Integrand size = 250, antiderivative size = 29 \[ \int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx=5+e^x+\frac {\log (x)}{\left (e^x+x\right ) \left (25+x+x^2\right )}-\log \left (x^2\right ) \]
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx=e^x-2 \log (x)+\frac {\log (x)}{\left (e^x+x\right ) \left (25+x+x^2\right )} \]
Integrate[(25*x - 1249*x^2 - 99*x^3 - 102*x^4 - 4*x^5 - 2*x^6 + E^(3*x)*(6 25*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + E^(2*x)*(-1250 - 100*x + 1148*x^2 + 96*x^3 + 100*x^4 + 4*x^5 + 2*x^6) + E^x*(25 - 2499*x - 199*x^2 + 421*x^3 + 42*x^4 + 47*x^5 + 2*x^6 + x^7) + (-25*x - 2*x^2 - 3*x^3 + E^x*(-26*x - 3*x^2 - x^3))*Log[x])/(625*x^3 + 50*x^4 + 51*x^5 + 2*x^6 + x^7 + E^(2*x)*( 625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + E^x*(1250*x^2 + 100*x^3 + 102*x^4 + 4*x^5 + 2*x^6)),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.
| \(\displaystyle \int \frac {-2 x^6-4 x^5-102 x^4-99 x^3-1249 x^2+\left (-3 x^3-2 x^2+e^x \left (-x^3-3 x^2-26 x\right )-25 x\right ) \log (x)+e^{3 x} \left (x^5+2 x^4+51 x^3+50 x^2+625 x\right )+e^{2 x} \left (2 x^6+4 x^5+100 x^4+96 x^3+1148 x^2-100 x-1250\right )+e^x \left (x^7+2 x^6+47 x^5+42 x^4+421 x^3-199 x^2-2499 x+25\right )+25 x}{x^7+2 x^6+51 x^5+50 x^4+625 x^3+e^{2 x} \left (x^5+2 x^4+51 x^3+50 x^2+625 x\right )+e^x \left (2 x^6+4 x^5+102 x^4+100 x^3+1250 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^2+x+25\right ) \left (e^{3 x} x \left (x^2+x+25\right )-x \left (2 x^3+2 x^2+50 x-1\right )+2 e^{2 x} \left (x^4+x^3+24 x^2-x-25\right )+e^x \left (x^5+x^4+21 x^3-4 x^2-100 x+1\right )\right )-x \left (3 x^2+e^x \left (x^2+3 x+26\right )+2 x+25\right ) \log (x)}{x \left (x+e^x\right )^2 \left (x^2+x+25\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x-1) \log (x)}{\left (x+e^x\right )^2 \left (x^2+x+25\right )}-\frac {x^3 \log (x)-x^2+3 x^2 \log (x)-x+26 x \log (x)-25}{x \left (x+e^x\right ) \left (x^2+x+25\right )^2}+e^x-\frac {2}{x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {(x-1) \log (x)}{\left (x+e^x\right )^2 \left (x^2+x+25\right )}-\frac {x^3 \log (x)-x^2+3 x^2 \log (x)-x+26 x \log (x)-25}{x \left (x+e^x\right ) \left (x^2+x+25\right )^2}+e^x-\frac {2}{x}\right )dx\) |
Int[(25*x - 1249*x^2 - 99*x^3 - 102*x^4 - 4*x^5 - 2*x^6 + E^(3*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + E^(2*x)*(-1250 - 100*x + 1148*x^2 + 96*x ^3 + 100*x^4 + 4*x^5 + 2*x^6) + E^x*(25 - 2499*x - 199*x^2 + 421*x^3 + 42* x^4 + 47*x^5 + 2*x^6 + x^7) + (-25*x - 2*x^2 - 3*x^3 + E^x*(-26*x - 3*x^2 - x^3))*Log[x])/(625*x^3 + 50*x^4 + 51*x^5 + 2*x^6 + x^7 + E^(2*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + E^x*(1250*x^2 + 100*x^3 + 102*x^4 + 4*x ^5 + 2*x^6)),x]
3.1.39.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86
| method | result | size |
