\(\int \frac {e^{-2-\frac {1+4 x}{-1+x}} (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 (16-112 x+16 x^2)+e^{2+\frac {1+4 x}{-1+x}} (x-2 x^2+x^3)+e (40 x^2+16 x^3+8 x^4+16 x^5))}{x^2-2 x^3+x^4} \, dx\) [3883]

Optimal result

Integrand size = 130, antiderivative size = 35 \[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=-\frac {e^{-4+\frac {5}{1-x}} \left (-4+\frac {x \left (1+x^2\right )}{e}\right )^2}{x}+\log (x) \]

[Out]

ln(x)-(x/exp(1)*(x^2+1)-4)^2/exp(4-5/(1-x))/x

Rubi [F]

\[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=\int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx \]

[In]

Int[(E^(-2 - (1 + 4*x)/(-1 + x))*(-x^2 - 3*x^3 - 7*x^4 + 2*x^5 - 11*x^6 + 5*x^7 - 5*x^8 + E^2*(16 - 112*x + 16
*x^2) + E^(2 + (1 + 4*x)/(-1 + x))*(x - 2*x^2 + x^3) + E*(40*x^2 + 16*x^3 + 8*x^4 + 16*x^5)))/(x^2 - 2*x^3 + x
^4),x]

[Out]

