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

Optimal. Leaf size=29 \[ \left (\log (x)+\frac {x+\frac {5 \left (x-x \log ^2(-2+\log (x))\right )}{x}}{\log (x)}\right )^2 \]

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Rubi [F]  time = 4.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100 + 40*x + 4*x^2 + (-50 - 40*x - 6*x^2)*Log[x] + (10*x + 2*x^2)*Log[x]^2 - 4*x*Log[x]^3 + (-4 + 2*x)*Lo
g[x]^4 + 2*Log[x]^5 + ((-100 - 20*x)*Log[x] - 20*Log[x]^3)*Log[-2 + Log[x]] + (-200 - 40*x + (100 + 40*x)*Log[
x] - 10*x*Log[x]^2)*Log[-2 + Log[x]]^2 + 100*Log[x]*Log[-2 + Log[x]]^3 + (100 - 50*Log[x])*Log[-2 + Log[x]]^4)
/(-2*x*Log[x]^3 + x*Log[x]^4),x]

[Out]

2*x + E^4*ExpIntegralEi[-2*(2 - Log[x])] + 10*E^2*ExpIntegralEi[-2 + Log[x]] - ExpIntegralEi[2*Log[x]] + 25/Lo
g[x]^2 + (10*x)/Log[x]^2 + 25/Log[x] + (20*x)/Log[x] + Log[x]^2 + (25*Log[2 - Log[x]])/2 - 10*Log[-2 + Log[x]]
^2 - (50*Log[-2 + Log[x]]^2)/Log[x]^2 + (25*Log[-2 + Log[x]]^4)/Log[x]^2 - (25*Log[Log[x]])/2 - 30*LogIntegral
[x] - Defer[Int][(25 + 20*x + 3*x^2)/(x*(-2 + Log[x])), x]/2 + 4*Defer[Int][x/((-2 + Log[x])*Log[x]^3), x] - D
efer[Int][(-25 - 20*x - 3*x^2)/(x*Log[x]^2), x] - Defer[Int][(-25 - 20*x - 3*x^2)/(x*Log[x]), x]/2 - 20*Defer[
Int][Log[-2 + Log[x]]/((-2 + Log[x])*Log[x]^2), x] + 20*Defer[Int][Log[-2 + Log[x]]^2/Log[x]^3, x] - 10*Defer[
Int][Log[-2 + Log[x]]^2/Log[x]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \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 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {\log ^4(-2+x)}{x^3} \, dx,x,\log (x)\right )+100 \operatorname {Subst}\left (\int \frac {1}{(-2+x) x^3} \, dx,x,\log (x)\right )+100 \operatorname {Subst}\left (\int \frac {\log ^3(-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right )\\ &=-4 e^2 \text {Ei}(-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 \operatorname {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 \operatorname {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 \operatorname {Subst}\left (\int \frac {\log ^3(-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right )+100 \operatorname {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 \text {Ei}(-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 \text {li}(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 \operatorname {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 \operatorname {Subst}\left (\int \frac {\log (-2+x)}{-2+x} \, dx,x,\log (x)\right )-50 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )+50 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x (2+x)} \, dx,x,-2+\log (x)\right )-100 \operatorname {Subst}\left (\int \frac {\log (-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right )+100 \operatorname {Subst}\left (\int \frac {\log ^2(-2+x)}{x^3} \, dx,x,\log (x)\right )-100 \operatorname {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 \text {Ei}(-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)}+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {25}{2} \log (\log (x))-15 \text {li}(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 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+\log (x)\right )+25 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x} \, dx,x,-2+\log (x)\right )-25 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{2+x} \, dx,x,-2+\log (x)\right )+50 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )-50 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x (2+x)} \, dx,x,-2+\log (x)\right )+75 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{2+x} \, dx,x,-2+\log (x)\right )+100 \operatorname {Subst}\left (\int \frac {\log (-2+x)}{(-2+x) x^2} \, dx,x,\log (x)\right )-100 \operatorname {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 \text {Ei}(-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))-25 \log \left (1+\frac {1}{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))-30 \text {li}(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+4 \operatorname {Subst}\left (\int \frac {e^x}{-2+x} \, dx,x,\log (x)\right )+5 \operatorname {Subst}\left (\int \frac {e^x}{-2+x} \, dx,x,\log (x)\right )-8 \operatorname {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+25 \operatorname {Subst}\left (\int x^3 \, dx,x,\log (-2+\log (x))\right )-25 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x} \, dx,x,-2+\log (x)\right )+25 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{2+x} \, dx,x,-2+\log (x)\right )+50 \operatorname {Subst}\left (\int \frac {\log (x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )-50 \operatorname {Subst}\left (\int \frac {\log (x)}{x (2+x)} \, dx,x,-2+\log (x)\right )-75 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{2+x} \, dx,x,-2+\log (x)\right )+75 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right ) \log ^2(x)}{x} \, dx,x,-2+\log (x)\right )+100 \operatorname {Subst}\left (\int \frac {\log (x)}{x (2+x)^2} \, dx,x,-2+\log (x)\right )-150 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right ) \log (x)}{x} \, dx,x,-2+\log (x)\right )-\int \frac {-25-20 x-3 x^2}{x \log ^2(x)} \, dx+\operatorname {Subst}\left (\int \frac {e^{2 x}}{-2+x} \, dx,x,\log (x)\right )-\operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=2 x+e^4 \text {Ei}(-2 (2-\log (x)))+10 e^2 \text {Ei}(-2+\log (x))-\text {Ei}(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)}-10 \log ^2(-2+\log (x))-\frac {50 \log ^2(-2+\log (x))}{\log ^2(x)}+\frac {25}{4} \log ^4(-2+\log (x))+\frac {25 \log ^4(-2+\log (x))}{\log ^2(x)}-\frac {25}{2} \log (\log (x))-30 \text {li}(x)+150 \log (-2+\log (x)) \text {Li}_2\left (\frac {1}{2} (2-\log (x))\right )-75 \log ^2(-2+\log (x)) \text {Li}_2\left (\frac {1}{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 \operatorname {Subst}\left (\int x^3 \, dx,x,\log (-2+\log (x))\right )-25 \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,-2+\log (x)\right )-25 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+\log (x)\right )+25 \operatorname {Subst}\left (\int \frac {\log (x)}{2+x} \, dx,x,-2+\log (x)\right )-50 \operatorname {Subst}\left (\int \frac {\log (x)}{(2+x)^2} \, dx,x,-2+\log (x)\right )+50 \operatorname {Subst}\left (\int \frac {\log (x)}{x (2+x)} \, dx,x,-2+\log (x)\right )-75 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right ) \log ^2(x)}{x} \, dx,x,-2+\log (x)\right )+150 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right ) \log (x)}{x} \, dx,x,-2+\log (x)\right )-150 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx,x,-2+\log (x)\right )+150 \operatorname {Subst}\left (\int \frac {\log (x) \text {Li}_2\left (-\frac {x}{2}\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 \text {Ei}(-2 (2-\log (x)))+10 e^2 \text {Ei}(-2+\log (x))-\text {Ei}(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))+25 \log \left (1+\frac {1}{2} (-2+\log (x))\right ) \log (-2+\log (x))-\frac {45}{2} \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 \text {li}(x)-150 \text {Li}_3\left (\frac {1}{2} (2-\log (x))\right )+150 \log (-2+\log (x)) \text {Li}_3\left (\frac {1}{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 \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,-2+\log (x)\right )-25 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,-2+\log (x)\right )+25 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-2+\log (x)\right )-25 \operatorname {Subst}\left (\int \frac {\log (x)}{2+x} \, dx,x,-2+\log (x)\right )+150 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx,x,-2+\log (x)\right )-150 \operatorname {Subst}\left (\int \frac {\log (x) \text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx,x,-2+\log (x)\right )-150 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {x}{2}\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 \text {Ei}(-2 (2-\log (x)))+10 e^2 \text {Ei}(-2+\log (x))-\text {Ei}(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 \text {li}(x)+25 \text {Li}_2\left (\frac {1}{2} (2-\log (x))\right )-150 \text {Li}_4\left (\frac {1}{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 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {x}{2}\right )}{x} \, dx,x,-2+\log (x)\right )+150 \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {x}{2}\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 \text {Ei}(-2 (2-\log (x)))+10 e^2 \text {Ei}(-2+\log (x))-\text {Ei}(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 \text {li}(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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 43, normalized size = 1.48 \begin {gather*} 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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100 + 40*x + 4*x^2 + (-50 - 40*x - 6*x^2)*Log[x] + (10*x + 2*x^2)*Log[x]^2 - 4*x*Log[x]^3 + (-4 + 2
*x)*Log[x]^4 + 2*Log[x]^5 + ((-100 - 20*x)*Log[x] - 20*Log[x]^3)*Log[-2 + Log[x]] + (-200 - 40*x + (100 + 40*x
)*Log[x] - 10*x*Log[x]^2)*Log[-2 + Log[x]]^2 + 100*Log[x]*Log[-2 + Log[x]]^3 + (100 - 50*Log[x])*Log[-2 + Log[
x]]^4)/(-2*x*Log[x]^3 + x*Log[x]^4),x]

