3.12.41 \(\int \frac {(120+10 e^x-2 x+10 x^2+(2 x-10 e^x x-20 x^2) \log (x)+40 x \log ^2(x)) \log (\frac {-60-5 e^x+x-5 x^2+(120+20 x) \log (x)}{20 \log (x)})}{(-60 x-5 e^x x+x^2-5 x^3) \log (x)+(120 x+20 x^2) \log ^2(x)} \, dx\)

Optimal. Leaf size=32 \[ -1+\log ^2\left (6+x-\frac {3+\frac {1}{4} \left (e^x-\frac {x}{5}+x^2\right )}{\log (x)}\right ) \]

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Rubi [F]  time = 15.43, 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 {\left (120+10 e^x-2 x+10 x^2+\left (2 x-10 e^x x-20 x^2\right ) \log (x)+40 x \log ^2(x)\right ) \log \left (\frac {-60-5 e^x+x-5 x^2+(120+20 x) \log (x)}{20 \log (x)}\right )}{\left (-60 x-5 e^x x+x^2-5 x^3\right ) \log (x)+\left (120 x+20 x^2\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((120 + 10*E^x - 2*x + 10*x^2 + (2*x - 10*E^x*x - 20*x^2)*Log[x] + 40*x*Log[x]^2)*Log[(-60 - 5*E^x + x - 5
*x^2 + (120 + 20*x)*Log[x])/(20*Log[x])])/((-60*x - 5*E^x*x + x^2 - 5*x^3)*Log[x] + (120*x + 20*x^2)*Log[x]^2)
,x]

[Out]