-4*E^(-6 + 5/(1 - x)) + 16*E^(-5 + 5/(1 - x)) - (16*E^((1 + 4*x)/(1 - x)))/x + (60*ExpIntegralEi[5/(1 - x)])/E
^6 - (80*ExpIntegralEi[5/(1 - x)])/E^5 + Log[x] - 41*Defer[Int][E^((1 - 6*x)/(-1 + x)), x] + 40*Defer[Int][E^(
(-5*x)/(-1 + x)), x] - 25*Defer[Int][E^((1 - 6*x)/(-1 + x))*x, x] + 16*Defer[Int][x/E^((5*x)/(-1 + x)), x] - 1
6*Defer[Int][E^((1 - 6*x)/(-1 + x))*x^2, x] - 5*Defer[Int][E^((1 - 6*x)/(-1 + x))*x^3, x] - 5*Defer[Int][E^((1
- 6*x)/(-1 + x))*x^4, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2 \left (1-2 x+x^2\right )} \, dx \\ & = \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{(-1+x)^2 x^2} \, dx \\ & = \int \frac {e^{\frac {1-6 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{(1-x)^2 x^2} \, dx \\ & = \int \left (-\frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2}+\frac {e^{\frac {1-6 x}{-1+x}+\frac {-1+6 x}{-1+x}}}{x}-\frac {3 e^{\frac {1-6 x}{-1+x}} x}{(-1+x)^2}-\frac {7 e^{\frac {1-6 x}{-1+x}} x^2}{(-1+x)^2}+\frac {2 e^{\frac {1-6 x}{-1+x}} x^3}{(-1+x)^2}-\frac {11 e^{\frac {1-6 x}{-1+x}} x^4}{(-1+x)^2}+\frac {5 e^{\frac {1-6 x}{-1+x}} x^5}{(-1+x)^2}-\frac {5 e^{\frac {1-6 x}{-1+x}} x^6}{(-1+x)^2}+\frac {16 e^{2+\frac {1-6 x}{-1+x}} \left (1-7 x+x^2\right )}{(-1+x)^2 x^2}+\frac {8 e^{1+\frac {1-6 x}{-1+x}} \left (5+2 x+x^2+2 x^3\right )}{(-1+x)^2}\right ) \, dx \\ & = 2 \int \frac {e^{\frac {1-6 x}{-1+x}} x^3}{(-1+x)^2} \, dx-3 \int \frac {e^{\frac {1-6 x}{-1+x}} x}{(-1+x)^2} \, dx+5 \int \frac {e^{\frac {1-6 x}{-1+x}} x^5}{(-1+x)^2} \, dx-5 \int \frac {e^{\frac {1-6 x}{-1+x}} x^6}{(-1+x)^2} \, dx-7 \int \frac {e^{\frac {1-6 x}{-1+x}} x^2}{(-1+x)^2} \, dx+8 \int \frac {e^{1+\frac {1-6 x}{-1+x}} \left (5+2 x+x^2+2 x^3\right )}{(-1+x)^2} \, dx-11 \int \frac {e^{\frac {1-6 x}{-1+x}} x^4}{(-1+x)^2} \, dx+16 \int \frac {e^{2+\frac {1-6 x}{-1+x}} \left (1-7 x+x^2\right )}{(-1+x)^2 x^2} \, dx-\int \frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2} \, dx+\int \frac {e^{\frac {1-6 x}{-1+x}+\frac {-1+6 x}{-1+x}}}{x} \, dx \\ & = 2 \int \left (2 e^{\frac {1-6 x}{-1+x}}+\frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2}+\frac {3 e^{\frac {1-6 x}{-1+x}}}{-1+x}+e^{\frac {1-6 x}{-1+x}} x\right ) \, dx-3 \int \left (\frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2}+\frac {e^{\frac {1-6 x}{-1+x}}}{-1+x}\right ) \, dx+5 \int \left (4 e^{\frac {1-6 x}{-1+x}}+\frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2}+\frac {5 e^{\frac {1-6 x}{-1+x}}}{-1+x}+3 e^{\frac {1-6 x}{-1+x}} x+2 e^{\frac {1-6 x}{-1+x}} x^2+e^{\frac {1-6 x}{-1+x}} x^3\right ) \, dx-5 \int \left (5 e^{\frac {1-6 x}{-1+x}}+\frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2}+\frac {6 e^{\frac {1-6 x}{-1+x}}}{-1+x}+4 e^{\frac {1-6 x}{-1+x}} x+3 e^{\frac {1-6 x}{-1+x}} x^2+2 e^{\frac {1-6 x}{-1+x}} x^3+e^{\frac {1-6 x}{-1+x}} x^4\right ) \, dx-7 \int \left (e^{\frac {1-6 x}{-1+x}}+\frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2}+\frac {2 e^{\frac {1-6 x}{-1+x}}}{-1+x}\right ) \, dx+8 \int \frac {e^{-\frac {5 x}{-1+x}} \left (5+2 x+x^2+2 x^3\right )}{(1-x)^2} \, dx-11 \int \left (3 e^{\frac {1-6 x}{-1+x}}+\frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2}+\frac {4 e^{\frac {1-6 x}{-1+x}}}{-1+x}+2 e^{\frac {1-6 x}{-1+x}} x+e^{\frac {1-6 x}{-1+x}} x^2\right ) \, dx+16 \int \frac {e^{\frac {-1-4 x}{-1+x}} \left (1-7 x+x^2\right )}{(1-x)^2 x^2} \, dx-\int \frac {e^{-6-\frac {5}{-1+x}}}{(-1+x)^2} \, dx+\int \frac {1}{x} \, dx \\ & = -\frac {1}{5} e^{-6+\frac {5}{1-x}}+\log (x)+2 \int \frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2} \, dx+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx-3 \int \frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2} \, dx-3 \int \frac {e^{\frac {1-6 x}{-1+x}}}{-1+x} \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx+6 \int \frac {e^{\frac {1-6 x}{-1+x}}}{-1+x} \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx-7 \int \frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2} \, dx+8 \int \left (5 e^{-\frac {5 x}{-1+x}}+\frac {10 e^{-\frac {5 x}{-1+x}}}{(-1+x)^2}+\frac {10 e^{-\frac {5 x}{-1+x}}}{-1+x}+2 e^{-\frac {5 x}{-1+x}} x\right ) \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int \frac {e^{\frac {1-6 x}{-1+x}}}{(-1+x)^2} \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-14 \int \frac {e^{\frac {1-6 x}{-1+x}}}{-1+x} \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int \left (-\frac {5 e^{\frac {-1-4 x}{-1+x}}}{(-1+x)^2}+\frac {5 e^{\frac {-1-4 x}{-1+x}}}{-1+x}+\frac {e^{\frac {-1-4 x}{-1+x}}}{x^2}-\frac {5 e^{\frac {-1-4 x}{-1+x}}}{x}\right ) \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx+25 \int \frac {e^{\frac {1-6 x}{-1+x}}}{-1+x} \, dx-30 \int \frac {e^{\frac {1-6 x}{-1+x}}}{-1+x} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx-44 \int \frac {e^{\frac {1-6 x}{-1+x}}}{-1+x} \, dx \\ & = -\frac {1}{5} e^{-6+\frac {5}{1-x}}+\log (x)+2 \int \frac {e^{-6-\frac {5}{-1+x}}}{(-1+x)^2} \, dx+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx-3 \int \frac {e^{-6-\frac {5}{-1+x}}}{(-1+x)^2} \, dx-3 \int \frac {e^{-6-\frac {5}{-1+x}}}{-1+x} \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx+6 \int \frac {e^{-6-\frac {5}{-1+x}}}{-1+x} \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx-7 \int \frac {e^{-6-\frac {5}{-1+x}}}{(-1+x)^2} \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int \frac {e^{-6-\frac {5}{-1+x}}}{(-1+x)^2} \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-14 \int \frac {e^{-6-\frac {5}{-1+x}}}{-1+x} \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{x^2} \, dx+16 \int e^{-\frac {5 x}{-1+x}} x \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx+25 \int \frac {e^{-6-\frac {5}{-1+x}}}{-1+x} \, dx-30 \int \frac {e^{-6-\frac {5}{-1+x}}}{-1+x} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx+40 \int e^{-\frac {5 x}{-1+x}} \, dx-44 \int \frac {e^{-6-\frac {5}{-1+x}}}{-1+x} \, dx-80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{(-1+x)^2} \, dx+80 \int \frac {e^{-\frac {5 x}{-1+x}}}{(-1+x)^2} \, dx+80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{-1+x} \, dx+80 \int \frac {e^{-\frac {5 x}{-1+x}}}{-1+x} \, dx-80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{x} \, dx \\ & = -4 e^{-6+\frac {5}{1-x}}-\frac {16 e^{\frac {1+4 x}{1-x}}}{x}+\frac {60 \text {Ei}\left (\frac {5}{1-x}\right )}{e^6}+\log (x)+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int e^{-\frac {5 x}{-1+x}} x \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx+40 \int e^{-\frac {5 x}{-1+x}} \, dx+80 \int \frac {e^{-5-\frac {5}{-1+x}}}{(-1+x)^2} \, dx-80 \int \frac {e^{-4-\frac {5}{-1+x}}}{(-1+x)^2} \, dx+80 \int \frac {e^{-5-\frac {5}{-1+x}}}{-1+x} \, dx+80 \int \frac {e^{-4-\frac {5}{-1+x}}}{-1+x} \, dx-80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{-1+x} \, dx+80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{(-1+x)^2 x} \, dx+80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{(-1+x) x} \, dx \\ & = -4 e^{-6+\frac {5}{1-x}}+16 e^{-5+\frac {5}{1-x}}-16 e^{-4+\frac {5}{1-x}}-\frac {16 e^{\frac {1+4 x}{1-x}}}{x}+\frac {60 \text {Ei}\left (\frac {5}{1-x}\right )}{e^6}-\frac {80 \text {Ei}\left (\frac {5}{1-x}\right )}{e^5}-\frac {80 \text {Ei}\left (\frac {5}{1-x}\right )}{e^4}+\log (x)+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int e^{-\frac {5 x}{-1+x}} x \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx+40 \int e^{-\frac {5 x}{-1+x}} \, dx+80 \int \left (\frac {e^{\frac {-1-4 x}{-1+x}}}{1-x}+\frac {e^{\frac {-1-4 x}{-1+x}}}{(-1+x)^2}+\frac {e^{\frac {-1-4 x}{-1+x}}}{x}\right ) \, dx-80 \int \frac {e^{-4-\frac {5}{-1+x}}}{-1+x} \, dx-80 \text {Subst}\left (\int \frac {e^{1-5 x}}{x} \, dx,x,\frac {x}{-1+x}\right ) \\ & = -4 e^{-6+\frac {5}{1-x}}+16 e^{-5+\frac {5}{1-x}}-16 e^{-4+\frac {5}{1-x}}-\frac {16 e^{\frac {1+4 x}{1-x}}}{x}+\frac {60 \text {Ei}\left (\frac {5}{1-x}\right )}{e^6}-\frac {80 \text {Ei}\left (\frac {5}{1-x}\right )}{e^5}-80 e \text {Ei}\left (\frac {5 x}{1-x}\right )+\log (x)+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int e^{-\frac {5 x}{-1+x}} x \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx+40 \int e^{-\frac {5 x}{-1+x}} \, dx+80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{1-x} \, dx+80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{(-1+x)^2} \, dx+80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{x} \, dx \\ & = -4 e^{-6+\frac {5}{1-x}}+16 e^{-5+\frac {5}{1-x}}-16 e^{-4+\frac {5}{1-x}}-\frac {16 e^{\frac {1+4 x}{1-x}}}{x}+\frac {60 \text {Ei}\left (\frac {5}{1-x}\right )}{e^6}-\frac {80 \text {Ei}\left (\frac {5}{1-x}\right )}{e^5}-80 e \text {Ei}\left (\frac {5 x}{1-x}\right )+\log (x)+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int e^{-\frac {5 x}{-1+x}} x \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx+40 \int e^{-\frac {5 x}{-1+x}} \, dx+80 \int \frac {e^{-4-\frac {5}{-1+x}}}{1-x} \, dx+80 \int \frac {e^{-4-\frac {5}{-1+x}}}{(-1+x)^2} \, dx+80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{-1+x} \, dx-80 \int \frac {e^{\frac {-1-4 x}{-1+x}}}{(-1+x) x} \, dx \\ & = -4 e^{-6+\frac {5}{1-x}}+16 e^{-5+\frac {5}{1-x}}-\frac {16 e^{\frac {1+4 x}{1-x}}}{x}+\frac {60 \text {Ei}\left (\frac {5}{1-x}\right )}{e^6}-\frac {80 \text {Ei}\left (\frac {5}{1-x}\right )}{e^5}+\frac {80 \text {Ei}\left (\frac {5}{1-x}\right )}{e^4}-80 e \text {Ei}\left (\frac {5 x}{1-x}\right )+\log (x)+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int e^{-\frac {5 x}{-1+x}} x \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx+40 \int e^{-\frac {5 x}{-1+x}} \, dx+80 \int \frac {e^{-4-\frac {5}{-1+x}}}{-1+x} \, dx+80 \text {Subst}\left (\int \frac {e^{1-5 x}}{x} \, dx,x,\frac {x}{-1+x}\right ) \\ & = -4 e^{-6+\frac {5}{1-x}}+16 e^{-5+\frac {5}{1-x}}-\frac {16 e^{\frac {1+4 x}{1-x}}}{x}+\frac {60 \text {Ei}\left (\frac {5}{1-x}\right )}{e^6}-\frac {80 \text {Ei}\left (\frac {5}{1-x}\right )}{e^5}+\log (x)+2 \int e^{\frac {1-6 x}{-1+x}} x \, dx+4 \int e^{\frac {1-6 x}{-1+x}} \, dx+5 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-5 \int e^{\frac {1-6 x}{-1+x}} x^4 \, dx-7 \int e^{\frac {1-6 x}{-1+x}} \, dx+10 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx-10 \int e^{\frac {1-6 x}{-1+x}} x^3 \, dx-11 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+15 \int e^{\frac {1-6 x}{-1+x}} x \, dx-15 \int e^{\frac {1-6 x}{-1+x}} x^2 \, dx+16 \int e^{-\frac {5 x}{-1+x}} x \, dx+20 \int e^{\frac {1-6 x}{-1+x}} \, dx-20 \int e^{\frac {1-6 x}{-1+x}} x \, dx-22 \int e^{\frac {1-6 x}{-1+x}} x \, dx-25 \int e^{\frac {1-6 x}{-1+x}} \, dx-33 \int e^{\frac {1-6 x}{-1+x}} \, dx+40 \int e^{-\frac {5 x}{-1+x}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=\frac {e^{-6-\frac {5}{-1+x}} \left (-\left (-4 e+x+x^3\right )^2+e^{6+\frac {5}{-1+x}} x \log (x)\right )}{x} \]

