3.14.0 ∫ 100+40x+4x2+(−50−40x−6x2) log(x)+(10x+2x2) log2(x)−4x log3(x)+(−4+2x) log4(x)+2 log5(x)+((−100−20x) log(x)−20 log3(x)) log(−2+log(x))+(−200−40x+(100+40x) log(x)−10x log2(x)) log2(−2+log(x))+100 log(x) log3(−2+log(x))+(100−50 log(x)) log4(−2+log(x)) −2x log3(x)+x log4(x) dx [1331]

Integrand size = 151, antiderivative size = 29 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\left (\log (x)+\frac {x+\frac {5 \left (x-x \log ^2(-2+\log (x))\right )}{x}}{\log (x)}\right )^2 \]

Rubi [F]

\[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx \]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-100-40 x-4 x^2-\left (-50-40 x-6 x^2\right ) \log (x)-\left (10 x+2 x^2\right ) \log ^2(x)+4 x \log ^3(x)-(-4+2 x) \log ^4(x)-2 \log ^5(x)-\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))-\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))-100 \log (x) \log ^3(-2+\log (x))-(100-50 \log (x)) \log ^4(-2+\log (x))}{x (2-\log (x)) \log ^3(x)} \, dx \\ & = \int \left (-\frac {4}{-2+\log (x)}+\frac {40}{(-2+\log (x)) \log ^3(x)}+\frac {100}{x (-2+\log (x)) \log ^3(x)}+\frac {4 x}{(-2+\log (x)) \log ^3(x)}-\frac {2 (5+x) (5+3 x)}{x (-2+\log (x)) \log ^2(x)}+\frac {2 (5+x)}{(-2+\log (x)) \log (x)}+\frac {2 (-2+x) \log (x)}{x (-2+\log (x))}+\frac {2 \log ^2(x)}{x (-2+\log (x))}-\frac {20 \left (5+x+\log ^2(x)\right ) \log (-2+\log (x))}{x (-2+\log (x)) \log ^2(x)}-\frac {10 (-10-2 x+x \log (x)) \log ^2(-2+\log (x))}{x \log ^3(x)}+\frac {100 \log ^3(-2+\log (x))}{x (-2+\log (x)) \log ^2(x)}-\frac {50 \log ^4(-2+\log (x))}{x \log ^3(x)}\right ) \, dx \\ & = -\left (2 \int \frac {(5+x) (5+3 x)}{x (-2+\log (x)) \log ^2(x)} \, dx\right )+2 \int \frac {5+x}{(-2+\log (x)) \log (x)} \, dx+2 \int \frac {(-2+x) \log (x)}{x (-2+\log (x))} \, dx+2 \int \frac {\log ^2(x)}{x (-2+\log (x))} \, dx-4 \int \frac {1}{-2+\log (x)} \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx-10 \int \frac {(-10-2 x+x \log (x)) \log ^2(-2+\log (x))}{x \log ^3(x)} \, dx-20 \int \frac {\left (5+x+\log ^2(x)\right ) \log (-2+\log (x))}{x (-2+\log (x)) \log ^2(x)} \, dx+40 \int \frac {1}{(-2+\log (x)) \log ^3(x)} \, dx-50 \int \frac {\log ^4(-2+\log (x))}{x \log ^3(x)} \, dx+100 \int \frac {1}{x (-2+\log (x)) \log ^3(x)} \, dx+100 \int \frac {\log ^3(-2+\log (x))}{x (-2+\log (x)) \log ^2(x)} \, dx \\ & = 2 \int \left (\frac {-2+x}{x}+\frac {2 (-2+x)}{x (-2+\log (x))}\right ) \, dx+2 \int \left (\frac {5+x}{2 (-2+\log (x))}+\frac {-5-x}{2 \log (x)}\right ) \, dx-2 \int \left (\frac {25+20 x+3 x^2}{4 x (-2+\log (x))}+\frac {-25-20 x-3 x^2}{2 x \log ^2(x)}+\frac {-25-20 x-3 x^2}{4 x \log (x)}\right ) \, dx+2 \text {Subst}\left (\int \frac {x^2}{-2+x} \, dx,x,\log (x)\right )+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx-4 \text {Subst}\left (\int \frac {e^x}{-2+x} \, dx,x,\log (x)\right )-10 \int \left (-\frac {2 \log ^2(-2+\log (x))}{\log ^3(x)}-\frac {10 \log ^2(-2+\log (x))}{x \log ^3(x)}+\frac {\log ^2(-2+\log (x))}{\log ^2(x)}\right ) \, dx-20 \int \left (\frac {\log (-2+\log (x))}{x (-2+\log (x))}+\frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)}+\frac {5 \log (-2+\log (x))}{x (-2+\log (x)) \log ^2(x)}\right ) \, dx+40 \int \left (\frac {1}{8 (-2+\log (x))}-\frac {1}{2 \log ^3(x)}-\frac {1}{4 \log ^2(x)}-\frac {1}{8 \log (x)}\right ) \, dx-50 \text {Subst}\left (\int \frac {\log ^4(-2+x)}{x^3} \, dx,x,\log (x)\right )+100 \text {Subst}\left (\int \frac {1}{(-2+x) x^3} \, dx,x,\log (x)\right )+100 \text {Subst}\left (\int \frac {\log ^3(-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right ) \\ & = -4 e^2 \operatorname {ExpIntegralEi}(-2+\log (x))+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {1}{2} \int \frac {25+20 x+3 x^2}{x (-2+\log (x))} \, dx-\frac {1}{2} \int \frac {-25-20 x-3 x^2}{x \log (x)} \, dx+2 \int \frac {-2+x}{x} \, dx+2 \text {Subst}\left (\int \left (2+\frac {4}{-2+x}+x\right ) \, dx,x,\log (x)\right )+4 \int \frac {-2+x}{x (-2+\log (x))} \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx+5 \int \frac {1}{-2+\log (x)} \, dx-5 \int \frac {1}{\log (x)} \, dx-10 \int \frac {1}{\log ^2(x)} \, dx-10 \int \frac {\log ^2(-2+\log (x))}{\log ^2(x)} \, dx-20 \int \frac {1}{\log ^3(x)} \, dx-20 \int \frac {\log (-2+\log (x))}{x (-2+\log (x))} \, dx-20 \int \frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)} \, dx+20 \int \frac {\log ^2(-2+\log (x))}{\log ^3(x)} \, dx-100 \int \frac {\log (-2+\log (x))}{x (-2+\log (x)) \log ^2(x)} \, dx+100 \int \frac {\log ^2(-2+\log (x))}{x \log ^3(x)} \, dx+100 \text {Subst}\left (\int \left (\frac {1}{8 (-2+x)}-\frac {1}{2 x^3}-\frac {1}{4 x^2}-\frac {1}{8 x}\right ) \, dx,x,\log (x)\right )-100 \text {Subst}\left (\int \frac {\log ^3(-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right )+100 \text {Subst}\left (\int \frac {\log ^3(x)}{x (2+x)^2} \, dx,x,-2+\log (x)\right )+\int \frac {5+x}{-2+\log (x)} \, dx-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx+\int \frac {-5-x}{\log (x)} \, dx \\ & = -4 e^2 \operatorname {ExpIntegralEi}(-2+\log (x))+\frac {25}{\log ^2(x)}+\frac {10 x}{\log ^2(x)}+\frac {25}{\log (x)}+\frac {10 x}{\log (x)}+4 \log (x)+\log ^2(x)+\frac {41}{2} \log (2-\log (x))+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {25}{2} \log (\log (x))-5 \operatorname {LogIntegral}(x)-\frac {1}{2} \int \frac {25+20 x+3 x^2}{x (-2+\log (x))} \, dx-\frac {1}{2} \int \frac {-25-20 x-3 x^2}{x \log (x)} \, dx+2 \int \left (1-\frac {2}{x}\right ) \, dx+4 \int \left (\frac {1}{-2+\log (x)}-\frac {2}{x (-2+\log (x))}\right ) \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx+5 \text {Subst}\left (\int \frac {e^x}{-2+x} \, dx,x,\log (x)\right )-10 \int \frac {1}{\log ^2(x)} \, dx-10 \int \frac {1}{\log (x)} \, dx-10 \int \frac {\log ^2(-2+\log (x))}{\log ^2(x)} \, dx-20 \int \frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)} \, dx+20 \int \frac {\log ^2(-2+\log (x))}{\log ^3(x)} \, dx-20 \text {Subst}\left (\int \frac {\log (-2+x)}{-2+x} \, dx,x,\log (x)\right )-50 \text {Subst}\left (\int \frac {\log ^3(x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )+50 \text {Subst}\left (\int \frac {\log ^3(x)}{x (2+x)} \, dx,x,-2+\log (x)\right )-100 \text {Subst}\left (\int \frac {\log (-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right )+100 \text {Subst}\left (\int \frac {\log ^2(-2+x)}{x^3} \, dx,x,\log (x)\right )-100 \text {Subst}\left (\int \frac {\log ^3(x)}{x (2+x)^2} \, dx,x,-2+\log (x)\right )+\int \left (\frac {5}{-2+\log (x)}+\frac {x}{-2+\log (x)}\right ) \, dx+\int \left (-\frac {5}{\log (x)}-\frac {x}{\log (x)}\right ) \, dx-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx \\ & = 2 x+e^2 \operatorname {ExpIntegralEi}(-2+\log (x))+\frac {25}{\log ^2(x)}+\frac {10 x}{\log ^2(x)}+\frac {25}{\log (x)}+\frac {20 x}{\log (x)}+\log ^2(x)+\frac {41}{2} \log (2-\log (x))-\frac {50 \log ^2(-2+\log (x))}{\log ^2(x)}+\frac {25 (2-\log (x)) \log ^3(-2+\log (x))}{\log (x)}-25 \log \left (1-\frac {2}{2-\log (x)}\right ) \log ^3(-2+\log (x))+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {25}{2} \log (\log (x))-15 \operatorname {LogIntegral}(x)-\frac {1}{2} \int \frac {25+20 x+3 x^2}{x (-2+\log (x))} \, dx-\frac {1}{2} \int \frac {-25-20 x-3 x^2}{x \log (x)} \, dx+4 \int \frac {1}{-2+\log (x)} \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx+5 \int \frac {1}{-2+\log (x)} \, dx-5 \int \frac {1}{\log (x)} \, dx-8 \int \frac {1}{x (-2+\log (x))} \, dx-10 \int \frac {1}{\log (x)} \, dx-10 \int \frac {\log ^2(-2+\log (x))}{\log ^2(x)} \, dx-20 \int \frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)} \, dx+20 \int \frac {\log ^2(-2+\log (x))}{\log ^3(x)} \, dx-20 \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+\log (x)\right )+50 \text {Subst}\left (\int \frac {\log ^3(x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )-50 \text {Subst}\left (\int \frac {\log ^3(x)}{x (2+x)} \, dx,x,-2+\log (x)\right )+75 \text {Subst}\left (\int \frac {\log ^2(x)}{2+x} \, dx,x,-2+\log (x)\right )+75 \text {Subst}\left (\int \frac {\log \left (1+\frac {2}{x}\right ) \log ^2(x)}{x} \, dx,x,-2+\log (x)\right )+100 \text {Subst}\left (\int \frac {\log (-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right )-100 \text {Subst}\left (\int \frac {\log (x)}{x (2+x)^2} \, dx,x,-2+\log (x)\right )+\int \frac {x}{-2+\log (x)} \, dx-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx-\int \frac {x}{\log (x)} \, dx \\ & = 2 x+e^2 \operatorname {ExpIntegralEi}(-2+\log (x))+\frac {25}{\log ^2(x)}+\frac {10 x}{\log ^2(x)}+\frac {25}{\log (x)}+\frac {20 x}{\log (x)}+\log ^2(x)+\frac {41}{2} \log (2-\log (x))-10 \log ^2(-2+\log (x))-\frac {50 \log ^2(-2+\log (x))}{\log ^2(x)}+75 \log \left (1+\frac {1}{2} (-2+\log (x))\right ) \log ^2(-2+\log (x))+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {25}{2} \log (\log (x))-30 \operatorname {LogIntegral}(x)+75 \log ^2(-2+\log (x)) \operatorname {PolyLog}\left (2,\frac {2}{2-\log (x)}\right )-\frac {1}{2} \int \frac {25+20 x+3 x^2}{x (-2+\log (x))} \, dx-\frac {1}{2} \int \frac {-25-20 x-3 x^2}{x \log (x)} \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx+4 \text {Subst}\left (\int \frac {e^x}{-2+x} \, dx,x,\log (x)\right )+5 \text {Subst}\left (\int \frac {e^x}{-2+x} \, dx,x,\log (x)\right )-8 \text {Subst}\left (\int \frac {1}{x} \, dx,x,-2+\log (x)\right )-10 \int \frac {\log ^2(-2+\log (x))}{\log ^2(x)} \, dx-20 \int \frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)} \, dx+20 \int \frac {\log ^2(-2+\log (x))}{\log ^3(x)} \, dx+50 \text {Subst}\left (\int \frac {\log (x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )-50 \text {Subst}\left (\int \frac {\log (x)}{x (2+x)} \, dx,x,-2+\log (x)\right )-75 \text {Subst}\left (\int \frac {\log ^2(x)}{2+x} \, dx,x,-2+\log (x)\right )-75 \text {Subst}\left (\int \frac {\log \left (1+\frac {2}{x}\right ) \log ^2(x)}{x} \, dx,x,-2+\log (x)\right )+100 \text {Subst}\left (\int \frac {\log (x)}{x (2+x)^2} \, dx,x,-2+\log (x)\right )-150 \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right ) \log (x)}{x} \, dx,x,-2+\log (x)\right )-150 \text {Subst}\left (\int \frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {2}{x}\right )}{x} \, dx,x,-2+\log (x)\right )-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx+\text {Subst}\left (\int \frac {e^{2 x}}{-2+x} \, dx,x,\log (x)\right )-\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = 2 x+e^4 \operatorname {ExpIntegralEi}(-2 (2-\log (x)))+10 e^2 \operatorname {ExpIntegralEi}(-2+\log (x))-\operatorname {ExpIntegralEi}(2 \log (x))+\frac {25}{\log ^2(x)}+\frac {10 x}{\log ^2(x)}+\frac {25}{\log (x)}+\frac {20 x}{\log (x)}+\log ^2(x)+\frac {25}{2} \log (2-\log (x))-\frac {25 (2-\log (x)) \log (-2+\log (x))}{\log (x)}+25 \log \left (1-\frac {2}{2-\log (x)}\right ) \log (-2+\log (x))-10 \log ^2(-2+\log (x))-\frac {50 \log ^2(-2+\log (x))}{\log ^2(x)}+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {25}{2} \log (\log (x))-30 \operatorname {LogIntegral}(x)+150 \log (-2+\log (x)) \operatorname {PolyLog}\left (2,\frac {1}{2} (2-\log (x))\right )+150 \log (-2+\log (x)) \operatorname {PolyLog}\left (3,\frac {2}{2-\log (x)}\right )-\frac {1}{2} \int \frac {25+20 x+3 x^2}{x (-2+\log (x))} \, dx-\frac {1}{2} \int \frac {-25-20 x-3 x^2}{x \log (x)} \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx-10 \int \frac {\log ^2(-2+\log (x))}{\log ^2(x)} \, dx-20 \int \frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)} \, dx+20 \int \frac {\log ^2(-2+\log (x))}{\log ^3(x)} \, dx-25 \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,-2+\log (x)\right )-25 \text {Subst}\left (\int \frac {\log \left (1+\frac {2}{x}\right )}{x} \, dx,x,-2+\log (x)\right )-50 \text {Subst}\left (\int \frac {\log (x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )+50 \text {Subst}\left (\int \frac {\log (x)}{x (2+x)} \, dx,x,-2+\log (x)\right )+150 \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right ) \log (x)}{x} \, dx,x,-2+\log (x)\right )+150 \text {Subst}\left (\int \frac {\log (x) \operatorname {PolyLog}\left (2,-\frac {2}{x}\right )}{x} \, dx,x,-2+\log (x)\right )-150 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {x}{2}\right )}{x} \, dx,x,-2+\log (x)\right )-150 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {2}{x}\right )}{x} \, dx,x,-2+\log (x)\right )-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx \\ & = 2 x+e^4 \operatorname {ExpIntegralEi}(-2 (2-\log (x)))+10 e^2 \operatorname {ExpIntegralEi}(-2+\log (x))-\operatorname {ExpIntegralEi}(2 \log (x))+\frac {25}{\log ^2(x)}+\frac {10 x}{\log ^2(x)}+\frac {25}{\log (x)}+\frac {20 x}{\log (x)}+\log ^2(x)+\frac {25}{2} \log (2-\log (x))-10 \log ^2(-2+\log (x))-\frac {50 \log ^2(-2+\log (x))}{\log ^2(x)}+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {75}{2} \log (\log (x))-30 \operatorname {LogIntegral}(x)-25 \operatorname {PolyLog}\left (2,\frac {2}{2-\log (x)}\right )-150 \operatorname {PolyLog}\left (3,\frac {1}{2} (2-\log (x))\right )+150 \operatorname {PolyLog}\left (4,\frac {2}{2-\log (x)}\right )-\frac {1}{2} \int \frac {25+20 x+3 x^2}{x (-2+\log (x))} \, dx-\frac {1}{2} \int \frac {-25-20 x-3 x^2}{x \log (x)} \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx-10 \int \frac {\log ^2(-2+\log (x))}{\log ^2(x)} \, dx-20 \int \frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)} \, dx+20 \int \frac {\log ^2(-2+\log (x))}{\log ^3(x)} \, dx+25 \text {Subst}\left (\int \frac {1}{2+x} \, dx,x,-2+\log (x)\right )+25 \text {Subst}\left (\int \frac {\log \left (1+\frac {2}{x}\right )}{x} \, dx,x,-2+\log (x)\right )+150 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {x}{2}\right )}{x} \, dx,x,-2+\log (x)\right )+150 \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {2}{x}\right )}{x} \, dx,x,-2+\log (x)\right )-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx \\ & = 2 x+e^4 \operatorname {ExpIntegralEi}(-2 (2-\log (x)))+10 e^2 \operatorname {ExpIntegralEi}(-2+\log (x))-\operatorname {ExpIntegralEi}(2 \log (x))+\frac {25}{\log ^2(x)}+\frac {10 x}{\log ^2(x)}+\frac {25}{\log (x)}+\frac {20 x}{\log (x)}+\log ^2(x)+\frac {25}{2} \log (2-\log (x))-10 \log ^2(-2+\log (x))-\frac {50 \log ^2(-2+\log (x))}{\log ^2(x)}+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {25}{2} \log (\log (x))-30 \operatorname {LogIntegral}(x)-\frac {1}{2} \int \frac {25+20 x+3 x^2}{x (-2+\log (x))} \, dx-\frac {1}{2} \int \frac {-25-20 x-3 x^2}{x \log (x)} \, dx+4 \int \frac {x}{(-2+\log (x)) \log ^3(x)} \, dx-10 \int \frac {\log ^2(-2+\log (x))}{\log ^2(x)} \, dx-20 \int \frac {\log (-2+\log (x))}{(-2+\log (x)) \log ^2(x)} \, dx+20 \int \frac {\log ^2(-2+\log (x))}{\log ^3(x)} \, dx-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.13 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=2 \left (x+\frac {\log ^2(x)}{2}-5 \log ^2(-2+\log (x))+\frac {\left (5+x-5 \log ^2(-2+\log (x))\right )^2}{2 \log ^2(x)}\right ) \]

