exp(x))*exp(exp(x)))/((x**4-48*x**3+576*x**2
)*exp(exp(x))**2+(4*x**2-96*x)*exp(x)*exp(exp(x))+4*exp(x)**2),x)

[Out]

3*x*exp(x)/(2*x*exp(x) + (x**3 - 48*x**2 + 576*x)*exp(exp(x)) - 48*exp(x)) - 36/(x - 24)

________________________________________________________________________________________