Integrand size = 147, antiderivative size = 26 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{e^5 x \left (5+x+\frac {1+x}{e^{4 x}+x+x^2}\right )} \]
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{5 e^5 x+e^5 x^2+\frac {e^5 x+e^5 x^2}{e^{4 x}+x+x^2}} \]
Integrate[(E^((E^(5 + 4*x)*(5*x + x^2) + E^5*(x + 6*x^2 + 6*x^3 + x^4))/(E ^(4*x) + x + x^2))*(E^(5 + 8*x)*(5 + 2*x) + E^(5 + 4*x)*(1 + 8*x + 10*x^2 + 4*x^3) + E^5*(5*x^2 + 12*x^3 + 9*x^4 + 2*x^5)))/(E^(8*x) + x^2 + 2*x^3 + x^4 + E^(4*x)*(2*x + 2*x^2)),x]
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.
\(\displaystyle \int \frac {\left (e^{4 x+5} \left (4 x^3+10 x^2+8 x+1\right )+e^5 \left (2 x^5+9 x^4+12 x^3+5 x^2\right )+e^{8 x+5} (2 x+5)\right ) \exp \left (\frac {e^{4 x+5} \left (x^2+5 x\right )+e^5 \left (x^4+6 x^3+6 x^2+x\right )}{x^2+x+e^{4 x}}\right )}{x^4+2 x^3+x^2+e^{4 x} \left (2 x^2+2 x\right )+e^{8 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (e^{4 x+5} \left (4 x^3+10 x^2+8 x+1\right )+e^5 \left (2 x^5+9 x^4+12 x^3+5 x^2\right )+e^{8 x+5} (2 x+5)\right ) \exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}\right )}{\left (x^2+x+e^{4 x}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left ((2 x+5) \exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right )-\frac {\left (4 x^2+2 x-1\right ) \exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right )}{x^2+x+e^{4 x}}+\frac {x \left (4 x^3+6 x^2+x-1\right ) \exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right )}{\left (x^2+x+e^{4 x}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5 \int \exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right )dx+2 \int \exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right ) xdx-\int \frac {\exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right ) x}{\left (x^2+x+e^{4 x}\right )^2}dx+\int \frac {\exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right ) x^2}{\left (x^2+x+e^{4 x}\right )^2}dx+6 \int \frac {\exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right ) x^3}{\left (x^2+x+e^{4 x}\right )^2}dx+\int \frac {\exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right )}{x^2+x+e^{4 x}}dx-2 \int \frac {\exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right ) x}{x^2+x+e^{4 x}}dx-4 \int \frac {\exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right ) x^2}{x^2+x+e^{4 x}}dx+4 \int \frac {\exp \left (\frac {e^5 x \left (x^3+6 x^2+e^{4 x} x+6 x+5 e^{4 x}+1\right )}{x^2+x+e^{4 x}}+5\right ) x^4}{\left (x^2+x+e^{4 x}\right )^2}dx\) |
Int[(E^((E^(5 + 4*x)*(5*x + x^2) + E^5*(x + 6*x^2 + 6*x^3 + x^4))/(E^(4*x) + x + x^2))*(E^(5 + 8*x)*(5 + 2*x) + E^(5 + 4*x)*(1 + 8*x + 10*x^2 + 4*x^ 3) + E^5*(5*x^2 + 12*x^3 + 9*x^4 + 2*x^5)))/(E^(8*x) + x^2 + 2*x^3 + x^4 + E^(4*x)*(2*x + 2*x^2)),x]
3.30.70.3.1 Defintions of rubi rules used
Time = 5.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {{\mathrm e}^{5} x \left (x^{3}+x \,{\mathrm e}^{4 x}+6 x^{2}+5 \,{\mathrm e}^{4 x}+6 x +1\right )}{{\mathrm e}^{4 x}+x^{2}+x}}\) | \(42\) |
risch | \({\mathrm e}^{\frac {x \left (x^{3} {\mathrm e}^{5}+6 x^{2} {\mathrm e}^{5}+{\mathrm e}^{5+4 x} x +6 x \,{\mathrm e}^{5}+5 \,{\mathrm e}^{5+4 x}+{\mathrm e}^{5}\right )}{{\mathrm e}^{4 x}+x^{2}+x}}\) | \(52\) |
