Integrand size = 176, antiderivative size = 27 \begin {dmath*} \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\frac {1}{9} x^2 \left (5+3 \left (x+\log \left (\frac {\log (x)}{x}\right )+\log \left (\log \left (x^2\right )\right )\right )\right )^2 \end {dmath*}
Time = 0.95 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \begin {dmath*} \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\frac {1}{9} x^2 \left (5+3 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )\right )^2 \end {dmath*}
Integrate[((60*x + 36*x^2)*Log[x] + (30*x + 18*x^2 + (20*x + 72*x^2 + 36*x ^3)*Log[x])*Log[x^2] + (36*x*Log[x] + (18*x + (42*x + 54*x^2)*Log[x])*Log[ x^2])*Log[Log[x]/x] + 18*x*Log[x]*Log[x^2]*Log[Log[x]/x]^2 + (36*x*Log[x] + (18*x + (42*x + 54*x^2)*Log[x])*Log[x^2] + 36*x*Log[x]*Log[x^2]*Log[Log[ x]/x])*Log[Log[x^2]] + 18*x*Log[x]*Log[x^2]*Log[Log[x^2]]^2)/(9*Log[x]*Log [x^2]),x]
Time = 0.64 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {27, 27, 7239, 7238}
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.
| \(\displaystyle \int \frac {18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+\left (\left (\left (54 x^2+42 x\right ) \log (x)+18 x\right ) \log \left (x^2\right )+36 x \log (x)\right ) \log \left (\frac {\log (x)}{x}\right )+\left (36 x^2+60 x\right ) \log (x)+\left (36 x \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right ) \log (x)+\left (\left (54 x^2+42 x\right ) \log (x)+18 x\right ) \log \left (x^2\right )+36 x \log (x)\right ) \log \left (\log \left (x^2\right )\right )+\left (18 x^2+\left (36 x^3+72 x^2+20 x\right ) \log (x)+30 x\right ) \log \left (x^2\right )}{9 \log (x) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {2 \left (9 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+3 \left (6 x \log (x)+\left (3 x+\left (9 x^2+7 x\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+9 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+6 \left (3 x^2+5 x\right ) \log (x)+\left (9 x^2+15 x+2 \left (9 x^3+18 x^2+5 x\right ) \log (x)\right ) \log \left (x^2\right )+3 \left (6 x \log (x)+6 x \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right ) \log (x)+\left (3 x+\left (9 x^2+7 x\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )\right )}{\log (x) \log \left (x^2\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \int \frac {9 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+3 \left (6 x \log (x)+\left (3 x+\left (9 x^2+7 x\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+9 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )+6 \left (3 x^2+5 x\right ) \log (x)+\left (9 x^2+15 x+2 \left (9 x^3+18 x^2+5 x\right ) \log (x)\right ) \log \left (x^2\right )+3 \left (6 x \log (x)+6 x \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right ) \log (x)+\left (3 x+\left (9 x^2+7 x\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\log \left (x^2\right )\right )}{\log (x) \log \left (x^2\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2}{9} \int \frac {x \left (3 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )+5\right ) \left (3 \log \left (x^2\right )+\log (x) \left (\log \left (x^2\right ) \left (6 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )+2\right )+6\right )\right )}{\log (x) \log \left (x^2\right )}dx\) |
| \(\Big \downarrow \) 7238 |
| \(\displaystyle \frac {1}{9} x^2 \left (3 \log \left (\log \left (x^2\right )\right )+3 x+3 \log \left (\frac {\log (x)}{x}\right )+5\right )^2\) |
Int[((60*x + 36*x^2)*Log[x] + (30*x + 18*x^2 + (20*x + 72*x^2 + 36*x^3)*Lo g[x])*Log[x^2] + (36*x*Log[x] + (18*x + (42*x + 54*x^2)*Log[x])*Log[x^2])* Log[Log[x]/x] + 18*x*Log[x]*Log[x^2]*Log[Log[x]/x]^2 + (36*x*Log[x] + (18* x + (42*x + 54*x^2)*Log[x])*Log[x^2] + 36*x*Log[x]*Log[x^2]*Log[Log[x]/x]) *Log[Log[x^2]] + 18*x*Log[x]*Log[x^2]*Log[Log[x^2]]^2)/(9*Log[x]*Log[x^2]) ,x]
3.4.14.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y* z, u*z^(n - m), x]}, Simp[q*y^(m + 1)*(z^(m + 1)/(m + 1)), x] /; !FalseQ[q ]] /; FreeQ[{m, n}, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(28)=56\).