[In]

Integrate[(E^(-2 - (1 + 4*x)/(-1 + x))*(-x^2 - 3*x^3 - 7*x^4 + 2*x^5 - 11*x^6 + 5*x^7 - 5*x^8 + E^2*(16 - 112*
x + 16*x^2) + E^(2 + (1 + 4*x)/(-1 + x))*(x - 2*x^2 + x^3) + E*(40*x^2 + 16*x^3 + 8*x^4 + 16*x^5)))/(x^2 - 2*x
^3 + x^4),x]

[Out]

(E^(-6 - 5/(-1 + x))*(-(-4*E + x + x^3)^2 + E^(6 + 5/(-1 + x))*x*Log[x]))/x

Maple [A] (verified)

Time = 3.51 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43

method	result	size
risch	\(\ln \left (x \right )-\frac {\left (x^{6}-8 x^{3} {\mathrm e}+2 x^{4}+16 \,{\mathrm e}^{2}-8 x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{-\frac {-1+6 x}{-1+x}}}{x}\)	\(50\)
norman	\(\frac {\left (\left (-16 \,{\mathrm e}-8\right ) x +{\mathrm e}^{-1} x^{6}+\left (1+8 \,{\mathrm e}\right ) {\mathrm e}^{-1} x^{2}-2 \,{\mathrm e}^{-1} x^{5}-{\mathrm e}^{-1} x^{7}-\left (1+8 \,{\mathrm e}\right ) {\mathrm e}^{-1} x^{3}+2 \left (4 \,{\mathrm e}+1\right ) {\mathrm e}^{-1} x^{4}+16 \,{\mathrm e}\right ) {\mathrm e}^{-1} {\mathrm e}^{-\frac {1+4 x}{-1+x}}}{x \left (-1+x \right )}+\ln \left (x \right )\)	\(114\)
parallelrisch	\(\frac {{\mathrm e}^{-2} \left (-x^{7}+{\mathrm e}^{2} \ln \left (x \right ) x^{2} {\mathrm e}^{\frac {1+4 x}{-1+x}}+x^{6}-{\mathrm e}^{2} \ln \left (x \right ) {\mathrm e}^{\frac {1+4 x}{-1+x}} x +8 x^{4} {\mathrm e}-2 x^{5}-8 x^{3} {\mathrm e}+2 x^{4}-16 \,{\mathrm e}^{2} x +8 x^{2} {\mathrm e}-x^{3}+16 \,{\mathrm e}^{2}-8 x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{-\frac {1+4 x}{-1+x}}}{x \left (-1+x \right )}\)	\(137\)
parts	\(\text {Expression too large to display}\)	\(847\)
default	\(\text {Expression too large to display}\)	\(907\)
derivativedivides	\(\text {Expression too large to display}\)	\(908\)