| risch | \(\frac {\ln \left (x \right )}{\left (x^{2}+x +25\right ) \left ({\mathrm e}^{x}+x \right )}-2 \ln \left (x \right )+{\mathrm e}^{x}\) | \(25\) |
| parallelrisch | \(\frac {-1250 \,{\mathrm e}^{x} \ln \left (x \right )+25 \,{\mathrm e}^{2 x} x^{2}+25 \,{\mathrm e}^{x} x^{2}+25 x \,{\mathrm e}^{2 x}+25 \,{\mathrm e}^{x} x^{3}+625 \,{\mathrm e}^{x} x -1250 x \ln \left (x \right )-50 x \,{\mathrm e}^{x} \ln \left (x \right )-50 x^{2} {\mathrm e}^{x} \ln \left (x \right )+625 \,{\mathrm e}^{2 x}+25 \ln \left (x \right )-50 x^{3} \ln \left (x \right )-50 x^{2} \ln \left (x \right )}{25 \,{\mathrm e}^{x} x^{2}+25 x^{3}+25 \,{\mathrm e}^{x} x +25 x^{2}+625 \,{\mathrm e}^{x}+625 x}\) | \(116\) |
int((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*ln(x)+(x^5+2*x^4+51*x^3+5 0*x^2+625*x)*exp(x)^3+(2*x^6+4*x^5+100*x^4+96*x^3+1148*x^2-100*x-1250)*exp (x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-2*x^6-4*x ^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*exp(x)^2 +(2*x^6+4*x^5+102*x^4+100*x^3+1250*x^2)*exp(x)+x^7+2*x^6+51*x^5+50*x^4+625 *x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx=\frac {{\left (x^{2} + x + 25\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + x^{2} + 25 \, x\right )} e^{x} - {\left (2 \, x^{3} + 2 \, x^{2} + 2 \, {\left (x^{2} + x + 25\right )} e^{x} + 50 \, x - 1\right )} \log \left (x\right )}{x^{3} + x^{2} + {\left (x^{2} + x + 25\right )} e^{x} + 25 \, x} \]
integrate((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*log(x)+(x^5+2*x^4+5 1*x^3+50*x^2+625*x)*exp(x)^3+(2*x^6+4*x^5+100*x^4+96*x^3+1148*x^2-100*x-12 50)*exp(x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-2* x^6-4*x^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*e xp(x)^2+(2*x^6+4*x^5+102*x^4+100*x^3+1250*x^2)*exp(x)+x^7+2*x^6+51*x^5+50* x^4+625*x^3),x, algorithm=\
((x^2 + x + 25)*e^(2*x) + (x^3 + x^2 + 25*x)*e^x - (2*x^3 + 2*x^2 + 2*(x^2 + x + 25)*e^x + 50*x - 1)*log(x))/(x^3 + x^2 + (x^2 + x + 25)*e^x + 25*x)
Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx=e^{x} - 2 \log {\left (x \right )} + \frac {\log {\left (x \right )}}{x^{3} + x^{2} + 25 x + \left (x^{2} + x + 25\right ) e^{x}} \]
integrate((((-x**3-3*x**2-26*x)*exp(x)-3*x**3-2*x**2-25*x)*ln(x)+(x**5+2*x **4+51*x**3+50*x**2+625*x)*exp(x)**3+(2*x**6+4*x**5+100*x**4+96*x**3+1148* x**2-100*x-1250)*exp(x)**2+(x**7+2*x**6+47*x**5+42*x**4+421*x**3-199*x**2- 2499*x+25)*exp(x)-2*x**6-4*x**5-102*x**4-99*x**3-1249*x**2+25*x)/((x**5+2* x**4+51*x**3+50*x**2+625*x)*exp(x)**2+(2*x**6+4*x**5+102*x**4+100*x**3+125 0*x**2)*exp(x)+x**7+2*x**6+51*x**5+50*x**4+625*x**3),x)
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx=\frac {{\left (x^{2} + x + 25\right )} e^{\left (2 \, x\right )} + {\left (x^{3} + x^{2} + 25 \, x\right )} e^{x} + \log \left (x\right )}{x^{3} + x^{2} + {\left (x^{2} + x + 25\right )} e^{x} + 25 \, x} - 2 \, \log \left (x\right ) \]