Time = 4.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.70
| method | result | size |
| parallelrisch | \(x^{2} {\ln \left (\ln \left (x^{2}\right )\right )}^{2}+\frac {10 x^{2} \ln \left (\ln \left (x^{2}\right )\right )}{3}+\frac {10 x^{2} \ln \left (\frac {\ln \left (x \right )}{x}\right )}{3}+x^{4}+\frac {10 x^{3}}{3}+\frac {25 x^{2}}{9}+2 \ln \left (\frac {\ln \left (x \right )}{x}\right ) \ln \left (\ln \left (x^{2}\right )\right ) x^{2}+2 x^{3} \ln \left (\ln \left (x^{2}\right )\right )+2 \ln \left (\frac {\ln \left (x \right )}{x}\right ) x^{3}+\ln \left (\frac {\ln \left (x \right )}{x}\right )^{2} x^{2}\) | \(100\) |
| risch | \(\text {Expression too large to display}\) | \(935\) |
int(1/9*(18*x*ln(x)*ln(x^2)*ln(ln(x^2))^2+(36*x*ln(x)*ln(x^2)*ln(ln(x)/x)+ ((54*x^2+42*x)*ln(x)+18*x)*ln(x^2)+36*x*ln(x))*ln(ln(x^2))+18*x*ln(x)*ln(x ^2)*ln(ln(x)/x)^2+(((54*x^2+42*x)*ln(x)+18*x)*ln(x^2)+36*x*ln(x))*ln(ln(x) /x)+((36*x^3+72*x^2+20*x)*ln(x)+18*x^2+30*x)*ln(x^2)+(36*x^2+60*x)*ln(x))/ ln(x)/ln(x^2),x,method=_RETURNVERBOSE)
x^2*ln(ln(x^2))^2+10/3*x^2*ln(ln(x^2))+10/3*x^2*ln(ln(x)/x)+x^4+10/3*x^3+2 5/9*x^2+2*ln(ln(x)/x)*ln(ln(x^2))*x^2+2*x^3*ln(ln(x^2))+2*ln(ln(x)/x)*x^3+ ln(ln(x)/x)^2*x^2
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.26 \begin {dmath*} \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=x^{4} + x^{2} \log \left (2\right )^{2} + x^{2} \log \left (x\right )^{2} + 4 \, x^{2} \log \left (\frac {\log \left (x\right )}{x}\right )^{2} + \frac {10}{3} \, x^{3} + \frac {25}{9} \, x^{2} + \frac {2}{3} \, {\left (3 \, x^{3} + 5 \, x^{2}\right )} \log \left (2\right ) + \frac {2}{3} \, {\left (3 \, x^{3} + 3 \, x^{2} \log \left (2\right ) + 5 \, x^{2}\right )} \log \left (x\right ) + \frac {4}{3} \, {\left (3 \, x^{3} + 3 \, x^{2} \log \left (2\right ) + 3 \, x^{2} \log \left (x\right ) + 5 \, x^{2}\right )} \log \left (\frac {\log \left (x\right )}{x}\right ) \end {dmath*}
integrate(1/9*(18*x*log(x)*log(x^2)*log(log(x^2))^2+(36*x*log(x)*log(x^2)* log(log(x)/x)+((54*x^2+42*x)*log(x)+18*x)*log(x^2)+36*x*log(x))*log(log(x^ 2))+18*x*log(x)*log(x^2)*log(log(x)/x)^2+(((54*x^2+42*x)*log(x)+18*x)*log( x^2)+36*x*log(x))*log(log(x)/x)+((36*x^3+72*x^2+20*x)*log(x)+18*x^2+30*x)* log(x^2)+(36*x^2+60*x)*log(x))/log(x)/log(x^2),x, algorithm=\
x^4 + x^2*log(2)^2 + x^2*log(x)^2 + 4*x^2*log(log(x)/x)^2 + 10/3*x^3 + 25/ 9*x^2 + 2/3*(3*x^3 + 5*x^2)*log(2) + 2/3*(3*x^3 + 3*x^2*log(2) + 5*x^2)*lo g(x) + 4/3*(3*x^3 + 3*x^2*log(2) + 3*x^2*log(x) + 5*x^2)*log(log(x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (27) = 54\).
