Integrand size = 210, antiderivative size = 33 \[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=\frac {5-x+\frac {x}{\log (2)}}{-2+e^{-\frac {x}{1+x}} x+81 x^4} \] Output:
(x/ln(2)+5-x)/(x/exp(x/(1+x))-2+81*x^4)
\[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=\int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx \] Input:
Integrate[(E^(x/(1 + x))*(x^2 + (-5 - 5*x - 6*x^2)*Log[2]) + E^((2*x)/(1 + x))*(-2 - 4*x - 2*x^2 - 243*x^4 - 486*x^5 - 243*x^6 + (2 + 4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6)*Log[2]))/((x^2 + 2*x^3 + x^4)*Lo g[2] + E^(x/(1 + x))*(-4*x - 8*x^2 - 4*x^3 + 162*x^5 + 324*x^6 + 162*x^7)* Log[2] + E^((2*x)/(1 + x))*(4 + 8*x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10)*Log[2]),x]
Output:
Integrate[(E^(x/(1 + x))*(x^2 + (-5 - 5*x - 6*x^2)*Log[2]) + E^((2*x)/(1 + x))*(-2 - 4*x - 2*x^2 - 243*x^4 - 486*x^5 - 243*x^6 + (2 + 4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6)*Log[2]))/((x^2 + 2*x^3 + x^4)*Lo g[2] + E^(x/(1 + x))*(-4*x - 8*x^2 - 4*x^3 + 162*x^5 + 324*x^6 + 162*x^7)* Log[2] + E^((2*x)/(1 + x))*(4 + 8*x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10)*Log[2]), 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.
| \(\displaystyle \int \frac {e^{\frac {x}{x+1}} \left (x^2+\left (-6 x^2-5 x-5\right ) \log (2)\right )+e^{\frac {2 x}{x+1}} \left (-243 x^6-486 x^5-243 x^4-2 x^2+\left (243 x^6-1134 x^5-2997 x^4-1620 x^3+2 x^2+4 x+2\right ) \log (2)-4 x-2\right )}{\left (x^4+2 x^3+x^2\right ) \log (2)+e^{\frac {x}{x+1}} \left (162 x^7+324 x^6+162 x^5-4 x^3-8 x^2-4 x\right ) \log (2)+e^{\frac {2 x}{x+1}} \left (6561 x^{10}+13122 x^9+6561 x^8-324 x^6-648 x^5-324 x^4+4 x^2+8 x+4\right ) \log (2)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{\frac {x}{x+1}} \left (x^2 (1-6 \log (2))+e^{\frac {x}{x+1}} \left (243 x^6 (\log (2)-1)-162 x^5 (3+\log (128))-81 x^4 (3+37 \log (2))-1620 x^3 \log (2)+x^2 (\log (4)-2)+x (\log (16)-4)-2+\log (4)\right )-5 x \log (2)-5 \log (2)\right )}{(x+1)^2 \left (e^{\frac {x}{x+1}} \left (81 x^4-2\right )+x\right )^2 \log (2)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {e^{\frac {x}{x+1}} \left ((1-6 \log (2)) x^2-5 \log (2) x-e^{\frac {x}{x+1}} \left (243 (1-\log (2)) x^6+162 (3+\log (128)) x^5+81 (3+37 \log (2)) x^4+1620 \log (2) x^3+(2-\log (4)) x^2+(4-\log (16)) x-\log (4)+2\right )-5 \log (2)\right )}{(x+1)^2 \left (x-e^{\frac {x}{x+1}} \left (2-81 x^4\right )\right )^2}dx}{\log (2)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {e^{\frac {x}{x+1}} \left (-243 (1-\log (2)) x^7-567 \left (1+\frac {8 \log (2)}{7}\right ) x^6-243 \left (1+\frac {32 \log (2)}{3}\right ) x^5-1215 \log (2) x^4-2 (1-\log (2)) x^3-2 (1+\log (16)) x^2-2 (1+\log (16)) x-\log (1024)\right )}{(x+1)^2 \left (2-81 x^4\right ) \left (-81 e^{\frac {x}{x+1}} x^4-x+2 e^{\frac {x}{x+1}}\right )^2}+\frac {e^{\frac {x}{x+1}} \left (-243 (1-\log (2)) x^6-486 \left (1+\log \left (4 \sqrt [3]{2}\right )\right ) x^5-243 \left (1+\frac {37 \log (2)}{3}\right ) x^4-1620 \log (2) x^3-2 (1-\log (2)) x^2-4 (1-\log (2)) x-2 (1-\log (2))\right )}{(x+1)^2 \left (2-81 x^4\right ) \left (-81 e^{\frac {x}{x+1}} x^4-x+2 e^{\frac {x}{x+1}}\right )}\right )dx}{\log (2)}\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {\int \left (\frac {e^{\frac {x}{x+1}} \left (-243 (1-\log (2)) x^7-567 \left (1+\frac {8 \log (2)}{7}\right ) x^6-243 \left (1+\frac {32 \log (2)}{3}\right ) x^5-1215 \log (2) x^4-2 (1-\log (2)) x^3-2 (1+\log (16)) x^2-2 (1+\log (16)) x-\log (1024)\right )}{(x+1)^2 \left (2-81 x^4\right ) \left (-81 e^{\frac {x}{x+1}} x^4-x+2 e^{\frac {x}{x+1}}\right )^2}+\frac {e^{\frac {x}{x+1}} \left (-243 (1-\log (2)) x^6-486 \left (1+\log \left (4 \sqrt [3]{2}\right )\right ) x^5-243 \left (1+\frac {37 \log (2)}{3}\right ) x^4-1620 \log (2) x^3-2 (1-\log (2)) x^2-4 (1-\log (2)) x-2 (1-\log (2))\right )}{(x+1)^2 \left (2-81 x^4\right ) \left (-81 e^{\frac {x}{x+1}} x^4-x+2 e^{\frac {x}{x+1}}\right )}\right )dx}{\log (2)}\) |
Input:
Int[(E^(x/(1 + x))*(x^2 + (-5 - 5*x - 6*x^2)*Log[2]) + E^((2*x)/(1 + x))*( -2 - 4*x - 2*x^2 - 243*x^4 - 486*x^5 - 243*x^6 + (2 + 4*x + 2*x^2 - 1620*x ^3 - 2997*x^4 - 1134*x^5 + 243*x^6)*Log[2]))/((x^2 + 2*x^3 + x^4)*Log[2] + E^(x/(1 + x))*(-4*x - 8*x^2 - 4*x^3 + 162*x^5 + 324*x^6 + 162*x^7)*Log[2] + E^((2*x)/(1 + x))*(4 + 8*x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561 *x^8 + 13122*x^9 + 6561*x^10)*Log[2]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(33)=66\).
Time = 2.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.55
| method | result | size |
| risch | \(\frac {81 \left (-\frac {\ln \left (2\right )}{81}+\frac {1}{81}\right ) x +5 \ln \left (2\right )}{\ln \left (2\right ) \left (81 x^{4}-2\right )}+\frac {\left (x \ln \left (2\right )-5 \ln \left (2\right )-x \right ) x}{\ln \left (2\right ) \left (81 x^{4}-2\right ) \left (81 \,{\mathrm e}^{\frac {x}{1+x}} x^{4}-2 \,{\mathrm e}^{\frac {x}{1+x}}+x \right )}\) | \(84\) |
| norman | \(\frac {5 \,{\mathrm e}^{\frac {x}{1+x}}+\frac {\left (1+4 \ln \left (2\right )\right ) x \,{\mathrm e}^{\frac {x}{1+x}}}{\ln \left (2\right )}-\frac {\left (\ln \left (2\right )-1\right ) x^{2} {\mathrm e}^{\frac {x}{1+x}}}{\ln \left (2\right )}}{81 \,{\mathrm e}^{\frac {x}{1+x}} x^{5}+81 \,{\mathrm e}^{\frac {x}{1+x}} x^{4}+x^{2}-2 x \,{\mathrm e}^{\frac {x}{1+x}}+x -2 \,{\mathrm e}^{\frac {x}{1+x}}}\) | \(108\) |
| parallelrisch | \(\frac {1134 \ln \left (2\right ) {\mathrm e}^{\frac {x}{1+x}} x^{4}+486 \,{\mathrm e}^{\frac {x}{1+x}} x^{4}-8 \ln \left (2\right ) {\mathrm e}^{\frac {x}{1+x}} x +14 x \ln \left (2\right )+12 \,{\mathrm e}^{\frac {x}{1+x}} \ln \left (2\right )+8 x \,{\mathrm e}^{\frac {x}{1+x}}+6 x -12 \,{\mathrm e}^{\frac {x}{1+x}}}{8 \left (81 \,{\mathrm e}^{\frac {x}{1+x}} x^{4}-2 \,{\mathrm e}^{\frac {x}{1+x}}+x \right ) \ln \left (2\right )}\) | \(117\) |
Input:
int((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*ln(2)-243*x^6-486*x ^5-243*x^4-2*x^2-4*x-2)*exp(x/(1+x))^2+((-6*x^2-5*x-5)*ln(2)+x^2)*exp(x/(1 +x)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x^2+8*x+4)* ln(2)*exp(x/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*ln(2)*exp(x /(1+x))+(x^4+2*x^3+x^2)*ln(2)),x,method=_RETURNVERBOSE)
