Integrand size = 525, antiderivative size = 27 \[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=e^{-x+\left (1+x-\frac {1}{e^x+x}\right )^4} x-\log (x) \] Output:
x*exp((1+x-1/(exp(x)+x))^4-x)-ln(x)
Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(27)=54\).
Time = 4.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.74 \[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=e^{1+3 x+6 x^2+4 x^3+x^4+\frac {1}{\left (e^x+x\right )^4}-\frac {4 (1+x)}{\left (e^x+x\right )^3}+\frac {6 (1+x)^2}{\left (e^x+x\right )^2}-\frac {4 (1+x)^3}{e^x+x}} x-\log (x) \] Input:
Integrate[(-E^(5*x) - 5*E^(4*x)*x - 10*E^(3*x)*x^2 - 10*E^(2*x)*x^3 - 5*E^ x*x^4 - x^5 + E^((1 - 4*x + 2*x^2 + 8*x^3 - 5*x^4 - 9*x^5 + 2*x^6 + 4*x^7 + x^8 + E^(4*x)*(1 + 3*x + 6*x^2 + 4*x^3 + x^4) + E^(3*x)*(-4 - 8*x + 20*x ^3 + 16*x^4 + 4*x^5) + E^(2*x)*(6 - 24*x^2 - 18*x^3 + 24*x^4 + 24*x^5 + 6* x^6) + E^x*(-4 + 8*x + 12*x^2 - 20*x^3 - 24*x^4 + 12*x^5 + 16*x^6 + 4*x^7) )/(E^(4*x) + 4*E^(3*x)*x + 6*E^(2*x)*x^2 + 4*E^x*x^3 + x^4))*(-4*x^2 + 12* x^3 - 4*x^4 - 8*x^5 + x^6 - 9*x^7 + 4*x^8 + 12*x^9 + 4*x^10 + E^(5*x)*(x + 3*x^2 + 12*x^3 + 12*x^4 + 4*x^5) + E^(4*x)*(-3*x^2 + 3*x^3 + 60*x^4 + 64* x^5 + 20*x^6) + E^(3*x)*(4*x^2 - 26*x^3 - 30*x^4 + 112*x^5 + 132*x^6 + 40* x^7) + E^(2*x)*(-4*x^2 + 12*x^3 - 38*x^4 - 66*x^5 + 96*x^6 + 132*x^7 + 40* x^8) + E^x*(8*x^2 - 8*x^3 - 15*x^5 - 45*x^6 + 36*x^7 + 64*x^8 + 20*x^9)))/ (E^(5*x)*x + 5*E^(4*x)*x^2 + 10*E^(3*x)*x^3 + 10*E^(2*x)*x^4 + 5*E^x*x^5 + x^6),x]
Output:
E^(1 + 3*x + 6*x^2 + 4*x^3 + x^4 + (E^x + x)^(-4) - (4*(1 + x))/(E^x + x)^ 3 + (6*(1 + x)^2)/(E^x + x)^2 - (4*(1 + x)^3)/(E^x + x))*x - Log[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.
| \(\displaystyle \int \frac {\left (4 x^{10}+12 x^9+4 x^8-9 x^7+x^6-8 x^5-4 x^4+12 x^3-4 x^2+e^{5 x} \left (4 x^5+12 x^4+12 x^3+3 x^2+x\right )+e^{4 x} \left (20 x^6+64 x^5+60 x^4+3 x^3-3 x^2\right )+e^{3 x} \left (40 x^7+132 x^6+112 x^5-30 x^4-26 x^3+4 x^2\right )+e^x \left (20 x^9+64 x^8+36 x^7-45 x^6-15 x^5-8 x^3+8 x^2\right )+e^{2 x} \left (40 x^8+132 x^7+96 x^6-66 x^5-38 x^4+12 x^3-4 x^2\right )\right ) \exp \left (\frac {x^8+4 x^7+2 x^6-9 x^5-5 x^4+8 x^3+2 x^2+e^{3 x} \left (4 x^5+16 x^4+20 x^3-8 x-4\right )+e^{4 x} \left (x^4+4 x^3+6 x^2+3 x+1\right )+e^{2 x} \left (6 x^6+24 x^5+24 x^4-18 x^3-24 x^2+6\right )+e^x \left (4 x^7+16 x^6+12 x^5-24 x^4-20 x^3+12 x^2+8 x-4\right )-4 x+1}{x^4+4 e^x x^3+6 e^{2 x} x^2+4 e^{3 x} x+e^{4 x}}\right )-x^5-5 e^x x^4-10 e^{2 x} x^3-10 e^{3 x} x^2-5 e^{4 x} x-e^{5 x}}{x^6+5 e^x x^5+10 e^{2 x} x^4+10 e^{3 x} x^3+5 e^{4 x} x^2+e^{5 x} x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (4 x^{10}+12 x^9+4 x^8-9 x^7+x^6-8 x^5-4 x^4+12 x^3-4 x^2+e^{5 x} \left (4 x^5+12 x^4+12 x^3+3 x^2+x\right )+e^{4 x} \left (20 x^6+64 x^5+60 x^4+3 x^3-3 x^2\right )+e^{3 x} \left (40 x^7+132 x^6+112 x^5-30 x^4-26 x^3+4 x^2\right )+e^x \left (20 x^9+64 x^8+36 x^7-45 x^6-15 x^5-8 x^3+8 x^2\right )+e^{2 x} \left (40 x^8+132 x^7+96 x^6-66 x^5-38 x^4+12 x^3-4 x^2\right )\right ) \exp \left (\frac {x^8+4 x^7+2 x^6-9 x^5-5 x^4+8 x^3+2 x^2+e^{3 x} \left (4 x^5+16 x^4+20 x^3-8 x-4\right )+e^{4 x} \left (x^4+4 x^3+6 x^2+3 x+1\right )+e^{2 x} \left (6 x^6+24 x^5+24 x^4-18 x^3-24 x^2+6\right )+e^x \left (4 x^7+16 x^6+12 x^5-24 x^4-20 x^3+12 x^2+8 x-4\right )-4 x+1}{x^4+4 e^x x^3+6 e^{2 x} x^2+4 e^{3 x} x+e^{4 x}}\right )-x^5-5 e^x x^4-10 e^{2 x} x^3-10 e^{3 x} x^2-5 e^{4 x} x-e^{5 x}}{x \left (x+e^x\right )^5}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (4 x^9+20 e^x x^8+12 x^8+64 e^x x^7+40 e^{2 x} x^7+4 x^7+36 e^x x^6+132 e^{2 x} x^6+40 e^{3 x} x^6-9 x^6-45 e^x x^5+96 e^{2 x} x^5+132 e^{3 x} x^5+20 e^{4 x} x^5+x^5-15 e^x x^4-66 e^{2 x} x^4+112 e^{3 x} x^4+64 e^{4 x} x^4+4 e^{5 x} x^4-8 x^4-38 e^{2 x} x^3-30 e^{3 x} x^3+60 e^{4 x} x^3+12 e^{5 x} x^3-4 x^3-8 e^x x^2+12 e^{2 x} x^2-26 e^{3 x} x^2+3 e^{4 x} x^2+12 e^{5 x} x^2+12 x^2+8 e^x x-4 e^{2 x} x+4 e^{3 x} x-3 e^{4 x} x+3 e^{5 x} x-4 x+e^{5 x}\right ) \exp \left (\frac {x^8+4 e^x x^7+4 x^7+16 e^x x^6+6 e^{2 x} x^6+2 x^6+12 e^x x^5+24 e^{2 x} x^5+4 e^{3 x} x^5-9 x^5-24 e^x x^4+24 e^{2 x} x^4+16 e^{3 x} x^4+e^{4 x} x^4-5 x^4-20 e^x x^3-18 e^{2 x} x^3+20 e^{3 x} x^3+4 e^{4 x} x^3+8 x^3+12 e^x x^2-24 e^{2 x} x^2+6 e^{4 x} x^2+2 x^2+8 e^x x-8 e^{3 x} x+3 e^{4 x} x-4 x-4 e^x+6 e^{2 x}-4 e^{3 x}+e^{4 x}+1}{\left (x+e^x\right )^4}\right )}{\left (x+e^x\right )^5}-\frac {x^4}{\left (x+e^x\right )^5}-\frac {5 e^x x^3}{\left (x+e^x\right )^5}-\frac {10 e^{2 x} x^2}{\left (x+e^x\right )^5}-\frac {10 e^{3 x} x}{\left (x+e^x\right )^5}-\frac {5 e^{4 x}}{\left (x+e^x\right )^5}-\frac {e^{5 x}}{\left (x+e^x\right )^5 x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {\left (4 x^9+20 e^x x^8+12 x^8+64 e^x x^7+40 e^{2 x} x^7+4 x^7+36 e^x x^6+132 e^{2 x} x^6+40 e^{3 x} x^6-9 x^6-45 e^x x^5+96 e^{2 x} x^5+132 e^{3 x} x^5+20 e^{4 x} x^5+x^5-15 e^x x^4-66 e^{2 x} x^4+112 e^{3 x} x^4+64 e^{4 x} x^4+4 e^{5 x} x^4-8 x^4-38 e^{2 x} x^3-30 e^{3 x} x^3+60 e^{4 x} x^3+12 e^{5 x} x^3-4 x^3-8 e^x x^2+12 e^{2 x} x^2-26 e^{3 x} x^2+3 e^{4 x} x^2+12 e^{5 x} x^2+12 x^2+8 e^x x-4 e^{2 x} x+4 e^{3 x} x-3 e^{4 x} x+3 e^{5 x} x-4 x+e^{5 x}\right ) \exp \left (\frac {x^8+4 e^x x^7+4 x^7+16 e^x x^6+6 e^{2 x} x^6+2 x^6+12 e^x x^5+24 e^{2 x} x^5+4 e^{3 x} x^5-9 x^5-24 e^x x^4+24 e^{2 x} x^4+16 e^{3 x} x^4+e^{4 x} x^4-5 x^4-20 e^x x^3-18 e^{2 x} x^3+20 e^{3 x} x^3+4 e^{4 x} x^3+8 x^3+12 e^x x^2-24 e^{2 x} x^2+6 e^{4 x} x^2+2 x^2+8 e^x x-8 e^{3 x} x+3 e^{4 x} x-4 x-4 e^x+6 e^{2 x}-4 e^{3 x}+e^{4 x}+1}{\left (x+e^x\right )^4}\right )}{\left (x+e^x\right )^5}-\frac {x^4}{\left (x+e^x\right )^5}-\frac {5 e^x x^3}{\left (x+e^x\right )^5}-\frac {10 e^{2 x} x^2}{\left (x+e^x\right )^5}-\frac {10 e^{3 x} x}{\left (x+e^x\right )^5}-\frac {5 e^{4 x}}{\left (x+e^x\right )^5}-\frac {e^{5 x}}{\left (x+e^x\right )^5 x}\right )dx\) |
