\(\int \frac {(60 x+36 x^2) \log (x)+(30 x+18 x^2+(20 x+72 x^2+36 x^3) \log (x)) \log (x^2)+(36 x \log (x)+(18 x+(42 x+54 x^2) \log (x)) \log (x^2)) \log (\frac {\log (x)}{x})+18 x \log (x) \log (x^2) \log ^2(\frac {\log (x)}{x})+(36 x \log (x)+(18 x+(42 x+54 x^2) \log (x)) \log (x^2)+36 x \log (x) \log (x^2) \log (\frac {\log (x)}{x})) \log (\log (x^2))+18 x \log (x) \log (x^2) \log ^2(\log (x^2))}{9 \log (x) \log (x^2)} \, dx\) [9314]

Optimal result

Integrand size = 176, antiderivative size = 27 \[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\frac {1}{9} x^2 \left (5+3 \left (x+\log \left (\frac {\log (x)}{x}\right )+\log \left (\log \left (x^2\right )\right )\right )\right )^2 \]

[Out]

1/9*(3*x+3*ln(ln(x)/x)+3*ln(ln(x^2))+5)^2*x^2

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {12, 6820, 6819} \[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\frac {1}{9} x^2 \left (3 \log \left (\log \left (x^2\right )\right )+3 x+3 \log \left (\frac {\log (x)}{x}\right )+5\right )^2 \]

[In]

Int[((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] + (1
8*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]

[Out]

(x^2*(5 + 3*x + 3*Log[Log[x]/x] + 3*Log[Log[x^2]])^2)/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6819

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]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \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 )}{\log (x) \log \left (x^2\right )} \, dx \\ & = \frac {1}{9} \int \frac {2 x \left (5+3 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )\right ) \left (3 \log \left (x^2\right )+\log (x) \left (6+\log \left (x^2\right ) \left (2+6 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )\right )\right )\right )}{\log (x) \log \left (x^2\right )} \, dx \\ & = \frac {2}{9} \int \frac {x \left (5+3 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )\right ) \left (3 \log \left (x^2\right )+\log (x) \left (6+\log \left (x^2\right ) \left (2+6 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )\right )\right )\right )}{\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{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\frac {1}{9} x^2 \left (5+3 x+3 \log \left (\frac {\log (x)}{x}\right )+3 \log \left (\log \left (x^2\right )\right )\right )^2 \]

[In]

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]

[Out]

(x^2*(5 + 3*x + 3*Log[Log[x]/x] + 3*Log[Log[x^2]])^2)/9

Maple [B] (verified)

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	\(\text {Expression too large to display}\)	\(935\)

[In]

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)+3
6*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=_RETURNVERB
OSE)

[Out]

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+25/9*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

Fricas [B] (verification not implemented)

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 \[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=x^{4} + x^{2} \log \left (2\right )^{2} + x^{2} \log \left (x\right )^{2} + 4 \, x^{2} \log \left (\frac {\log \left (x\right )}{x}\right )^{2} + \frac {10}{3} \, x^{3} + \frac {25}{9} \, x^{2} + \frac {2}{3} \, {\left (3 \, x^{3} + 5 \, x^{2}\right )} \log \left (2\right ) + \frac {2}{3} \, {\left (3 \, x^{3} + 3 \, x^{2} \log \left (2\right ) + 5 \, x^{2}\right )} \log \left (x\right ) + \frac {4}{3} \, {\left (3 \, x^{3} + 3 \, x^{2} \log \left (2\right ) + 3 \, x^{2} \log \left (x\right ) + 5 \, x^{2}\right )} \log \left (\frac {\log \left (x\right )}{x}\right ) \]

[In]

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)*lo
g(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="fricas")

[Out]

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)*log(x) + 4/3*(3*x^3 + 3*x^2*log(2) + 3*x^2*log(x) + 5*x^2)*log(log(x)/x)

Sympy [B] (verification not implemented)