Output:
81*((-1/81*ln(2)+1/81)*x+5/81*ln(2))/ln(2)/(81*x^4-2)+(x*ln(2)-5*ln(2)-x)* x/ln(2)/(81*x^4-2)/(81*exp(x/(1+x))*x^4-2*exp(x/(1+x))+x)
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=-\frac {{\left ({\left (x - 5\right )} \log \left (2\right ) - x\right )} e^{\left (\frac {x}{x + 1}\right )}}{{\left (81 \, x^{4} - 2\right )} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right ) + x \log \left (2\right )} \] Input:
integrate((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*log(2)-243*x^ 6-486*x^5-243*x^4-2*x^2-4*x-2)*exp(x/(1+x))^2+((-6*x^2-5*x-5)*log(2)+x^2)* exp(x/(1+x)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x^2 +8*x+4)*log(2)*exp(x/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*lo g(2)*exp(x/(1+x))+(x^4+2*x^3+x^2)*log(2)),x, algorithm="fricas")
Output:
-((x - 5)*log(2) - x)*e^(x/(x + 1))/((81*x^4 - 2)*e^(x/(x + 1))*log(2) + x *log(2))
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (22) = 44\).
Time = 0.89 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.58 \[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=- \frac {x \left (-1 + \log {\left (2 \right )}\right ) - 5 \log {\left (2 \right )}}{81 x^{4} \log {\left (2 \right )} - 2 \log {\left (2 \right )}} + \frac {- x^{2} + x^{2} \log {\left (2 \right )} - 5 x \log {\left (2 \right )}}{81 x^{5} \log {\left (2 \right )} - 2 x \log {\left (2 \right )} + \left (6561 x^{8} \log {\left (2 \right )} - 324 x^{4} \log {\left (2 \right )} + 4 \log {\left (2 \right )}\right ) e^{\frac {x}{x + 1}}} \] Input:
integrate((((243*x**6-1134*x**5-2997*x**4-1620*x**3+2*x**2+4*x+2)*ln(2)-24 3*x**6-486*x**5-243*x**4-2*x**2-4*x-2)*exp(x/(1+x))**2+((-6*x**2-5*x-5)*ln (2)+x**2)*exp(x/(1+x)))/((6561*x**10+13122*x**9+6561*x**8-324*x**6-648*x** 5-324*x**4+4*x**2+8*x+4)*ln(2)*exp(x/(1+x))**2+(162*x**7+324*x**6+162*x**5 -4*x**3-8*x**2-4*x)*ln(2)*exp(x/(1+x))+(x**4+2*x**3+x**2)*ln(2)),x)
Output:
-(x*(-1 + log(2)) - 5*log(2))/(81*x**4*log(2) - 2*log(2)) + (-x**2 + x**2* log(2) - 5*x*log(2))/(81*x**5*log(2) - 2*x*log(2) + (6561*x**8*log(2) - 32 4*x**4*log(2) + 4*log(2))*exp(x/(x + 1)))
Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=-\frac {x {\left (\log \left (2\right ) - 1\right )} e - 5 \, e \log \left (2\right )}{81 \, x^{4} e \log \left (2\right ) + x e^{\left (\frac {1}{x + 1}\right )} \log \left (2\right ) - 2 \, e \log \left (2\right )} \] Input:
integrate((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*log(2)-243*x^ 6-486*x^5-243*x^4-2*x^2-4*x-2)*exp(x/(1+x))^2+((-6*x^2-5*x-5)*log(2)+x^2)* exp(x/(1+x)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x^2 +8*x+4)*log(2)*exp(x/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*lo g(2)*exp(x/(1+x))+(x^4+2*x^3+x^2)*log(2)),x, algorithm="maxima")
Output:
-(x*(log(2) - 1)*e - 5*e*log(2))/(81*x^4*e*log(2) + x*e^(1/(x + 1))*log(2) - 2*e*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (32) = 64\).
