3.21.49 \(\int \frac {-36 x+54 x^2-18 x^3+(-36 x+72 x^2-18 x^3) \log (x)+(36 x-54 x^2+18 x^3) \log (x) \log (\frac {(2 x-2 x^2) \log (x)}{-2+x})}{(2-3 x+x^2) \log (x) \log ^3(\frac {(2 x-2 x^2) \log (x)}{-2+x})} \, dx\) [2049]

3.21.49.1 Optimal result

Integrand size = 101, antiderivative size = 24 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 x^2}{\log ^2\left (\frac {2 \left (x-x^2\right ) \log (x)}{-2+x}\right )} \]

output

9*x^2/ln(2*ln(x)/(-2+x)*(-x^2+x))^2
 

3.21.49.2 Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 x^2}{\log ^2\left (-\frac {2 (-1+x) x \log (x)}{-2+x}\right )} \]

input

Integrate[(-36*x + 54*x^2 - 18*x^3 + (-36*x + 72*x^2 - 18*x^3)*Log[x] + (3 
6*x - 54*x^2 + 18*x^3)*Log[x]*Log[((2*x - 2*x^2)*Log[x])/(-2 + x)])/((2 - 
3*x + x^2)*Log[x]*Log[((2*x - 2*x^2)*Log[x])/(-2 + x)]^3),x]
 

output

(9*x^2)/Log[(-2*(-1 + x)*x*Log[x])/(-2 + x)]^2
 

3.21.49.3 Rubi [F]

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.

\(\displaystyle \int \frac {-18 x^3+54 x^2+\left (-18 x^3+72 x^2-36 x\right ) \log (x)+\left (18 x^3-54 x^2+36 x\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{x-2}\right )-36 x}{\left (x^2-3 x+2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{x-2}\right )} \, dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {18 x}{\log ^2\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}-\frac {18 x \left (x^2+x^2 \log (x)-3 x-4 x \log (x)+2 \log (x)+2\right )}{(x-2) (x-1) \log (x) \log ^3\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 18 \int \frac {1}{\log ^3\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}dx+72 \int \frac {1}{(x-2) \log ^3\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}dx-18 \int \frac {1}{(x-1) \log ^3\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}dx-18 \int \frac {x}{\log ^3\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}dx-18 \int \frac {x}{\log (x) \log ^3\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}dx+18 \int \frac {x}{\log ^2\left (-\frac {2 (x-1) x \log (x)}{x-2}\right )}dx\)

input

Int[(-36*x + 54*x^2 - 18*x^3 + (-36*x + 72*x^2 - 18*x^3)*Log[x] + (36*x - 
54*x^2 + 18*x^3)*Log[x]*Log[((2*x - 2*x^2)*Log[x])/(-2 + x)])/((2 - 3*x + 
x^2)*Log[x]*Log[((2*x - 2*x^2)*Log[x])/(-2 + x)]^3),x]
 

output

$Aborted
 

3.21.49.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7279

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]
 

3.21.49.4 Maple [A] (verified)

Time = 24.99 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08

method	result	size
parallelrisch	\(\frac {9 x^{2}}{{\ln \left (\frac {\left (-2 x^{2}+2 x \right ) \ln \left (x \right )}{-2+x}\right )}^{2}}\)	\(26\)
risch	\(-\frac {36 x^{2}}{\left (\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )-\pi \,\operatorname {csgn}\left (i \ln \left (x \right )\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )^{2}+2 \pi \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{2}+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )^{2}+\pi \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (\frac {i}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )}{-2+x}\right )+\pi \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right )}{-2+x}\right ) \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i \ln \left (x \right ) \left (-1+x \right ) x}{-2+x}\right )^{3}-2 \pi +2 i \ln \left (\ln \left (x \right )\right )+2 i \ln \left (x \right )+2 i \ln \left (2\right )+2 i \ln \left (-1+x \right )-2 i \ln \left (-2+x \right )\right )^{2}}\)	\(381\)
default	\(\text {Expression too large to display}\)	\(13908\)

input

int(((18*x^3-54*x^2+36*x)*ln(x)*ln((-2*x^2+2*x)*ln(x)/(-2+x))+(-18*x^3+72* 
x^2-36*x)*ln(x)-18*x^3+54*x^2-36*x)/(x^2-3*x+2)/ln(x)/ln((-2*x^2+2*x)*ln(x 
)/(-2+x))^3,x,method=_RETURNVERBOSE)
 

output

9*x^2/ln((-2*x^2+2*x)*ln(x)/(-2+x))^2
 

3.21.49.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 \, x^{2}}{\log \left (-\frac {2 \, {\left (x^{2} - x\right )} \log \left (x\right )}{x - 2}\right )^{2}} \]