Input:
Int[(-E^(5*x) - 5*E^(4*x)*x - 10*E^(3*x)*x^2 - 10*E^(2*x)*x^3 - 5*E^x*x^4 - x^5 + E^((1 - 4*x + 2*x^2 + 8*x^3 - 5*x^4 - 9*x^5 + 2*x^6 + 4*x^7 + x^8 + E^(4*x)*(1 + 3*x + 6*x^2 + 4*x^3 + x^4) + E^(3*x)*(-4 - 8*x + 20*x^3 + 1 6*x^4 + 4*x^5) + E^(2*x)*(6 - 24*x^2 - 18*x^3 + 24*x^4 + 24*x^5 + 6*x^6) + E^x*(-4 + 8*x + 12*x^2 - 20*x^3 - 24*x^4 + 12*x^5 + 16*x^6 + 4*x^7))/(E^( 4*x) + 4*E^(3*x)*x + 6*E^(2*x)*x^2 + 4*E^x*x^3 + x^4))*(-4*x^2 + 12*x^3 - 4*x^4 - 8*x^5 + x^6 - 9*x^7 + 4*x^8 + 12*x^9 + 4*x^10 + E^(5*x)*(x + 3*x^2 + 12*x^3 + 12*x^4 + 4*x^5) + E^(4*x)*(-3*x^2 + 3*x^3 + 60*x^4 + 64*x^5 + 20*x^6) + E^(3*x)*(4*x^2 - 26*x^3 - 30*x^4 + 112*x^5 + 132*x^6 + 40*x^7) + E^(2*x)*(-4*x^2 + 12*x^3 - 38*x^4 - 66*x^5 + 96*x^6 + 132*x^7 + 40*x^8) + E^x*(8*x^2 - 8*x^3 - 15*x^5 - 45*x^6 + 36*x^7 + 64*x^8 + 20*x^9)))/(E^(5* x)*x + 5*E^(4*x)*x^2 + 10*E^(3*x)*x^3 + 10*E^(2*x)*x^4 + 5*E^x*x^5 + x^6), x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(198\) vs. \(2(25)=50\).
Time = 29.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 7.37
| method | result | size |
| parallelrisch | \(-\ln \left (x \right )+x \,{\mathrm e}^{\frac {\left (x^{4}+4 x^{3}+6 x^{2}+3 x +1\right ) {\mathrm e}^{4 x}+\left (4 x^{5}+16 x^{4}+20 x^{3}-8 x -4\right ) {\mathrm e}^{3 x}+\left (6 x^{6}+24 x^{5}+24 x^{4}-18 x^{3}-24 x^{2}+6\right ) {\mathrm e}^{2 x}+\left (4 x^{7}+16 x^{6}+12 x^{5}-24 x^{4}-20 x^{3}+12 x^{2}+8 x -4\right ) {\mathrm e}^{x}+x^{8}+4 x^{7}+2 x^{6}-9 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1}{{\mathrm e}^{4 x}+4 x \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{2 x} x^{2}+4 \,{\mathrm e}^{x} x^{3}+x^{4}}}\) | \(199\) |
| risch | \(-\ln \left (x \right )+x \,{\mathrm e}^{\frac {1-4 x -24 \,{\mathrm e}^{2 x} x^{2}-18 x^{3} {\mathrm e}^{2 x}+20 x^{3} {\mathrm e}^{3 x}+24 \,{\mathrm e}^{2 x} x^{5}+16 x^{6} {\mathrm e}^{x}+3 x \,{\mathrm e}^{4 x}-24 \,{\mathrm e}^{x} x^{4}+12 \,{\mathrm e}^{x} x^{2}+8 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{7}-8 x \,{\mathrm e}^{3 x}-20 \,{\mathrm e}^{x} x^{3}+12 x^{5} {\mathrm e}^{x}+6 \,{\mathrm e}^{2 x}+x^{8}-4 \,{\mathrm e}^{3 x}-9 x^{5}+{\mathrm e}^{4 x}+4 x^{7}+2 x^{6}+2 x^{2}-4 \,{\mathrm e}^{x}+8 x^{3}-5 x^{4}+16 \,{\mathrm e}^{3 x} x^{4}+x^{4} {\mathrm e}^{4 x}+4 \,{\mathrm e}^{3 x} x^{5}+4 x^{3} {\mathrm e}^{4 x}+6 x^{2} {\mathrm e}^{4 x}+6 \,{\mathrm e}^{2 x} x^{6}+24 \,{\mathrm e}^{2 x} x^{4}}{{\mathrm e}^{4 x}+4 x \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{2 x} x^{2}+4 \,{\mathrm e}^{x} x^{3}+x^{4}}}\) | \(260\) |
Input:
