\(\int \frac {5 x^2+2 x^3+x^4+(-10 x-4 x^2-2 x^3) \log (\frac {5+2 x+x^2}{5 x+e^3 x})+(5+2 x+x^2) \log ^2(\frac {5+2 x+x^2}{5 x+e^3 x})+(5-x^2+(5+2 x+x^2) \log (\frac {5+2 x+x^2}{5 x+e^3 x})) \log (\log (3))}{5 x^2+2 x^3+x^4+(-10 x-4 x^2-2 x^3) \log (\frac {5+2 x+x^2}{5 x+e^3 x})+(5+2 x+x^2) \log ^2(\frac {5+2 x+x^2}{5 x+e^3 x})} \, dx\) [6257]

Optimal result

Integrand size = 208, antiderivative size = 35 \[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=x+\frac {x \log (\log (3))}{x-\frac {x^2}{\log \left (\frac {2+\frac {5}{x}+x}{5+e^3}\right )}} \]

[Out]

ln(ln(3))/(x-x^2/ln((x+5/x+2)/(exp(3)+5)))*x+x

Rubi [F]

\[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=\int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx \]

[In]

Int[(5*x^2 + 2*x^3 + x^4 + (-10*x - 4*x^2 - 2*x^3)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)] + (5 + 2*x + x^2)*Log[(5
+ 2*x + x^2)/(5*x + E^3*x)]^2 + (5 - x^2 + (5 + 2*x + x^2)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)])*Log[Log[3]])/(
5*x^2 + 2*x^3 + x^4 + (-10*x - 4*x^2 - 2*x^3)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)] + (5 + 2*x + x^2)*Log[(5 + 2*
x + x^2)/(5*x + E^3*x)]^2),x]

[Out]

