Integrand size = 359, antiderivative size = 33 \[ \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx=\frac {x-\log (x)}{x \left (e^5+x\right ) \log (5) \left (5+e^x+2 x+\log (5)\right )} \]
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx=-\frac {-x+\log (x)}{x \left (e^5+x\right ) \log (5) \left (5+e^x+2 x+\log (5)\right )} \]
Integrate[(-5*x - 7*x^2 - 4*x^3 + E^5*(-5 - 2*x - 2*x^2) + E^x*(-x - x^2 - x^3 + E^5*(-1 - x^2)) + (-E^5 - x - x^2)*Log[5] + (10*x + 6*x^2 + E^5*(5 + 4*x) + E^x*(2*x + x^2 + E^5*(1 + x)) + (E^5 + 2*x)*Log[5])*Log[x])/(E^(2 *x)*(E^10*x^2 + 2*E^5*x^3 + x^4)*Log[5] + (25*x^4 + 20*x^5 + 4*x^6 + E^10* (25*x^2 + 20*x^3 + 4*x^4) + E^5*(50*x^3 + 40*x^4 + 8*x^5))*Log[5] + (10*x^ 4 + 4*x^5 + E^10*(10*x^2 + 4*x^3) + E^5*(20*x^3 + 8*x^4))*Log[5]^2 + (E^10 *x^2 + 2*E^5*x^3 + x^4)*Log[5]^3 + E^x*((10*x^4 + 4*x^5 + E^10*(10*x^2 + 4 *x^3) + E^5*(20*x^3 + 8*x^4))*Log[5] + (2*E^10*x^2 + 4*E^5*x^3 + 2*x^4)*Lo g[5]^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.
| \(\displaystyle \int \frac {-4 x^3-7 x^2+e^5 \left (-2 x^2-2 x-5\right )+\left (6 x^2+e^x \left (x^2+2 x+e^5 (x+1)\right )+10 x+e^5 (4 x+5)+\left (2 x+e^5\right ) \log (5)\right ) \log (x)+\left (-x^2-x-e^5\right ) \log (5)+e^x \left (-x^3-x^2+e^5 \left (-x^2-1\right )-x\right )-5 x}{\left (x^4+2 e^5 x^3+e^{10} x^2\right ) \log ^3(5)+e^{2 x} \left (x^4+2 e^5 x^3+e^{10} x^2\right ) \log (5)+e^x \left (\left (2 x^4+4 e^5 x^3+2 e^{10} x^2\right ) \log ^2(5)+\left (4 x^5+10 x^4+e^5 \left (8 x^4+20 x^3\right )+e^{10} \left (4 x^3+10 x^2\right )\right ) \log (5)\right )+\left (4 x^5+10 x^4+e^5 \left (8 x^4+20 x^3\right )+e^{10} \left (4 x^3+10 x^2\right )\right ) \log ^2(5)+\left (4 x^6+20 x^5+25 x^4+e^5 \left (8 x^5+40 x^4+50 x^3\right )+e^{10} \left (4 x^4+20 x^3+25 x^2\right )\right ) \log (5)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-e^{x+5} \left (x^2+1\right )-e^x x \left (x^2+x+1\right )-e^5 \left (2 x^2+2 x+5+\log (5)\right )-x \left (4 x^2+x (7+\log (5))+5+\log (5)\right )+\left (e^{x+5} (x+1)+e^x x (x+2)+2 x (3 x+5+\log (5))+e^5 (4 x+5+\log (5))\right ) \log (x)}{x^2 \left (x+e^5\right )^2 \log (5) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {e^{x+5} \left (x^2+1\right )+e^x x \left (x^2+x+1\right )+e^5 \left (2 x^2+2 x+\log (5)+5\right )+x \left (4 x^2+(7+\log (5)) x+\log (5)+5\right )-\left (e^{x+5} (x+1)+e^x x (x+2)+2 x (3 x+\log (5)+5)+e^5 (4 x+\log (5)+5)\right ) \log (x)}{x^2 \left (x+e^5\right )^2 \left (2 x+e^x+\log (5)+5\right )^2}dx}{\log (5)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {e^{x+5} \left (x^2+1\right )+e^x x \left (x^2+x+1\right )+e^5 \left (2 x^2+2 x+\log (5)+5\right )+x \left (4 x^2+(7+\log (5)) x+\log (5)+5\right )-\left (e^{x+5} (x+1)+e^x x (x+2)+2 x (3 x+\log (5)+5)+e^5 (4 x+\log (5)+5)\right ) \log (x)}{x^2 \left (x+e^5\right )^2 \left (2 x+e^x+\log (5)+5\right )^2}dx}{\log (5)}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {\int \frac {e^{x+5} \left (x^2+1\right )+e^x x \left (x^2+x+1\right )+e^5 \left (2 x^2+2 x+\log (5)+5\right )+x \left (4 x^2+(7+\log (5)) x+\log (5)+5\right )-\left (e^{x+5} (x+1)+e^x x (x+2)+2 x (3 x+\log (5)+5)+e^5 (4 x+\log (5)+5)\right ) \log (x)}{x^2 \left (x+e^5\right )^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx}{\log (5)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (\frac {(2 x+\log (5)+3) (\log (x)-x)}{x \left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}+\frac {x^3-\log (x) x^2+\left (1+e^5\right ) x^2-2 \left (1+\frac {e^5}{2}\right ) \log (x) x+x-e^5 \log (x)+e^5}{x^2 \left (x+e^5\right )^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}\right )dx}{\log (5)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {2 \log (x) \int \frac {1}{-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{10}}+\frac {2 \left (1+e^5\right ) \int \frac {1}{-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{10}}+\frac {2 \int \frac {1}{-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^5}+\frac {2 \int \frac {1}{-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{15}}+\frac {\log (x) \int \frac {1}{x^2 \left (-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}+\frac {2 \int \frac {1}{x \left (-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+\frac {2 \log (x) \int \frac {x}{-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{15}}+\frac {2 \int \frac {x}{-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{10}}+\frac {2 \int \frac {x^2}{-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{15}}-\frac {\left (2+e^5\right ) \log (x) \int \frac {1}{\left (x+e^5\right )^2 \left (-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}+\frac {\log (x) \int \frac {1}{\left (x+e^5\right )^2 \left (-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}+\log (x) \int \frac {1}{\left (x+e^5\right )^2 \left (-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )\right )}dx+e^5 \int \frac {1}{\left (x+e^5\right )^2 \left (-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )\right )}dx+\frac {\int \frac {1}{\left (x+e^5\right )^2 \left (-2 x-e^x-5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}+\frac {\left (3-2 e^5+\log (5)\right ) \int \frac {1}{\left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx}{e^5}-\frac {(3+\log (5)) \int \frac {1}{\left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx}{e^5}+\frac {\left (3-2 e^5+\log (5)\right ) \log (x) \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx}{e^5}+\left (3-2 e^5+\log (5)\right ) \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx+\frac {(3+\log (5)) \log (x) \int \frac {1}{x \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx}{e^5}+\frac {2 \log (x) \int \frac {1}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{10}}+\frac {2 \left (1+e^5\right ) \int \frac {1}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{10}}+\frac {2 \int \frac {1}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^5}+\frac {2 \int \frac {1}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{15}}-\frac {2 \left (2+e^5\right ) \log (x) \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+\frac {2 \log (x) \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}+\frac {2 \log (x) \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+\frac {2 \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+2 \left (1+\frac {1}{e^5}\right ) \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx+2 \int \frac {1}{\left (-x-e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx+\frac {\int \frac {1}{x^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}-\frac {\left (2+e^5\right ) \log (x) \int \frac {1}{x \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+\frac {2 \log (x) \int \frac {1}{x \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+\frac {\int \frac {1}{x \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+\frac {2 \log (x) \int \frac {x}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{15}}+\frac {2 \int \frac {x}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{10}}+\frac {2 \int \frac {x^2}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{e^{15}}+\left (1+e^5\right ) \int \frac {1}{\left (x+e^5\right )^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx+\frac {\int \frac {1}{\left (x+e^5\right )^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}-\frac {\left (2+e^5\right ) \log (x) \int \frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+\frac {2 \log (x) \int \frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^5}+\frac {3 \int \frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{e^{10}}+2 \left (1+\frac {1}{e^5}\right ) \int \frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx+3 \int \frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx-\frac {(3+\log (5)) \int \frac {\int \frac {1}{x \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx}{x}dx}{e^5}-\frac {\left (3-2 e^5+\log (5)\right ) \int \frac {\int -\frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )^2}dx}{x}dx}{e^5}-\frac {2 \int \frac {\int -\frac {1}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{x}dx}{e^{10}}-\frac {2 \int \frac {\int \frac {1}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{x}dx}{e^{10}}-\frac {\int \frac {\int -\frac {1}{x^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^5}+\frac {\left (2+e^5\right ) \int \frac {\int \frac {1}{x \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^{10}}-\frac {2 \int \frac {\int \frac {1}{x \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^{10}}-\frac {2 \int \frac {\int -\frac {x}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{x}dx}{e^{15}}-\frac {2 \int \frac {\int \frac {x}{2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )}dx}{x}dx}{e^{15}}+\frac {\left (2+e^5\right ) \int \frac {\int -\frac {1}{\left (x+e^5\right )^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^5}-\frac {\int \frac {\int -\frac {1}{\left (x+e^5\right )^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^5}-\int \frac {\int -\frac {1}{\left (x+e^5\right )^2 \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx+\frac {2 \left (2+e^5\right ) \int \frac {\int -\frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^{10}}-\frac {2 \int \frac {\int -\frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^5}-\frac {2 \int \frac {\int -\frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^{10}}+\frac {\left (2+e^5\right ) \int \frac {\int \frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^{10}}-\frac {2 \int \frac {\int \frac {1}{\left (x+e^5\right ) \left (2 x+e^x+5 \left (1+\frac {\log (5)}{5}\right )\right )}dx}{x}dx}{e^5}}{\log (5)}\) |
Int[(-5*x - 7*x^2 - 4*x^3 + E^5*(-5 - 2*x - 2*x^2) + E^x*(-x - x^2 - x^3 + E^5*(-1 - x^2)) + (-E^5 - x - x^2)*Log[5] + (10*x + 6*x^2 + E^5*(5 + 4*x) + E^x*(2*x + x^2 + E^5*(1 + x)) + (E^5 + 2*x)*Log[5])*Log[x])/(E^(2*x)*(E ^10*x^2 + 2*E^5*x^3 + x^4)*Log[5] + (25*x^4 + 20*x^5 + 4*x^6 + E^10*(25*x^ 2 + 20*x^3 + 4*x^4) + E^5*(50*x^3 + 40*x^4 + 8*x^5))*Log[5] + (10*x^4 + 4* x^5 + E^10*(10*x^2 + 4*x^3) + E^5*(20*x^3 + 8*x^4))*Log[5]^2 + (E^10*x^2 + 2*E^5*x^3 + x^4)*Log[5]^3 + E^x*((10*x^4 + 4*x^5 + E^10*(10*x^2 + 4*x^3) + E^5*(20*x^3 + 8*x^4))*Log[5] + (2*E^10*x^2 + 4*E^5*x^3 + 2*x^4)*Log[5]^2 )),x]
3.30.54.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.68 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58
| method | result | size |
| risch | \(-\frac {\ln \left (x \right )}{x \ln \left (5\right ) \left ({\mathrm e}^{5}+x \right ) \left (2 x +{\mathrm e}^{x}+\ln \left (5\right )+5\right )}+\frac {1}{\ln \left (5\right ) \left ({\mathrm e}^{5}+x \right ) \left (2 x +{\mathrm e}^{x}+\ln \left (5\right )+5\right )}\) | \(52\) |
| parallelrisch | \(\frac {x -\ln \left (x \right )}{x \ln \left (5\right ) \left ({\mathrm e}^{5} {\mathrm e}^{x}+{\mathrm e}^{5} \ln \left (5\right )+2 x \,{\mathrm e}^{5}+{\mathrm e}^{x} x +x \ln \left (5\right )+2 x^{2}+5 \,{\mathrm e}^{5}+5 x \right )}\) | \(53\) |
int(((((1+x)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*ln(5)+(5+4*x)*exp(5)+6*x^ 2+10*x)*ln(x)+((-x^2-1)*exp(5)-x^3-x^2-x)*exp(x)+(-exp(5)-x^2-x)*ln(5)+(-2 *x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)+x^4)*ln(5) *exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*ln(5)^2+((4*x^3+10*x^2)*exp (5)^2+(8*x^4+20*x^3)*exp(5)+4*x^5+10*x^4)*ln(5))*exp(x)+(x^2*exp(5)^2+2*x^ 3*exp(5)+x^4)*ln(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3)*exp(5)+4*x^5 +10*x^4)*ln(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x^4+50*x^3)*exp (5)+4*x^6+20*x^5+25*x^4)*ln(5)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx=\frac {x - \log \left (x\right )}{{\left (x^{2} + x e^{5}\right )} e^{x} \log \left (5\right ) + {\left (x^{2} + x e^{5}\right )} \log \left (5\right )^{2} + {\left (2 \, x^{3} + 5 \, x^{2} + {\left (2 \, x^{2} + 5 \, x\right )} e^{5}\right )} \log \left (5\right )} \]
integrate(((((1+x)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*log(5)+(5+4*x)*exp( 5)+6*x^2+10*x)*log(x)+((-x^2-1)*exp(5)-x^3-x^2-x)*exp(x)+(-exp(5)-x^2-x)*l og(5)+(-2*x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)+x ^4)*log(5)*exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*log(5)^2+((4*x^3+ 10*x^2)*exp(5)^2+(8*x^4+20*x^3)*exp(5)+4*x^5+10*x^4)*log(5))*exp(x)+(x^2*e xp(5)^2+2*x^3*exp(5)+x^4)*log(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3) *exp(5)+4*x^5+10*x^4)*log(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x ^4+50*x^3)*exp(5)+4*x^6+20*x^5+25*x^4)*log(5)),x, algorithm=\
(x - log(x))/((x^2 + x*e^5)*e^x*log(5) + (x^2 + x*e^5)*log(5)^2 + (2*x^3 + 5*x^2 + (2*x^2 + 5*x)*e^5)*log(5))
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (26) = 52\).
Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx=\frac {x - \log {\left (x \right )}}{2 x^{3} \log {\left (5 \right )} + x^{2} \log {\left (5 \right )}^{2} + 5 x^{2} \log {\left (5 \right )} + 2 x^{2} e^{5} \log {\left (5 \right )} + x e^{5} \log {\left (5 \right )}^{2} + 5 x e^{5} \log {\left (5 \right )} + \left (x^{2} \log {\left (5 \right )} + x e^{5} \log {\left (5 \right )}\right ) e^{x}} \]
integrate(((((1+x)*exp(5)+x**2+2*x)*exp(x)+(2*x+exp(5))*ln(5)+(5+4*x)*exp( 5)+6*x**2+10*x)*ln(x)+((-x**2-1)*exp(5)-x**3-x**2-x)*exp(x)+(-exp(5)-x**2- x)*ln(5)+(-2*x**2-2*x-5)*exp(5)-4*x**3-7*x**2-5*x)/((x**2*exp(5)**2+2*x**3 *exp(5)+x**4)*ln(5)*exp(x)**2+((2*x**2*exp(5)**2+4*x**3*exp(5)+2*x**4)*ln( 5)**2+((4*x**3+10*x**2)*exp(5)**2+(8*x**4+20*x**3)*exp(5)+4*x**5+10*x**4)* ln(5))*exp(x)+(x**2*exp(5)**2+2*x**3*exp(5)+x**4)*ln(5)**3+((4*x**3+10*x** 2)*exp(5)**2+(8*x**4+20*x**3)*exp(5)+4*x**5+10*x**4)*ln(5)**2+((4*x**4+20* x**3+25*x**2)*exp(5)**2+(8*x**5+40*x**4+50*x**3)*exp(5)+4*x**6+20*x**5+25* x**4)*ln(5)),x)
(x - log(x))/(2*x**3*log(5) + x**2*log(5)**2 + 5*x**2*log(5) + 2*x**2*exp( 5)*log(5) + x*exp(5)*log(5)**2 + 5*x*exp(5)*log(5) + (x**2*log(5) + x*exp( 5)*log(5))*exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (31) = 62\).
Time = 0.55 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.97 \[ \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx=\frac {x - \log \left (x\right )}{2 \, x^{3} \log \left (5\right ) + {\left (2 \, e^{5} \log \left (5\right ) + \log \left (5\right )^{2} + 5 \, \log \left (5\right )\right )} x^{2} + {\left (\log \left (5\right )^{2} + 5 \, \log \left (5\right )\right )} x e^{5} + {\left (x^{2} \log \left (5\right ) + x e^{5} \log \left (5\right )\right )} e^{x}} \]
integrate(((((1+x)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*log(5)+(5+4*x)*exp( 5)+6*x^2+10*x)*log(x)+((-x^2-1)*exp(5)-x^3-x^2-x)*exp(x)+(-exp(5)-x^2-x)*l og(5)+(-2*x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)+x ^4)*log(5)*exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*log(5)^2+((4*x^3+ 10*x^2)*exp(5)^2+(8*x^4+20*x^3)*exp(5)+4*x^5+10*x^4)*log(5))*exp(x)+(x^2*e xp(5)^2+2*x^3*exp(5)+x^4)*log(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3) *exp(5)+4*x^5+10*x^4)*log(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x ^4+50*x^3)*exp(5)+4*x^6+20*x^5+25*x^4)*log(5)),x, algorithm=\
(x - log(x))/(2*x^3*log(5) + (2*e^5*log(5) + log(5)^2 + 5*log(5))*x^2 + (l og(5)^2 + 5*log(5))*x*e^5 + (x^2*log(5) + x*e^5*log(5))*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (31) = 62\).