int((((4*x^5+12*x^4+12*x^3+3*x^2+x)*exp(x)^5+(20*x^6+64*x^5+60*x^4+3*x^3-3 *x^2)*exp(x)^4+(40*x^7+132*x^6+112*x^5-30*x^4-26*x^3+4*x^2)*exp(x)^3+(40*x ^8+132*x^7+96*x^6-66*x^5-38*x^4+12*x^3-4*x^2)*exp(x)^2+(20*x^9+64*x^8+36*x ^7-45*x^6-15*x^5-8*x^3+8*x^2)*exp(x)+4*x^10+12*x^9+4*x^8-9*x^7+x^6-8*x^5-4 *x^4+12*x^3-4*x^2)*exp(((x^4+4*x^3+6*x^2+3*x+1)*exp(x)^4+(4*x^5+16*x^4+20* x^3-8*x-4)*exp(x)^3+(6*x^6+24*x^5+24*x^4-18*x^3-24*x^2+6)*exp(x)^2+(4*x^7+ 16*x^6+12*x^5-24*x^4-20*x^3+12*x^2+8*x-4)*exp(x)+x^8+4*x^7+2*x^6-9*x^5-5*x ^4+8*x^3+2*x^2-4*x+1)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x)*x^3+x ^4))-exp(x)^5-5*x*exp(x)^4-10*x^2*exp(x)^3-10*exp(x)^2*x^3-5*exp(x)*x^4-x^ 5)/(x*exp(x)^5+5*x^2*exp(x)^4+10*x^3*exp(x)^3+10*exp(x)^2*x^4+5*x^5*exp(x) +x^6),x,method=_RETURNVERBOSE)
Output:
-ln(x)+x*exp(((x^4+4*x^3+6*x^2+3*x+1)*exp(x)^4+(4*x^5+16*x^4+20*x^3-8*x-4) *exp(x)^3+(6*x^6+24*x^5+24*x^4-18*x^3-24*x^2+6)*exp(x)^2+(4*x^7+16*x^6+12* x^5-24*x^4-20*x^3+12*x^2+8*x-4)*exp(x)+x^8+4*x^7+2*x^6-9*x^5-5*x^4+8*x^3+2 *x^2-4*x+1)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x)*x^3+x^4))
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 7.22 \[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=x e^{\left (\frac {x^{8} + 4 \, x^{7} + 2 \, x^{6} - 9 \, x^{5} - 5 \, x^{4} + 8 \, x^{3} + 2 \, x^{2} + {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 3 \, x + 1\right )} e^{\left (4 \, x\right )} + 4 \, {\left (x^{5} + 4 \, x^{4} + 5 \, x^{3} - 2 \, x - 1\right )} e^{\left (3 \, x\right )} + 6 \, {\left (x^{6} + 4 \, x^{5} + 4 \, x^{4} - 3 \, x^{3} - 4 \, x^{2} + 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{7} + 4 \, x^{6} + 3 \, x^{5} - 6 \, x^{4} - 5 \, x^{3} + 3 \, x^{2} + 2 \, x - 1\right )} e^{x} - 4 \, x + 1}{x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}}\right )} - \log \left (x\right ) \] Input:
integrate((((4*x^5+12*x^4+12*x^3+3*x^2+x)*exp(x)^5+(20*x^6+64*x^5+60*x^4+3 *x^3-3*x^2)*exp(x)^4+(40*x^7+132*x^6+112*x^5-30*x^4-26*x^3+4*x^2)*exp(x)^3 +(40*x^8+132*x^7+96*x^6-66*x^5-38*x^4+12*x^3-4*x^2)*exp(x)^2+(20*x^9+64*x^ 8+36*x^7-45*x^6-15*x^5-8*x^3+8*x^2)*exp(x)+4*x^10+12*x^9+4*x^8-9*x^7+x^6-8 *x^5-4*x^4+12*x^3-4*x^2)*exp(((x^4+4*x^3+6*x^2+3*x+1)*exp(x)^4+(4*x^5+16*x ^4+20*x^3-8*x-4)*exp(x)^3+(6*x^6+24*x^5+24*x^4-18*x^3-24*x^2+6)*exp(x)^2+( 4*x^7+16*x^6+12*x^5-24*x^4-20*x^3+12*x^2+8*x-4)*exp(x)+x^8+4*x^7+2*x^6-9*x ^5-5*x^4+8*x^3+2*x^2-4*x+1)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x) *x^3+x^4))-exp(x)^5-5*x*exp(x)^4-10*x^2*exp(x)^3-10*exp(x)^2*x^3-5*exp(x)* x^4-x^5)/(x*exp(x)^5+5*x^2*exp(x)^4+10*x^3*exp(x)^3+10*exp(x)^2*x^4+5*x^5* exp(x)+x^6),x, algorithm="fricas")
Output:
x*e^((x^8 + 4*x^7 + 2*x^6 - 9*x^5 - 5*x^4 + 8*x^3 + 2*x^2 + (x^4 + 4*x^3 + 6*x^2 + 3*x + 1)*e^(4*x) + 4*(x^5 + 4*x^4 + 5*x^3 - 2*x - 1)*e^(3*x) + 6* (x^6 + 4*x^5 + 4*x^4 - 3*x^3 - 4*x^2 + 1)*e^(2*x) + 4*(x^7 + 4*x^6 + 3*x^5 - 6*x^4 - 5*x^3 + 3*x^2 + 2*x - 1)*e^x - 4*x + 1)/(x^4 + 4*x^3*e^x + 6*x^ 2*e^(2*x) + 4*x*e^(3*x) + e^(4*x))) - log(x)
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (19) = 38\).
