Integrand size = 141, antiderivative size = 35 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=\frac {e^2 x}{2 \left (5-e^x-x^2\right )}+\frac {1}{3} \log \left (-3+\frac {5}{x}\right ) \] Output:
1/2*exp(2)*x/(5-x^2-exp(x))+1/3*ln(5/x-3)
Time = 3.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=\frac {1}{6} \left (-\frac {3 e^2 x}{-5+e^x+x^2}+2 \log (5-3 x)-2 \log (x)\right ) \] Input:
Integrate[(250 + 10*E^(2*x) - 100*x^2 + 10*x^4 + E^2*(-75*x + 45*x^2 - 15* x^3 + 9*x^4) + E^x*(-100 + 20*x^2 + E^2*(15*x - 24*x^2 + 9*x^3)))/(-750*x + 450*x^2 + 300*x^3 - 180*x^4 - 30*x^5 + 18*x^6 + E^(2*x)*(-30*x + 18*x^2) + E^x*(300*x - 180*x^2 - 60*x^3 + 36*x^4)),x]
Output:
((-3*E^2*x)/(-5 + E^x + x^2) + 2*Log[5 - 3*x] - 2*Log[x])/6
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.
| \(\displaystyle \int \frac {10 x^4-100 x^2+e^x \left (20 x^2+e^2 \left (9 x^3-24 x^2+15 x\right )-100\right )+e^2 \left (9 x^4-15 x^3+45 x^2-75 x\right )+10 e^{2 x}+250}{18 x^6-30 x^5-180 x^4+300 x^3+450 x^2+e^{2 x} \left (18 x^2-30 x\right )+e^x \left (36 x^4-60 x^3-180 x^2+300 x\right )-750 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-10 x^4+100 x^2-e^x \left (20 x^2+e^2 \left (9 x^3-24 x^2+15 x\right )-100\right )-e^2 \left (9 x^4-15 x^3+45 x^2-75 x\right )-10 e^{2 x}-250}{6 (5-3 x) x \left (-x^2-e^x+5\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \int -\frac {10 x^4-100 x^2+10 e^{2 x}-3 e^2 \left (-3 x^4+5 x^3-15 x^2+25 x\right )-e^x \left (-20 x^2-3 e^2 \left (3 x^3-8 x^2+5 x\right )+100\right )+250}{(5-3 x) x \left (-x^2-e^x+5\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{6} \int \frac {10 x^4-100 x^2+10 e^{2 x}-3 e^2 \left (-3 x^4+5 x^3-15 x^2+25 x\right )-e^x \left (-20 x^2-3 e^2 \left (3 x^3-8 x^2+5 x\right )+100\right )+250}{(5-3 x) x \left (-x^2-e^x+5\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{6} \int \left (-\frac {3 e^2 (x-1)}{x^2+e^x-5}+\frac {3 e^2 x \left (x^2-2 x-5\right )}{\left (x^2+e^x-5\right )^2}-\frac {10}{x (3 x-5)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left (15 e^2 \int \frac {x}{\left (x^2+e^x-5\right )^2}dx+6 e^2 \int \frac {x^2}{\left (x^2+e^x-5\right )^2}dx-3 e^2 \int \frac {1}{x^2+e^x-5}dx+3 e^2 \int \frac {x}{x^2+e^x-5}dx-3 e^2 \int \frac {x^3}{\left (x^2+e^x-5\right )^2}dx+2 \log (5-3 x)-2 \log (x)\right )\) |
Input:
Int[(250 + 10*E^(2*x) - 100*x^2 + 10*x^4 + E^2*(-75*x + 45*x^2 - 15*x^3 + 9*x^4) + E^x*(-100 + 20*x^2 + E^2*(15*x - 24*x^2 + 9*x^3)))/(-750*x + 450* x^2 + 300*x^3 - 180*x^4 - 30*x^5 + 18*x^6 + E^(2*x)*(-30*x + 18*x^2) + E^x *(300*x - 180*x^2 - 60*x^3 + 36*x^4)),x]
Output:
$Aborted