integrate((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*log(x)+(x^5+2*x^4+5 1*x^3+50*x^2+625*x)*exp(x)^3+(2*x^6+4*x^5+100*x^4+96*x^3+1148*x^2-100*x-12 50)*exp(x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-2* x^6-4*x^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*e xp(x)^2+(2*x^6+4*x^5+102*x^4+100*x^3+1250*x^2)*exp(x)+x^7+2*x^6+51*x^5+50* x^4+625*x^3),x, algorithm=\
((x^2 + x + 25)*e^(2*x) + (x^3 + x^2 + 25*x)*e^x + log(x))/(x^3 + x^2 + (x ^2 + x + 25)*e^x + 25*x) - 2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.72 \[ \int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx=\frac {x^{3} e^{x} - 2 \, x^{3} \log \left (x\right ) - 2 \, x^{2} e^{x} \log \left (x\right ) + x^{2} e^{\left (2 \, x\right )} + x^{2} e^{x} - 2 \, x^{2} \log \left (x\right ) - 2 \, x e^{x} \log \left (x\right ) + x e^{\left (2 \, x\right )} + 25 \, x e^{x} - 50 \, x \log \left (x\right ) - 50 \, e^{x} \log \left (x\right ) + 25 \, e^{\left (2 \, x\right )} + \log \left (x\right )}{x^{3} + x^{2} e^{x} + x^{2} + x e^{x} + 25 \, x + 25 \, e^{x}} \]
integrate((((-x^3-3*x^2-26*x)*exp(x)-3*x^3-2*x^2-25*x)*log(x)+(x^5+2*x^4+5 1*x^3+50*x^2+625*x)*exp(x)^3+(2*x^6+4*x^5+100*x^4+96*x^3+1148*x^2-100*x-12 50)*exp(x)^2+(x^7+2*x^6+47*x^5+42*x^4+421*x^3-199*x^2-2499*x+25)*exp(x)-2* x^6-4*x^5-102*x^4-99*x^3-1249*x^2+25*x)/((x^5+2*x^4+51*x^3+50*x^2+625*x)*e xp(x)^2+(2*x^6+4*x^5+102*x^4+100*x^3+1250*x^2)*exp(x)+x^7+2*x^6+51*x^5+50* x^4+625*x^3),x, algorithm=\
(x^3*e^x - 2*x^3*log(x) - 2*x^2*e^x*log(x) + x^2*e^(2*x) + x^2*e^x - 2*x^2 *log(x) - 2*x*e^x*log(x) + x*e^(2*x) + 25*x*e^x - 50*x*log(x) - 50*e^x*log (x) + 25*e^(2*x) + log(x))/(x^3 + x^2*e^x + x^2 + x*e^x + 25*x + 25*e^x)
Time = 8.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {25 x-1249 x^2-99 x^3-102 x^4-4 x^5-2 x^6+e^{3 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^{2 x} \left (-1250-100 x+1148 x^2+96 x^3+100 x^4+4 x^5+2 x^6\right )+e^x \left (25-2499 x-199 x^2+421 x^3+42 x^4+47 x^5+2 x^6+x^7\right )+\left (-25 x-2 x^2-3 x^3+e^x \left (-26 x-3 x^2-x^3\right )\right ) \log (x)}{625 x^3+50 x^4+51 x^5+2 x^6+x^7+e^{2 x} \left (625 x+50 x^2+51 x^3+2 x^4+x^5\right )+e^x \left (1250 x^2+100 x^3+102 x^4+4 x^5+2 x^6\right )} \, dx={\mathrm {e}}^x-2\,\ln \left (x\right )+\frac {\ln \left (x\right )}{\left (x+{\mathrm {e}}^x\right )\,\left (x^2+x+25\right )} \]
int(-(log(x)*(25*x + exp(x)*(26*x + 3*x^2 + x^3) + 2*x^2 + 3*x^3) - exp(3* x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) - exp(x)*(421*x^3 - 199*x^2 - 2 499*x + 42*x^4 + 47*x^5 + 2*x^6 + x^7 + 25) - 25*x + 1249*x^2 + 99*x^3 + 1 02*x^4 + 4*x^5 + 2*x^6 - exp(2*x)*(1148*x^2 - 100*x + 96*x^3 + 100*x^4 + 4 *x^5 + 2*x^6 - 1250))/(exp(2*x)*(625*x + 50*x^2 + 51*x^3 + 2*x^4 + x^5) + exp(x)*(1250*x^2 + 100*x^3 + 102*x^4 + 4*x^5 + 2*x^6) + 625*x^3 + 50*x^4 + 51*x^5 + 2*x^6 + x^7),x)