Time = 37.23 (sec) , antiderivative size = 201, normalized size of antiderivative = 7.44 \[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=x e^{\frac {x^{8} + 4 x^{7} + 2 x^{6} - 9 x^{5} - 5 x^{4} + 8 x^{3} + 2 x^{2} - 4 x + \left (x^{4} + 4 x^{3} + 6 x^{2} + 3 x + 1\right ) e^{4 x} + \left (4 x^{5} + 16 x^{4} + 20 x^{3} - 8 x - 4\right ) e^{3 x} + \left (6 x^{6} + 24 x^{5} + 24 x^{4} - 18 x^{3} - 24 x^{2} + 6\right ) e^{2 x} + \left (4 x^{7} + 16 x^{6} + 12 x^{5} - 24 x^{4} - 20 x^{3} + 12 x^{2} + 8 x - 4\right ) e^{x} + 1}{x^{4} + 4 x^{3} e^{x} + 6 x^{2} e^{2 x} + 4 x e^{3 x} + e^{4 x}}} - \log {\left (x \right )} \] Input:
integrate((((4*x**5+12*x**4+12*x**3+3*x**2+x)*exp(x)**5+(20*x**6+64*x**5+6 0*x**4+3*x**3-3*x**2)*exp(x)**4+(40*x**7+132*x**6+112*x**5-30*x**4-26*x**3 +4*x**2)*exp(x)**3+(40*x**8+132*x**7+96*x**6-66*x**5-38*x**4+12*x**3-4*x** 2)*exp(x)**2+(20*x**9+64*x**8+36*x**7-45*x**6-15*x**5-8*x**3+8*x**2)*exp(x )+4*x**10+12*x**9+4*x**8-9*x**7+x**6-8*x**5-4*x**4+12*x**3-4*x**2)*exp(((x **4+4*x**3+6*x**2+3*x+1)*exp(x)**4+(4*x**5+16*x**4+20*x**3-8*x-4)*exp(x)** 3+(6*x**6+24*x**5+24*x**4-18*x**3-24*x**2+6)*exp(x)**2+(4*x**7+16*x**6+12* x**5-24*x**4-20*x**3+12*x**2+8*x-4)*exp(x)+x**8+4*x**7+2*x**6-9*x**5-5*x** 4+8*x**3+2*x**2-4*x+1)/(exp(x)**4+4*x*exp(x)**3+6*exp(x)**2*x**2+4*exp(x)* x**3+x**4))-exp(x)**5-5*x*exp(x)**4-10*x**2*exp(x)**3-10*exp(x)**2*x**3-5* exp(x)*x**4-x**5)/(x*exp(x)**5+5*x**2*exp(x)**4+10*x**3*exp(x)**3+10*exp(x )**2*x**4+5*x**5*exp(x)+x**6),x)
Output:
x*exp((x**8 + 4*x**7 + 2*x**6 - 9*x**5 - 5*x**4 + 8*x**3 + 2*x**2 - 4*x + (x**4 + 4*x**3 + 6*x**2 + 3*x + 1)*exp(4*x) + (4*x**5 + 16*x**4 + 20*x**3 - 8*x - 4)*exp(3*x) + (6*x**6 + 24*x**5 + 24*x**4 - 18*x**3 - 24*x**2 + 6) *exp(2*x) + (4*x**7 + 16*x**6 + 12*x**5 - 24*x**4 - 20*x**3 + 12*x**2 + 8* x - 4)*exp(x) + 1)/(x**4 + 4*x**3*exp(x) + 6*x**2*exp(2*x) + 4*x*exp(3*x) + exp(4*x))) - log(x)
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (25) = 50\).