Time = 0.44 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.22 \begin {dmath*} \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=x^{4} + x^{3} \cdot \left (\frac {10}{3} - 2 \log {\left (2 \right )}\right ) + x^{2} \log {\left (x \right )}^{2} + 4 x^{2} \log {\left (2 \log {\left (x \right )} \right )}^{2} + x^{2} \left (- \frac {10 \log {\left (2 \right )}}{3} + \log {\left (2 \right )}^{2} + \frac {25}{9}\right ) + \left (- 2 x^{3} - \frac {10 x^{2}}{3} + 2 x^{2} \log {\left (2 \right )}\right ) \log {\left (x \right )} + \left (4 x^{3} - 4 x^{2} \log {\left (x \right )} - 4 x^{2} \log {\left (2 \right )} + \frac {20 x^{2}}{3}\right ) \log {\left (2 \log {\left (x \right )} \right )} \end {dmath*}
integrate(1/9*(18*x*ln(x)*ln(x**2)*ln(ln(x**2))**2+(36*x*ln(x)*ln(x**2)*ln (ln(x)/x)+((54*x**2+42*x)*ln(x)+18*x)*ln(x**2)+36*x*ln(x))*ln(ln(x**2))+18 *x*ln(x)*ln(x**2)*ln(ln(x)/x)**2+(((54*x**2+42*x)*ln(x)+18*x)*ln(x**2)+36* x*ln(x))*ln(ln(x)/x)+((36*x**3+72*x**2+20*x)*ln(x)+18*x**2+30*x)*ln(x**2)+ (36*x**2+60*x)*ln(x))/ln(x)/ln(x**2),x)
x**4 + x**3*(10/3 - 2*log(2)) + x**2*log(x)**2 + 4*x**2*log(2*log(x))**2 + x**2*(-10*log(2)/3 + log(2)**2 + 25/9) + (-2*x**3 - 10*x**2/3 + 2*x**2*lo g(2))*log(x) + (4*x**3 - 4*x**2*log(x) - 4*x**2*log(2) + 20*x**2/3)*log(2* log(x))
\begin {dmath*} \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\int { \frac {2 \, {\left (9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right )^{2} + 9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right )^{2} + {\left (9 \, x^{2} + 2 \, {\left (9 \, x^{3} + 18 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 15 \, x\right )} \log \left (x^{2}\right ) + 6 \, {\left (3 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 3 \, {\left ({\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\frac {\log \left (x\right )}{x}\right ) + 3 \, {\left (6 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right ) + {\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\log \left (x^{2}\right )\right )\right )}}{9 \, \log \left (x^{2}\right ) \log \left (x\right )} \,d x } \end {dmath*}
integrate(1/9*(18*x*log(x)*log(x^2)*log(log(x^2))^2+(36*x*log(x)*log(x^2)* log(log(x)/x)+((54*x^2+42*x)*log(x)+18*x)*log(x^2)+36*x*log(x))*log(log(x^ 2))+18*x*log(x)*log(x^2)*log(log(x)/x)^2+(((54*x^2+42*x)*log(x)+18*x)*log( x^2)+36*x*log(x))*log(log(x)/x)+((36*x^3+72*x^2+20*x)*log(x)+18*x^2+30*x)* log(x^2)+(36*x^2+60*x)*log(x))/log(x)/log(x^2),x, algorithm=\
x^4 + 2/3*x^3*(3*log(2) + 1) + x^2*log(x)^2 + 4*x^2*log(log(x))^2 + 1/3*(3 *log(2)^2 + 10*log(2) + 5)*x^2 + 8/3*x^3 + 10/9*x^2 - 2/3*(3*x^3 + x^2*(3* log(2) + 5))*log(x) + 4/3*(3*x^3 + x^2*(3*log(2) + 5) - 3*x^2*log(x))*log( log(x)) + 2*Ei(3*log(x)) + 10/3*Ei(2*log(x)) - 2/9*integrate(3*(3*x^2 + 5* x)/log(x), x)