Maple [A] (veriﬁed)

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03


method	result	size

risch	\(\frac {25 \ln \left (\ln \left (x \right )-2\right )^{4}}{\ln \left (x \right )^{2}}-\frac {10 \left (\ln \left (x \right )^{2}+x +5\right ) \ln \left (\ln \left (x \right )-2\right )^{2}}{\ln \left (x \right )^{2}}+\frac {\ln \left (x \right )^{4}+2 x \ln \left (x \right )^{2}+x^{2}+10 x +25}{\ln \left (x \right )^{2}}\)	\(59\)
parallelrisch	\(\frac {4 \ln \left (x \right )^{4}-40 \ln \left (x \right )^{2} \ln \left (\ln \left (x \right )-2\right )^{2}+100 \ln \left (\ln \left (x \right )-2\right )^{4}+8 x \ln \left (x \right )^{2}-40 x \ln \left (\ln \left (x \right )-2\right )^{2}+100+4 x^{2}-200 \ln \left (\ln \left (x \right )-2\right )^{2}+40 x}{4 \ln \left (x \right )^{2}}\)	\(71\)

Fricas [A] (veriﬁcation not implemented)

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\frac {\log \left (x\right )^{4} + 25 \, \log \left (\log \left (x\right ) - 2\right )^{4} + 2 \, x \log \left (x\right )^{2} - 10 \, {\left (\log \left (x\right )^{2} + x + 5\right )} \log \left (\log \left (x\right ) - 2\right )^{2} + x^{2} + 10 \, x + 25}{\log \left (x\right )^{2}} \]

Sympy [F(-2)]

Exception generated. \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\text {Exception raised: TypeError} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (30) = 60\).

Time = 0.22 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=\frac {\log \left (x\right )^{4} + 25 \, \log \left (\log \left (x\right ) - 2\right )^{4} + 2 \, x \log \left (x\right )^{2} - 10 \, {\left (\log \left (x\right )^{2} + x + 5\right )} \log \left (\log \left (x\right ) - 2\right )^{2} + x^{2} + 10 \, x - 25 \, \log \left (x\right )}{\log \left (x\right )^{2}} + \frac {25 \, {\left (\log \left (x\right ) + 1\right )}}{\log \left (x\right )^{2}} \]

Giac [A] (veriﬁcation not implemented)

Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=-10 \, {\left (\frac {x + 5}{\log \left (x\right )^{2}} + 1\right )} \log \left (\log \left (x\right ) - 2\right )^{2} + \log \left (x\right )^{2} + \frac {25 \, \log \left (\log \left (x\right ) - 2\right )^{4}}{\log \left (x\right )^{2}} + 2 \, x + \frac {x^{2} + 10 \, x + 25}{\log \left (x\right )^{2}} \]

Mupad [B] (veriﬁcation not implemented)

Time = 8.57 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.00 \[ \int \frac {100+40 x+4 x^2+\left (-50-40 x-6 x^2\right ) \log (x)+\left (10 x+2 x^2\right ) \log ^2(x)-4 x \log ^3(x)+(-4+2 x) \log ^4(x)+2 \log ^5(x)+\left ((-100-20 x) \log (x)-20 \log ^3(x)\right ) \log (-2+\log (x))+\left (-200-40 x+(100+40 x) \log (x)-10 x \log ^2(x)\right ) \log ^2(-2+\log (x))+100 \log (x) \log ^3(-2+\log (x))+(100-50 \log (x)) \log ^4(-2+\log (x))}{-2 x \log ^3(x)+x \log ^4(x)} \, dx=7\,x+{\ln \left (x\right )}^2+\frac {x\,\left (x+5\right )-x\,\ln \left (x\right )\,\left (2\,x+5\right )}{\ln \left (x\right )}-{\ln \left (\ln \left (x\right )-2\right )}^2\,\left (\frac {10\,x+50}{{\ln \left (x\right )}^2}+10\right )+\frac {25\,{\ln \left (\ln \left (x\right )-2\right )}^4}{{\ln \left (x\right )}^2}+\frac {{\left (x+5\right )}^2-x\,\ln \left (x\right )\,\left (x+5\right )}{{\ln \left (x\right )}^2}+2\,x^2 \]

Optimal result