-5*Defer[Int][(x + Log[5 + E^3] - Log[2 + 5/x + x])^(-2), x] + (5 - Log[Log[3]])*Defer[Int][(x + Log[5 + E^3]
- Log[2 + 5/x + x])^(-2), x] + ((25*I)/2)*Defer[Int][1/(((-2 + 4*I) - 2*x)*(x + Log[5 + E^3] - Log[2 + 5/x + x
])^2), x] - ((5*I)/2)*(5 - Log[Log[3]])*Defer[Int][1/(((-2 + 4*I) - 2*x)*(x + Log[5 + E^3] - Log[2 + 5/x + x])
^2), x] + ((5*I)/2)*Log[Log[3]]*Defer[Int][1/(((-2 + 4*I) - 2*x)*(x + Log[5 + E^3] - Log[2 + 5/x + x])^2), x]
+ Defer[Int][x^2/(x + Log[5 + E^3] - Log[2 + 5/x + x])^2, x] + (10 + 5*I)*Defer[Int][1/(((2 - 4*I) + 2*x)*(x +
Log[5 + E^3] - Log[2 + 5/x + x])^2), x] - (2 + I)*(5 - Log[Log[3]])*Defer[Int][1/(((2 - 4*I) + 2*x)*(x + Log[
5 + E^3] - Log[2 + 5/x + x])^2), x] + (10 + (15*I)/2)*Defer[Int][1/(((2 + 4*I) + 2*x)*(x + Log[5 + E^3] - Log[
2 + 5/x + x])^2), x] - (2 + (3*I)/2)*(5 - Log[Log[3]])*Defer[Int][1/(((2 + 4*I) + 2*x)*(x + Log[5 + E^3] - Log
[2 + 5/x + x])^2), x] + ((5*I)/2)*Log[Log[3]]*Defer[Int][1/(((2 + 4*I) + 2*x)*(x + Log[5 + E^3] - Log[2 + 5/x
+ x])^2), x] + Log[Log[3]]*Defer[Int][Log[(5 + 2*x + x^2)/((5 + E^3)*x)]/(x + Log[5 + E^3] - Log[2 + 5/x + x])
^2, x] - 2*Defer[Int][(x*Log[(5 + 2*x + x^2)/((5 + E^3)*x)])/(x + Log[5 + E^3] - Log[2 + 5/x + x])^2, x] + Def
er[Int][Log[(5 + 2*x + x^2)/((5 + E^3)*x)]^2/(x + Log[5 + E^3] - Log[2 + 5/x + x])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 x^3+x^4+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )-\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right ) (2 x-\log (\log (3)))-x^2 (-5+\log (\log (3)))+5 \log (\log (3))}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx \\ & = \int \left (\frac {2 x^3}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {x^4}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {x^2 (-5+\log (\log (3)))}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {5 \log (\log (3))}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right ) (-2 x+\log (\log (3)))}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx \\ & = 2 \int \frac {x^3}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {x^2}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5 \log (\log (3))) \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^4}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right ) (-2 x+\log (\log (3)))}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx \\ & = 2 \int \left (-\frac {2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {x}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {10-x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+(5-\log (\log (3))) \int \left (\frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {-5-2 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+(5 \log (\log (3))) \int \left (\frac {i}{2 ((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {i}{2 ((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+\int \left (-\frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {2 x}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {5+12 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \left (-\frac {2 x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right ) \log (\log (3))}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx \\ & = 2 \int \frac {10-x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {-5-2 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {5+12 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx \\ & = 2 \int \left (\frac {10}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \left (-\frac {5}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}-\frac {2 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \left (\frac {5}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {12 x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx \\ & = -\left (2 \int \frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx\right )-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+5 \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+12 \int \frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+20 \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-(2 (5-\log (\log (3)))) \int \frac {x}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-(5 (5-\log (\log (3)))) \int \frac {1}{\left (5+2 x+x^2\right ) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx \\ & = -\left (2 \int \left (\frac {1+\frac {i}{2}}{((2-4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {1-\frac {i}{2}}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx\right )-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+5 \int \left (\frac {i}{2 ((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {i}{2 ((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+12 \int \left (\frac {1+\frac {i}{2}}{((2-4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {1-\frac {i}{2}}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+20 \int \left (\frac {i}{2 ((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {i}{2 ((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-(2 (5-\log (\log (3)))) \int \left (\frac {1+\frac {i}{2}}{((2-4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {1-\frac {i}{2}}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx-(5 (5-\log (\log (3)))) \int \left (\frac {i}{2 ((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}+\frac {i}{2 ((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2}\right ) \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx \\ & = \frac {5}{2} i \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {5}{2} i \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+10 i \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+10 i \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-2 \int \frac {x \log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-(2-i) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-(2+i) \int \frac {1}{((2-4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-4 \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(12-6 i) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(12+6 i) \int \frac {1}{((2-4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\frac {1}{2} (5 i (5-\log (\log (3)))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\frac {1}{2} (5 i (5-\log (\log (3)))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+(5-\log (\log (3))) \int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-((2-i) (5-\log (\log (3)))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-((2+i) (5-\log (\log (3)))) \int \frac {1}{((2-4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((-2+4 i)-2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\frac {1}{2} (5 i \log (\log (3))) \int \frac {1}{((2+4 i)+2 x) \left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\log (\log (3)) \int \frac {\log \left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx-\int \frac {1}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {x^2}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx+\int \frac {\log ^2\left (\frac {5+2 x+x^2}{\left (5+e^3\right ) x}\right )}{\left (x+\log \left (5+e^3\right )-\log \left (2+\frac {5}{x}+x\right )\right )^2} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=\int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx \]

[In]

Integrate[(5*x^2 + 2*x^3 + x^4 + (-10*x - 4*x^2 - 2*x^3)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)] + (5 + 2*x + x^2)*
Log[(5 + 2*x + x^2)/(5*x + E^3*x)]^2 + (5 - x^2 + (5 + 2*x + x^2)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)])*Log[Log[
3]])/(5*x^2 + 2*x^3 + x^4 + (-10*x - 4*x^2 - 2*x^3)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)] + (5 + 2*x + x^2)*Log[(
5 + 2*x + x^2)/(5*x + E^3*x)]^2),x]

[Out]