Time = 1.66 (sec) , antiderivative size = 356, normalized size of antiderivative = 10.79 \[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=-\frac {\frac {1707 \, x e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{x + 1} - \frac {2679 \, x^{2} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {1865 \, x^{3} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{3}} - 407 \, e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right ) - \frac {87 \, x e^{\left (\frac {x}{x + 1}\right )}}{x + 1} + \frac {249 \, x^{2} e^{\left (\frac {x}{x + 1}\right )}}{{\left (x + 1\right )}^{2}} - \frac {245 \, x^{3} e^{\left (\frac {x}{x + 1}\right )}}{{\left (x + 1\right )}^{3}} + \frac {6 \, x \log \left (2\right )}{x + 1} - \frac {18 \, x^{2} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {18 \, x^{3} \log \left (2\right )}{{\left (x + 1\right )}^{3}} - \frac {6 \, x^{4} \log \left (2\right )}{{\left (x + 1\right )}^{4}} - \frac {x}{x + 1} + \frac {3 \, x^{2}}{{\left (x + 1\right )}^{2}} - \frac {3 \, x^{3}}{{\left (x + 1\right )}^{3}} + \frac {x^{4}}{{\left (x + 1\right )}^{4}} + 2 \, e^{\left (\frac {x}{x + 1}\right )}}{79 \, {\left (\frac {8 \, x e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{x + 1} - \frac {12 \, x^{2} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {8 \, x^{3} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{3}} + \frac {79 \, x^{4} e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right )}{{\left (x + 1\right )}^{4}} - 2 \, e^{\left (\frac {x}{x + 1}\right )} \log \left (2\right ) + \frac {x \log \left (2\right )}{x + 1} - \frac {3 \, x^{2} \log \left (2\right )}{{\left (x + 1\right )}^{2}} + \frac {3 \, x^{3} \log \left (2\right )}{{\left (x + 1\right )}^{3}} - \frac {x^{4} \log \left (2\right )}{{\left (x + 1\right )}^{4}}\right )}} \] Input:
integrate((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*log(2)-243*x^ 6-486*x^5-243*x^4-2*x^2-4*x-2)*exp(x/(1+x))^2+((-6*x^2-5*x-5)*log(2)+x^2)* exp(x/(1+x)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x^2 +8*x+4)*log(2)*exp(x/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*lo g(2)*exp(x/(1+x))+(x^4+2*x^3+x^2)*log(2)),x, algorithm="giac")
Output:
-1/79*(1707*x*e^(x/(x + 1))*log(2)/(x + 1) - 2679*x^2*e^(x/(x + 1))*log(2) /(x + 1)^2 + 1865*x^3*e^(x/(x + 1))*log(2)/(x + 1)^3 - 407*e^(x/(x + 1))*l og(2) - 87*x*e^(x/(x + 1))/(x + 1) + 249*x^2*e^(x/(x + 1))/(x + 1)^2 - 245 *x^3*e^(x/(x + 1))/(x + 1)^3 + 6*x*log(2)/(x + 1) - 18*x^2*log(2)/(x + 1)^ 2 + 18*x^3*log(2)/(x + 1)^3 - 6*x^4*log(2)/(x + 1)^4 - x/(x + 1) + 3*x^2/( x + 1)^2 - 3*x^3/(x + 1)^3 + x^4/(x + 1)^4 + 2*e^(x/(x + 1)))/(8*x*e^(x/(x + 1))*log(2)/(x + 1) - 12*x^2*e^(x/(x + 1))*log(2)/(x + 1)^2 + 8*x^3*e^(x /(x + 1))*log(2)/(x + 1)^3 + 79*x^4*e^(x/(x + 1))*log(2)/(x + 1)^4 - 2*e^( x/(x + 1))*log(2) + x*log(2)/(x + 1) - 3*x^2*log(2)/(x + 1)^2 + 3*x^3*log( 2)/(x + 1)^3 - x^4*log(2)/(x + 1)^4)
