\(\int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 (9+6 x+x^2))}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 (27+18 x+3 x^2)+e^4 (-54 x+180 x^2+138 x^3+24 x^4)+e^x (-18 x+66 x^2+24 x^3+e^4 (18+6 x))} \, dx\) [353]

Optimal result

Integrand size = 156, antiderivative size = 33 \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=\frac {2+\frac {e^x}{3}}{e^4-x+4 x^2+\frac {e^x}{3+x}} \] Output:

(2+1/3*exp(x))/(4*x^2+exp(4)-x+exp(x)/(3+x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 8.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.82 \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=\frac {e^x x-3 e^4 (3+x)+3 \left (6+5 x-11 x^2-4 x^3\right )}{3 \left (e^x+e^4 (3+x)+x \left (-3+11 x+4 x^2\right )\right )} \] Input:

Integrate[(54 + E^(2*x) - 396*x - 282*x^2 - 48*x^3 + E^x*(-3 - 81*x - 17*x 
^2 + 15*x^3 + 4*x^4 + E^4*(9 + 6*x + x^2)))/(3*E^(2*x) + 27*x^2 - 198*x^3 
+ 291*x^4 + 264*x^5 + 48*x^6 + E^8*(27 + 18*x + 3*x^2) + E^4*(-54*x + 180* 
x^2 + 138*x^3 + 24*x^4) + E^x*(-18*x + 66*x^2 + 24*x^3 + E^4*(18 + 6*x))), 
x]
 

Output:

(E^x*x - 3*E^4*(3 + x) + 3*(6 + 5*x - 11*x^2 - 4*x^3))/(3*(E^x + E^4*(3 + 
x) + x*(-3 + 11*x + 4*x^2)))
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {-48 x^3-282 x^2+e^x \left (4 x^4+15 x^3-17 x^2+e^4 \left (x^2+6 x+9\right )-81 x-3\right )-396 x+e^{2 x}+54}{48 x^6+264 x^5+291 x^4-198 x^3+27 x^2+e^8 \left (3 x^2+18 x+27\right )+e^x \left (24 x^3+66 x^2-18 x+e^4 (6 x+18)\right )+e^4 \left (24 x^4+138 x^3+180 x^2-54 x\right )+3 e^{2 x}} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^x \left (4 x^4+15 x^3-17 x^2-81 x-3\right )+e^{x+4} (x+3)^2-6 (8 x-1) (x+3)^2+e^{2 x}}{3 \left (x \left (4 x^2+11 x-3\right )+e^4 (x+3)+e^x\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {e^{x+4} (x+3)^2+6 (1-8 x) (x+3)^2+e^{2 x}-e^x \left (-4 x^4-15 x^3+17 x^2+81 x+3\right )}{\left (e^4 (x+3)+e^x-x \left (-4 x^2-11 x+3\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {1}{3} \int \left (\frac {4 x^4+7 x^3-\left (39-e^4\right ) x^2-\left (75-4 e^4\right ) x-3 \left (1-e^4\right )}{4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4}+\frac {(x+3) \left (-16 x^6-40 x^5+123 \left (1-\frac {8 e^4}{123}\right ) x^4+284 \left (1-\frac {15 e^4}{142}\right ) x^3-114 \left (1+\frac {1}{114} e^4 \left (-9+e^4\right )\right ) x^2-141 \left (1+\frac {1}{141} e^4 \left (-84+5 e^4\right )\right ) x+18 \left (1+\frac {1}{6} \left (e^4-2 e^8\right )\right )\right )}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (9 \left (6+e^4-2 e^8\right ) \int \frac {1}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx-3 \left (135-85 e^4+7 e^8\right ) \int \frac {x}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx-\left (483-111 e^4+8 e^8\right ) \int \frac {x^2}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx+\left (738-81 e^4-e^8\right ) \int \frac {x^3}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx-3 \left (1-e^4\right ) \int \frac {1}{4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4}dx-\left (75-4 e^4\right ) \int \frac {x}{4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4}dx-\left (39-e^4\right ) \int \frac {x^2}{4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4}dx+7 \int \frac {x^3}{4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4}dx-16 \int \frac {x^7}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx-88 \int \frac {x^6}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx+\left (3-8 e^4\right ) \int \frac {x^5}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx+\left (653-54 e^4\right ) \int \frac {x^4}{\left (4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4\right )^2}dx+4 \int \frac {x^4}{4 x^3+11 x^2-3 \left (1-\frac {e^4}{3}\right ) x+e^x+3 e^4}dx+x\right )\)

Input:

Int[(54 + E^(2*x) - 396*x - 282*x^2 - 48*x^3 + E^x*(-3 - 81*x - 17*x^2 + 1 
5*x^3 + 4*x^4 + E^4*(9 + 6*x + x^2)))/(3*E^(2*x) + 27*x^2 - 198*x^3 + 291* 
x^4 + 264*x^5 + 48*x^6 + E^8*(27 + 18*x + 3*x^2) + E^4*(-54*x + 180*x^2 + 
138*x^3 + 24*x^4) + E^x*(-18*x + 66*x^2 + 24*x^3 + E^4*(18 + 6*x))),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73

method	result	size
norman	\(\frac {-11 x^{2}-4 x^{3}+\left (5-{\mathrm e}^{4}\right ) x +\frac {{\mathrm e}^{x} x}{3}-3 \,{\mathrm e}^{4}+6}{4 x^{3}+x \,{\mathrm e}^{4}+11 x^{2}+3 \,{\mathrm e}^{4}+{\mathrm e}^{x}-3 x}\)	\(57\)
parallelrisch	\(\frac {18-12 x^{3}-3 x \,{\mathrm e}^{4}-33 x^{2}+{\mathrm e}^{x} x -9 \,{\mathrm e}^{4}+15 x}{12 x^{3}+3 x \,{\mathrm e}^{4}+33 x^{2}+9 \,{\mathrm e}^{4}+3 \,{\mathrm e}^{x}-9 x}\)	\(57\)
risch	\(\frac {x}{3}-\frac {4 x^{4}+x^{2} {\mathrm e}^{4}+23 x^{3}+6 x \,{\mathrm e}^{4}+30 x^{2}+9 \,{\mathrm e}^{4}-15 x -18}{3 \left (4 x^{3}+x \,{\mathrm e}^{4}+11 x^{2}+3 \,{\mathrm e}^{4}+{\mathrm e}^{x}-3 x \right )}\)	\(68\)

Input:

int((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x)-48*x^ 
3-282*x^2-396*x+54)/(3*exp(x)^2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x)*exp(x 
)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48*x^6+264 
*x^5+291*x^4-198*x^3+27*x^2),x,method=_RETURNVERBOSE)
 

Output:

(-11*x^2-4*x^3+(5-exp(4))*x+1/3*exp(x)*x-3*exp(4)+6)/(4*x^3+x*exp(4)+11*x^ 
2+3*exp(4)+exp(x)-3*x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=-\frac {12 \, x^{3} + 33 \, x^{2} + 3 \, {\left (x + 3\right )} e^{4} - x e^{x} - 15 \, x - 18}{3 \, {\left (4 \, x^{3} + 11 \, x^{2} + {\left (x + 3\right )} e^{4} - 3 \, x + e^{x}\right )}} \] Input:

integrate((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x) 
-48*x^3-282*x^2-396*x+54)/(3*exp(x)^2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x) 
*exp(x)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48*x 
^6+264*x^5+291*x^4-198*x^3+27*x^2),x, algorithm="fricas")
 

Output:

-1/3*(12*x^3 + 33*x^2 + 3*(x + 3)*e^4 - x*e^x - 15*x - 18)/(4*x^3 + 11*x^2 
 + (x + 3)*e^4 - 3*x + e^x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).

Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.15 \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=\frac {x}{3} + \frac {- 4 x^{4} - 23 x^{3} - x^{2} e^{4} - 30 x^{2} - 6 x e^{4} + 15 x - 9 e^{4} + 18}{12 x^{3} + 33 x^{2} - 9 x + 3 x e^{4} + 3 e^{x} + 9 e^{4}} \] Input:

integrate((exp(x)**2+((x**2+6*x+9)*exp(4)+4*x**4+15*x**3-17*x**2-81*x-3)*e 
xp(x)-48*x**3-282*x**2-396*x+54)/(3*exp(x)**2+((18+6*x)*exp(4)+24*x**3+66* 
x**2-18*x)*exp(x)+(3*x**2+18*x+27)*exp(4)**2+(24*x**4+138*x**3+180*x**2-54 
*x)*exp(4)+48*x**6+264*x**5+291*x**4-198*x**3+27*x**2),x)
 

Output:

x/3 + (-4*x**4 - 23*x**3 - x**2*exp(4) - 30*x**2 - 6*x*exp(4) + 15*x - 9*e 
xp(4) + 18)/(12*x**3 + 33*x**2 - 9*x + 3*x*exp(4) + 3*exp(x) + 9*exp(4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=-\frac {12 \, x^{3} + 33 \, x^{2} + 3 \, x {\left (e^{4} - 5\right )} - x e^{x} + 9 \, e^{4} - 18}{3 \, {\left (4 \, x^{3} + 11 \, x^{2} + x {\left (e^{4} - 3\right )} + 3 \, e^{4} + e^{x}\right )}} \] Input:

integrate((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x) 
-48*x^3-282*x^2-396*x+54)/(3*exp(x)^2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x) 
*exp(x)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48*x 
^6+264*x^5+291*x^4-198*x^3+27*x^2),x, algorithm="maxima")
 

Output:

-1/3*(12*x^3 + 33*x^2 + 3*x*(e^4 - 5) - x*e^x + 9*e^4 - 18)/(4*x^3 + 11*x^ 
2 + x*(e^4 - 3) + 3*e^4 + e^x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).

Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73 \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=-\frac {12 \, x^{3} + 33 \, x^{2} + 3 \, x e^{4} - x e^{x} - 15 \, x + 9 \, e^{4} - 18}{3 \, {\left (4 \, x^{3} + 11 \, x^{2} + x e^{4} - 3 \, x + 3 \, e^{4} + e^{x}\right )}} \] Input:

integrate((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x) 
-48*x^3-282*x^2-396*x+54)/(3*exp(x)^2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x) 
*exp(x)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48*x 
^6+264*x^5+291*x^4-198*x^3+27*x^2),x, algorithm="giac")
 

Output:

-1/3*(12*x^3 + 33*x^2 + 3*x*e^4 - x*e^x - 15*x + 9*e^4 - 18)/(4*x^3 + 11*x 
^2 + x*e^4 - 3*x + 3*e^4 + e^x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=-\int \frac {396\,x-{\mathrm {e}}^{2\,x}+282\,x^2+48\,x^3+{\mathrm {e}}^x\,\left (81\,x-{\mathrm {e}}^4\,\left (x^2+6\,x+9\right )+17\,x^2-15\,x^3-4\,x^4+3\right )-54}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^8\,\left (3\,x^2+18\,x+27\right )+{\mathrm {e}}^4\,\left (24\,x^4+138\,x^3+180\,x^2-54\,x\right )+27\,x^2-198\,x^3+291\,x^4+264\,x^5+48\,x^6+{\mathrm {e}}^x\,\left (66\,x^2-18\,x+24\,x^3+{\mathrm {e}}^4\,\left (6\,x+18\right )\right )} \,d x \] Input:

int(-(396*x - exp(2*x) + 282*x^2 + 48*x^3 + exp(x)*(81*x - exp(4)*(6*x + x 
^2 + 9) + 17*x^2 - 15*x^3 - 4*x^4 + 3) - 54)/(3*exp(2*x) + exp(8)*(18*x + 
3*x^2 + 27) + exp(4)*(180*x^2 - 54*x + 138*x^3 + 24*x^4) + 27*x^2 - 198*x^ 
3 + 291*x^4 + 264*x^5 + 48*x^6 + exp(x)*(66*x^2 - 18*x + 24*x^3 + exp(4)*( 
6*x + 18))),x)
 

Output:

-int((396*x - exp(2*x) + 282*x^2 + 48*x^3 + exp(x)*(81*x - exp(4)*(6*x + x 
^2 + 9) + 17*x^2 - 15*x^3 - 4*x^4 + 3) - 54)/(3*exp(2*x) + exp(8)*(18*x + 
3*x^2 + 27) + exp(4)*(180*x^2 - 54*x + 138*x^3 + 24*x^4) + 27*x^2 - 198*x^ 
3 + 291*x^4 + 264*x^5 + 48*x^6 + exp(x)*(66*x^2 - 18*x + 24*x^3 + exp(4)*( 
6*x + 18))), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {54+e^{2 x}-396 x-282 x^2-48 x^3+e^x \left (-3-81 x-17 x^2+15 x^3+4 x^4+e^4 \left (9+6 x+x^2\right )\right )}{3 e^{2 x}+27 x^2-198 x^3+291 x^4+264 x^5+48 x^6+e^8 \left (27+18 x+3 x^2\right )+e^4 \left (-54 x+180 x^2+138 x^3+24 x^4\right )+e^x \left (-18 x+66 x^2+24 x^3+e^4 (18+6 x)\right )} \, dx=\frac {e^{x} x +3 e^{x}+6 x +18}{3 e^{x}+3 e^{4} x +9 e^{4}+12 x^{3}+33 x^{2}-9 x} \] Input:

int((exp(x)^2+((x^2+6*x+9)*exp(4)+4*x^4+15*x^3-17*x^2-81*x-3)*exp(x)-48*x^ 
3-282*x^2-396*x+54)/(3*exp(x)^2+((18+6*x)*exp(4)+24*x^3+66*x^2-18*x)*exp(x 
)+(3*x^2+18*x+27)*exp(4)^2+(24*x^4+138*x^3+180*x^2-54*x)*exp(4)+48*x^6+264 
*x^5+291*x^4-198*x^3+27*x^2),x)
 

Output:

(e**x*x + 3*e**x + 6*x + 18)/(3*(e**x + e**4*x + 3*e**4 + 4*x**3 + 11*x**2 
 - 3*x))