-x^2 + 2*x*Log[-1/20*(60 + 5*E^x - x + 5*x^2 - 20*(6 + x)*Log[x])/Log[x]] + 2*LogIntegral[x] + 240*Defer[Int][
(60 + 5*E^x - x + 5*x^2 - 120*Log[x] - 20*x*Log[x])^(-1), x] + 162*Defer[Int][x/(60 + 5*E^x - x + 5*x^2 - 120*
Log[x] - 20*x*Log[x]), x] - 22*Defer[Int][x^2/(60 + 5*E^x - x + 5*x^2 - 120*Log[x] - 20*x*Log[x]), x] + 10*Def
er[Int][x^3/(60 + 5*E^x - x + 5*x^2 - 120*Log[x] - 20*x*Log[x]), x] - 200*Defer[Int][(x*Log[x])/(60 + 5*E^x -
x + 5*x^2 - 120*Log[x] - 20*x*Log[x]), x] - 40*Defer[Int][(x^2*Log[x])/(60 + 5*E^x - x + 5*x^2 - 120*Log[x] -
20*x*Log[x]), x] - 2*Defer[Int][Log[(-60 - 5*E^x + x - 5*x^2 + 20*(6 + x)*Log[x])/(20*Log[x])]/(x*Log[x]), x]
- 162*Defer[Int][Log[(-60 - 5*E^x + x - 5*x^2 + 20*(6 + x)*Log[x])/(20*Log[x])]/(60 + 5*E^x - x + 5*x^2 - 120*
Log[x] - 20*x*Log[x]), x] - 240*Defer[Int][Log[(-60 - 5*E^x + x - 5*x^2 + 20*(6 + x)*Log[x])/(20*Log[x])]/(x*(
60 + 5*E^x - x + 5*x^2 - 120*Log[x] - 20*x*Log[x])), x] + 22*Defer[Int][(x*Log[(-60 - 5*E^x + x - 5*x^2 + 20*(
6 + x)*Log[x])/(20*Log[x])])/(60 + 5*E^x - x + 5*x^2 - 120*Log[x] - 20*x*Log[x]), x] - 10*Defer[Int][(x^2*Log[
(-60 - 5*E^x + x - 5*x^2 + 20*(6 + x)*Log[x])/(20*Log[x])])/(60 + 5*E^x - x + 5*x^2 - 120*Log[x] - 20*x*Log[x]
), x] + 40*Defer[Int][(x*Log[x]*Log[(-60 - 5*E^x + x - 5*x^2 + 20*(6 + x)*Log[x])/(20*Log[x])])/(60 + 5*E^x -
x + 5*x^2 - 120*Log[x] - 20*x*Log[x]), x] - 200*Defer[Int][(Log[x]*Log[(-60 - 5*E^x + x - 5*x^2 + 20*(6 + x)*L
og[x])/(20*Log[x])])/(-60 - 5*E^x + x - 5*x^2 + 120*Log[x] + 20*x*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-60-5 e^x+x-5 x^2-x \log (x)+5 e^x x \log (x)+10 x^2 \log (x)-20 x \log ^2(x)\right ) \log \left (\frac {-60-5 e^x+x-5 x^2+(120+20 x) \log (x)}{20 \log (x)}\right )}{x \log (x) \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=2 \int \frac {\left (-60-5 e^x+x-5 x^2-x \log (x)+5 e^x x \log (x)+10 x^2 \log (x)-20 x \log ^2(x)\right ) \log \left (\frac {-60-5 e^x+x-5 x^2+(120+20 x) \log (x)}{20 \log (x)}\right )}{x \log (x) \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=2 \int \left (\frac {(-1+x \log (x)) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)}-\frac {\left (120+81 x-11 x^2+5 x^3-100 x \log (x)-20 x^2 \log (x)\right ) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )}\right ) \, dx\\ &=2 \int \frac {(-1+x \log (x)) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx-2 \int \frac {\left (120+81 x-11 x^2+5 x^3-100 x \log (x)-20 x^2 \log (x)\right ) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=2 \int \left (\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )-\frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)}\right ) \, dx-2 \int \left (\frac {81 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}+\frac {120 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )}-\frac {11 x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}+\frac {5 x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}-\frac {20 x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}+\frac {100 \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)}\right ) \, dx\\ &=2 \int \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right ) \, dx-2 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx-10 \int \frac {x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+22 \int \frac {x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+40 \int \frac {x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-162 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {\log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)} \, dx-240 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=2 x \log \left (-\frac {60+5 e^x-x+5 x^2-20 (6+x) \log (x)}{20 \log (x)}\right )-2 \int \frac {-60-5 e^x+x-5 x^2+x \left (-1+5 e^x+10 x\right ) \log (x)-20 x \log ^2(x)}{\log (x) \left (60+5 e^x-x+5 x^2-20 (6+x) \log (x)\right )} \, dx-2 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx-10 \int \frac {x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+22 \int \frac {x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+40 \int \frac {x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-162 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {\log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)} \, dx-240 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=2 x \log \left (-\frac {60+5 e^x-x+5 x^2-20 (6+x) \log (x)}{20 \log (x)}\right )-2 \int \left (\frac {-1+x \log (x)}{\log (x)}-\frac {120+81 x-11 x^2+5 x^3-100 x \log (x)-20 x^2 \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}\right ) \, dx-2 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx-10 \int \frac {x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+22 \int \frac {x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+40 \int \frac {x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-162 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {\log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)} \, dx-240 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=2 x \log \left (-\frac {60+5 e^x-x+5 x^2-20 (6+x) \log (x)}{20 \log (x)}\right )-2 \int \frac {-1+x \log (x)}{\log (x)} \, dx+2 \int \frac {120+81 x-11 x^2+5 x^3-100 x \log (x)-20 x^2 \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-2 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx-10 \int \frac {x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+22 \int \frac {x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+40 \int \frac {x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-162 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {\log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)} \, dx-240 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=2 x \log \left (-\frac {60+5 e^x-x+5 x^2-20 (6+x) \log (x)}{20 \log (x)}\right )-2 \int \left (x-\frac {1}{\log (x)}\right ) \, dx+2 \int \left (\frac {120}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}+\frac {81 x}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}-\frac {11 x^2}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}+\frac {5 x^3}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}-\frac {100 x \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}-\frac {20 x^2 \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)}\right ) \, dx-2 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx-10 \int \frac {x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+22 \int \frac {x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+40 \int \frac {x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-162 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {\log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)} \, dx-240 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=-x^2+2 x \log \left (-\frac {60+5 e^x-x+5 x^2-20 (6+x) \log (x)}{20 \log (x)}\right )+2 \int \frac {1}{\log (x)} \, dx-2 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx+10 \int \frac {x^3}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-10 \int \frac {x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-22 \int \frac {x^2}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+22 \int \frac {x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-40 \int \frac {x^2 \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+40 \int \frac {x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+162 \int \frac {x}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-162 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {x \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {\log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)} \, dx+240 \int \frac {1}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-240 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ &=-x^2+2 x \log \left (-\frac {60+5 e^x-x+5 x^2-20 (6+x) \log (x)}{20 \log (x)}\right )+2 \text {li}(x)-2 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \log (x)} \, dx+10 \int \frac {x^3}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-10 \int \frac {x^2 \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-22 \int \frac {x^2}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+22 \int \frac {x \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-40 \int \frac {x^2 \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+40 \int \frac {x \log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx+162 \int \frac {x}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-162 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {x \log (x)}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-200 \int \frac {\log (x) \log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{-60-5 e^x+x-5 x^2+120 \log (x)+20 x \log (x)} \, dx+240 \int \frac {1}{60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)} \, dx-240 \int \frac {\log \left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right )}{x \left (60+5 e^x-x+5 x^2-120 \log (x)-20 x \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 31, normalized size = 0.97 \begin {gather*} \log ^2\left (\frac {-60-5 e^x+x-5 x^2+20 (6+x) \log (x)}{20 \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((120 + 10*E^x - 2*x + 10*x^2 + (2*x - 10*E^x*x - 20*x^2)*Log[x] + 40*x*Log[x]^2)*Log[(-60 - 5*E^x +
x - 5*x^2 + (120 + 20*x)*Log[x])/(20*Log[x])])/((-60*x - 5*E^x*x + x^2 - 5*x^3)*Log[x] + (120*x + 20*x^2)*Log
[x]^2),x]