Integrate[(5*x^2 + 2*x^3 + x^4 + (-10*x - 4*x^2 - 2*x^3)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)] + (5 + 2*x + x^2)*
Log[(5 + 2*x + x^2)/(5*x + E^3*x)]^2 + (5 - x^2 + (5 + 2*x + x^2)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)])*Log[Log[
3]])/(5*x^2 + 2*x^3 + x^4 + (-10*x - 4*x^2 - 2*x^3)*Log[(5 + 2*x + x^2)/(5*x + E^3*x)] + (5 + 2*x + x^2)*Log[(
5 + 2*x + x^2)/(5*x + E^3*x)]^2), x]

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00

method	result	size
risch	\(x -\frac {x \ln \left (\ln \left (3\right )\right )}{x -\ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )}\)	\(35\)
default	\(x +\frac {-\ln \left (\ln \left (3\right )\right ) \ln \left (\frac {x^{2}+2 x +5}{x}\right )+\ln \left (\ln \left (3\right )\right ) \ln \left ({\mathrm e}^{3}+5\right )}{\ln \left ({\mathrm e}^{3}+5\right )+x -\ln \left (\frac {x^{2}+2 x +5}{x}\right )}\)	\(56\)
norman	\(\frac {x^{2}-\ln \left (\ln \left (3\right )\right ) \ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )-x \ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )}{x -\ln \left (\frac {x^{2}+2 x +5}{x \,{\mathrm e}^{3}+5 x}\right )}\)	\(80\)
parallelrisch	\(-\frac {\ln \left (\ln \left (3\right )\right ) x -x^{2}+\ln \left (\frac {x^{2}+2 x +5}{x \left ({\mathrm e}^{3}+5\right )}\right ) x +4 x -4 \ln \left (\frac {x^{2}+2 x +5}{x \left ({\mathrm e}^{3}+5\right )}\right )}{x -\ln \left (\frac {x^{2}+2 x +5}{x \left ({\mathrm e}^{3}+5\right )}\right )}\)	\(84\)

[In]

int((((x^2+2*x+5)*ln((x^2+2*x+5)/(x*exp(3)+5*x))-x^2+5)*ln(ln(3))+(x^2+2*x+5)*ln((x^2+2*x+5)/(x*exp(3)+5*x))^2
+(-2*x^3-4*x^2-10*x)*ln((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2)/((x^2+2*x+5)*ln((x^2+2*x+5)/(x*exp(3)+5*x
))^2+(-2*x^3-4*x^2-10*x)*ln((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2),x,method=_RETURNVERBOSE)

[Out]

x-x*ln(ln(3))/(x-ln((x^2+2*x+5)/(x*exp(3)+5*x)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.71 \[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=\frac {x^{2} - x \log \left (\frac {x^{2} + 2 \, x + 5}{x e^{3} + 5 \, x}\right ) - x \log \left (\log \left (3\right )\right )}{x - \log \left (\frac {x^{2} + 2 \, x + 5}{x e^{3} + 5 \, x}\right )} \]

[In]

integrate((((x^2+2*x+5)*log((x^2+2*x+5)/(x*exp(3)+5*x))-x^2+5)*log(log(3))+(x^2+2*x+5)*log((x^2+2*x+5)/(x*exp(
3)+5*x))^2+(-2*x^3-4*x^2-10*x)*log((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2)/((x^2+2*x+5)*log((x^2+2*x+5)/(
x*exp(3)+5*x))^2+(-2*x^3-4*x^2-10*x)*log((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2),x, algorithm="fricas")

[Out]

(x^2 - x*log((x^2 + 2*x + 5)/(x*e^3 + 5*x)) - x*log(log(3)))/(x - log((x^2 + 2*x + 5)/(x*e^3 + 5*x)))

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=x + \frac {x \log {\left (\log {\left (3 \right )} \right )}}{- x + \log {\left (\frac {x^{2} + 2 x + 5}{5 x + x e^{3}} \right )}} \]

