Integrand size = 222, antiderivative size = 33 \[ \int \frac {-1250+1300 x-440 x^2+48 x^3+e^{3-x} \left (100-112 x+38 x^2-4 x^3\right )+\left (-250+250 x-80 x^2+8 x^3+e^{3-x} \left (20-18 x+4 x^2\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )}{-625 x+625 x^2-200 x^3+20 x^4+e^{3-x} \left (50 x-45 x^2+10 x^3\right )+\left (-125 x+125 x^2-40 x^3+4 x^4+e^{3-x} \left (10 x-9 x^2+2 x^3\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )} \, dx=2 \log \left (x \left (5+\log \left (5-x+\frac {e^{3-x} (2-x)}{-5+2 x}\right )\right )\right ) \]
Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-1250+1300 x-440 x^2+48 x^3+e^{3-x} \left (100-112 x+38 x^2-4 x^3\right )+\left (-250+250 x-80 x^2+8 x^3+e^{3-x} \left (20-18 x+4 x^2\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )}{-625 x+625 x^2-200 x^3+20 x^4+e^{3-x} \left (50 x-45 x^2+10 x^3\right )+\left (-125 x+125 x^2-40 x^3+4 x^4+e^{3-x} \left (10 x-9 x^2+2 x^3\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )} \, dx=2 \left (\log (x)+\log \left (5+\log \left (-\frac {25+e^{3-x} (-2+x)-15 x+2 x^2}{-5+2 x}\right )\right )\right ) \]
Integrate[(-1250 + 1300*x - 440*x^2 + 48*x^3 + E^(3 - x)*(100 - 112*x + 38 *x^2 - 4*x^3) + (-250 + 250*x - 80*x^2 + 8*x^3 + E^(3 - x)*(20 - 18*x + 4* x^2))*Log[(-25 + E^(3 - x)*(2 - x) + 15*x - 2*x^2)/(-5 + 2*x)])/(-625*x + 625*x^2 - 200*x^3 + 20*x^4 + E^(3 - x)*(50*x - 45*x^2 + 10*x^3) + (-125*x + 125*x^2 - 40*x^3 + 4*x^4 + E^(3 - x)*(10*x - 9*x^2 + 2*x^3))*Log[(-25 + E^(3 - x)*(2 - x) + 15*x - 2*x^2)/(-5 + 2*x)]),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.
| \(\displaystyle \int \frac {48 x^3-440 x^2+e^{3-x} \left (-4 x^3+38 x^2-112 x+100\right )+\left (8 x^3-80 x^2+e^{3-x} \left (4 x^2-18 x+20\right )+250 x-250\right ) \log \left (\frac {-2 x^2+15 x+e^{3-x} (2-x)-25}{2 x-5}\right )+1300 x-1250}{20 x^4-200 x^3+625 x^2+e^{3-x} \left (10 x^3-45 x^2+50 x\right )+\left (4 x^4-40 x^3+125 x^2+e^{3-x} \left (2 x^3-9 x^2+10 x\right )-125 x\right ) \log \left (\frac {-2 x^2+15 x+e^{3-x} (2-x)-25}{2 x-5}\right )-625 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (48 x^3-440 x^2+e^{3-x} \left (-4 x^3+38 x^2-112 x+100\right )+\left (8 x^3-80 x^2+e^{3-x} \left (4 x^2-18 x+20\right )+250 x-250\right ) \log \left (\frac {-2 x^2+15 x+e^{3-x} (2-x)-25}{2 x-5}\right )+1300 x-1250\right )}{(5-2 x) x \left (-2 e^x x^2+15 e^x x-e^3 x-25 e^x+2 e^3\right ) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^x \left (2 x^3-17 x^2+47 x-45\right )}{(x-2) \left (2 e^x x^2-15 e^x x+e^3 x+25 e^x-2 e^3\right ) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}-\frac {2 \left (2 x^3-19 x^2-2 x^2 \log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+9 x \log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )-10 \log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+56 x-50\right )}{(x-2) x (2 x-5) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {1}{\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5}dx+2 \int \frac {1}{(x-2) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}dx-4 \int \frac {1}{(2 x-5) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}dx+42 \int \frac {e^x}{\left (2 e^x x^2-15 e^x x+e^3 x+25 e^x-2 e^3\right ) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}dx-6 \int \frac {e^x}{(x-2) \left (2 e^x x^2-15 e^x x+e^3 x+25 e^x-2 e^3\right ) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}dx-26 \int \frac {e^x x}{\left (2 e^x x^2-15 e^x x+e^3 x+25 e^x-2 e^3\right ) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}dx+4 \int \frac {e^x x^2}{\left (2 e^x x^2-15 e^x x+e^3 x+25 e^x-2 e^3\right ) \left (\log \left (-\frac {2 x^2-15 x+e^{3-x} (x-2)+25}{2 x-5}\right )+5\right )}dx+2 \log (x)\) |
Int[(-1250 + 1300*x - 440*x^2 + 48*x^3 + E^(3 - x)*(100 - 112*x + 38*x^2 - 4*x^3) + (-250 + 250*x - 80*x^2 + 8*x^3 + E^(3 - x)*(20 - 18*x + 4*x^2))* Log[(-25 + E^(3 - x)*(2 - x) + 15*x - 2*x^2)/(-5 + 2*x)])/(-625*x + 625*x^ 2 - 200*x^3 + 20*x^4 + E^(3 - x)*(50*x - 45*x^2 + 10*x^3) + (-125*x + 125* x^2 - 40*x^3 + 4*x^4 + E^(3 - x)*(10*x - 9*x^2 + 2*x^3))*Log[(-25 + E^(3 - x)*(2 - x) + 15*x - 2*x^2)/(-5 + 2*x)]),x]
3.30.26.3.1 Defintions of rubi rules used
Time = 1.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27
| method | result | size |
| norman | \(2 \ln \left (x \right )+2 \ln \left (\ln \left (\frac {\left (2-x \right ) {\mathrm e}^{-x +3}-2 x^{2}+15 x -25}{-5+2 x}\right )+5\right )\) | \(42\) |
| parallelrisch | \(2 \ln \left (x \right )+2 \ln \left (\ln \left (\frac {\left (2-x \right ) {\mathrm e}^{-x +3}-2 x^{2}+15 x -25}{-5+2 x}\right )+5\right )\) | \(42\) |
| risch | \(2 \ln \left (x \right )+2 \ln \left (\ln \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )+\frac {i \left (-2 \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )}{x -\frac {5}{2}}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{x -\frac {5}{2}}\right ) \operatorname {csgn}\left (i \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )}{x -\frac {5}{2}}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x -\frac {5}{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )}{x -\frac {5}{2}}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )}{x -\frac {5}{2}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}+\left (\frac {{\mathrm e}^{-x +3}}{2}-\frac {15}{2}\right ) x -{\mathrm e}^{-x +3}+\frac {25}{2}\right )}{x -\frac {5}{2}}\right )}^{3}+2 \pi +2 i \ln \left (x -\frac {5}{2}\right )-10 i\right )}{2}\right )\) | \(318\) |
