Integrand size = 231, antiderivative size = 31 \[ \int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx=5+\frac {x}{2 \left (e^2-e^{\frac {1}{-4+x}}-4 \left (25-\frac {2}{x^2}\right )\right )} \]
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx=-\frac {x^3}{2 \left (-8+\left (100-e^2+e^{\frac {1}{-4+x}}\right ) x^2\right )} \]
Integrate[(384*x^2 - 192*x^3 - 1576*x^4 + 800*x^5 - 100*x^6 + E^(-4 + x)^( -1)*(-16*x^4 + 7*x^5 - x^6) + E^2*(16*x^4 - 8*x^5 + x^6))/(2048 - 1024*x - 51072*x^2 + 25600*x^3 + 316800*x^4 - 160000*x^5 + 20000*x^6 + E^2*(512*x^ 2 - 256*x^3 - 6368*x^4 + 3200*x^5 - 400*x^6) + E^4*(32*x^4 - 16*x^5 + 2*x^ 6) + E^(2/(-4 + x))*(32*x^4 - 16*x^5 + 2*x^6) + E^(-4 + x)^(-1)*(-512*x^2 + 256*x^3 + 6368*x^4 - 3200*x^5 + 400*x^6 + E^2*(-64*x^4 + 32*x^5 - 4*x^6) )),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.
| \(\displaystyle \int \frac {-100 x^6+800 x^5-1576 x^4-192 x^3+384 x^2+e^{\frac {1}{x-4}} \left (-x^6+7 x^5-16 x^4\right )+e^2 \left (x^6-8 x^5+16 x^4\right )}{20000 x^6-160000 x^5+316800 x^4+25600 x^3-51072 x^2+e^{\frac {2}{x-4}} \left (2 x^6-16 x^5+32 x^4\right )+e^4 \left (2 x^6-16 x^5+32 x^4\right )+e^2 \left (-400 x^6+3200 x^5-6368 x^4-256 x^3+512 x^2\right )+e^{\frac {1}{x-4}} \left (400 x^6-3200 x^5+6368 x^4+256 x^3-512 x^2+e^2 \left (-4 x^6+32 x^5-64 x^4\right )\right )-1024 x+2048} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^2 \left (-\left (\left (e^{\frac {1}{x-4}}+100-e^2\right ) x^4\right )-\left (-7 e^{\frac {1}{x-4}}-800+8 e^2\right ) x^3-8 \left (2 e^{\frac {1}{x-4}}+197-2 e^2\right ) x^2-192 x+384\right )}{2 (4-x)^2 \left (8-\left (e^{\frac {1}{x-4}}+100-e^2\right ) x^2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2 \left (-\left (\left (100-e^2+e^{\frac {1}{x-4}}\right ) x^4\right )+\left (7 e^{\frac {1}{x-4}}+8 \left (100-e^2\right )\right ) x^3-8 \left (197-2 e^2+2 e^{\frac {1}{x-4}}\right ) x^2-192 x+384\right )}{(4-x)^2 \left (8-\left (100-e^2+e^{\frac {1}{x-4}}\right ) x^2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\left (\left (100-e^2\right ) x^3+16 x^2-136 x+256\right ) x^2}{(4-x)^2 \left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )^2}+\frac {\left (x^2-7 x+16\right ) x^2}{(4-x)^2 \left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (64 \left (399-4 e^2\right ) \int \frac {1}{\left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )^2}dx+512 \left (199-2 e^2\right ) \int \frac {1}{(4-x)^2 \left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )^2}dx-128 \left (997-10 e^2\right ) \int \frac {1}{(4-x) \left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )^2}dx+8 \left (599-6 e^2\right ) \int \frac {x}{\left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )^2}dx+8 \left (102-e^2\right ) \int \frac {x^2}{\left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )^2}dx+8 \int \frac {1}{-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8}dx+64 \int \frac {1}{(4-x)^2 \left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )}dx+48 \int \frac {1}{(x-4) \left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )}dx+\int \frac {x}{-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8}dx+\int \frac {x^2}{-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8}dx+\left (100-e^2\right ) \int \frac {x^3}{\left (-e^{\frac {1}{x-4}} x^2-100 \left (1-\frac {e^2}{100}\right ) x^2+8\right )^2}dx\right )\) |
Int[(384*x^2 - 192*x^3 - 1576*x^4 + 800*x^5 - 100*x^6 + E^(-4 + x)^(-1)*(- 16*x^4 + 7*x^5 - x^6) + E^2*(16*x^4 - 8*x^5 + x^6))/(2048 - 1024*x - 51072 *x^2 + 25600*x^3 + 316800*x^4 - 160000*x^5 + 20000*x^6 + E^2*(512*x^2 - 25 6*x^3 - 6368*x^4 + 3200*x^5 - 400*x^6) + E^4*(32*x^4 - 16*x^5 + 2*x^6) + E ^(2/(-4 + x))*(32*x^4 - 16*x^5 + 2*x^6) + E^(-4 + x)^(-1)*(-512*x^2 + 256* x^3 + 6368*x^4 - 3200*x^5 + 400*x^6 + E^2*(-64*x^4 + 32*x^5 - 4*x^6))),x]
3.7.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.34 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {x^{3}}{2 x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}^{\frac {1}{x -4}}-200 x^{2}+16}\) | \(32\) |
| parallelrisch | \(\frac {x^{3}}{2 x^{2} {\mathrm e}^{2}-2 x^{2} {\mathrm e}^{\frac {1}{x -4}}-200 x^{2}+16}\) | \(32\) |
| norman | \(\frac {-2 x^{3}+\frac {1}{2} x^{4}}{\left (x -4\right ) \left (x^{2} {\mathrm e}^{2}-x^{2} {\mathrm e}^{\frac {1}{x -4}}-100 x^{2}+8\right )}\) | \(44\) |
int(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100*x^6+80 0*x^5-1576*x^4-192*x^3+384*x^2)/((2*x^6-16*x^5+32*x^4)*exp(1/(x-4))^2+((-4 *x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x^2)*exp( 1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4-256*x^ 3+512*x^2)*exp(2)+20000*x^6-160000*x^5+316800*x^4+25600*x^3-51072*x^2-1024 *x+2048),x,method=_RETURNVERBOSE)
Time = 0.31 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx=\frac {x^{3}}{2 \, {\left (x^{2} e^{2} - x^{2} e^{\left (\frac {1}{x - 4}\right )} - 100 \, x^{2} + 8\right )}} \]
integrate(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100* x^6+800*x^5-1576*x^4-192*x^3+384*x^2)/((2*x^6-16*x^5+32*x^4)*exp(1/(x-4))^ 2+((-4*x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x^2 )*exp(1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4- 256*x^3+512*x^2)*exp(2)+20000*x^6-160000*x^5+316800*x^4+25600*x^3-51072*x^ 2-1024*x+2048),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx=- \frac {x^{3}}{2 x^{2} e^{\frac {1}{x - 4}} - 2 x^{2} e^{2} + 200 x^{2} - 16} \]
integrate(((-x**6+7*x**5-16*x**4)*exp(1/(x-4))+(x**6-8*x**5+16*x**4)*exp(2 )-100*x**6+800*x**5-1576*x**4-192*x**3+384*x**2)/((2*x**6-16*x**5+32*x**4) *exp(1/(x-4))**2+((-4*x**6+32*x**5-64*x**4)*exp(2)+400*x**6-3200*x**5+6368 *x**4+256*x**3-512*x**2)*exp(1/(x-4))+(2*x**6-16*x**5+32*x**4)*exp(2)**2+( -400*x**6+3200*x**5-6368*x**4-256*x**3+512*x**2)*exp(2)+20000*x**6-160000* x**5+316800*x**4+25600*x**3-51072*x**2-1024*x+2048),x)