[In]

int(((x^3-2*x^2+x)*exp(1)^2*exp((1+4*x)/(-1+x))+(16*x^2-112*x+16)*exp(1)^2+(16*x^5+8*x^4+16*x^3+40*x^2)*exp(1)
-5*x^8+5*x^7-11*x^6+2*x^5-7*x^4-3*x^3-x^2)/(x^4-2*x^3+x^2)/exp(1)^2/exp((1+4*x)/(-1+x)),x,method=_RETURNVERBOS
E)

[Out]

ln(x)-1/x*(x^6-8*x^3*exp(1)+2*x^4+16*exp(2)-8*x*exp(1)+x^2)*exp(-(-1+6*x)/(-1+x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.71 \[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=-\frac {{\left (x^{6} + 2 \, x^{4} - x e^{\left (\frac {6 \, x - 1}{x - 1}\right )} \log \left (x\right ) + x^{2} - 8 \, {\left (x^{3} + x\right )} e + 16 \, e^{2}\right )} e^{\left (-\frac {6 \, x - 1}{x - 1}\right )}}{x} \]

[In]

integrate(((x^3-2*x^2+x)*exp(1)^2*exp((1+4*x)/(-1+x))+(16*x^2-112*x+16)*exp(1)^2+(16*x^5+8*x^4+16*x^3+40*x^2)*
exp(1)-5*x^8+5*x^7-11*x^6+2*x^5-7*x^4-3*x^3-x^2)/(x^4-2*x^3+x^2)/exp(1)^2/exp((1+4*x)/(-1+x)),x, algorithm="fr
icas")

[Out]

-(x^6 + 2*x^4 - x*e^((6*x - 1)/(x - 1))*log(x) + x^2 - 8*(x^3 + x)*e + 16*e^2)*e^(-(6*x - 1)/(x - 1))/x

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.40 \[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=\log {\left (x \right )} + \frac {\left (- x^{6} - 2 x^{4} + 8 e x^{3} - x^{2} + 8 e x - 16 e^{2}\right ) e^{- \frac {4 x + 1}{x - 1}}}{x e^{2}} \]

[In]

integrate(((x**3-2*x**2+x)*exp(1)**2*exp((1+4*x)/(-1+x))+(16*x**2-112*x+16)*exp(1)**2+(16*x**5+8*x**4+16*x**3+
40*x**2)*exp(1)-5*x**8+5*x**7-11*x**6+2*x**5-7*x**4-3*x**3-x**2)/(x**4-2*x**3+x**2)/exp(1)**2/exp((1+4*x)/(-1+
x)),x)

[Out]

log(x) + (-x**6 - 2*x**4 + 8*E*x**3 - x**2 + 8*E*x - 16*exp(2))*exp(-2)*exp(-(4*x + 1)/(x - 1))/x

Maxima [F]

\[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=\int { -\frac {{\left (5 \, x^{8} - 5 \, x^{7} + 11 \, x^{6} - 2 \, x^{5} + 7 \, x^{4} + 3 \, x^{3} + x^{2} - 16 \, {\left (x^{2} - 7 \, x + 1\right )} e^{2} - 8 \, {\left (2 \, x^{5} + x^{4} + 2 \, x^{3} + 5 \, x^{2}\right )} e - {\left (x^{3} - 2 \, x^{2} + x\right )} e^{\left (\frac {4 \, x + 1}{x - 1} + 2\right )}\right )} e^{\left (-\frac {4 \, x + 1}{x - 1} - 2\right )}}{x^{4} - 2 \, x^{3} + x^{2}} \,d x } \]

[In]

integrate(((x^3-2*x^2+x)*exp(1)^2*exp((1+4*x)/(-1+x))+(16*x^2-112*x+16)*exp(1)^2+(16*x^5+8*x^4+16*x^3+40*x^2)*
exp(1)-5*x^8+5*x^7-11*x^6+2*x^5-7*x^4-3*x^3-x^2)/(x^4-2*x^3+x^2)/exp(1)^2/exp((1+4*x)/(-1+x)),x, algorithm="ma
xima")

[Out]

-(x^6 + 2*x^4 - 8*x^3*e + x^2 + 16*e^2 + 40*e + 1)*e^(-5/(x - 1) - 6)/x - 1/5*e^(-5/(x - 1) - 6) + integrate((
7*x*(40*e + 1) - 40*e - 1)*e^(-5/(x - 1))/(x^4*e^6 - 2*x^3*e^6 + x^2*e^6), x) + log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (31) = 62\).

Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.80 \[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=-\frac {{\left (x^{6} e^{\left (-\frac {5 \, x}{x - 1}\right )} + 2 \, x^{4} e^{\left (-\frac {5 \, x}{x - 1}\right )} - 8 \, x^{3} e^{\left (-\frac {5 \, x}{x - 1} + 1\right )} + x^{2} e^{\left (-\frac {5 \, x}{x - 1}\right )} - x e \log \left (x\right ) - 8 \, x e^{\left (-\frac {5 \, x}{x - 1} + 1\right )} + 16 \, e^{\left (-\frac {5 \, x}{x - 1} + 2\right )}\right )} e^{\left (-1\right )}}{x} \]

[In]

integrate(((x^3-2*x^2+x)*exp(1)^2*exp((1+4*x)/(-1+x))+(16*x^2-112*x+16)*exp(1)^2+(16*x^5+8*x^4+16*x^3+40*x^2)*
exp(1)-5*x^8+5*x^7-11*x^6+2*x^5-7*x^4-3*x^3-x^2)/(x^4-2*x^3+x^2)/exp(1)^2/exp((1+4*x)/(-1+x)),x, algorithm="gi
ac")

[Out]

-(x^6*e^(-5*x/(x - 1)) + 2*x^4*e^(-5*x/(x - 1)) - 8*x^3*e^(-5*x/(x - 1) + 1) + x^2*e^(-5*x/(x - 1)) - x*e*log(
x) - 8*x*e^(-5*x/(x - 1) + 1) + 16*e^(-5*x/(x - 1) + 2))*e^(-1)/x

Mupad [B] (verification not implemented)

Time = 9.62 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.31 \[ \int \frac {e^{-2-\frac {1+4 x}{-1+x}} \left (-x^2-3 x^3-7 x^4+2 x^5-11 x^6+5 x^7-5 x^8+e^2 \left (16-112 x+16 x^2\right )+e^{2+\frac {1+4 x}{-1+x}} \left (x-2 x^2+x^3\right )+e \left (40 x^2+16 x^3+8 x^4+16 x^5\right )\right )}{x^2-2 x^3+x^4} \, dx=-\frac {16\,{\mathrm {e}}^{\frac {1}{x-1}-\frac {6\,x}{x-1}+2}}{x}-{\mathrm {e}}^{\frac {1}{x-1}-\frac {6\,x}{x-1}}\,\left (x-8\,\mathrm {e}-8\,x^2\,\mathrm {e}-{\mathrm {e}}^{\frac {6\,x}{x-1}-\frac {1}{x-1}}\,\ln \left (x\right )+2\,x^3+x^5\right ) \]

[In]

int((exp(-(4*x + 1)/(x - 1))*exp(-2)*(exp(2)*(16*x^2 - 112*x + 16) - x^2 - 3*x^3 - 7*x^4 + 2*x^5 - 11*x^6 + 5*
x^7 - 5*x^8 + exp(1)*(40*x^2 + 16*x^3 + 8*x^4 + 16*x^5) + exp((4*x + 1)/(x - 1))*exp(2)*(x - 2*x^2 + x^3)))/(x
^2 - 2*x^3 + x^4),x)

[Out]

- (16*exp(1/(x - 1) - (6*x)/(x - 1) + 2))/x - exp(1/(x - 1) - (6*x)/(x - 1))*(x - 8*exp(1) - 8*x^2*exp(1) - ex
p((6*x)/(x - 1) - 1/(x - 1))*log(x) + 2*x^3 + x^5)