Time = 0.67 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80
| method | result | size |
| norman | \(-\frac {{\mathrm e}^{2} x}{2 \left (x^{2}+{\mathrm e}^{x}-5\right )}-\frac {\ln \left (x \right )}{3}+\frac {\ln \left (3 x -5\right )}{3}\) | \(28\) |
| risch | \(-\frac {{\mathrm e}^{2} x}{2 \left (x^{2}+{\mathrm e}^{x}-5\right )}-\frac {\ln \left (x \right )}{3}+\frac {\ln \left (3 x -5\right )}{3}\) | \(28\) |
| parallelrisch | \(-\frac {2 x^{2} \ln \left (x \right )-2 \ln \left (x -\frac {5}{3}\right ) x^{2}+3 \,{\mathrm e}^{2} x +2 \,{\mathrm e}^{x} \ln \left (x \right )-2 \ln \left (x -\frac {5}{3}\right ) {\mathrm e}^{x}-10 \ln \left (x \right )+10 \ln \left (x -\frac {5}{3}\right )}{6 \left (x^{2}+{\mathrm e}^{x}-5\right )}\) | \(58\) |
Input:
int((10*exp(x)^2+((9*x^3-24*x^2+15*x)*exp(2)+20*x^2-100)*exp(x)+(9*x^4-15* x^3+45*x^2-75*x)*exp(2)+10*x^4-100*x^2+250)/((18*x^2-30*x)*exp(x)^2+(36*x^ 4-60*x^3-180*x^2+300*x)*exp(x)+18*x^6-30*x^5-180*x^4+300*x^3+450*x^2-750*x ),x,method=_RETURNVERBOSE)
Output:
-1/2*exp(2)*x/(x^2+exp(x)-5)-1/3*ln(x)+1/3*ln(3*x-5)
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=-\frac {3 \, x e^{2} - 2 \, {\left (x^{2} + e^{x} - 5\right )} \log \left (3 \, x - 5\right ) + 2 \, {\left (x^{2} + e^{x} - 5\right )} \log \left (x\right )}{6 \, {\left (x^{2} + e^{x} - 5\right )}} \] Input:
integrate((10*exp(x)^2+((9*x^3-24*x^2+15*x)*exp(2)+20*x^2-100)*exp(x)+(9*x ^4-15*x^3+45*x^2-75*x)*exp(2)+10*x^4-100*x^2+250)/((18*x^2-30*x)*exp(x)^2+ (36*x^4-60*x^3-180*x^2+300*x)*exp(x)+18*x^6-30*x^5-180*x^4+300*x^3+450*x^2 -750*x),x, algorithm="fricas")
Output:
-1/6*(3*x*e^2 - 2*(x^2 + e^x - 5)*log(3*x - 5) + 2*(x^2 + e^x - 5)*log(x)) /(x^2 + e^x - 5)
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=- \frac {x e^{2}}{2 x^{2} + 2 e^{x} - 10} - \frac {\log {\left (x \right )}}{3} + \frac {\log {\left (x - \frac {5}{3} \right )}}{3} \] Input:
integrate((10*exp(x)**2+((9*x**3-24*x**2+15*x)*exp(2)+20*x**2-100)*exp(x)+ (9*x**4-15*x**3+45*x**2-75*x)*exp(2)+10*x**4-100*x**2+250)/((18*x**2-30*x) *exp(x)**2+(36*x**4-60*x**3-180*x**2+300*x)*exp(x)+18*x**6-30*x**5-180*x** 4+300*x**3+450*x**2-750*x),x)
Output:
-x*exp(2)/(2*x**2 + 2*exp(x) - 10) - log(x)/3 + log(x - 5/3)/3
Time = 0.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=-\frac {x e^{2}}{2 \, {\left (x^{2} + e^{x} - 5\right )}} + \frac {1}{3} \, \log \left (3 \, x - 5\right ) - \frac {1}{3} \, \log \left (x\right ) \] Input:
integrate((10*exp(x)^2+((9*x^3-24*x^2+15*x)*exp(2)+20*x^2-100)*exp(x)+(9*x ^4-15*x^3+45*x^2-75*x)*exp(2)+10*x^4-100*x^2+250)/((18*x^2-30*x)*exp(x)^2+ (36*x^4-60*x^3-180*x^2+300*x)*exp(x)+18*x^6-30*x^5-180*x^4+300*x^3+450*x^2 -750*x),x, algorithm="maxima")
