Integrand size = 89, antiderivative size = 26 \[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx=\left (e^x+\left (x-x^2\right )^2\right ) \log \left (e^2+\frac {1}{2} (-5+x)\right ) \] Output:
ln(exp(2)+1/2*x-5/2)*((-x^2+x)^2+exp(x))
Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(26)=52\).
Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx=\frac {\left (2 e^{2+x}+e^x x+2 e^2 (-1+x)^2 x^2+(-5+x) (-1+x)^2 x^2\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )+e^x \left (\log (32)-5 \log \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \] Input:
Integrate[(E^x + x^2 - 2*x^3 + x^4 + (-10*x + 32*x^2 - 26*x^3 + 4*x^4 + E^ x*(-5 + 2*E^2 + x) + E^2*(4*x - 12*x^2 + 8*x^3))*Log[(-5 + 2*E^2 + x)/2])/ (-5 + 2*E^2 + x),x]
Output:
((2*E^(2 + x) + E^x*x + 2*E^2*(-1 + x)^2*x^2 + (-5 + x)*(-1 + x)^2*x^2)*Lo g[(-5 + 2*E^2 + x)/2] + E^x*(Log[32] - 5*Log[-5 + 2*E^2 + x]))/(-5 + 2*E^2 + x)
Leaf count is larger than twice the leaf count of optimal. \(1053\) vs. \(2(26)=52\).
Time = 2.61 (sec) , antiderivative size = 1053, normalized size of antiderivative = 40.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {7293, 2009}
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.
| \(\displaystyle \int \frac {x^4-2 x^3+x^2+\left (4 x^4-26 x^3+32 x^2+e^2 \left (8 x^3-12 x^2+4 x\right )-10 x+e^x \left (x+2 e^2-5\right )\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+e^x}{x+2 e^2-5} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^4}{x+2 e^2-5}+\frac {4 x^4 \log \left (\frac {x}{2}+\frac {1}{2} \left (2 e^2-5\right )\right )}{x+2 e^2-5}-\frac {2 x^3}{x+2 e^2-5}+\frac {26 x^3 \log \left (\frac {x}{2}+\frac {1}{2} \left (2 e^2-5\right )\right )}{-x-2 e^2+5}+\frac {x^2}{x+2 e^2-5}+\frac {32 x^2 \log \left (\frac {x}{2}+\frac {1}{2} \left (2 e^2-5\right )\right )}{x+2 e^2-5}+\frac {4 e^2 (1-2 x) (x-1) x \log \left (\frac {x}{2}+\frac {1}{2} \left (2 e^2-5\right )\right )}{-x-2 e^2+5}+\frac {10 x \log \left (\frac {x}{2}+\frac {1}{2} \left (2 e^2-5\right )\right )}{-x-2 e^2+5}+\frac {e^x \left (x \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+2 e^2 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-5 \log \left (x+2 e^2-5\right )+1+\log (32)\right )}{x+2 e^2-5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^4-\frac {1}{4} \left (-x-2 e^2+5\right )^4-\frac {16}{3} \left (5-2 e^2\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^3+\frac {26}{3} \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^3+\frac {16}{9} \left (5-2 e^2\right ) \left (-x-2 e^2+5\right )^3-\frac {26}{9} \left (-x-2 e^2+5\right )^3+12 \left (5-2 e^2\right )^2 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^2-39 \left (5-2 e^2\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^2+16 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )^2-6 \left (5-2 e^2\right )^2 \left (-x-2 e^2+5\right )^2+\frac {39}{2} \left (5-2 e^2\right ) \left (-x-2 e^2+5\right )^2-8 \left (-x-2 e^2+5\right )^2-8 e^2 \left (18-17 e^2+4 e^4\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )-16 \left (5-2 e^2\right )^3 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )+78 \left (5-2 e^2\right )^2 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )-64 \left (5-2 e^2\right ) \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )+10 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right ) \left (-x-2 e^2+5\right )+\frac {x^4}{4}+\frac {1}{3} \left (5-2 e^2\right ) x^3-\frac {8 e^2 x^3}{9}-\frac {2 x^3}{3}+\frac {1}{2} \left (5-2 e^2\right )^2 x^2-\frac {4}{3} e^2 \left (5-2 e^2\right ) x^2-\left (5-2 e^2\right ) x^2-e^2 \left (7-4 e^2\right ) x^2+\frac {x^2}{2}+4 e^2 \left (90-121 e^2+54 e^4-8 e^6\right ) \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+2 \left (5-2 e^2\right )^4 \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-13 \left (5-2 e^2\right )^3 \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+16 \left (5-2 e^2\right )^2 \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-5 \left (5-2 e^2\right ) \log ^2\left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-8 e^2 \left (18-17 e^2+4 e^4\right ) x-15 \left (5-2 e^2\right )^3 x-\frac {8}{3} e^2 \left (5-2 e^2\right )^2 x+76 \left (5-2 e^2\right )^2 x-2 e^2 \left (7-4 e^2\right ) \left (5-2 e^2\right ) x-63 \left (5-2 e^2\right ) x+10 x+\left (5-2 e^2\right )^4 \log \left (-x-2 e^2+5\right )-\frac {8}{3} e^2 \left (5-2 e^2\right )^3 \log \left (-x-2 e^2+5\right )-2 \left (5-2 e^2\right )^3 \log \left (-x-2 e^2+5\right )-2 e^2 \left (7-4 e^2\right ) \left (5-2 e^2\right )^2 \log \left (-x-2 e^2+5\right )+\left (5-2 e^2\right )^2 \log \left (-x-2 e^2+5\right )+\frac {8}{3} e^2 x^3 \log \left (\frac {x}{2}+\frac {1}{2} \left (-5+2 e^2\right )\right )+2 e^2 \left (7-4 e^2\right ) x^2 \log \left (\frac {x}{2}+\frac {1}{2} \left (-5+2 e^2\right )\right )-\frac {e^x \left (x \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right )+2 e^2 \log \left (\frac {1}{2} \left (x+2 e^2-5\right )\right )-5 \log \left (x+2 e^2-5\right )+\log (32)\right )}{-x-2 e^2+5}\) |
Input:
Int[(E^x + x^2 - 2*x^3 + x^4 + (-10*x + 32*x^2 - 26*x^3 + 4*x^4 + E^x*(-5 + 2*E^2 + x) + E^2*(4*x - 12*x^2 + 8*x^3))*Log[(-5 + 2*E^2 + x)/2])/(-5 + 2*E^2 + x),x]
Output:
-8*(5 - 2*E^2 - x)^2 + (39*(5 - 2*E^2)*(5 - 2*E^2 - x)^2)/2 - 6*(5 - 2*E^2 )^2*(5 - 2*E^2 - x)^2 - (26*(5 - 2*E^2 - x)^3)/9 + (16*(5 - 2*E^2)*(5 - 2* E^2 - x)^3)/9 - (5 - 2*E^2 - x)^4/4 + 10*x - 63*(5 - 2*E^2)*x - 2*E^2*(7 - 4*E^2)*(5 - 2*E^2)*x + 76*(5 - 2*E^2)^2*x - (8*E^2*(5 - 2*E^2)^2*x)/3 - 1 5*(5 - 2*E^2)^3*x - 8*E^2*(18 - 17*E^2 + 4*E^4)*x + x^2/2 - E^2*(7 - 4*E^2 )*x^2 - (5 - 2*E^2)*x^2 - (4*E^2*(5 - 2*E^2)*x^2)/3 + ((5 - 2*E^2)^2*x^2)/ 2 - (2*x^3)/3 - (8*E^2*x^3)/9 + ((5 - 2*E^2)*x^3)/3 + x^4/4 + (5 - 2*E^2)^ 2*Log[5 - 2*E^2 - x] - 2*E^2*(7 - 4*E^2)*(5 - 2*E^2)^2*Log[5 - 2*E^2 - x] - 2*(5 - 2*E^2)^3*Log[5 - 2*E^2 - x] - (8*E^2*(5 - 2*E^2)^3*Log[5 - 2*E^2 - x])/3 + (5 - 2*E^2)^4*Log[5 - 2*E^2 - x] + 2*E^2*(7 - 4*E^2)*x^2*Log[(-5 + 2*E^2)/2 + x/2] + (8*E^2*x^3*Log[(-5 + 2*E^2)/2 + x/2])/3 + 10*(5 - 2*E ^2 - x)*Log[(-5 + 2*E^2 + x)/2] - 64*(5 - 2*E^2)*(5 - 2*E^2 - x)*Log[(-5 + 2*E^2 + x)/2] + 78*(5 - 2*E^2)^2*(5 - 2*E^2 - x)*Log[(-5 + 2*E^2 + x)/2] - 16*(5 - 2*E^2)^3*(5 - 2*E^2 - x)*Log[(-5 + 2*E^2 + x)/2] - 8*E^2*(18 - 1 7*E^2 + 4*E^4)*(5 - 2*E^2 - x)*Log[(-5 + 2*E^2 + x)/2] + 16*(5 - 2*E^2 - x )^2*Log[(-5 + 2*E^2 + x)/2] - 39*(5 - 2*E^2)*(5 - 2*E^2 - x)^2*Log[(-5 + 2 *E^2 + x)/2] + 12*(5 - 2*E^2)^2*(5 - 2*E^2 - x)^2*Log[(-5 + 2*E^2 + x)/2] + (26*(5 - 2*E^2 - x)^3*Log[(-5 + 2*E^2 + x)/2])/3 - (16*(5 - 2*E^2)*(5 - 2*E^2 - x)^3*Log[(-5 + 2*E^2 + x)/2])/3 + (5 - 2*E^2 - x)^4*Log[(-5 + 2*E^ 2 + x)/2] - 5*(5 - 2*E^2)*Log[(-5 + 2*E^2 + x)/2]^2 + 16*(5 - 2*E^2)^2*...
Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs. \(2(21)=42\).
Time = 0.10 (sec) , antiderivative size = 299, normalized size of antiderivative = 11.50
\[-488 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+{\mathrm e}^{x} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+720 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )-304 \,{\mathrm e}^{6}+\frac {100 \,{\mathrm e}^{8}}{3}+1036 \,{\mathrm e}^{4}-\frac {4690 \,{\mathrm e}^{2}}{3}+\frac {3525}{4}-32 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x -24 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{2}-8 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{3}-32 \,{\mathrm e}^{8} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+160 \,{\mathrm e}^{6} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+484 \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+720 \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+144 \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )^{3} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+16 \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right )+\left (-64 \,{\mathrm e}^{6}+16 \left ({\mathrm e}^{4}\right )^{2}+484 \,{\mathrm e}^{4}-80 \,{\mathrm e}^{2} {\mathrm e}^{4}-720 \,{\mathrm e}^{2}+400\right ) \ln \left (2 \,{\mathrm e}^{2}+x -5\right )+24 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x +12 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{2}-4 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{2}+\frac {x}{2}-\frac {5}{2}\right ) x\]
Input:
int((((2*exp(2)+x-5)*exp(x)+(8*x^3-12*x^2+4*x)*exp(2)+4*x^4-26*x^3+32*x^2- 10*x)*ln(exp(2)+1/2*x-5/2)+exp(x)+x^4-2*x^3+x^2)/(2*exp(2)+x-5),x)
