Integrand size = 325, antiderivative size = 25 \[ \int \frac {-45 x-78 x^2-48 x^3-12 x^4-x^5+\left (-45-51 x-21 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log \left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )}{\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log ^2\left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )} \, dx=\frac {3+x}{\log \left (\log \left (-\frac {3}{x}+\log (x+\log (5+x (4+x)))\right )\right )} \]
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-45 x-78 x^2-48 x^3-12 x^4-x^5+\left (-45-51 x-21 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log \left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )}{\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log ^2\left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )} \, dx=-\frac {-3-x}{\log \left (\log \left (-\frac {3}{x}+\log \left (x+\log \left (5+4 x+x^2\right )\right )\right )\right )} \]
Integrate[(-45*x - 78*x^2 - 48*x^3 - 12*x^4 - x^5 + (-45 - 51*x - 21*x^2 - 3*x^3)*Log[5 + 4*x + x^2] + (-15*x^2 - 12*x^3 - 3*x^4 + (-15*x - 12*x^2 - 3*x^3)*Log[5 + 4*x + x^2] + (5*x^3 + 4*x^4 + x^5 + (5*x^2 + 4*x^3 + x^4)* Log[5 + 4*x + x^2])*Log[x + Log[5 + 4*x + x^2]])*Log[(-3 + x*Log[x + Log[5 + 4*x + x^2]])/x]*Log[Log[(-3 + x*Log[x + Log[5 + 4*x + x^2]])/x]])/((-15 *x^2 - 12*x^3 - 3*x^4 + (-15*x - 12*x^2 - 3*x^3)*Log[5 + 4*x + x^2] + (5*x ^3 + 4*x^4 + x^5 + (5*x^2 + 4*x^3 + x^4)*Log[5 + 4*x + x^2])*Log[x + Log[5 + 4*x + x^2]])*Log[(-3 + x*Log[x + Log[5 + 4*x + x^2]])/x]*Log[Log[(-3 + x*Log[x + Log[5 + 4*x + x^2]])/x]]^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-x^5-12 x^4-48 x^3-78 x^2+\left (-3 x^3-21 x^2-51 x-45\right ) \log \left (x^2+4 x+5\right )+\left (-3 x^4-12 x^3-15 x^2+\left (-3 x^3-12 x^2-15 x\right ) \log \left (x^2+4 x+5\right )+\left (x^5+4 x^4+5 x^3+\left (x^4+4 x^3+5 x^2\right ) \log \left (x^2+4 x+5\right )\right ) \log \left (\log \left (x^2+4 x+5\right )+x\right )\right ) \log \left (\frac {x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3}{x}\right ) \log \left (\log \left (\frac {x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3}{x}\right )\right )-45 x}{\left (-3 x^4-12 x^3-15 x^2+\left (-3 x^3-12 x^2-15 x\right ) \log \left (x^2+4 x+5\right )+\left (x^5+4 x^4+5 x^3+\left (x^4+4 x^3+5 x^2\right ) \log \left (x^2+4 x+5\right )\right ) \log \left (\log \left (x^2+4 x+5\right )+x\right )\right ) \log \left (\frac {x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3}{x}\right ) \log ^2\left (\log \left (\frac {x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3}{x}\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x^5+12 x^4+48 x^3+78 x^2-\left (-3 x^3-21 x^2-51 x-45\right ) \log \left (x^2+4 x+5\right )-\left (-3 x^4-12 x^3-15 x^2+\left (-3 x^3-12 x^2-15 x\right ) \log \left (x^2+4 x+5\right )+\left (x^5+4 x^4+5 x^3+\left (x^4+4 x^3+5 x^2\right ) \log \left (x^2+4 x+5\right )\right ) \log \left (\log \left (x^2+4 x+5\right )+x\right )\right ) \log \left (\frac {x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3}{x}\right ) \log \left (\log \left (\frac {x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3}{x}\right )\right )+45 x}{x \left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (3-x \log \left (\log \left (x^2+4 x+5\right )+x\right )\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (-\frac {48 x^2}{\left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}-\frac {78 x}{\left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}-\frac {45}{\left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}-\frac {3 (x+3) \log \left (x^2+4 x+5\right )}{x \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}+\frac {1}{\log \left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}-\frac {x^4}{\left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}-\frac {12 x^3}{\left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-\left (\left (x^2+4 x+5\right ) \log \left (x^2+4 x+5\right ) \left (x \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log \left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )-3 (x+3)\right )\right )-x \left (-x^4-12 x^3-48 x^2+\left (x^2+4 x+5\right ) x \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log \left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )-78 x-45\right )}{x \left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (3-x \log \left (\log \left (x^2+4 x+5\right )+x\right )\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {1}{\log \left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}-\frac {(x+3) \left (x^4+9 x^3+21 x^2+3 x^2 \log \left (x^2+4 x+5\right )+12 x \log \left (x^2+4 x+5\right )+15 \log \left (x^2+4 x+5\right )+15 x\right )}{x \left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {1}{\log \left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}-\frac {(x+3) \left (x^4+9 x^3+21 x^2+3 x^2 \log \left (x^2+4 x+5\right )+12 x \log \left (x^2+4 x+5\right )+15 \log \left (x^2+4 x+5\right )+15 x\right )}{x \left (x^2+4 x+5\right ) \left (\log \left (x^2+4 x+5\right )+x\right ) \left (x \log \left (\log \left (x^2+4 x+5\right )+x\right )-3\right ) \log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right ) \log ^2\left (\log \left (\log \left (\log \left (x^2+4 x+5\right )+x\right )-\frac {3}{x}\right )\right )}\right )dx\) |
Int[(-45*x - 78*x^2 - 48*x^3 - 12*x^4 - x^5 + (-45 - 51*x - 21*x^2 - 3*x^3 )*Log[5 + 4*x + x^2] + (-15*x^2 - 12*x^3 - 3*x^4 + (-15*x - 12*x^2 - 3*x^3 )*Log[5 + 4*x + x^2] + (5*x^3 + 4*x^4 + x^5 + (5*x^2 + 4*x^3 + x^4)*Log[5 + 4*x + x^2])*Log[x + Log[5 + 4*x + x^2]])*Log[(-3 + x*Log[x + Log[5 + 4*x + x^2]])/x]*Log[Log[(-3 + x*Log[x + Log[5 + 4*x + x^2]])/x]])/((-15*x^2 - 12*x^3 - 3*x^4 + (-15*x - 12*x^2 - 3*x^3)*Log[5 + 4*x + x^2] + (5*x^3 + 4 *x^4 + x^5 + (5*x^2 + 4*x^3 + x^4)*Log[5 + 4*x + x^2])*Log[x + Log[5 + 4*x + x^2]])*Log[(-3 + x*Log[x + Log[5 + 4*x + x^2]])/x]*Log[Log[(-3 + x*Log[ x + Log[5 + 4*x + x^2]])/x]]^2),x]
3.15.12.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.44
\[\frac {3+x}{\ln \left (-\ln \left (x \right )+\ln \left (x \ln \left (\ln \left (x^{2}+4 x +5\right )+x \right )-3\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x^{2}+4 x +5\right )+x \right )-3\right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x^{2}+4 x +5\right )+x \right )-3\right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x \ln \left (\ln \left (x^{2}+4 x +5\right )+x \right )-3\right )}{x}\right )+\operatorname {csgn}\left (i \left (x \ln \left (\ln \left (x^{2}+4 x +5\right )+x \right )-3\right )\right )\right )}{2}\right )}\]