Time = 2.38 (sec) , antiderivative size = 217, normalized size of antiderivative = 8.04 \[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=x e^{\left (x^{4} + 4 \, x^{3} + 2 \, x^{2} + 4 \, x e^{x} - 9 \, x + \frac {4 \, e^{\left (3 \, x\right )}}{x + e^{x}} + \frac {6 \, e^{\left (2 \, x\right )}}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} - \frac {12 \, e^{\left (2 \, x\right )}}{x + e^{x}} + \frac {4 \, e^{x}}{x^{3} + 3 \, x^{2} e^{x} + 3 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}} - \frac {12 \, e^{x}}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} + \frac {1}{x^{4} + 4 \, x^{3} e^{x} + 6 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (4 \, x\right )}} - \frac {4}{x^{3} + 3 \, x^{2} e^{x} + 3 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}} + \frac {2}{x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}} + \frac {8}{x + e^{x}} - 4 \, e^{\left (2 \, x\right )} + 12 \, e^{x} - 5\right )} - \log \left (x\right ) \] Input:
integrate((((4*x^5+12*x^4+12*x^3+3*x^2+x)*exp(x)^5+(20*x^6+64*x^5+60*x^4+3 *x^3-3*x^2)*exp(x)^4+(40*x^7+132*x^6+112*x^5-30*x^4-26*x^3+4*x^2)*exp(x)^3 +(40*x^8+132*x^7+96*x^6-66*x^5-38*x^4+12*x^3-4*x^2)*exp(x)^2+(20*x^9+64*x^ 8+36*x^7-45*x^6-15*x^5-8*x^3+8*x^2)*exp(x)+4*x^10+12*x^9+4*x^8-9*x^7+x^6-8 *x^5-4*x^4+12*x^3-4*x^2)*exp(((x^4+4*x^3+6*x^2+3*x+1)*exp(x)^4+(4*x^5+16*x ^4+20*x^3-8*x-4)*exp(x)^3+(6*x^6+24*x^5+24*x^4-18*x^3-24*x^2+6)*exp(x)^2+( 4*x^7+16*x^6+12*x^5-24*x^4-20*x^3+12*x^2+8*x-4)*exp(x)+x^8+4*x^7+2*x^6-9*x ^5-5*x^4+8*x^3+2*x^2-4*x+1)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x) *x^3+x^4))-exp(x)^5-5*x*exp(x)^4-10*x^2*exp(x)^3-10*exp(x)^2*x^3-5*exp(x)* x^4-x^5)/(x*exp(x)^5+5*x^2*exp(x)^4+10*x^3*exp(x)^3+10*exp(x)^2*x^4+5*x^5* exp(x)+x^6),x, algorithm="maxima")
Output:
x*e^(x^4 + 4*x^3 + 2*x^2 + 4*x*e^x - 9*x + 4*e^(3*x)/(x + e^x) + 6*e^(2*x) /(x^2 + 2*x*e^x + e^(2*x)) - 12*e^(2*x)/(x + e^x) + 4*e^x/(x^3 + 3*x^2*e^x + 3*x*e^(2*x) + e^(3*x)) - 12*e^x/(x^2 + 2*x*e^x + e^(2*x)) + 1/(x^4 + 4* x^3*e^x + 6*x^2*e^(2*x) + 4*x*e^(3*x) + e^(4*x)) - 4/(x^3 + 3*x^2*e^x + 3* x*e^(2*x) + e^(3*x)) + 2/(x^2 + 2*x*e^x + e^(2*x)) + 8/(x + e^x) - 4*e^(2* x) + 12*e^x - 5) - log(x)
Exception generated. \[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((4*x^5+12*x^4+12*x^3+3*x^2+x)*exp(x)^5+(20*x^6+64*x^5+60*x^4+3 *x^3-3*x^2)*exp(x)^4+(40*x^7+132*x^6+112*x^5-30*x^4-26*x^3+4*x^2)*exp(x)^3 +(40*x^8+132*x^7+96*x^6-66*x^5-38*x^4+12*x^3-4*x^2)*exp(x)^2+(20*x^9+64*x^ 8+36*x^7-45*x^6-15*x^5-8*x^3+8*x^2)*exp(x)+4*x^10+12*x^9+4*x^8-9*x^7+x^6-8 *x^5-4*x^4+12*x^3-4*x^2)*exp(((x^4+4*x^3+6*x^2+3*x+1)*exp(x)^4+(4*x^5+16*x ^4+20*x^3-8*x-4)*exp(x)^3+(6*x^6+24*x^5+24*x^4-18*x^3-24*x^2+6)*exp(x)^2+( 4*x^7+16*x^6+12*x^5-24*x^4-20*x^3+12*x^2+8*x-4)*exp(x)+x^8+4*x^7+2*x^6-9*x ^5-5*x^4+8*x^3+2*x^2-4*x+1)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x) *x^3+x^4))-exp(x)^5-5*x*exp(x)^4-10*x^2*exp(x)^3-10*exp(x)^2*x^3-5*exp(x)* x^4-x^5)/(x*exp(x)^5+5*x^2*exp(x)^4+10*x^3*exp(x)^3+10*exp(x)^2*x^4+5*x^5* exp(x)+x^6),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{-167772160,[4,36]%%%}+%%%{4697620480,[4,35]%%%}+%%%{-62914 560000,[4
Time = 4.22 (sec) , antiderivative size = 1346, normalized size of antiderivative = 49.85 \[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=\text {Too large to display} \] Input:
int(-(exp(5*x) + 5*x*exp(4*x) + 5*x^4*exp(x) - exp((exp(2*x)*(24*x^4 - 18* x^3 - 24*x^2 + 24*x^5 + 6*x^6 + 6) - 4*x + exp(4*x)*(3*x + 6*x^2 + 4*x^3 + x^4 + 1) + exp(3*x)*(20*x^3 - 8*x + 16*x^4 + 4*x^5 - 4) + exp(x)*(8*x + 1 2*x^2 - 20*x^3 - 24*x^4 + 12*x^5 + 16*x^6 + 4*x^7 - 4) + 2*x^2 + 8*x^3 - 5 *x^4 - 9*x^5 + 2*x^6 + 4*x^7 + x^8 + 1)/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*e xp(x) + 6*x^2*exp(2*x) + x^4))*(exp(4*x)*(3*x^3 - 3*x^2 + 60*x^4 + 64*x^5 + 20*x^6) + exp(x)*(8*x^2 - 8*x^3 - 15*x^5 - 45*x^6 + 36*x^7 + 64*x^8 + 20 *x^9) + exp(3*x)*(4*x^2 - 26*x^3 - 30*x^4 + 112*x^5 + 132*x^6 + 40*x^7) + exp(5*x)*(x + 3*x^2 + 12*x^3 + 12*x^4 + 4*x^5) + exp(2*x)*(12*x^3 - 4*x^2 - 38*x^4 - 66*x^5 + 96*x^6 + 132*x^7 + 40*x^8) - 4*x^2 + 12*x^3 - 4*x^4 - 8*x^5 + x^6 - 9*x^7 + 4*x^8 + 12*x^9 + 4*x^10) + 10*x^2*exp(3*x) + 10*x^3* exp(2*x) + x^5)/(x*exp(5*x) + 5*x^5*exp(x) + 5*x^2*exp(4*x) + 10*x^3*exp(3 *x) + 10*x^4*exp(2*x) + x^6),x)
Output:
x*exp((2*x^2)/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x ^4))*exp(x^8/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^ 4))*exp((2*x^6)/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp(-(5*x^4)/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2* x) + x^4))*exp((8*x^3)/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp (2*x) + x^4))*exp((4*x^7)/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2* exp(2*x) + x^4))*exp(-(9*x^5)/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6* x^2*exp(2*x) + x^4))*exp(-(4*exp(x))/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp( x) + 6*x^2*exp(2*x) + x^4))*exp((x^4*exp(4*x))/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp((4*x^3*exp(4*x))/(exp(4*x) + 4*x *exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp((6*x^2*exp(4*x))/(ex p(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp((4*x^5*e xp(3*x))/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))* exp((6*x^6*exp(2*x))/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2 *x) + x^4))*exp(-(18*x^3*exp(2*x))/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp((16*x^4*exp(3*x))/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp((20*x^3*exp(3*x))/(exp(4*x) + 4 *x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp(-(24*x^2*exp(2*x)) /(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) + x^4))*exp((24* x^4*exp(2*x))/(exp(4*x) + 4*x*exp(3*x) + 4*x^3*exp(x) + 6*x^2*exp(2*x) ...
\[ \int \frac {-e^{5 x}-5 e^{4 x} x-10 e^{3 x} x^2-10 e^{2 x} x^3-5 e^x x^4-x^5+e^{\frac {1-4 x+2 x^2+8 x^3-5 x^4-9 x^5+2 x^6+4 x^7+x^8+e^{4 x} \left (1+3 x+6 x^2+4 x^3+x^4\right )+e^{3 x} \left (-4-8 x+20 x^3+16 x^4+4 x^5\right )+e^{2 x} \left (6-24 x^2-18 x^3+24 x^4+24 x^5+6 x^6\right )+e^x \left (-4+8 x+12 x^2-20 x^3-24 x^4+12 x^5+16 x^6+4 x^7\right )}{e^{4 x}+4 e^{3 x} x+6 e^{2 x} x^2+4 e^x x^3+x^4}} \left (-4 x^2+12 x^3-4 x^4-8 x^5+x^6-9 x^7+4 x^8+12 x^9+4 x^{10}+e^{5 x} \left (x+3 x^2+12 x^3+12 x^4+4 x^5\right )+e^{4 x} \left (-3 x^2+3 x^3+60 x^4+64 x^5+20 x^6\right )+e^{3 x} \left (4 x^2-26 x^3-30 x^4+112 x^5+132 x^6+40 x^7\right )+e^{2 x} \left (-4 x^2+12 x^3-38 x^4-66 x^5+96 x^6+132 x^7+40 x^8\right )+e^x \left (8 x^2-8 x^3-15 