[Out]

Log[(-60 - 5*E^x + x - 5*x^2 + 20*(6 + x)*Log[x])/(20*Log[x])]^2

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fricas [A]  time = 0.51, size = 30, normalized size = 0.94 \begin {gather*} \log \left (-\frac {5 \, x^{2} - 20 \, {\left (x + 6\right )} \log \relax (x) - x + 5 \, e^{x} + 60}{20 \, \log \relax (x)}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x*log(x)^2+(-10*exp(x)*x-20*x^2+2*x)*log(x)+10*exp(x)+10*x^2-2*x+120)*log(1/20*((20*x+120)*log(x
)-5*exp(x)-5*x^2+x-60)/log(x))/((20*x^2+120*x)*log(x)^2+(-5*exp(x)*x-5*x^3+x^2-60*x)*log(x)),x, algorithm="fri
cas")

[Out]

log(-1/20*(5*x^2 - 20*(x + 6)*log(x) - x + 5*e^x + 60)/log(x))^2

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x*log(x)^2+(-10*exp(x)*x-20*x^2+2*x)*log(x)+10*exp(x)+10*x^2-2*x+120)*log(1/20*((20*x+120)*log(x
)-5*exp(x)-5*x^2+x-60)/log(x))/((20*x^2+120*x)*log(x)^2+(-5*exp(x)*x-5*x^3+x^2-60*x)*log(x)),x, algorithm="gia
c")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Sign error %%%{ln
(` w`),0%%%}Sign error %%%{ln(` w`),0%%%}2*(ln(-5*sageVARx^2+20*sageVARx*ln(sageVARx)+sageVARx-5*exp(sageVARx)
+120*ln(sageVA

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (40 x \ln \relax (x )^{2}+\left (-10 \,{\mathrm e}^{x} x -20 x^{2}+2 x \right ) \ln \relax (x )+10 \,{\mathrm e}^{x}+10 x^{2}-2 x +120\right ) \ln \left (\frac {\left (20 x +120\right ) \ln \relax (x )-5 \,{\mathrm e}^{x}-5 x^{2}+x -60}{20 \ln \relax (x )}\right )}{\left (20 x^{2}+120 x \right ) \ln \relax (x )^{2}+\left (-5 \,{\mathrm e}^{x} x -5 x^{3}+x^{2}-60 x \right ) \ln \relax (x )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x*ln(x)^2+(-10*exp(x)*x-20*x^2+2*x)*ln(x)+10*exp(x)+10*x^2-2*x+120)*ln(1/20*((20*x+120)*ln(x)-5*exp(x)
-5*x^2+x-60)/ln(x))/((20*x^2+120*x)*ln(x)^2+(-5*exp(x)*x-5*x^3+x^2-60*x)*ln(x)),x)