Timed out. \[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=\int -\frac {{\mathrm {e}}^{\frac {x}{x+1}}\,\left (\ln \left (2\right )\,\left (6\,x^2+5\,x+5\right )-x^2\right )+{\mathrm {e}}^{\frac {2\,x}{x+1}}\,\left (4\,x-\ln \left (2\right )\,\left (243\,x^6-1134\,x^5-2997\,x^4-1620\,x^3+2\,x^2+4\,x+2\right )+2\,x^2+243\,x^4+486\,x^5+243\,x^6+2\right )}{\ln \left (2\right )\,\left (x^4+2\,x^3+x^2\right )-{\mathrm {e}}^{\frac {x}{x+1}}\,\ln \left (2\right )\,\left (-162\,x^7-324\,x^6-162\,x^5+4\,x^3+8\,x^2+4\,x\right )+{\mathrm {e}}^{\frac {2\,x}{x+1}}\,\ln \left (2\right )\,\left (6561\,x^{10}+13122\,x^9+6561\,x^8-324\,x^6-648\,x^5-324\,x^4+4\,x^2+8\,x+4\right )} \,d x \] Input:
int(-(exp(x/(x + 1))*(log(2)*(5*x + 6*x^2 + 5) - x^2) + exp((2*x)/(x + 1)) *(4*x - log(2)*(4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6 + 2 ) + 2*x^2 + 243*x^4 + 486*x^5 + 243*x^6 + 2))/(log(2)*(x^2 + 2*x^3 + x^4) - exp(x/(x + 1))*log(2)*(4*x + 8*x^2 + 4*x^3 - 162*x^5 - 324*x^6 - 162*x^7 ) + exp((2*x)/(x + 1))*log(2)*(8*x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10 + 4)),x)
Output:
int(-(exp(x/(x + 1))*(log(2)*(5*x + 6*x^2 + 5) - x^2) + exp((2*x)/(x + 1)) *(4*x - log(2)*(4*x + 2*x^2 - 1620*x^3 - 2997*x^4 - 1134*x^5 + 243*x^6 + 2 ) + 2*x^2 + 243*x^4 + 486*x^5 + 243*x^6 + 2))/(log(2)*(x^2 + 2*x^3 + x^4) - exp(x/(x + 1))*log(2)*(4*x + 8*x^2 + 4*x^3 - 162*x^5 - 324*x^6 - 162*x^7 ) + exp((2*x)/(x + 1))*log(2)*(8*x + 4*x^2 - 324*x^4 - 648*x^5 - 324*x^6 + 6561*x^8 + 13122*x^9 + 6561*x^10 + 4)), x)
Time = 0.26 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {x}{1+x}} \left (x^2+\left (-5-5 x-6 x^2\right ) \log (2)\right )+e^{\frac {2 x}{1+x}} \left (-2-4 x-2 x^2-243 x^4-486 x^5-243 x^6+\left (2+4 x+2 x^2-1620 x^3-2997 x^4-1134 x^5+243 x^6\right ) \log (2)\right )}{\left (x^2+2 x^3+x^4\right ) \log (2)+e^{\frac {x}{1+x}} \left (-4 x-8 x^2-4 x^3+162 x^5+324 x^6+162 x^7\right ) \log (2)+e^{\frac {2 x}{1+x}} \left (4+8 x+4 x^2-324 x^4-648 x^5-324 x^6+6561 x^8+13122 x^9+6561 x^{10}\right ) \log (2)} \, dx=\frac {e \left (-\mathrm {log}\left (2\right ) x +5 \,\mathrm {log}\left (2\right )+x \right )}{\mathrm {log}\left (2\right ) \left (e^{\frac {1}{x +1}} x +81 e \,x^{4}-2 e \right )} \] Input:
int((((243*x^6-1134*x^5-2997*x^4-1620*x^3+2*x^2+4*x+2)*log(2)-243*x^6-486* x^5-243*x^4-2*x^2-4*x-2)*exp(x/(1+x))^2+((-6*x^2-5*x-5)*log(2)+x^2)*exp(x/ (1+x)))/((6561*x^10+13122*x^9+6561*x^8-324*x^6-648*x^5-324*x^4+4*x^2+8*x+4 )*log(2)*exp(x/(1+x))^2+(162*x^7+324*x^6+162*x^5-4*x^3-8*x^2-4*x)*log(2)*e xp(x/(1+x))+(x^4+2*x^3+x^2)*log(2)),x)
Output:
(e*( - log(2)*x + 5*log(2) + x))/(log(2)*(e**(1/(x + 1))*x + 81*e*x**4 - 2 *e))