int((((4*x^2-18*x+20)*exp(-x+3)+8*x^3-80*x^2+250*x-250)*ln(((2-x)*exp(-x+3 )-2*x^2+15*x-25)/(-5+2*x))+(-4*x^3+38*x^2-112*x+100)*exp(-x+3)+48*x^3-440* x^2+1300*x-1250)/(((2*x^3-9*x^2+10*x)*exp(-x+3)+4*x^4-40*x^3+125*x^2-125*x )*ln(((2-x)*exp(-x+3)-2*x^2+15*x-25)/(-5+2*x))+(10*x^3-45*x^2+50*x)*exp(-x +3)+20*x^4-200*x^3+625*x^2-625*x),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {-1250+1300 x-440 x^2+48 x^3+e^{3-x} \left (100-112 x+38 x^2-4 x^3\right )+\left (-250+250 x-80 x^2+8 x^3+e^{3-x} \left (20-18 x+4 x^2\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )}{-625 x+625 x^2-200 x^3+20 x^4+e^{3-x} \left (50 x-45 x^2+10 x^3\right )+\left (-125 x+125 x^2-40 x^3+4 x^4+e^{3-x} \left (10 x-9 x^2+2 x^3\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )} \, dx=2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (-\frac {2 \, x^{2} + {\left (x - 2\right )} e^{\left (-x + 3\right )} - 15 \, x + 25}{2 \, x - 5}\right ) + 5\right ) \]
integrate((((4*x^2-18*x+20)*exp(-x+3)+8*x^3-80*x^2+250*x-250)*log(((2-x)*e xp(-x+3)-2*x^2+15*x-25)/(-5+2*x))+(-4*x^3+38*x^2-112*x+100)*exp(-x+3)+48*x ^3-440*x^2+1300*x-1250)/(((2*x^3-9*x^2+10*x)*exp(-x+3)+4*x^4-40*x^3+125*x^ 2-125*x)*log(((2-x)*exp(-x+3)-2*x^2+15*x-25)/(-5+2*x))+(10*x^3-45*x^2+50*x )*exp(-x+3)+20*x^4-200*x^3+625*x^2-625*x),x, algorithm=\
Time = 0.53 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {-1250+1300 x-440 x^2+48 x^3+e^{3-x} \left (100-112 x+38 x^2-4 x^3\right )+\left (-250+250 x-80 x^2+8 x^3+e^{3-x} \left (20-18 x+4 x^2\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )}{-625 x+625 x^2-200 x^3+20 x^4+e^{3-x} \left (50 x-45 x^2+10 x^3\right )+\left (-125 x+125 x^2-40 x^3+4 x^4+e^{3-x} \left (10 x-9 x^2+2 x^3\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )} \, dx=2 \log {\left (x \right )} + 2 \log {\left (\log {\left (\frac {- 2 x^{2} + 15 x + \left (2 - x\right ) e^{3 - x} - 25}{2 x - 5} \right )} + 5 \right )} \]
integrate((((4*x**2-18*x+20)*exp(-x+3)+8*x**3-80*x**2+250*x-250)*ln(((2-x) *exp(-x+3)-2*x**2+15*x-25)/(-5+2*x))+(-4*x**3+38*x**2-112*x+100)*exp(-x+3) +48*x**3-440*x**2+1300*x-1250)/(((2*x**3-9*x**2+10*x)*exp(-x+3)+4*x**4-40* x**3+125*x**2-125*x)*ln(((2-x)*exp(-x+3)-2*x**2+15*x-25)/(-5+2*x))+(10*x** 3-45*x**2+50*x)*exp(-x+3)+20*x**4-200*x**3+625*x**2-625*x),x)
Time = 0.45 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-1250+1300 x-440 x^2+48 x^3+e^{3-x} \left (100-112 x+38 x^2-4 x^3\right )+\left (-250+250 x-80 x^2+8 x^3+e^{3-x} \left (20-18 x+4 x^2\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )}{-625 x+625 x^2-200 x^3+20 x^4+e^{3-x} \left (50 x-45 x^2+10 x^3\right )+\left (-125 x+125 x^2-40 x^3+4 x^4+e^{3-x} \left (10 x-9 x^2+2 x^3\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )} \, dx=2 \, \log \left (x\right ) + 2 \, \log \left (-x + \log \left (-x e^{3} - {\left (2 \, x^{2} - 15 \, x + 25\right )} e^{x} + 2 \, e^{3}\right ) - \log \left (2 \, x - 5\right ) + 5\right ) \]