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 \[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=x^{4} + x^{3} \cdot \left (\frac {10}{3} - 2 \log {\left (2 \right )}\right ) + x^{2} \log {\left (x \right )}^{2} + 4 x^{2} \log {\left (2 \log {\left (x \right )} \right )}^{2} + x^{2} \left (- \frac {10 \log {\left (2 \right )}}{3} + \log {\left (2 \right )}^{2} + \frac {25}{9}\right ) + \left (- 2 x^{3} - \frac {10 x^{2}}{3} + 2 x^{2} \log {\left (2 \right )}\right ) \log {\left (x \right )} + \left (4 x^{3} - 4 x^{2} \log {\left (x \right )} - 4 x^{2} \log {\left (2 \right )} + \frac {20 x^{2}}{3}\right ) \log {\left (2 \log {\left (x \right )} \right )} \]

[In]

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)+3
6*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)

[Out]

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

Maxima [F]

\[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\int { \frac {2 \, {\left (9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right )^{2} + 9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right )^{2} + {\left (9 \, x^{2} + 2 \, {\left (9 \, x^{3} + 18 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 15 \, x\right )} \log \left (x^{2}\right ) + 6 \, {\left (3 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 3 \, {\left ({\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\frac {\log \left (x\right )}{x}\right ) + 3 \, {\left (6 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right ) + {\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\log \left (x^{2}\right )\right )\right )}}{9 \, \log \left (x^{2}\right ) \log \left (x\right )} \,d x } \]

[In]

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)*lo
g(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="maxima")

[Out]

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))*lo
g(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)

Giac [F]

\[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=\int { \frac {2 \, {\left (9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right )^{2} + 9 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\log \left (x^{2}\right )\right )^{2} + {\left (9 \, x^{2} + 2 \, {\left (9 \, x^{3} + 18 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 15 \, x\right )} \log \left (x^{2}\right ) + 6 \, {\left (3 \, x^{2} + 5 \, x\right )} \log \left (x\right ) + 3 \, {\left ({\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\frac {\log \left (x\right )}{x}\right ) + 3 \, {\left (6 \, x \log \left (x^{2}\right ) \log \left (x\right ) \log \left (\frac {\log \left (x\right )}{x}\right ) + {\left ({\left (9 \, x^{2} + 7 \, x\right )} \log \left (x\right ) + 3 \, x\right )} \log \left (x^{2}\right ) + 6 \, x \log \left (x\right )\right )} \log \left (\log \left (x^{2}\right )\right )\right )}}{9 \, \log \left (x^{2}\right ) \log \left (x\right )} \,d x } \]

[In]

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)*lo
g(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="giac")

[Out]

integrate(2/9*(9*x*log(x^2)*log(x)*log(log(x)/x)^2 + 9*x*log(x^2)*log(x)*log(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)

Mupad [B] (verification not implemented)

Time = 12.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.33 \[ \int \frac {\left (60 x+36 x^2\right ) \log (x)+\left (30 x+18 x^2+\left (20 x+72 x^2+36 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {\log (x)}{x}\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\frac {\log (x)}{x}\right )+\left (36 x \log (x)+\left (18 x+\left (42 x+54 x^2\right ) \log (x)\right ) \log \left (x^2\right )+36 x \log (x) \log \left (x^2\right ) \log \left (\frac {\log (x)}{x}\right )\right ) \log \left (\log \left (x^2\right )\right )+18 x \log (x) \log \left (x^2\right ) \log ^2\left (\log \left (x^2\right )\right )}{9 \log (x) \log \left (x^2\right )} \, dx=x^2\,{\ln \left (\frac {\ln \left (x\right )}{x}\right )}^2+\ln \left (\frac {\ln \left (x\right )}{x}\right )\,\left (\frac {6\,x^4+10\,x^3}{3\,x}+2\,x^2\,\ln \left (\ln \left (x^2\right )\right )\right )+x^2\,{\ln \left (\ln \left (x^2\right )\right )}^2+\frac {25\,x^2}{9}+\frac {10\,x^3}{3}+x^4+\ln \left (\ln \left (x^2\right )\right )\,\left (2\,x^3+\frac {10\,x^2}{3}\right ) \]

[In]

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*log(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)

[Out]

x^2*log(log(x)/x)^2 + log(log(x)/x)*((10*x^3 + 6*x^4)/(3*x) + 2*x^2*log(log(x^2))) + x^2*log(log(x^2))^2 + (25
*x^2)/9 + (10*x^3)/3 + x^4 + log(log(x^2))*((10*x^2)/3 + 2*x^3)