Output:
-1/2*x*e^2/(x^2 + e^x - 5) + 1/3*log(3*x - 5) - 1/3*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=\frac {2 \, x^{2} \log \left (3 \, x - 5\right ) - 2 \, x^{2} \log \left (x\right ) - 3 \, x e^{2} + 2 \, e^{x} \log \left (3 \, x - 5\right ) - 2 \, e^{x} \log \left (x\right ) - 10 \, \log \left (3 \, x - 5\right ) + 10 \, \log \left (x\right )}{6 \, {\left (x^{2} + e^{x} - 5\right )}} \] Input:
integrate((10*exp(x)^2+((9*x^3-24*x^2+15*x)*exp(2)+20*x^2-100)*exp(x)+(9*x ^4-15*x^3+45*x^2-75*x)*exp(2)+10*x^4-100*x^2+250)/((18*x^2-30*x)*exp(x)^2+ (36*x^4-60*x^3-180*x^2+300*x)*exp(x)+18*x^6-30*x^5-180*x^4+300*x^3+450*x^2 -750*x),x, algorithm="giac")
Output:
1/6*(2*x^2*log(3*x - 5) - 2*x^2*log(x) - 3*x*e^2 + 2*e^x*log(3*x - 5) - 2* e^x*log(x) - 10*log(3*x - 5) + 10*log(x))/(x^2 + e^x - 5)
Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.49 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {6\,x}{5}-1\right )}{3}-\frac {-{\mathrm {e}}^2\,x^3+2\,{\mathrm {e}}^2\,x^2+5\,{\mathrm {e}}^2\,x}{2\,\left ({\mathrm {e}}^x+x^2-5\right )\,\left (-x^2+2\,x+5\right )} \] Input:
int(-(10*exp(2*x) + exp(x)*(exp(2)*(15*x - 24*x^2 + 9*x^3) + 20*x^2 - 100) - exp(2)*(75*x - 45*x^2 + 15*x^3 - 9*x^4) - 100*x^2 + 10*x^4 + 250)/(750* x + exp(2*x)*(30*x - 18*x^2) - exp(x)*(300*x - 180*x^2 - 60*x^3 + 36*x^4) - 450*x^2 - 300*x^3 + 180*x^4 + 30*x^5 - 18*x^6),x)
Output:
- (2*atanh((6*x)/5 - 1))/3 - (5*x*exp(2) + 2*x^2*exp(2) - x^3*exp(2))/(2*( exp(x) + x^2 - 5)*(2*x - x^2 + 5))
Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.00 \[ \int \frac {250+10 e^{2 x}-100 x^2+10 x^4+e^2 \left (-75 x+45 x^2-15 x^3+9 x^4\right )+e^x \left (-100+20 x^2+e^2 \left (15 x-24 x^2+9 x^3\right )\right )}{-750 x+450 x^2+300 x^3-180 x^4-30 x^5+18 x^6+e^{2 x} \left (-30 x+18 x^2\right )+e^x \left (300 x-180 x^2-60 x^3+36 x^4\right )} \, dx=\frac {2 e^{x} \mathrm {log}\left (3 x -5\right )-2 e^{x} \mathrm {log}\left (x \right )+2 \,\mathrm {log}\left (3 x -5\right ) x^{2}-10 \,\mathrm {log}\left (3 x -5\right )-2 \,\mathrm {log}\left (x \right ) x^{2}+10 \,\mathrm {log}\left (x \right )-3 e^{2} x}{6 e^{x}+6 x^{2}-30} \] Input:
int((10*exp(x)^2+((9*x^3-24*x^2+15*x)*exp(2)+20*x^2-100)*exp(x)+(9*x^4-15* x^3+45*x^2-75*x)*exp(2)+10*x^4-100*x^2+250)/((18*x^2-30*x)*exp(x)^2+(36*x^ 4-60*x^3-180*x^2+300*x)*exp(x)+18*x^6-30*x^5-180*x^4+300*x^3+450*x^2-750*x ),x)
Output:
(2*e**x*log(3*x - 5) - 2*e**x*log(x) + 2*log(3*x - 5)*x**2 - 10*log(3*x - 5) - 2*log(x)*x**2 + 10*log(x) - 3*e**2*x)/(6*(e**x + x**2 - 5))