\begin {dmath*} \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\int { \frac {2 \, {\left (9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right )^{2} + 9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right )^{2} + {\left (9 \, x^{2} + 2 \, {\left (9 \, x^{3} + 18 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 15 \, x\right )} \log \left (x^{2}\right ) + 6 \, {\left (3 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 3 \, {\left ({\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\frac {\log \left (x\right )}{x}\right ) + 3 \, {\left (6 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right ) + {\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\log \left (x^{2}\right )\right )\right )}}{9 \, \log \left (x^{2}\right ) \log \left (x\right )} \,d x } \end {dmath*}
integrate(1/9*(18*x*log(x)*log(x^2)*log(log(x^2))^2+(36*x*log(x)*log(x^2)* log(log(x)/x)+((54*x^2+42*x)*log(x)+18*x)*log(x^2)+36*x*log(x))*log(log(x^ 2))+18*x*log(x)*log(x^2)*log(log(x)/x)^2+(((54*x^2+42*x)*log(x)+18*x)*log( x^2)+36*x*log(x))*log(log(x)/x)+((36*x^3+72*x^2+20*x)*log(x)+18*x^2+30*x)* log(x^2)+(36*x^2+60*x)*log(x))/log(x)/log(x^2),x, algorithm=\
integrate(2/9*(9*x*log(x^2)*log(x)*log(log(x)/x)^2 + 9*x*log(x^2)*log(x)*l og(log(x^2))^2 + (9*x^2 + 2*(9*x^3 + 18*x^2 + 5*x)*log(x) + 15*x)*log(x^2) + 6*(3*x^2 + 5*x)*log(x) + 3*(((9*x^2 + 7*x)*log(x) + 3*x)*log(x^2) + 6*x *log(x))*log(log(x)/x) + 3*(6*x*log(x^2)*log(x)*log(log(x)/x) + ((9*x^2 + 7*x)*log(x) + 3*x)*log(x^2) + 6*x*log(x))*log(log(x^2)))/(log(x^2)*log(x)) , x)
Time = 12.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.33 \begin {dmath*} \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=x^2\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^2+\ln \left (\frac {\ln \left (x\right )}{x}\right )\,\left (\frac {6\,x^4+10\,x^3}{3\,x}+2\,x^2\,\ln \left (\ln \left (x^2\right )\right )\right )+x^2\,{\ln \left (\ln \left (x^2\right )\right )}^2+\frac {25\,x^2}{9}+\frac {10\,x^3}{3}+x^4+\ln \left (\ln \left (x^2\right )\right )\,\left (2\,x^3+\frac {10\,x^2}{3}\right ) \end {dmath*}
int(((log(log(x^2))*(log(x^2)*(18*x + log(x)*(42*x + 54*x^2)) + 36*x*log(x ) + 36*x*log(x^2)*log(log(x)/x)*log(x)))/9 + (log(log(x)/x)*(log(x^2)*(18* x + log(x)*(42*x + 54*x^2)) + 36*x*log(x)))/9 + (log(x)*(60*x + 36*x^2))/9 + (log(x^2)*(30*x + 18*x^2 + log(x)*(20*x + 72*x^2 + 36*x^3)))/9 + 2*x*lo g(x^2)*log(log(x^2))^2*log(x) + 2*x*log(x^2)*log(log(x)/x)^2*log(x))/(log( x^2)*log(x)),x)