integrate((((4*x^2-18*x+20)*exp(-x+3)+8*x^3-80*x^2+250*x-250)*log(((2-x)*e xp(-x+3)-2*x^2+15*x-25)/(-5+2*x))+(-4*x^3+38*x^2-112*x+100)*exp(-x+3)+48*x ^3-440*x^2+1300*x-1250)/(((2*x^3-9*x^2+10*x)*exp(-x+3)+4*x^4-40*x^3+125*x^ 2-125*x)*log(((2-x)*exp(-x+3)-2*x^2+15*x-25)/(-5+2*x))+(10*x^3-45*x^2+50*x )*exp(-x+3)+20*x^4-200*x^3+625*x^2-625*x),x, algorithm=\
Time = 0.63 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {-1250+1300 x-440 x^2+48 x^3+e^{3-x} \left (100-112 x+38 x^2-4 x^3\right )+\left (-250+250 x-80 x^2+8 x^3+e^{3-x} \left (20-18 x+4 x^2\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )}{-625 x+625 x^2-200 x^3+20 x^4+e^{3-x} \left (50 x-45 x^2+10 x^3\right )+\left (-125 x+125 x^2-40 x^3+4 x^4+e^{3-x} \left (10 x-9 x^2+2 x^3\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )} \, dx=2 \, \log \left (x\right ) + 2 \, \log \left (\log \left (-\frac {2 \, x^{2} + x e^{\left (-x + 3\right )} - 15 \, x - 2 \, e^{\left (-x + 3\right )} + 25}{2 \, x - 5}\right ) + 5\right ) \]
integrate((((4*x^2-18*x+20)*exp(-x+3)+8*x^3-80*x^2+250*x-250)*log(((2-x)*e xp(-x+3)-2*x^2+15*x-25)/(-5+2*x))+(-4*x^3+38*x^2-112*x+100)*exp(-x+3)+48*x ^3-440*x^2+1300*x-1250)/(((2*x^3-9*x^2+10*x)*exp(-x+3)+4*x^4-40*x^3+125*x^ 2-125*x)*log(((2-x)*exp(-x+3)-2*x^2+15*x-25)/(-5+2*x))+(10*x^3-45*x^2+50*x )*exp(-x+3)+20*x^4-200*x^3+625*x^2-625*x),x, algorithm=\
Time = 13.57 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \[ \int \frac {-1250+1300 x-440 x^2+48 x^3+e^{3-x} \left (100-112 x+38 x^2-4 x^3\right )+\left (-250+250 x-80 x^2+8 x^3+e^{3-x} \left (20-18 x+4 x^2\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )}{-625 x+625 x^2-200 x^3+20 x^4+e^{3-x} \left (50 x-45 x^2+10 x^3\right )+\left (-125 x+125 x^2-40 x^3+4 x^4+e^{3-x} \left (10 x-9 x^2+2 x^3\right )\right ) \log \left (\frac {-25+e^{3-x} (2-x)+15 x-2 x^2}{-5+2 x}\right )} \, dx=2\,\ln \left (\ln \left (-\frac {2\,x^2-15\,x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^3\,\left (x-2\right )+25}{2\,x-5}\right )+5\right )+2\,\ln \left (x\right ) \]
int((1300*x + log(-(exp(3 - x)*(x - 2) - 15*x + 2*x^2 + 25)/(2*x - 5))*(25 0*x + exp(3 - x)*(4*x^2 - 18*x + 20) - 80*x^2 + 8*x^3 - 250) - exp(3 - x)* (112*x - 38*x^2 + 4*x^3 - 100) - 440*x^2 + 48*x^3 - 1250)/(log(-(exp(3 - x )*(x - 2) - 15*x + 2*x^2 + 25)/(2*x - 5))*(exp(3 - x)*(10*x - 9*x^2 + 2*x^ 3) - 125*x + 125*x^2 - 40*x^3 + 4*x^4) - 625*x + exp(3 - x)*(50*x - 45*x^2 + 10*x^3) + 625*x^2 - 200*x^3 + 20*x^4),x)