int(((((x^4+4*x^3+5*x^2)*ln(x^2+4*x+5)+x^5+4*x^4+5*x^3)*ln(ln(x^2+4*x+5)+x )+(-3*x^3-12*x^2-15*x)*ln(x^2+4*x+5)-3*x^4-12*x^3-15*x^2)*ln((x*ln(ln(x^2+ 4*x+5)+x)-3)/x)*ln(ln((x*ln(ln(x^2+4*x+5)+x)-3)/x))+(-3*x^3-21*x^2-51*x-45 )*ln(x^2+4*x+5)-x^5-12*x^4-48*x^3-78*x^2-45*x)/(((x^4+4*x^3+5*x^2)*ln(x^2+ 4*x+5)+x^5+4*x^4+5*x^3)*ln(ln(x^2+4*x+5)+x)+(-3*x^3-12*x^2-15*x)*ln(x^2+4* x+5)-3*x^4-12*x^3-15*x^2)/ln((x*ln(ln(x^2+4*x+5)+x)-3)/x)/ln(ln((x*ln(ln(x ^2+4*x+5)+x)-3)/x))^2,x)
(3+x)/ln(-ln(x)+ln(x*ln(ln(x^2+4*x+5)+x)-3)-1/2*I*Pi*csgn(I/x*(x*ln(ln(x^2 +4*x+5)+x)-3))*(-csgn(I/x*(x*ln(ln(x^2+4*x+5)+x)-3))+csgn(I/x))*(-csgn(I/x *(x*ln(ln(x^2+4*x+5)+x)-3))+csgn(I*(x*ln(ln(x^2+4*x+5)+x)-3))))
Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-45 x-78 x^2-48 x^3-12 x^4-x^5+\left (-45-51 x-21 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log \left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )}{\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log ^2\left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )} \, dx=\frac {x + 3}{\log \left (\log \left (\frac {x \log \left (x + \log \left (x^{2} + 4 \, x + 5\right )\right ) - 3}{x}\right )\right )} \]
integrate(((((x^4+4*x^3+5*x^2)*log(x^2+4*x+5)+x^5+4*x^4+5*x^3)*log(log(x^2 +4*x+5)+x)+(-3*x^3-12*x^2-15*x)*log(x^2+4*x+5)-3*x^4-12*x^3-15*x^2)*log((x *log(log(x^2+4*x+5)+x)-3)/x)*log(log((x*log(log(x^2+4*x+5)+x)-3)/x))+(-3*x ^3-21*x^2-51*x-45)*log(x^2+4*x+5)-x^5-12*x^4-48*x^3-78*x^2-45*x)/(((x^4+4* x^3+5*x^2)*log(x^2+4*x+5)+x^5+4*x^4+5*x^3)*log(log(x^2+4*x+5)+x)+(-3*x^3-1 2*x^2-15*x)*log(x^2+4*x+5)-3*x^4-12*x^3-15*x^2)/log((x*log(log(x^2+4*x+5)+ x)-3)/x)/log(log((x*log(log(x^2+4*x+5)+x)-3)/x))^2,x, algorithm=\
Timed out. \[ \int \frac {-45 x-78 x^2-48 x^3-12 x^4-x^5+\left (-45-51 x-21 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log \left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )}{\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log ^2\left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )} \, dx=\text {Timed out} \]
integrate(((((x**4+4*x**3+5*x**2)*ln(x**2+4*x+5)+x**5+4*x**4+5*x**3)*ln(ln (x**2+4*x+5)+x)+(-3*x**3-12*x**2-15*x)*ln(x**2+4*x+5)-3*x**4-12*x**3-15*x* *2)*ln((x*ln(ln(x**2+4*x+5)+x)-3)/x)*ln(ln((x*ln(ln(x**2+4*x+5)+x)-3)/x))+ (-3*x**3-21*x**2-51*x-45)*ln(x**2+4*x+5)-x**5-12*x**4-48*x**3-78*x**2-45*x )/(((x**4+4*x**3+5*x**2)*ln(x**2+4*x+5)+x**5+4*x**4+5*x**3)*ln(ln(x**2+4*x +5)+x)+(-3*x**3-12*x**2-15*x)*ln(x**2+4*x+5)-3*x**4-12*x**3-15*x**2)/ln((x *ln(ln(x**2+4*x+5)+x)-3)/x)/ln(ln((x*ln(ln(x**2+4*x+5)+x)-3)/x))**2,x)
Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-45 x-78 x^2-48 x^3-12 x^4-x^5+\left (-45-51 x-21 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log \left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )}{\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log ^2\left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )} \, dx=\frac {x + 3}{\log \left (\log \left (x \log \left (x + \log \left (x^{2} + 4 \, x + 5\right )\right ) - 3\right ) - \log \left (x\right )\right )} \]
