Integrand size = 123, antiderivative size = 30 \[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\frac {1}{16} \left (5+e^{4+\frac {2}{x (1+x)}}-x+x (4+x)\right )^2 \] Output:
1/4*(exp(4+2/x/(1+x))-x+(4+x)*x+5)*(1/4*exp(4+2/x/(1+x))-1/4*x+1/4*(4+x)*x +5/4)
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(30)=60\).
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\frac {1}{16} e^{-\frac {4}{1+x}} \left (e^{8+\frac {4}{x}}+2 e^{2 \left (2+\frac {1}{x}+\frac {1}{1+x}\right )} \left (5+3 x+x^2\right )+e^{\frac {4}{1+x}} x \left (30+19 x+6 x^2+x^3\right )\right ) \] Input:
Integrate[(E^((2*(2 + 4*x + 4*x^2))/(x + x^2))*(-2 - 4*x) + 15*x^2 + 49*x^ 3 + 62*x^4 + 39*x^5 + 13*x^6 + 2*x^7 + E^((2 + 4*x + 4*x^2)/(x + x^2))*(-1 0 - 26*x - 11*x^2 + 4*x^3 + 7*x^4 + 2*x^5))/(8*x^2 + 16*x^3 + 8*x^4),x]
Output:
(E^(8 + 4/x) + 2*E^(2*(2 + x^(-1) + (1 + x)^(-1)))*(5 + 3*x + x^2) + E^(4/ (1 + x))*x*(30 + 19*x + 6*x^2 + x^3))/(16*E^(4/(1 + 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.
| \(\displaystyle \int \frac {2 x^7+13 x^6+39 x^5+62 x^4+49 x^3+15 x^2+e^{\frac {2 \left (4 x^2+4 x+2\right )}{x^2+x}} (-4 x-2)+e^{\frac {4 x^2+4 x+2}{x^2+x}} \left (2 x^5+7 x^4+4 x^3-11 x^2-26 x-10\right )}{8 x^4+16 x^3+8 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^7+13 x^6+39 x^5+62 x^4+49 x^3+15 x^2+e^{\frac {2 \left (4 x^2+4 x+2\right )}{x^2+x}} (-4 x-2)+e^{\frac {4 x^2+4 x+2}{x^2+x}} \left (2 x^5+7 x^4+4 x^3-11 x^2-26 x-10\right )}{x^2 \left (8 x^2+16 x+8\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {2 x^7+13 x^6+39 x^5+62 x^4+49 x^3+15 x^2+e^{\frac {2 \left (4 x^2+4 x+2\right )}{x^2+x}} (-4 x-2)+e^{\frac {4 x^2+4 x+2}{x^2+x}} \left (2 x^5+7 x^4+4 x^3-11 x^2-26 x-10\right )}{x^2 \left (2 \sqrt {2} x+2 \sqrt {2}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^5}{4 (x+1)^2}+\frac {13 x^4}{8 (x+1)^2}+\frac {39 x^3}{8 (x+1)^2}+\frac {31 x^2}{4 (x+1)^2}+\frac {e^{\frac {4 \left (2 x^2+2 x+1\right )}{x (x+1)}} (-2 x-1)}{4 (x+1)^2 x^2}+\frac {e^{\frac {2 \left (2 x^2+2 x+1\right )}{x (x+1)}} \left (2 x^5+7 x^4+4 x^3-11 x^2-26 x-10\right )}{8 (x+1)^2 x^2}+\frac {49 x}{8 (x+1)^2}+\frac {15}{8 (x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{8} \int e^{\frac {2 \left (2 x^2+2 x+1\right )}{x (x+1)}}dx-\frac {5}{4} \int \frac {e^{\frac {2 \left (2 x^2+2 x+1\right )}{x (x+1)}}}{x^2}dx-\frac {1}{4} \int \frac {e^{\frac {4 \left (2 x^2+2 x+1\right )}{x (x+1)}}}{x^2}dx-\frac {3}{4} \int \frac {e^{\frac {2 \left (2 x^2+2 x+1\right )}{x (x+1)}}}{x}dx+\frac {1}{4} \int e^{\frac {2 \left (2 x^2+2 x+1\right )}{x (x+1)}} xdx+\frac {3}{4} \int \frac {e^{\frac {2 \left (2 x^2+2 x+1\right )}{x (x+1)}}}{(x+1)^2}dx+\frac {1}{4} \int \frac {e^{\frac {4 \left (2 x^2+2 x+1\right )}{x (x+1)}}}{(x+1)^2}dx+\frac {1}{4} \int \frac {e^{\frac {2 \left (2 x^2+2 x+1\right )}{x (x+1)}}}{x+1}dx+\frac {x^4}{16}+\frac {3 x^3}{8}+\frac {19 x^2}{16}+\frac {15 x}{8}\) |