Output:
-488*exp(2)^2*ln(exp(2)+1/2*x-5/2)+exp(x)*ln(exp(2)+1/2*x-5/2)+720*exp(2)* ln(exp(2)+1/2*x-5/2)-304*exp(2)^3+100/3*exp(2)^4+1036*exp(2)^2-4690/3*exp( 2)+3525/4-32*exp(2)^3*ln(exp(2)+1/2*x-5/2)*x-24*exp(2)^2*ln(exp(2)+1/2*x-5 /2)*x^2-8*exp(2)*ln(exp(2)+1/2*x-5/2)*x^3-32*exp(2)^4*ln(exp(2)+1/2*x-5/2) +160*exp(2)^3*ln(exp(2)+1/2*x-5/2)+484*(exp(2)+1/2*x-5/2)^2*ln(exp(2)+1/2* x-5/2)+720*(exp(2)+1/2*x-5/2)*ln(exp(2)+1/2*x-5/2)+144*(exp(2)+1/2*x-5/2)^ 3*ln(exp(2)+1/2*x-5/2)+16*(exp(2)+1/2*x-5/2)^4*ln(exp(2)+1/2*x-5/2)+(-64*e xp(2)^3+16*exp(2)^2*exp(4)+384*exp(2)^2-80*exp(2)*exp(4)-720*exp(2)+100*ex p(4)+400)*ln(2*exp(2)+x-5)+24*exp(2)^2*ln(exp(2)+1/2*x-5/2)*x+12*exp(2)*ln (exp(2)+1/2*x-5/2)*x^2-4*exp(2)*ln(exp(2)+1/2*x-5/2)*x
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx={\left (x^{4} - 2 \, x^{3} + x^{2} + e^{x}\right )} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) \] Input:
integrate((((2*exp(2)+x-5)*exp(x)+(8*x^3-12*x^2+4*x)*exp(2)+4*x^4-26*x^3+3 2*x^2-10*x)*log(exp(2)+1/2*x-5/2)+exp(x)+x^4-2*x^3+x^2)/(2*exp(2)+x-5),x, algorithm="fricas")
Output:
(x^4 - 2*x^3 + x^2 + e^x)*log(1/2*x + e^2 - 5/2)
Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx=\left (x^{4} - 2 x^{3} + x^{2}\right ) \log {\left (\frac {x}{2} - \frac {5}{2} + e^{2} \right )} + e^{x} \log {\left (\frac {x}{2} - \frac {5}{2} + e^{2} \right )} \] Input:
integrate((((2*exp(2)+x-5)*exp(x)+(8*x**3-12*x**2+4*x)*exp(2)+4*x**4-26*x* *3+32*x**2-10*x)*ln(exp(2)+1/2*x-5/2)+exp(x)+x**4-2*x**3+x**2)/(2*exp(2)+x -5),x)
Output:
(x**4 - 2*x**3 + x**2)*log(x/2 - 5/2 + exp(2)) + exp(x)*log(x/2 - 5/2 + ex p(2))
\[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx=\int { \frac {x^{4} - 2 \, x^{3} + x^{2} + {\left (4 \, x^{4} - 26 \, x^{3} + 32 \, x^{2} + 4 \, {\left (2 \, x^{3} - 3 \, x^{2} + x\right )} e^{2} + {\left (x + 2 \, e^{2} - 5\right )} e^{x} - 10 \, x\right )} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + e^{x}}{x + 2 \, e^{2} - 5} \,d x } \] Input:
integrate((((2*exp(2)+x-5)*exp(x)+(8*x^3-12*x^2+4*x)*exp(2)+4*x^4-26*x^3+3 2*x^2-10*x)*log(exp(2)+1/2*x-5/2)+exp(x)+x^4-2*x^3+x^2)/(2*exp(2)+x-5),x, algorithm="maxima")
Output:
4/9*x^3*(2*e^2 - 5) + 20/9*x^3 - 5/3*x^2*(4*e^4 - 20*e^2 + 25) - 59/6*x^2* (2*e^2 - 5) - 2*(16*e^8 - 160*e^6 + 600*e^4 - 1000*e^2 + 625)*log(x + 2*e^ 2 - 5)^2 - 13*(8*e^6 - 60*e^4 + 150*e^2 - 125)*log(x + 2*e^2 - 5)^2 - 16*( 4*e^4 - 20*e^2 + 25)*log(x + 2*e^2 - 5)^2 - 5*(2*e^2 - 5)*log(x + 2*e^2 - 5)^2 + 4/3*(2*x^3 - 3*x^2*(2*e^2 - 5) + 6*x*(4*e^4 - 20*e^2 + 25) - 6*(8*e ^6 - 60*e^4 + 150*e^2 - 125)*log(x + 2*e^2 - 5))*e^2*log(1/2*x + e^2 - 5/2 ) - 6*(x^2 - 2*x*(2*e^2 - 5) + 2*(4*e^4 - 20*e^2 + 25)*log(x + 2*e^2 - 5)) *e^2*log(1/2*x + e^2 - 5/2) - 4*((2*e^2 - 5)*log(x + 2*e^2 - 5) - x)*e^2*l og(1/2*x + e^2 - 5/2) - 15/2*x^2 + 22/3*x*(8*e^6 - 60*e^4 + 150*e^2 - 125) + 137/3*x*(4*e^4 - 20*e^2 + 25) + 47*x*(2*e^2 - 5) - 2/9*(4*x^3 - 15*x^2* (2*e^2 - 5) - 18*(8*e^6 - 60*e^4 + 150*e^2 - 125)*log(x + 2*e^2 - 5)^2 + 6 6*x*(4*e^4 - 20*e^2 + 25) - 66*(8*e^6 - 60*e^4 + 150*e^2 - 125)*log(x + 2* e^2 - 5))*e^2 + 3*(2*(4*e^4 - 20*e^2 + 25)*log(x + 2*e^2 - 5)^2 + x^2 - 6* x*(2*e^2 - 5) + 6*(4*e^4 - 20*e^2 + 25)*log(x + 2*e^2 - 5))*e^2 + 2*((2*e^ 2 - 5)*log(x + 2*e^2 - 5)^2 + 2*(2*e^2 - 5)*log(x + 2*e^2 - 5) - 2*x)*e^2 - e^(-2*e^2 + 5)*exp_integral_e(1, -x - 2*e^2 + 5) - 22/3*(16*e^8 - 160*e^ 6 + 600*e^4 - 1000*e^2 + 625)*log(x + 2*e^2 - 5) - 137/3*(8*e^6 - 60*e^4 + 150*e^2 - 125)*log(x + 2*e^2 - 5) - 47*(4*e^4 - 20*e^2 + 25)*log(x + 2*e^ 2 - 5) - 10*(2*e^2 - 5)*log(x + 2*e^2 - 5) + 1/3*(3*x^4 - 4*x^3*(2*e^2 - 5 ) + 6*x^2*(4*e^4 - 20*e^2 + 25) - 12*x*(8*e^6 - 60*e^4 + 150*e^2 - 125)...
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (21) = 42\).
Time = 0.15 (sec) , antiderivative size = 263, normalized size of antiderivative = 10.12 \[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx={\left (x + 2 \, e^{2} - 5\right )}^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 8 \, {\left (x + 2 \, e^{2} - 5\right )}^{3} e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 18 \, {\left (x + 2 \, e^{2} - 5\right )}^{3} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 24 \, {\left (x + 2 \, e^{2} - 5\right )}^{2} e^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 108 \, {\left (x + 2 \, e^{2} - 5\right )}^{2} e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 121 \, {\left (x + 2 \, e^{2} - 5\right )}^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 32 \, {\left (x + 2 \, e^{2} - 5\right )} e^{6} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 216 \, {\left (x + 2 \, e^{2} - 5\right )} e^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 484 \, {\left (x + 2 \, e^{2} - 5\right )} e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 360 \, {\left (x + 2 \, e^{2} - 5\right )} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 16 \, e^{8} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 144 \, e^{6} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 484 \, e^{4} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) - 720 \, e^{2} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + e^{x} \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) + 400 \, \log \left (\frac {1}{2} \, x + e^{2} - \frac {5}{2}\right ) \] Input:
integrate((((2*exp(2)+x-5)*exp(x)+(8*x^3-12*x^2+4*x)*exp(2)+4*x^4-26*x^3+3 2*x^2-10*x)*log(exp(2)+1/2*x-5/2)+exp(x)+x^4-2*x^3+x^2)/(2*exp(2)+x-5),x, algorithm="giac")
Output:
(x + 2*e^2 - 5)^4*log(1/2*x + e^2 - 5/2) - 8*(x + 2*e^2 - 5)^3*e^2*log(1/2 *x + e^2 - 5/2) + 18*(x + 2*e^2 - 5)^3*log(1/2*x + e^2 - 5/2) + 24*(x + 2* e^2 - 5)^2*e^4*log(1/2*x + e^2 - 5/2) - 108*(x + 2*e^2 - 5)^2*e^2*log(1/2* x + e^2 - 5/2) + 121*(x + 2*e^2 - 5)^2*log(1/2*x + e^2 - 5/2) - 32*(x + 2* e^2 - 5)*e^6*log(1/2*x + e^2 - 5/2) + 216*(x + 2*e^2 - 5)*e^4*log(1/2*x + e^2 - 5/2) - 484*(x + 2*e^2 - 5)*e^2*log(1/2*x + e^2 - 5/2) + 360*(x + 2*e ^2 - 5)*log(1/2*x + e^2 - 5/2) + 16*e^8*log(1/2*x + e^2 - 5/2) - 144*e^6*l og(1/2*x + e^2 - 5/2) + 484*e^4*log(1/2*x + e^2 - 5/2) - 720*e^2*log(1/2*x + e^2 - 5/2) + e^x*log(1/2*x + e^2 - 5/2) + 400*log(1/2*x + e^2 - 5/2)
Time = 2.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx=\ln \left (\frac {x}{2}+{\mathrm {e}}^2-\frac {5}{2}\right )\,\left ({\mathrm {e}}^x+x^2-2\,x^3+x^4\right ) \] Input:
int((exp(x) + log(x/2 + exp(2) - 5/2)*(exp(2)*(4*x - 12*x^2 + 8*x^3) - 10* x + 32*x^2 - 26*x^3 + 4*x^4 + exp(x)*(x + 2*exp(2) - 5)) + x^2 - 2*x^3 + x ^4)/(x + 2*exp(2) - 5),x)
Output:
log(x/2 + exp(2) - 5/2)*(exp(x) + x^2 - 2*x^3 + x^4)
Time = 0.16 (sec) , antiderivative size = 188, normalized size of antiderivative = 7.23 \[ \int \frac {e^x+x^2-2 x^3+x^4+\left (-10 x+32 x^2-26 x^3+4 x^4+e^x \left (-5+2 e^2+x\right )+e^2 \left (4 x-12 x^2+8 x^3\right )\right ) \log \left (\frac {1}{2} \left (-5+2 e^2+x\right )\right )}{-5+2 e^2+x} \, dx=e^{x} \mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right )+16 \,\mathrm {log}\left (2 e^{2}+x -5\right ) e^{8}-144 \,\mathrm {log}\left (2 e^{2}+x -5\right ) e^{6}+484 \,\mathrm {log}\left (2 e^{2}+x -5\right ) e^{4}-720 \,\mathrm {log}\left (2 e^{2}+x -5\right ) e^{2}+400 \,\mathrm {log}\left (2 e^{2}+x -5\right )-16 \,\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) e^{8}+144 \,\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) e^{6}-484 \,\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) e^{4}+720 \,\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) e^{2}+\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{4}-2 \,\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{3}+\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) x^{2}-400 \,\mathrm {log}\left (e^{2}+\frac {x}{2}-\frac {5}{2}\right ) \] Input:
int((((2*exp(2)+x-5)*exp(x)+(8*x^3-12*x^2+4*x)*exp(2)+4*x^4-26*x^3+32*x^2- 10*x)*log(exp(2)+1/2*x-5/2)+exp(x)+x^4-2*x^3+x^2)/(2*exp(2)+x-5),x)
Output:
e**x*log((2*e**2 + x - 5)/2) + 16*log(2*e**2 + x - 5)*e**8 - 144*log(2*e** 2 + x - 5)*e**6 + 484*log(2*e**2 + x - 5)*e**4 - 720*log(2*e**2 + x - 5)*e **2 + 400*log(2*e**2 + x - 5) - 16*log((2*e**2 + x - 5)/2)*e**8 + 144*log( (2*e**2 + x - 5)/2)*e**6 - 484*log((2*e**2 + x - 5)/2)*e**4 + 720*log((2*e **2 + x - 5)/2)*e**2 + log((2*e**2 + x - 5)/2)*x**4 - 2*log((2*e**2 + x - 5)/2)*x**3 + log((2*e**2 + x - 5)/2)*x**2 - 400*log((2*e**2 + x - 5)/2)