Time = 0.40 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx=\frac {x - \log \left (x\right )}{2 \, x^{3} \log \left (5\right ) + 2 \, x^{2} e^{5} \log \left (5\right ) + x^{2} e^{x} \log \left (5\right ) + x^{2} \log \left (5\right )^{2} + x e^{5} \log \left (5\right )^{2} + 5 \, x^{2} \log \left (5\right ) + 5 \, x e^{5} \log \left (5\right ) + x e^{\left (x + 5\right )} \log \left (5\right )} \]
integrate(((((1+x)*exp(5)+x^2+2*x)*exp(x)+(2*x+exp(5))*log(5)+(5+4*x)*exp( 5)+6*x^2+10*x)*log(x)+((-x^2-1)*exp(5)-x^3-x^2-x)*exp(x)+(-exp(5)-x^2-x)*l og(5)+(-2*x^2-2*x-5)*exp(5)-4*x^3-7*x^2-5*x)/((x^2*exp(5)^2+2*x^3*exp(5)+x ^4)*log(5)*exp(x)^2+((2*x^2*exp(5)^2+4*x^3*exp(5)+2*x^4)*log(5)^2+((4*x^3+ 10*x^2)*exp(5)^2+(8*x^4+20*x^3)*exp(5)+4*x^5+10*x^4)*log(5))*exp(x)+(x^2*e xp(5)^2+2*x^3*exp(5)+x^4)*log(5)^3+((4*x^3+10*x^2)*exp(5)^2+(8*x^4+20*x^3) *exp(5)+4*x^5+10*x^4)*log(5)^2+((4*x^4+20*x^3+25*x^2)*exp(5)^2+(8*x^5+40*x ^4+50*x^3)*exp(5)+4*x^6+20*x^5+25*x^4)*log(5)),x, algorithm=\
(x - log(x))/(2*x^3*log(5) + 2*x^2*e^5*log(5) + x^2*e^x*log(5) + x^2*log(5 )^2 + x*e^5*log(5)^2 + 5*x^2*log(5) + 5*x*e^5*log(5) + x*e^(x + 5)*log(5))
Time = 11.82 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-5 x-7 x^2-4 x^3+e^5 \left (-5-2 x-2 x^2\right )+e^x \left (-x-x^2-x^3+e^5 \left (-1-x^2\right )\right )+\left (-e^5-x-x^2\right ) \log (5)+\left (10 x+6 x^2+e^5 (5+4 x)+e^x \left (2 x+x^2+e^5 (1+x)\right )+\left (e^5+2 x\right ) \log (5)\right ) \log (x)}{e^{2 x} \left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log (5)+\left (25 x^4+20 x^5+4 x^6+e^{10} \left (25 x^2+20 x^3+4 x^4\right )+e^5 \left (50 x^3+40 x^4+8 x^5\right )\right ) \log (5)+\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log ^2(5)+\left (e^{10} x^2+2 e^5 x^3+x^4\right ) \log ^3(5)+e^x \left (\left (10 x^4+4 x^5+e^{10} \left (10 x^2+4 x^3\right )+e^5 \left (20 x^3+8 x^4\right )\right ) \log (5)+\left (2 e^{10} x^2+4 e^5 x^3+2 x^4\right ) \log ^2(5)\right )} \, dx=\frac {x-\ln \left (x\right )}{x\,\ln \left (5\right )\,\left (x+{\mathrm {e}}^5\right )\,\left (2\,x+\ln \left (5\right )+{\mathrm {e}}^x+5\right )} \]
int(-(5*x + exp(5)*(2*x + 2*x^2 + 5) + exp(x)*(x + x^2 + x^3 + exp(5)*(x^2 + 1)) + 7*x^2 + 4*x^3 + log(5)*(x + exp(5) + x^2) - log(x)*(10*x + log(5) *(2*x + exp(5)) + 6*x^2 + exp(x)*(2*x + exp(5)*(x + 1) + x^2) + exp(5)*(4* x + 5)))/(log(5)^2*(exp(10)*(10*x^2 + 4*x^3) + exp(5)*(20*x^3 + 8*x^4) + 1 0*x^4 + 4*x^5) + exp(x)*(log(5)*(exp(10)*(10*x^2 + 4*x^3) + exp(5)*(20*x^3 + 8*x^4) + 10*x^4 + 4*x^5) + log(5)^2*(4*x^3*exp(5) + 2*x^2*exp(10) + 2*x ^4)) + log(5)*(exp(10)*(25*x^2 + 20*x^3 + 4*x^4) + exp(5)*(50*x^3 + 40*x^4 + 8*x^5) + 25*x^4 + 20*x^5 + 4*x^6) + log(5)^3*(2*x^3*exp(5) + x^2*exp(10 ) + x^4) + exp(2*x)*log(5)*(2*x^3*exp(5) + x^2*exp(10) + x^4)),x)