[Out]

int((40*x*ln(x)^2+(-10*exp(x)*x-20*x^2+2*x)*ln(x)+10*exp(x)+10*x^2-2*x+120)*ln(1/20*((20*x+120)*ln(x)-5*exp(x)
-5*x^2+x-60)/ln(x))/((20*x^2+120*x)*ln(x)^2+(-5*exp(x)*x-5*x^3+x^2-60*x)*ln(x)),x)

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maxima [B]  time = 0.74, size = 125, normalized size = 3.91 \begin {gather*} 2 \, {\left (\log \relax (5) + \log \left (\log \relax (x)\right )\right )} \log \left (5 \, x^{2} - 20 \, {\left (x + 6\right )} \log \relax (x) - x + 5 \, e^{x} + 60\right ) - \log \left (5 \, x^{2} - 20 \, {\left (x + 6\right )} \log \relax (x) - x + 5 \, e^{x} + 60\right )^{2} + 2 \, {\left (\log \left (x^{2} - 4 \, {\left (x + 6\right )} \log \relax (x) - \frac {1}{5} \, x + e^{x} + 12\right ) - \log \left (\log \relax (x)\right )\right )} \log \left (-\frac {5 \, x^{2} - 20 \, {\left (x + 6\right )} \log \relax (x) - x + 5 \, e^{x} + 60}{20 \, \log \relax (x)}\right ) - 2 \, \log \relax (5) \log \left (\log \relax (x)\right ) - \log \left (\log \relax (x)\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x*log(x)^2+(-10*exp(x)*x-20*x^2+2*x)*log(x)+10*exp(x)+10*x^2-2*x+120)*log(1/20*((20*x+120)*log(x
)-5*exp(x)-5*x^2+x-60)/log(x))/((20*x^2+120*x)*log(x)^2+(-5*exp(x)*x-5*x^3+x^2-60*x)*log(x)),x, algorithm="max
ima")

[Out]

2*(log(5) + log(log(x)))*log(5*x^2 - 20*(x + 6)*log(x) - x + 5*e^x + 60) - log(5*x^2 - 20*(x + 6)*log(x) - x +
5*e^x + 60)^2 + 2*(log(x^2 - 4*(x + 6)*log(x) - 1/5*x + e^x + 12) - log(log(x)))*log(-1/20*(5*x^2 - 20*(x + 6
)*log(x) - x + 5*e^x + 60)/log(x)) - 2*log(5)*log(log(x)) - log(log(x))^2

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mupad [B]  time = 1.52, size = 32, normalized size = 1.00 \begin {gather*} {\ln \left (-\frac {\frac {{\mathrm {e}}^x}{4}-\frac {x}{20}-\frac {\ln \relax (x)\,\left (20\,x+120\right )}{20}+\frac {x^2}{4}+3}{\ln \relax (x)}\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(-(exp(x)/4 - x/20 - (log(x)*(20*x + 120))/20 + x^2/4 + 3)/log(x))*(10*exp(x) - 2*x + 40*x*log(x)^2 -
log(x)*(10*x*exp(x) - 2*x + 20*x^2) + 10*x^2 + 120))/(log(x)^2*(120*x + 20*x^2) - log(x)*(60*x + 5*x*exp(x) -
x^2 + 5*x^3)),x)

[Out]

log(-(exp(x)/4 - x/20 - (log(x)*(20*x + 120))/20 + x^2/4 + 3)/log(x))^2

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sympy [A]  time = 7.26, size = 31, normalized size = 0.97 \begin {gather*} \log {\left (\frac {- \frac {x^{2}}{4} + \frac {x}{20} + \frac {\left (20 x + 120\right ) \log {\relax (x )}}{20} - \frac {e^{x}}{4} - 3}{\log {\relax (x )}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x*ln(x)**2+(-10*exp(x)*x-20*x**2+2*x)*ln(x)+10*exp(x)+10*x**2-2*x+120)*ln(1/20*((20*x+120)*ln(x)
-5*exp(x)-5*x**2+x-60)/ln(x))/((20*x**2+120*x)*ln(x)**2+(-5*exp(x)*x-5*x**3+x**2-60*x)*ln(x)),x)

[Out]

log((-x**2/4 + x/20 + (20*x + 120)*log(x)/20 - exp(x)/4 - 3)/log(x))**2

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