Integrand size = 237, antiderivative size = 27 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\log \left (2+e^{1-e^{x+\frac {x}{e (5+\log (3+x))}}+x}\right ) \] Output:
ln(2+exp(x-exp(x/exp(1)/(ln(3+x)+5)+x)+1))
Time = 0.79 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=-e^{x+\frac {x}{e (5+\log (3+x))}}+\log \left (2 e^{e^{x+\frac {x}{e (5+\log (3+x))}}}+e^{1+x}\right ) \] Input:
Integrate[(E^(1 - E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E*(75 + 25*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2 + E^ ((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x]))*(-15 + E*(-75 - 25*x) - 4*x + (-3 + E*(-30 - 10*x) - x)*Log[3 + x] + E*(-3 - x)*Log[3 + x]^2)))/ (E*(150 + 50*x) + E*(60 + 20*x)*Log[3 + x] + E*(6 + 2*x)*Log[3 + x]^2 + E^ (1 - E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E*(75 + 2 5*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2)),x]
Output:
-E^(x + x/(E*(5 + Log[3 + x]))) + Log[2*E^E^(x + x/(E*(5 + Log[3 + x]))) + E^(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.
| \(\displaystyle \int \frac {\exp \left (-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+1\right ) \left (\left (e (-25 x-75)-4 x+e (-x-3) \log ^2(x+3)+(e (-10 x-30)-x-3) \log (x+3)-15\right ) \exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+e (25 x+75)+e (x+3) \log ^2(x+3)+e (10 x+30) \log (x+3)\right )}{\left (e (25 x+75)+e (x+3) \log ^2(x+3)+e (10 x+30) \log (x+3)\right ) \exp \left (-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+1\right )+e (50 x+150)+e (2 x+6) \log ^2(x+3)+e (20 x+60) \log (x+3)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (-4 \left (1+\frac {25 e}{4}\right ) x-e x \log ^2(x+3)-3 e \log ^2(x+3)-(1+10 e) x \log (x+3)-3 (1+10 e) \log (x+3)-15 (1+5 e)\right ) (x+3)^{\frac {x-\log (x+3)-5}{\log (x+3)+5}} \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+\frac {(1+5 e) x}{e \log (x+3)+5 e}\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {\log ^2(x+3) \exp \left (-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+e^{\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}}+1\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {10 \log (x+3) \exp \left (-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+e^{\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}}+1\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 \exp \left (-\exp \left (\frac {5 e x+x+e x \log (x+3)}{e \log (x+3)+5 e}\right )+x+e^{\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}}+1\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^x \log ^2(x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 (x+3)^{\frac {x+1}{\log (x+3)+5}} \exp \left (x+\frac {(1+5 e) x+5 e}{e (\log (x+3)+5)}\right )}{(\log (x+3)+5)^2 \left (-2 \exp \left (e^{\frac {x}{e (\log (x+3)+5)}+\frac {5 x}{\log (x+3)+5}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )-e^{x+1}\right )}+\frac {(-4 x-15) e^{x+\frac {(1+5 e) x}{e (\log (x+3)+5)}} (x+3)^{\frac {x}{\log (x+3)+5}-1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}-\frac {e^x \log (x+3) \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )+10 (x+3)^{\frac {x}{\log (x+3)+5}+\frac {1}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}+\frac {5}{\log (x+3)+5}\right )-10 e\right )}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}+\frac {25 e^{x+1}}{(\log (x+3)+5)^2 \left (2 \exp \left ((x+3)^{\frac {x}{\log (x+3)+5}} \exp \left (\frac {5 e x}{e \log (x+3)+5 e}+\frac {x}{e \log (x+3)+5 e}\right )\right )+e^{x+1}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (25 e (x+3)-(x+3) \left (e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-e\right ) \log ^2(x+3)+(4 x+15) \left (-e^{\frac {5 e x+x}{e \log (x+3)+5 e}}\right ) (x+3)^{\frac {x}{\log (x+3)+5}}-25 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+\log (x+3)+6}{\log (x+3)+5}}-(x+3) \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}+10 e^{\frac {x+5 e (x+1)}{e (\log (x+3)+5)}} (x+3)^{\frac {x+1}{\log (x+3)+5}}-10 e\right ) \log (x+3)\right )}{(x+3) (\log (x+3)+5)^2 \left (2 \exp \left (e^{\frac {5 e x+x}{e \log (x+3)+5 e}} (x+3)^{\frac {x}{\log (x+3)+5}}\right )+e^{x+1}\right )}dx\) |
Input:
Int[(E^(1 - E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E* (75 + 25*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2 + E^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x]))*(-15 + E*(-75 - 25*x) - 4*x + (-3 + E*(-30 - 10*x) - x)*Log[3 + x] + E*(-3 - x)*Log[3 + x]^2)))/(E*(15 0 + 50*x) + E*(60 + 20*x)*Log[3 + x] + E*(6 + 2*x)*Log[3 + x]^2 + E^(1 - E ^((x + 5*E*x + E*x*Log[3 + x])/(5*E + E*Log[3 + x])) + x)*(E*(75 + 25*x) + E*(30 + 10*x)*Log[3 + x] + E*(3 + x)*Log[3 + x]^2)),x]
Output:
$Aborted
Time = 34.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41
| method | result | size |
| risch | \(-1+\ln \left ({\mathrm e}^{-{\mathrm e}^{\frac {x \left ({\mathrm e} \ln \left (3+x \right )+5 \,{\mathrm e}+1\right ) {\mathrm e}^{-1}}{\ln \left (3+x \right )+5}}+x +1}+2\right )\) | \(38\) |
| parallelrisch | \(\ln \left ({\mathrm e}^{-{\mathrm e}^{\frac {x \left ({\mathrm e} \ln \left (3+x \right )+5 \,{\mathrm e}+1\right ) {\mathrm e}^{-1}}{\ln \left (3+x \right )+5}}+x +1}+2\right )\) | \(38\) |
Input:
int((((-3-x)*exp(1)*ln(3+x)^2+((-10*x-30)*exp(1)-3-x)*ln(3+x)+(-25*x-75)*e xp(1)-4*x-15)*exp((x*exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1) ))+(3+x)*exp(1)*ln(3+x)^2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1))*exp(- exp((x*exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1)/(((3+x )*exp(1)*ln(3+x)^2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1))*exp(-exp((x* exp(1)*ln(3+x)+5*x*exp(1)+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1)+(2*x+6)*exp(1 )*ln(3+x)^2+(20*x+60)*exp(1)*ln(3+x)+(50*x+150)*exp(1)),x,method=_RETURNVE RBOSE)
Output:
-1+ln(exp(-exp(x*(exp(1)*ln(3+x)+5*exp(1)+1)*exp(-1)/(ln(3+x)+5))+x+1)+2)
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\log \left (e^{\left (x - e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e}\right )} + 1\right )} + 2\right ) \] Input:
integrate((((-3-x)*exp(1)*log(3+x)^2+((-10*x-30)*exp(1)-3-x)*log(3+x)+(-25 *x-75)*exp(1)-4*x-15)*exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x )+5*exp(1)))+(3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*e xp(1))*exp(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1) ))+x+1)/(((3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp( 1))*exp(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1)))+ x+1)+(2*x+6)*exp(1)*log(3+x)^2+(20*x+60)*exp(1)*log(3+x)+(50*x+150)*exp(1) ),x, algorithm="fricas")
Output:
log(e^(x - e^((x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e)) + 1) + 2)
Time = 18.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\log {\left (e^{x - e^{\frac {e x \log {\left (x + 3 \right )} + x + 5 e x}{e \log {\left (x + 3 \right )} + 5 e}} + 1} + 2 \right )} \] Input:
integrate((((-3-x)*exp(1)*ln(3+x)**2+((-10*x-30)*exp(1)-3-x)*ln(3+x)+(-25* x-75)*exp(1)-4*x-15)*exp((x*exp(1)*ln(3+x)+5*exp(1)*x+x)/(exp(1)*ln(3+x)+5 *exp(1)))+(3+x)*exp(1)*ln(3+x)**2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1 ))*exp(-exp((x*exp(1)*ln(3+x)+5*exp(1)*x+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1 )/(((3+x)*exp(1)*ln(3+x)**2+(10*x+30)*exp(1)*ln(3+x)+(25*x+75)*exp(1))*exp (-exp((x*exp(1)*ln(3+x)+5*exp(1)*x+x)/(exp(1)*ln(3+x)+5*exp(1)))+x+1)+(2*x +6)*exp(1)*ln(3+x)**2+(20*x+60)*exp(1)*ln(3+x)+(50*x+150)*exp(1)),x)
Output:
log(exp(x - exp((E*x*log(x + 3) + x + 5*E*x)/(E*log(x + 3) + 5*E)) + 1) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.63 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=-e^{\left (\frac {x \log \left (x + 3\right )}{\log \left (x + 3\right ) + 5} + \frac {x}{e \log \left (x + 3\right ) + 5 \, e} + \frac {5 \, x}{\log \left (x + 3\right ) + 5}\right )} + \log \left (\frac {1}{2} \, e^{\left (x + 1\right )} + e^{\left (e^{\left (\frac {x \log \left (x + 3\right )}{\log \left (x + 3\right ) + 5} + \frac {x}{e \log \left (x + 3\right ) + 5 \, e} + \frac {5 \, x}{\log \left (x + 3\right ) + 5}\right )}\right )}\right ) \] Input:
integrate((((-3-x)*exp(1)*log(3+x)^2+((-10*x-30)*exp(1)-3-x)*log(3+x)+(-25 *x-75)*exp(1)-4*x-15)*exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x )+5*exp(1)))+(3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*e xp(1))*exp(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1) ))+x+1)/(((3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp( 1))*exp(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1)))+ x+1)+(2*x+6)*exp(1)*log(3+x)^2+(20*x+60)*exp(1)*log(3+x)+(50*x+150)*exp(1) ),x, algorithm="maxima")
Output:
-e^(x*log(x + 3)/(log(x + 3) + 5) + x/(e*log(x + 3) + 5*e) + 5*x/(log(x + 3) + 5)) + log(1/2*e^(x + 1) + e^(e^(x*log(x + 3)/(log(x + 3) + 5) + x/(e* log(x + 3) + 5*e) + 5*x/(log(x + 3) + 5))))
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (24) = 48\).
Time = 3.27 (sec) , antiderivative size = 329, normalized size of antiderivative = 12.19 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=-\frac {x e \log \left (x + 3\right ) - e \log \left (x + 3\right ) \log \left (e^{\left (\frac {2 \, x e \log \left (x + 3\right ) + 10 \, x e - e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} \log \left (x + 3\right ) + x - 5 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )}}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} + 2 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e}\right )}\right ) + 5 \, x e - 5 \, e \log \left (e^{\left (\frac {2 \, x e \log \left (x + 3\right ) + 10 \, x e - e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} \log \left (x + 3\right ) + x - 5 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )}}{e \log \left (x + 3\right ) + 5 \, e} + 1\right )} + 2 \, e^{\left (\frac {x e \log \left (x + 3\right ) + 5 \, x e + x}{e \log \left (x + 3\right ) + 5 \, e}\right )}\right ) + x}{e \log \left (x + 3\right ) + 5 \, e} \] Input:
integrate((((-3-x)*exp(1)*log(3+x)^2+((-10*x-30)*exp(1)-3-x)*log(3+x)+(-25 *x-75)*exp(1)-4*x-15)*exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x )+5*exp(1)))+(3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*e xp(1))*exp(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1) ))+x+1)/(((3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp( 1))*exp(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1)))+ x+1)+(2*x+6)*exp(1)*log(3+x)^2+(20*x+60)*exp(1)*log(3+x)+(50*x+150)*exp(1) ),x, algorithm="giac")
Output:
-(x*e*log(x + 3) - e*log(x + 3)*log(e^((2*x*e*log(x + 3) + 10*x*e - e^((x* e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e) + 1)*log(x + 3) + x - 5*e^( (x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e) + 1))/(e*log(x + 3) + 5* e) + 1) + 2*e^((x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e))) + 5*x*e - 5*e*log(e^((2*x*e*log(x + 3) + 10*x*e - e^((x*e*log(x + 3) + 5*x*e + x) /(e*log(x + 3) + 5*e) + 1)*log(x + 3) + x - 5*e^((x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e) + 1))/(e*log(x + 3) + 5*e) + 1) + 2*e^((x*e*log(x + 3) + 5*x*e + x)/(e*log(x + 3) + 5*e))) + x)/(e*log(x + 3) + 5*e)
Time = 0.74 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=\ln \left (\mathrm {e}\,{\mathrm {e}}^x\,{\mathrm {e}}^{-{\mathrm {e}}^{\frac {5\,x}{\ln \left (x+3\right )+5}}\,{\mathrm {e}}^{\frac {x}{5\,\mathrm {e}+\ln \left (x+3\right )\,\mathrm {e}}}\,{\left (x+3\right )}^{\frac {x}{\ln \left (x+3\right )+5}}}+2\right ) \] Input:
int((exp(x - exp((x + 5*x*exp(1) + x*log(x + 3)*exp(1))/(5*exp(1) + log(x + 3)*exp(1))) + 1)*(exp(1)*(25*x + 75) - exp((x + 5*x*exp(1) + x*log(x + 3 )*exp(1))/(5*exp(1) + log(x + 3)*exp(1)))*(4*x + log(x + 3)*(x + exp(1)*(1 0*x + 30) + 3) + exp(1)*(25*x + 75) + log(x + 3)^2*exp(1)*(x + 3) + 15) + log(x + 3)*exp(1)*(10*x + 30) + log(x + 3)^2*exp(1)*(x + 3)))/(exp(x - exp ((x + 5*x*exp(1) + x*log(x + 3)*exp(1))/(5*exp(1) + log(x + 3)*exp(1))) + 1)*(exp(1)*(25*x + 75) + log(x + 3)*exp(1)*(10*x + 30) + log(x + 3)^2*exp( 1)*(x + 3)) + exp(1)*(50*x + 150) + log(x + 3)*exp(1)*(20*x + 60) + log(x + 3)^2*exp(1)*(2*x + 6)),x)
Output:
log(exp(1)*exp(x)*exp(-exp((5*x)/(log(x + 3) + 5))*exp(x/(5*exp(1) + log(x + 3)*exp(1)))*(x + 3)^(x/(log(x + 3) + 5))) + 2)
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.59 \[ \int \frac {e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)+e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}} \left (-15+e (-75-25 x)-4 x+(-3+e (-30-10 x)-x) \log (3+x)+e (-3-x) \log ^2(3+x)\right )\right )}{e (150+50 x)+e (60+20 x) \log (3+x)+e (6+2 x) \log ^2(3+x)+e^{1-e^{\frac {x+5 e x+e x \log (3+x)}{5 e+e \log (3+x)}}+x} \left (e (75+25 x)+e (30+10 x) \log (3+x)+e (3+x) \log ^2(3+x)\right )} \, dx=-e^{\frac {\mathrm {log}\left (x +3\right ) e x +5 e x +x}{\mathrm {log}\left (x +3\right ) e +5 e}}+\mathrm {log}\left (2 e^{e^{\frac {\mathrm {log}\left (x +3\right ) e x +5 e x +x}{\mathrm {log}\left (x +3\right ) e +5 e}}}+e^{x} e \right ) \] Input:
int((((-3-x)*exp(1)*log(3+x)^2+((-10*x-30)*exp(1)-3-x)*log(3+x)+(-25*x-75) *exp(1)-4*x-15)*exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*ex p(1)))+(3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp(1)) *exp(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1)))+x+1 )/(((3+x)*exp(1)*log(3+x)^2+(10*x+30)*exp(1)*log(3+x)+(25*x+75)*exp(1))*ex p(-exp((x*exp(1)*log(3+x)+5*exp(1)*x+x)/(exp(1)*log(3+x)+5*exp(1)))+x+1)+( 2*x+6)*exp(1)*log(3+x)^2+(20*x+60)*exp(1)*log(3+x)+(50*x+150)*exp(1)),x)
Output:
- e**((log(x + 3)*e*x + 5*e*x + x)/(log(x + 3)*e + 5*e)) + log(2*e**(e**( (log(x + 3)*e*x + 5*e*x + x)/(log(x + 3)*e + 5*e))) + e**x*e)