int(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+(2*x^5 +9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x^3+6*x ^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4+2*x^3 +x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (\frac {{\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + x\right )} e^{10} + {\left (x^{2} + 5 \, x\right )} e^{\left (4 \, x + 10\right )}}{{\left (x^{2} + x\right )} e^{5} + e^{\left (4 \, x + 5\right )}}\right )} \]
integrate(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+ (2*x^5+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x ^3+6*x^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4 +2*x^3+x^2),x, algorithm=\
e^(((x^4 + 6*x^3 + 6*x^2 + x)*e^10 + (x^2 + 5*x)*e^(4*x + 10))/((x^2 + x)* e^5 + e^(4*x + 5)))
Time = 0.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\frac {\left (x^{2} + 5 x\right ) e^{5} e^{4 x} + \left (x^{4} + 6 x^{3} + 6 x^{2} + x\right ) e^{5}}{x^{2} + x + e^{4 x}}} \]
integrate(((5+2*x)*exp(5)*exp(4*x)**2+(4*x**3+10*x**2+8*x+1)*exp(5)*exp(4* x)+(2*x**5+9*x**4+12*x**3+5*x**2)*exp(5))*exp(((x**2+5*x)*exp(5)*exp(4*x)+ (x**4+6*x**3+6*x**2+x)*exp(5))/(exp(4*x)+x**2+x))/(exp(4*x)**2+(2*x**2+2*x )*exp(4*x)+x**4+2*x**3+x**2),x)
exp(((x**2 + 5*x)*exp(5)*exp(4*x) + (x**4 + 6*x**3 + 6*x**2 + x)*exp(5))/( x**2 + x + exp(4*x)))
Time = 0.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (x^{2} e^{5} + 5 \, x e^{5} - \frac {e^{\left (4 \, x + 5\right )}}{x^{2} + x + e^{\left (4 \, x\right )}} + e^{5}\right )} \]
integrate(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+ (2*x^5+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x ^3+6*x^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4 +2*x^3+x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (23) = 46\).
Time = 0.40 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx=e^{\left (\frac {x^{4} e^{5} + 6 \, x^{3} e^{5} + 6 \, x^{2} e^{5} + x^{2} e^{\left (4 \, x + 5\right )} + 5 \, x^{2} + x e^{5} + 5 \, x e^{\left (4 \, x + 5\right )} + 5 \, x + 5 \, e^{\left (4 \, x\right )}}{x^{2} + x + e^{\left (4 \, x\right )}} - 5\right )} \]
integrate(((5+2*x)*exp(5)*exp(4*x)^2+(4*x^3+10*x^2+8*x+1)*exp(5)*exp(4*x)+ (2*x^5+9*x^4+12*x^3+5*x^2)*exp(5))*exp(((x^2+5*x)*exp(5)*exp(4*x)+(x^4+6*x ^3+6*x^2+x)*exp(5))/(exp(4*x)+x^2+x))/(exp(4*x)^2+(2*x^2+2*x)*exp(4*x)+x^4 +2*x^3+x^2),x, algorithm=\
e^((x^4*e^5 + 6*x^3*e^5 + 6*x^2*e^5 + x^2*e^(4*x + 5) + 5*x^2 + x*e^5 + 5* x*e^(4*x + 5) + 5*x + 5*e^(4*x))/(x^2 + x + e^(4*x)) - 5)
Time = 11.67 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.46 \[ \int \frac {e^{\frac {e^{5+4 x} \left (5 x+x^2\right )+e^5 \left (x+6 x^2+6 x^3+x^4\right )}{e^{4 x}+x+x^2}} \left (e^{5+8 x} (5+2 x)+e^{5+4 x} \left (1+8 x+10 x^2+4 x^3\right )+e^5 \left (5 x^2+12 x^3+9 x^4+2 x^5\right )\right )}{e^{8 x}+x^2+2 x^3+x^4+e^{4 x} \left (2 x+2 x^2\right )} \, dx={\mathrm {e}}^{\frac {5\,x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x^4\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {6\,x^2\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {6\,x^3\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x^2\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^5}{x+{\mathrm {e}}^{4\,x}+x^2}} \]
int((exp((exp(5)*(x + 6*x^2 + 6*x^3 + x^4) + exp(4*x)*exp(5)*(5*x + x^2))/ (x + exp(4*x) + x^2))*(exp(5)*(5*x^2 + 12*x^3 + 9*x^4 + 2*x^5) + exp(8*x)* exp(5)*(2*x + 5) + exp(4*x)*exp(5)*(8*x + 10*x^2 + 4*x^3 + 1)))/(exp(8*x) + exp(4*x)*(2*x + 2*x^2) + x^2 + 2*x^3 + x^4),x)