integrate(((((x^4+4*x^3+5*x^2)*log(x^2+4*x+5)+x^5+4*x^4+5*x^3)*log(log(x^2 +4*x+5)+x)+(-3*x^3-12*x^2-15*x)*log(x^2+4*x+5)-3*x^4-12*x^3-15*x^2)*log((x *log(log(x^2+4*x+5)+x)-3)/x)*log(log((x*log(log(x^2+4*x+5)+x)-3)/x))+(-3*x ^3-21*x^2-51*x-45)*log(x^2+4*x+5)-x^5-12*x^4-48*x^3-78*x^2-45*x)/(((x^4+4* x^3+5*x^2)*log(x^2+4*x+5)+x^5+4*x^4+5*x^3)*log(log(x^2+4*x+5)+x)+(-3*x^3-1 2*x^2-15*x)*log(x^2+4*x+5)-3*x^4-12*x^3-15*x^2)/log((x*log(log(x^2+4*x+5)+ x)-3)/x)/log(log((x*log(log(x^2+4*x+5)+x)-3)/x))^2,x, algorithm=\
Time = 5.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {-45 x-78 x^2-48 x^3-12 x^4-x^5+\left (-45-51 x-21 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log \left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )}{\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log ^2\left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )} \, dx=\frac {x + 3}{\log \left (\log \left (x \log \left (x + \log \left (x^{2} + 4 \, x + 5\right )\right ) - 3\right ) - \log \left (x\right )\right )} \]
integrate(((((x^4+4*x^3+5*x^2)*log(x^2+4*x+5)+x^5+4*x^4+5*x^3)*log(log(x^2 +4*x+5)+x)+(-3*x^3-12*x^2-15*x)*log(x^2+4*x+5)-3*x^4-12*x^3-15*x^2)*log((x *log(log(x^2+4*x+5)+x)-3)/x)*log(log((x*log(log(x^2+4*x+5)+x)-3)/x))+(-3*x ^3-21*x^2-51*x-45)*log(x^2+4*x+5)-x^5-12*x^4-48*x^3-78*x^2-45*x)/(((x^4+4* x^3+5*x^2)*log(x^2+4*x+5)+x^5+4*x^4+5*x^3)*log(log(x^2+4*x+5)+x)+(-3*x^3-1 2*x^2-15*x)*log(x^2+4*x+5)-3*x^4-12*x^3-15*x^2)/log((x*log(log(x^2+4*x+5)+ x)-3)/x)/log(log((x*log(log(x^2+4*x+5)+x)-3)/x))^2,x, algorithm=\
Time = 18.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-45 x-78 x^2-48 x^3-12 x^4-x^5+\left (-45-51 x-21 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log \left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )}{\left (-15 x^2-12 x^3-3 x^4+\left (-15 x-12 x^2-3 x^3\right ) \log \left (5+4 x+x^2\right )+\left (5 x^3+4 x^4+x^5+\left (5 x^2+4 x^3+x^4\right ) \log \left (5+4 x+x^2\right )\right ) \log \left (x+\log \left (5+4 x+x^2\right )\right )\right ) \log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right ) \log ^2\left (\log \left (\frac {-3+x \log \left (x+\log \left (5+4 x+x^2\right )\right )}{x}\right )\right )} \, dx=\frac {x+3}{\ln \left (\ln \left (\frac {x\,\ln \left (x+\ln \left (x^2+4\,x+5\right )\right )-3}{x}\right )\right )} \]
int((45*x + log(4*x + x^2 + 5)*(51*x + 21*x^2 + 3*x^3 + 45) + 78*x^2 + 48* x^3 + 12*x^4 + x^5 + log(log((x*log(x + log(4*x + x^2 + 5)) - 3)/x))*log(( x*log(x + log(4*x + x^2 + 5)) - 3)/x)*(log(4*x + x^2 + 5)*(15*x + 12*x^2 + 3*x^3) - log(x + log(4*x + x^2 + 5))*(log(4*x + x^2 + 5)*(5*x^2 + 4*x^3 + x^4) + 5*x^3 + 4*x^4 + x^5) + 15*x^2 + 12*x^3 + 3*x^4))/(log(log((x*log(x + log(4*x + x^2 + 5)) - 3)/x))^2*log((x*log(x + log(4*x + x^2 + 5)) - 3)/ x)*(log(4*x + x^2 + 5)*(15*x + 12*x^2 + 3*x^3) - log(x + log(4*x + x^2 + 5 ))*(log(4*x + x^2 + 5)*(5*x^2 + 4*x^3 + x^4) + 5*x^3 + 4*x^4 + x^5) + 15*x ^2 + 12*x^3 + 3*x^4)),x)