x^5-45 x^6+36 x^7+64 x^8+20 x^9\right )\right )}{e^{5 x} x+5 e^{4 x} x^2+10 e^{3 x} x^3+10 e^{2 x} x^4+5 e^x x^5+x^6} \, dx=\int \frac {\left (\left (4 x^{5}+12 x^{4}+12 x^{3}+3 x^{2}+x \right ) \left ({\mathrm e}^{x}\right )^{5}+\left (20 x^{6}+64 x^{5}+60 x^{4}+3 x^{3}-3 x^{2}\right ) \left ({\mathrm e}^{x}\right )^{4}+\left (40 x^{7}+132 x^{6}+112 x^{5}-30 x^{4}-26 x^{3}+4 x^{2}\right ) \left ({\mathrm e}^{x}\right )^{3}+\left (40 x^{8}+132 x^{7}+96 x^{6}-66 x^{5}-38 x^{4}+12 x^{3}-4 x^{2}\right ) \left ({\mathrm e}^{x}\right )^{2}+\left (20 x^{9}+64 x^{8}+36 x^{7}-45 x^{6}-15 x^{5}-8 x^{3}+8 x^{2}\right ) {\mathrm e}^{x}+4 x^{10}+12 x^{9}+4 x^{8}-9 x^{7}+x^{6}-8 x^{5}-4 x^{4}+12 x^{3}-4 x^{2}\right ) {\mathrm e}^{\frac {\left (x^{4}+4 x^{3}+6 x^{2}+3 x +1\right ) \left ({\mathrm e}^{x}\right )^{4}+\left (4 x^{5}+16 x^{4}+20 x^{3}-8 x -4\right ) \left ({\mathrm e}^{x}\right )^{3}+\left (6 x^{6}+24 x^{5}+24 x^{4}-18 x^{3}-24 x^{2}+6\right ) \left ({\mathrm e}^{x}\right )^{2}+\left (4 x^{7}+16 x^{6}+12 x^{5}-24 x^{4}-20 x^{3}+12 x^{2}+8 x -4\right ) {\mathrm e}^{x}+x^{8}+4 x^{7}+2 x^{6}-9 x^{5}-5 x^{4}+8 x^{3}+2 x^{2}-4 x +1}{\left ({\mathrm e}^{x}\right )^{4}+4 x \left ({\mathrm e}^{x}\right )^{3}+6 \left ({\mathrm e}^{x}\right )^{2} x^{2}+4 \,{\mathrm e}^{x} x^{3}+x^{4}}}-\left ({\mathrm e}^{x}\right )^{5}-5 x \left ({\mathrm e}^{x}\right )^{4}-10 x^{2} \left ({\mathrm e}^{x}\right )^{3}-10 \left ({\mathrm e}^{x}\right )^{2} x^{3}-5 \,{\mathrm e}^{x} x^{4}-x^{5}}{x \left ({\mathrm e}^{x}\right )^{5}+5 x^{2} \left ({\mathrm e}^{x}\right )^{4}+10 x^{3} \left ({\mathrm e}^{x}\right )^{3}+10 \left ({\mathrm e}^{x}\right )^{2} x^{4}+5 x^{5} {\mathrm e}^{x}+x^{6}}d x \] Input:
int((((4*x^5+12*x^4+12*x^3+3*x^2+x)*exp(x)^5+(20*x^6+64*x^5+60*x^4+3*x^3-3 *x^2)*exp(x)^4+(40*x^7+132*x^6+112*x^5-30*x^4-26*x^3+4*x^2)*exp(x)^3+(40*x ^8+132*x^7+96*x^6-66*x^5-38*x^4+12*x^3-4*x^2)*exp(x)^2+(20*x^9+64*x^8+36*x ^7-45*x^6-15*x^5-8*x^3+8*x^2)*exp(x)+4*x^10+12*x^9+4*x^8-9*x^7+x^6-8*x^5-4 *x^4+12*x^3-4*x^2)*exp(((x^4+4*x^3+6*x^2+3*x+1)*exp(x)^4+(4*x^5+16*x^4+20* x^3-8*x-4)*exp(x)^3+(6*x^6+24*x^5+24*x^4-18*x^3-24*x^2+6)*exp(x)^2+(4*x^7+ 16*x^6+12*x^5-24*x^4-20*x^3+12*x^2+8*x-4)*exp(x)+x^8+4*x^7+2*x^6-9*x^5-5*x ^4+8*x^3+2*x^2-4*x+1)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x)*x^3+x ^4))-exp(x)^5-5*x*exp(x)^4-10*x^2*exp(x)^3-10*exp(x)^2*x^3-5*exp(x)*x^4-x^ 5)/(x*exp(x)^5+5*x^2*exp(x)^4+10*x^3*exp(x)^3+10*exp(x)^2*x^4+5*x^5*exp(x) +x^6),x)
Output:
int((((4*x^5+12*x^4+12*x^3+3*x^2+x)*exp(x)^5+(20*x^6+64*x^5+60*x^4+3*x^3-3 *x^2)*exp(x)^4+(40*x^7+132*x^6+112*x^5-30*x^4-26*x^3+4*x^2)*exp(x)^3+(40*x ^8+132*x^7+96*x^6-66*x^5-38*x^4+12*x^3-4*x^2)*exp(x)^2+(20*x^9+64*x^8+36*x ^7-45*x^6-15*x^5-8*x^3+8*x^2)*exp(x)+4*x^10+12*x^9+4*x^8-9*x^7+x^6-8*x^5-4 *x^4+12*x^3-4*x^2)*exp(((x^4+4*x^3+6*x^2+3*x+1)*exp(x)^4+(4*x^5+16*x^4+20* x^3-8*x-4)*exp(x)^3+(6*x^6+24*x^5+24*x^4-18*x^3-24*x^2+6)*exp(x)^2+(4*x^7+ 16*x^6+12*x^5-24*x^4-20*x^3+12*x^2+8*x-4)*exp(x)+x^8+4*x^7+2*x^6-9*x^5-5*x ^4+8*x^3+2*x^2-4*x+1)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x)*x^3+x ^4))-exp(x)^5-5*x*exp(x)^4-10*x^2*exp(x)^3-10*exp(x)^2*x^3-5*exp(x)*x^4-x^ 5)/(x*exp(x)^5+5*x^2*exp(x)^4+10*x^3*exp(x)^3+10*exp(x)^2*x^4+5*x^5*exp(x) +x^6),x)