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx=\frac {x^{3}}{2 \, {\left (x^{2} {\left (e^{2} - 100\right )} - x^{2} e^{\left (\frac {1}{x - 4}\right )} + 8\right )}} \]
integrate(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100* x^6+800*x^5-1576*x^4-192*x^3+384*x^2)/((2*x^6-16*x^5+32*x^4)*exp(1/(x-4))^ 2+((-4*x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x^2 )*exp(1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4- 256*x^3+512*x^2)*exp(2)+20000*x^6-160000*x^5+316800*x^4+25600*x^3-51072*x^ 2-1024*x+2048),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx=\frac {x^{3} e^{\frac {1}{4}}}{2 \, {\left (x^{2} e^{\frac {9}{4}} - 100 \, x^{2} e^{\frac {1}{4}} - x^{2} e^{\left (\frac {x}{4 \, {\left (x - 4\right )}}\right )} + 8 \, e^{\frac {1}{4}}\right )}} \]
integrate(((-x^6+7*x^5-16*x^4)*exp(1/(x-4))+(x^6-8*x^5+16*x^4)*exp(2)-100* x^6+800*x^5-1576*x^4-192*x^3+384*x^2)/((2*x^6-16*x^5+32*x^4)*exp(1/(x-4))^ 2+((-4*x^6+32*x^5-64*x^4)*exp(2)+400*x^6-3200*x^5+6368*x^4+256*x^3-512*x^2 )*exp(1/(x-4))+(2*x^6-16*x^5+32*x^4)*exp(2)^2+(-400*x^6+3200*x^5-6368*x^4- 256*x^3+512*x^2)*exp(2)+20000*x^6-160000*x^5+316800*x^4+25600*x^3-51072*x^ 2-1024*x+2048),x, algorithm=\
Time = 13.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.16 \[ \int \frac {384 x^2-192 x^3-1576 x^4+800 x^5-100 x^6+e^{\frac {1}{-4+x}} \left (-16 x^4+7 x^5-x^6\right )+e^2 \left (16 x^4-8 x^5+x^6\right )}{2048-1024 x-51072 x^2+25600 x^3+316800 x^4-160000 x^5+20000 x^6+e^2 \left (512 x^2-256 x^3-6368 x^4+3200 x^5-400 x^6\right )+e^4 \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {2}{-4+x}} \left (32 x^4-16 x^5+2 x^6\right )+e^{\frac {1}{-4+x}} \left (-512 x^2+256 x^3+6368 x^4-3200 x^5+400 x^6+e^2 \left (-64 x^4+32 x^5-4 x^6\right )\right )} \, dx=-\frac {{\left (x^5-8\,x^4+16\,x^3\right )}^2\,\left (16\,x^2-x^3\,{\mathrm {e}}^2-136\,x+100\,x^3+256\right )}{2\,x^2\,\left ({\mathrm {e}}^{\frac {1}{x-4}}-\frac {x^2\,{\mathrm {e}}^2-100\,x^2+8}{x^2}\right )\,{\left (x-4\right )}^2\,\left (8\,x^7\,{\mathrm {e}}^2-16\,x^6\,{\mathrm {e}}^2-x^8\,{\mathrm {e}}^2+4096\,x^3-4224\,x^4+1600\,x^5+1336\,x^6-784\,x^7+100\,x^8\right )} \]
int(-(exp(1/(x - 4))*(16*x^4 - 7*x^5 + x^6) - exp(2)*(16*x^4 - 8*x^5 + x^6 ) - 384*x^2 + 192*x^3 + 1576*x^4 - 800*x^5 + 100*x^6)/(exp(2/(x - 4))*(32* x^4 - 16*x^5 + 2*x^6) - exp(1/(x - 4))*(exp(2)*(64*x^4 - 32*x^5 + 4*x^6) + 512*x^2 - 256*x^3 - 6368*x^4 + 3200*x^5 - 400*x^6) - exp(2)*(256*x^3 - 51 2*x^2 + 6368*x^4 - 3200*x^5 + 400*x^6) - 1024*x + exp(4)*(32*x^4 - 16*x^5 + 2*x^6) - 51072*x^2 + 25600*x^3 + 316800*x^4 - 160000*x^5 + 20000*x^6 + 2 048),x)