Input:
Int[(E^((2*(2 + 4*x + 4*x^2))/(x + x^2))*(-2 - 4*x) + 15*x^2 + 49*x^3 + 62 *x^4 + 39*x^5 + 13*x^6 + 2*x^7 + E^((2 + 4*x + 4*x^2)/(x + x^2))*(-10 - 26 *x - 11*x^2 + 4*x^3 + 7*x^4 + 2*x^5))/(8*x^2 + 16*x^3 + 8*x^4),x]
Output:
$Aborted
Time = 2.08 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.53
| method | result | size |
| risch | \(\frac {x^{4}}{16}+\frac {3 x^{3}}{8}+\frac {19 x^{2}}{16}+\frac {15 x}{8}+\frac {25}{16}+\frac {{\mathrm e}^{\frac {8 x^{2}+8 x +4}{\left (1+x \right ) x}}}{16}+\left (\frac {5}{8}+\frac {3}{8} x +\frac {1}{8} x^{2}\right ) {\mathrm e}^{\frac {4 x^{2}+4 x +2}{\left (1+x \right ) x}}\) | \(76\) |
| parallelrisch | \(\frac {x^{4}}{16}+\frac {3 x^{3}}{8}+\frac {{\mathrm e}^{\frac {4 x^{2}+4 x +2}{\left (1+x \right ) x}} x^{2}}{8}+\frac {19 x^{2}}{16}+\frac {3 \,{\mathrm e}^{\frac {4 x^{2}+4 x +2}{\left (1+x \right ) x}} x}{8}+\frac {{\mathrm e}^{\frac {8 x^{2}+8 x +4}{\left (1+x \right ) x}}}{16}+\frac {15 x}{8}+\frac {5 \,{\mathrm e}^{\frac {4 x^{2}+4 x +2}{\left (1+x \right ) x}}}{8}-\frac {131}{32}\) | \(119\) |
| norman | \(\frac {x^{2} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}-\frac {15 x}{8}+\frac {49 x^{3}}{16}+\frac {25 x^{4}}{16}+\frac {7 x^{5}}{16}+\frac {x^{6}}{16}+\frac {5 x \,{\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{8}+\frac {x \,{\mathrm e}^{\frac {8 x^{2}+8 x +4}{x^{2}+x}}}{16}+\frac {x^{2} {\mathrm e}^{\frac {8 x^{2}+8 x +4}{x^{2}+x}}}{16}+\frac {x^{3} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{2}+\frac {x^{4} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{8}}{\left (1+x \right ) x}\) | \(177\) |
| parts | \(\frac {15 x}{8}+\frac {19 x^{2}}{16}+\frac {3 x^{3}}{8}+\frac {x^{4}}{16}+\frac {\frac {x \,{\mathrm e}^{\frac {8 x^{2}+8 x +4}{x^{2}+x}}}{16}+\frac {x^{2} {\mathrm e}^{\frac {8 x^{2}+8 x +4}{x^{2}+x}}}{16}}{x \left (1+x \right )}+\frac {x^{2} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}+\frac {5 x \,{\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{8}+\frac {x^{3} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{2}+\frac {x^{4} {\mathrm e}^{\frac {4 x^{2}+4 x +2}{x^{2}+x}}}{8}}{x \left (1+x \right )}\) | \(183\) |
| orering | \(\text {Expression too large to display}\) | \(2231\) |
Input:
int(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2-26*x- 10)*exp((4*x^2+4*x+2)/(x^2+x))+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15*x^2)/( 8*x^4+16*x^3+8*x^2),x,method=_RETURNVERBOSE)
Output:
1/16*x^4+3/8*x^3+19/16*x^2+15/8*x+25/16+1/16*exp(4*(2*x^2+2*x+1)/(1+x)/x)+ (5/8+3/8*x+1/8*x^2)*exp(2*(2*x^2+2*x+1)/(1+x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (27) = 54\).
Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.37 \[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\frac {1}{16} \, x^{4} + \frac {3}{8} \, x^{3} + \frac {19}{16} \, x^{2} + \frac {1}{8} \, {\left (x^{2} + 3 \, x + 5\right )} e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} + \frac {15}{8} \, x + \frac {1}{16} \, e^{\left (\frac {4 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} \] Input:
integrate(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2 -26*x-10)*exp((4*x^2+4*x+2)/(x^2+x))+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15* x^2)/(8*x^4+16*x^3+8*x^2),x, algorithm="fricas")
Output:
1/16*x^4 + 3/8*x^3 + 19/16*x^2 + 1/8*(x^2 + 3*x + 5)*e^(2*(2*x^2 + 2*x + 1 )/(x^2 + x)) + 15/8*x + 1/16*e^(4*(2*x^2 + 2*x + 1)/(x^2 + x))
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (26) = 52\).
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\frac {x^{4}}{16} + \frac {3 x^{3}}{8} + \frac {19 x^{2}}{16} + \frac {15 x}{8} + \frac {\left (16 x^{2} + 48 x + 80\right ) e^{\frac {4 x^{2} + 4 x + 2}{x^{2} + x}}}{128} + \frac {e^{\frac {2 \cdot \left (4 x^{2} + 4 x + 2\right )}{x^{2} + x}}}{16} \] Input:
integrate(((-4*x-2)*exp((4*x**2+4*x+2)/(x**2+x))**2+(2*x**5+7*x**4+4*x**3- 11*x**2-26*x-10)*exp((4*x**2+4*x+2)/(x**2+x))+2*x**7+13*x**6+39*x**5+62*x* *4+49*x**3+15*x**2)/(8*x**4+16*x**3+8*x**2),x)
Output:
x**4/16 + 3*x**3/8 + 19*x**2/16 + 15*x/8 + (16*x**2 + 48*x + 80)*exp((4*x* *2 + 4*x + 2)/(x**2 + x))/128 + exp(2*(4*x**2 + 4*x + 2)/(x**2 + x))/16
\[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\int { \frac {2 \, x^{7} + 13 \, x^{6} + 39 \, x^{5} + 62 \, x^{4} + 49 \, x^{3} + 15 \, x^{2} - 2 \, {\left (2 \, x + 1\right )} e^{\left (\frac {4 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} + {\left (2 \, x^{5} + 7 \, x^{4} + 4 \, x^{3} - 11 \, x^{2} - 26 \, x - 10\right )} e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )}}{8 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}} \,d x } \] Input:
integrate(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2 -26*x-10)*exp((4*x^2+4*x+2)/(x^2+x))+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15* x^2)/(8*x^4+16*x^3+8*x^2),x, algorithm="maxima")
Output:
1/16*x^4 + 3/8*x^3 + 19/16*x^2 + 15/8*x + 1/16*e^(-4/(x + 1) + 4/x + 8) + 1/8*integrate((2*x^5*e^4 + 7*x^4*e^4 + 4*x^3*e^4 - 11*x^2*e^4 - 26*x*e^4 - 10*e^4)*e^(-2/(x + 1) + 2/x)/(x^4 + 2*x^3 + x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (27) = 54\).
Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.13 \[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\frac {1}{16} \, x^{4} + \frac {3}{8} \, x^{3} + \frac {1}{8} \, x^{2} e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} + \frac {19}{16} \, x^{2} + \frac {3}{8} \, x e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} + \frac {15}{8} \, x + \frac {1}{16} \, e^{\left (\frac {8 \, x^{2}}{x^{2} + x} + \frac {8 \, x}{x^{2} + x} + \frac {4}{x^{2} + x}\right )} + \frac {5}{8} \, e^{\left (\frac {2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}}{x^{2} + x}\right )} \] Input:
integrate(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2 -26*x-10)*exp((4*x^2+4*x+2)/(x^2+x))+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15* x^2)/(8*x^4+16*x^3+8*x^2),x, algorithm="giac")
Output:
1/16*x^4 + 3/8*x^3 + 1/8*x^2*e^(2*(2*x^2 + 2*x + 1)/(x^2 + x)) + 19/16*x^2 + 3/8*x*e^(2*(2*x^2 + 2*x + 1)/(x^2 + x)) + 15/8*x + 1/16*e^(8*x^2/(x^2 + x) + 8*x/(x^2 + x) + 4/(x^2 + x)) + 5/8*e^(2*(2*x^2 + 2*x + 1)/(x^2 + x))
Time = 3.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 5.37 \[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\frac {5\,{\mathrm {e}}^{\frac {2}{x^2+x}+\frac {4\,x^2}{x^2+x}+\frac {4\,x}{x^2+x}}}{8}+\frac {{\mathrm {e}}^{\frac {4}{x^2+x}+\frac {8\,x^2}{x^2+x}+\frac {8\,x}{x^2+x}}}{16}+x^2\,\left (\frac {{\mathrm {e}}^{\frac {2}{x^2+x}+\frac {4\,x^2}{x^2+x}+\frac {4\,x}{x^2+x}}}{8}+\frac {19}{16}\right )+\frac {3\,x^3}{8}+\frac {x^4}{16}+x\,\left (\frac {3\,{\mathrm {e}}^{\frac {2}{x^2+x}+\frac {4\,x^2}{x^2+x}+\frac {4\,x}{x^2+x}}}{8}+\frac {15}{8}\right ) \] Input:
int((15*x^2 - exp((2*(4*x + 4*x^2 + 2))/(x + x^2))*(4*x + 2) - exp((4*x + 4*x^2 + 2)/(x + x^2))*(26*x + 11*x^2 - 4*x^3 - 7*x^4 - 2*x^5 + 10) + 49*x^ 3 + 62*x^4 + 39*x^5 + 13*x^6 + 2*x^7)/(8*x^2 + 16*x^3 + 8*x^4),x)
Output:
(5*exp(2/(x + x^2) + (4*x^2)/(x + x^2) + (4*x)/(x + x^2)))/8 + exp(4/(x + x^2) + (8*x^2)/(x + x^2) + (8*x)/(x + x^2))/16 + x^2*(exp(2/(x + x^2) + (4 *x^2)/(x + x^2) + (4*x)/(x + x^2))/8 + 19/16) + (3*x^3)/8 + x^4/16 + x*((3 *exp(2/(x + x^2) + (4*x^2)/(x + x^2) + (4*x)/(x + x^2)))/8 + 15/8)
\[ \int \frac {e^{\frac {2 \left (2+4 x+4 x^2\right )}{x+x^2}} (-2-4 x)+15 x^2+49 x^3+62 x^4+39 x^5+13 x^6+2 x^7+e^{\frac {2+4 x+4 x^2}{x+x^2}} \left (-10-26 x-11 x^2+4 x^3+7 x^4+2 x^5\right )}{8 x^2+16 x^3+8 x^4} \, dx=\frac {150 e^{\frac {4}{x^{2}+x}} e^{8}}{2401}+\frac {e^{\frac {2}{x^{2}+x}} e^{4} x^{2}}{8}+\frac {3 e^{\frac {2}{x^{2}+x}} e^{4} x}{8}+\frac {5 e^{\frac {2}{x^{2}+x}} e^{4}}{8}-\frac {\left (\int \frac {e^{\frac {4}{x^{2}+x}}}{x^{4}+2 x^{3}+x^{2}}d x \right ) e^{8}}{9604}-\frac {\left (\int \frac {e^{\frac {4}{x^{2}+x}}}{x^{3}+2 x^{2}+x}d x \right ) e^{8}}{4802}+\frac {x^{4}}{16}+\frac {3 x^{3}}{8}+\frac {19 x^{2}}{16}+\frac {15 x}{8} \] Input:
int(((-4*x-2)*exp((4*x^2+4*x+2)/(x^2+x))^2+(2*x^5+7*x^4+4*x^3-11*x^2-26*x- 10)*exp((4*x^2+4*x+2)/(x^2+x))+2*x^7+13*x^6+39*x^5+62*x^4+49*x^3+15*x^2)/( 8*x^4+16*x^3+8*x^2),x)
Output:
(2400*e**(4/(x**2 + x))*e**8 + 4802*e**(2/(x**2 + x))*e**4*x**2 + 14406*e* *(2/(x**2 + x))*e**4*x + 24010*e**(2/(x**2 + x))*e**4 - 4*int(e**(4/(x**2 + x))/(x**4 + 2*x**3 + x**2),x)*e**8 - 8*int(e**(4/(x**2 + x))/(x**3 + 2*x **2 + x),x)*e**8 + 2401*x**4 + 14406*x**3 + 45619*x**2 + 72030*x)/38416