[Out]

2*(x + Log[x]^2/2 - 5*Log[-2 + Log[x]]^2 + (5 + x - 5*Log[-2 + Log[x]]^2)^2/(2*Log[x]^2))

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fricas [A]  time = 0.80, size = 49, normalized size = 1.69 \begin {gather*} \frac {\log \relax (x)^{4} + 25 \, \log \left (\log \relax (x) - 2\right )^{4} + 2 \, x \log \relax (x)^{2} - 10 \, {\left (\log \relax (x)^{2} + x + 5\right )} \log \left (\log \relax (x) - 2\right )^{2} + x^{2} + 10 \, x + 25}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*log(x)+100)*log(log(x)-2)^4+100*log(x)*log(log(x)-2)^3+(-10*x*log(x)^2+(40*x+100)*log(x)-40*x-
200)*log(log(x)-2)^2+(-20*log(x)^3+(-20*x-100)*log(x))*log(log(x)-2)+2*log(x)^5+(2*x-4)*log(x)^4-4*x*log(x)^3+
(2*x^2+10*x)*log(x)^2+(-6*x^2-40*x-50)*log(x)+4*x^2+40*x+100)/(x*log(x)^4-2*x*log(x)^3),x, algorithm="fricas")

[Out]

(log(x)^4 + 25*log(log(x) - 2)^4 + 2*x*log(x)^2 - 10*(log(x)^2 + x + 5)*log(log(x) - 2)^2 + x^2 + 10*x + 25)/l
og(x)^2

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giac [A]  time = 0.44, size = 53, normalized size = 1.83 \begin {gather*} -10 \, {\left (\frac {x + 5}{\log \relax (x)^{2}} + 1\right )} \log \left (\log \relax (x) - 2\right )^{2} + \log \relax (x)^{2} + \frac {25 \, \log \left (\log \relax (x) - 2\right )^{4}}{\log \relax (x)^{2}} + 2 \, x + \frac {x^{2} + 10 \, x + 25}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*log(x)+100)*log(log(x)-2)^4+100*log(x)*log(log(x)-2)^3+(-10*x*log(x)^2+(40*x+100)*log(x)-40*x-
200)*log(log(x)-2)^2+(-20*log(x)^3+(-20*x-100)*log(x))*log(log(x)-2)+2*log(x)^5+(2*x-4)*log(x)^4-4*x*log(x)^3+
(2*x^2+10*x)*log(x)^2+(-6*x^2-40*x-50)*log(x)+4*x^2+40*x+100)/(x*log(x)^4-2*x*log(x)^3),x, algorithm="giac")

[Out]

-10*((x + 5)/log(x)^2 + 1)*log(log(x) - 2)^2 + log(x)^2 + 25*log(log(x) - 2)^4/log(x)^2 + 2*x + (x^2 + 10*x +
25)/log(x)^2

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maple [A]  time = 0.05, size = 59, normalized size = 2.03

method	result	size
risch	\(\frac {25 \ln \left (\ln \relax (x )-2\right )^{4}}{\ln \relax (x )^{2}}-\frac {10 \left (\ln \relax (x )^{2}+x +5\right ) \ln \left (\ln \relax (x )-2\right )^{2}}{\ln \relax (x )^{2}}+\frac {\ln \relax (x )^{4}+2 x \ln \relax (x )^{2}+x^{2}+10 x +25}{\ln \relax (x )^{2}}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-50*ln(x)+100)*ln(ln(x)-2)^4+100*ln(x)*ln(ln(x)-2)^3+(-10*x*ln(x)^2+(40*x+100)*ln(x)-40*x-200)*ln(ln(x)-
2)^2+(-20*ln(x)^3+(-20*x-100)*ln(x))*ln(ln(x)-2)+2*ln(x)^5+(2*x-4)*ln(x)^4-4*x*ln(x)^3+(2*x^2+10*x)*ln(x)^2+(-
6*x^2-40*x-50)*ln(x)+4*x^2+40*x+100)/(x*ln(x)^4-2*x*ln(x)^3),x,method=_RETURNVERBOSE)

[Out]

25/ln(x)^2*ln(ln(x)-2)^4-10*(ln(x)^2+x+5)/ln(x)^2*ln(ln(x)-2)^2+(ln(x)^4+2*x*ln(x)^2+x^2+10*x+25)/ln(x)^2