[In]

integrate((((x**2+2*x+5)*ln((x**2+2*x+5)/(x*exp(3)+5*x))-x**2+5)*ln(ln(3))+(x**2+2*x+5)*ln((x**2+2*x+5)/(x*exp
(3)+5*x))**2+(-2*x**3-4*x**2-10*x)*ln((x**2+2*x+5)/(x*exp(3)+5*x))+x**4+2*x**3+5*x**2)/((x**2+2*x+5)*ln((x**2+
2*x+5)/(x*exp(3)+5*x))**2+(-2*x**3-4*x**2-10*x)*ln((x**2+2*x+5)/(x*exp(3)+5*x))+x**4+2*x**3+5*x**2),x)

[Out]

x + x*log(log(3))/(-x + log((x**2 + 2*x + 5)/(5*x + x*exp(3))))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=\frac {x^{2} + x {\left (\log \left (e^{3} + 5\right ) - \log \left (\log \left (3\right )\right )\right )} - x \log \left (x^{2} + 2 \, x + 5\right ) + x \log \left (x\right )}{x - \log \left (x^{2} + 2 \, x + 5\right ) + \log \left (x\right ) + \log \left (e^{3} + 5\right )} \]

[In]

integrate((((x^2+2*x+5)*log((x^2+2*x+5)/(x*exp(3)+5*x))-x^2+5)*log(log(3))+(x^2+2*x+5)*log((x^2+2*x+5)/(x*exp(
3)+5*x))^2+(-2*x^3-4*x^2-10*x)*log((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2)/((x^2+2*x+5)*log((x^2+2*x+5)/(
x*exp(3)+5*x))^2+(-2*x^3-4*x^2-10*x)*log((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2),x, algorithm="maxima")

[Out]

(x^2 + x*(log(e^3 + 5) - log(log(3))) - x*log(x^2 + 2*x + 5) + x*log(x))/(x - log(x^2 + 2*x + 5) + log(x) + lo
g(e^3 + 5))

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=\frac {x^{2} - x \log \left (x^{2} + 2 \, x + 5\right ) + x \log \left (x e^{3} + 5 \, x\right ) - x \log \left (\log \left (3\right )\right )}{x - \log \left (x^{2} + 2 \, x + 5\right ) + \log \left (x e^{3} + 5 \, x\right )} \]

[In]

integrate((((x^2+2*x+5)*log((x^2+2*x+5)/(x*exp(3)+5*x))-x^2+5)*log(log(3))+(x^2+2*x+5)*log((x^2+2*x+5)/(x*exp(
3)+5*x))^2+(-2*x^3-4*x^2-10*x)*log((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2)/((x^2+2*x+5)*log((x^2+2*x+5)/(
x*exp(3)+5*x))^2+(-2*x^3-4*x^2-10*x)*log((x^2+2*x+5)/(x*exp(3)+5*x))+x^4+2*x^3+5*x^2),x, algorithm="giac")

[Out]

(x^2 - x*log(x^2 + 2*x + 5) + x*log(x*e^3 + 5*x) - x*log(log(3)))/(x - log(x^2 + 2*x + 5) + log(x*e^3 + 5*x))

Mupad [B] (verification not implemented)

Time = 13.49 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.29 \[ \int \frac {5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5-x^2+\left (5+2 x+x^2\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )\right ) \log (\log (3))}{5 x^2+2 x^3+x^4+\left (-10 x-4 x^2-2 x^3\right ) \log \left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )+\left (5+2 x+x^2\right ) \log ^2\left (\frac {5+2 x+x^2}{5 x+e^3 x}\right )} \, dx=-\frac {x\,\ln \left (\frac {x^2+2\,x+5}{5\,x+x\,{\mathrm {e}}^3}\right )-x^2+\ln \left (\frac {x^2+2\,x+5}{5\,x+x\,{\mathrm {e}}^3}\right )\,\ln \left (\ln \left (3\right )\right )}{x-\ln \left (\frac {x^2+2\,x+5}{5\,x+x\,{\mathrm {e}}^3}\right )} \]

[In]

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

[Out]

-(x*log((2*x + x^2 + 5)/(5*x + x*exp(3))) - x^2 + log((2*x + x^2 + 5)/(5*x + x*exp(3)))*log(log(3)))/(x - log(
(2*x + x^2 + 5)/(5*x + x*exp(3))))