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maxima [B]  time = 0.91, size = 63, normalized size = 2.17 \begin {gather*} \frac {\log \relax (x)^{4} + 25 \, \log \left (\log \relax (x) - 2\right )^{4} + 2 \, x \log \relax (x)^{2} - 10 \, {\left (\log \relax (x)^{2} + x + 5\right )} \log \left (\log \relax (x) - 2\right )^{2} + x^{2} + 10 \, x - 25 \, \log \relax (x)}{\log \relax (x)^{2}} + \frac {25 \, {\left (\log \relax (x) + 1\right )}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*log(x)+100)*log(log(x)-2)^4+100*log(x)*log(log(x)-2)^3+(-10*x*log(x)^2+(40*x+100)*log(x)-40*x-
200)*log(log(x)-2)^2+(-20*log(x)^3+(-20*x-100)*log(x))*log(log(x)-2)+2*log(x)^5+(2*x-4)*log(x)^4-4*x*log(x)^3+
(2*x^2+10*x)*log(x)^2+(-6*x^2-40*x-50)*log(x)+4*x^2+40*x+100)/(x*log(x)^4-2*x*log(x)^3),x, algorithm="maxima")

[Out]

(log(x)^4 + 25*log(log(x) - 2)^4 + 2*x*log(x)^2 - 10*(log(x)^2 + x + 5)*log(log(x) - 2)^2 + x^2 + 10*x - 25*lo
g(x))/log(x)^2 + 25*(log(x) + 1)/log(x)^2

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mupad [B]  time = 1.16, size = 87, normalized size = 3.00 \begin {gather*} 7\,x+{\ln \relax (x)}^2+\frac {x\,\left (x+5\right )-x\,\ln \relax (x)\,\left (2\,x+5\right )}{\ln \relax (x)}-{\ln \left (\ln \relax (x)-2\right )}^2\,\left (\frac {10\,x+50}{{\ln \relax (x)}^2}+10\right )+\frac {25\,{\ln \left (\ln \relax (x)-2\right )}^4}{{\ln \relax (x)}^2}+\frac {{\left (x+5\right )}^2-x\,\ln \relax (x)\,\left (x+5\right )}{{\ln \relax (x)}^2}+2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(40*x + log(x)^2*(10*x + 2*x^2) - log(log(x) - 2)^4*(50*log(x) - 100) - 4*x*log(x)^3 - log(log(x) - 2)^2*
(40*x + 10*x*log(x)^2 - log(x)*(40*x + 100) + 200) + 2*log(x)^5 + 100*log(log(x) - 2)^3*log(x) - log(log(x) -
2)*(20*log(x)^3 + log(x)*(20*x + 100)) - log(x)*(40*x + 6*x^2 + 50) + 4*x^2 + log(x)^4*(2*x - 4) + 100)/(2*x*l
og(x)^3 - x*log(x)^4),x)

[Out]

7*x + log(x)^2 + (x*(x + 5) - x*log(x)*(2*x + 5))/log(x) - log(log(x) - 2)^2*((10*x + 50)/log(x)^2 + 10) + (25
*log(log(x) - 2)^4)/log(x)^2 + ((x + 5)^2 - x*log(x)*(x + 5))/log(x)^2 + 2*x^2

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sympy [B]  time = 0.53, size = 63, normalized size = 2.17 \begin {gather*} 2 x + \frac {\left (- 10 x - 10 \log {\relax (x )}^{2} - 50\right ) \log {\left (\log {\relax (x )} - 2 \right )}^{2}}{\log {\relax (x )}^{2}} + \frac {x^{2} + 10 x + 25}{\log {\relax (x )}^{2}} + \log {\relax (x )}^{2} + \frac {25 \log {\left (\log {\relax (x )} - 2 \right )}^{4}}{\log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*ln(x)+100)*ln(ln(x)-2)**4+100*ln(x)*ln(ln(x)-2)**3+(-10*x*ln(x)**2+(40*x+100)*ln(x)-40*x-200)*
ln(ln(x)-2)**2+(-20*ln(x)**3+(-20*x-100)*ln(x))*ln(ln(x)-2)+2*ln(x)**5+(2*x-4)*ln(x)**4-4*x*ln(x)**3+(2*x**2+1
0*x)*ln(x)**2+(-6*x**2-40*x-50)*ln(x)+4*x**2+40*x+100)/(x*ln(x)**4-2*x*ln(x)**3),x)

[Out]

2*x + (-10*x - 10*log(x)**2 - 50)*log(log(x) - 2)**2/log(x)**2 + (x**2 + 10*x + 25)/log(x)**2 + log(x)**2 + 25
*log(log(x) - 2)**4/log(x)**2

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