input

integrate(((18*x^3-54*x^2+36*x)*log(x)*log((-2*x^2+2*x)*log(x)/(-2+x))+(-1 
8*x^3+72*x^2-36*x)*log(x)-18*x^3+54*x^2-36*x)/(x^2-3*x+2)/log(x)/log((-2*x 
^2+2*x)*log(x)/(-2+x))^3,x, algorithm=\
 

output

9*x^2/log(-2*(x^2 - x)*log(x)/(x - 2))^2
 

3.21.49.6 Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=\frac {9 x^{2}}{\log {\left (\frac {\left (- 2 x^{2} + 2 x\right ) \log {\left (x \right )}}{x - 2} \right )}^{2}} \]

input

integrate(((18*x**3-54*x**2+36*x)*ln(x)*ln((-2*x**2+2*x)*ln(x)/(-2+x))+(-1 
8*x**3+72*x**2-36*x)*ln(x)-18*x**3+54*x**2-36*x)/(x**2-3*x+2)/ln(x)/ln((-2 
*x**2+2*x)*ln(x)/(-2+x))**3,x)
 

output

9*x**2/log((-2*x**2 + 2*x)*log(x)/(x - 2))**2
 

3.21.49.7 Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.43 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.17 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx=-\frac {9 \, x^{2}}{\pi ^{2} - 2 i \, \pi \log \left (2\right ) - \log \left (2\right )^{2} + 2 \, {\left (-i \, \pi - \log \left (2\right ) + \log \left (x - 2\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right )\right )} \log \left (x - 1\right ) - \log \left (x - 1\right )^{2} + 2 \, {\left (i \, \pi + \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (x\right )\right )\right )} \log \left (x - 2\right ) - \log \left (x - 2\right )^{2} + 2 \, {\left (-i \, \pi - \log \left (2\right )\right )} \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, {\left (-i \, \pi - \log \left (2\right ) - \log \left (x\right )\right )} \log \left (\log \left (x\right )\right ) - \log \left (\log \left (x\right )\right )^{2}} \]

input

integrate(((18*x^3-54*x^2+36*x)*log(x)*log((-2*x^2+2*x)*log(x)/(-2+x))+(-1 
8*x^3+72*x^2-36*x)*log(x)-18*x^3+54*x^2-36*x)/(x^2-3*x+2)/log(x)/log((-2*x 
^2+2*x)*log(x)/(-2+x))^3,x, algorithm=\
 

output

-9*x^2/(pi^2 - 2*I*pi*log(2) - log(2)^2 + 2*(-I*pi - log(2) + log(x - 2) - 
 log(x) - log(log(x)))*log(x - 1) - log(x - 1)^2 + 2*(I*pi + log(2) + log( 
x) + log(log(x)))*log(x - 2) - log(x - 2)^2 + 2*(-I*pi - log(2))*log(x) - 
log(x)^2 + 2*(-I*pi - log(2) - log(x))*log(log(x)) - log(log(x))^2)
 

3.21.49.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (24) = 48\).

Time = 0.38 (sec) , antiderivative size = 525, normalized size of antiderivative = 21.88 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx =\text {Too large to display} \]

input

integrate(((18*x^3-54*x^2+36*x)*log(x)*log((-2*x^2+2*x)*log(x)/(-2+x))+(-1 
8*x^3+72*x^2-36*x)*log(x)-18*x^3+54*x^2-36*x)/(x^2-3*x+2)/log(x)/log((-2*x 
^2+2*x)*log(x)/(-2+x))^3,x, algorithm=\
 

output

9*(x^4*log(x) + x^4 - 4*x^3*log(x) - 3*x^3 + 2*x^2*log(x) + 2*x^2)/(x^2*lo 
g(-2*x*log(x) + 2*log(x))^2*log(x) - 2*x^2*log(-2*x*log(x) + 2*log(x))*log 
(x - 2)*log(x) + x^2*log(x - 2)^2*log(x) + 2*x^2*log(-2*x*log(x) + 2*log(x 
))*log(x)^2 - 2*x^2*log(x - 2)*log(x)^2 + x^2*log(x)^3 + x^2*log(-2*x*log( 
x) + 2*log(x))^2 - 2*x^2*log(-2*x*log(x) + 2*log(x))*log(x - 2) + x^2*log( 
x - 2)^2 + 2*x^2*log(-2*x*log(x) + 2*log(x))*log(x) - 4*x*log(-2*x*log(x) 
+ 2*log(x))^2*log(x) - 2*x^2*log(x - 2)*log(x) + 8*x*log(-2*x*log(x) + 2*l 
og(x))*log(x - 2)*log(x) - 4*x*log(x - 2)^2*log(x) + x^2*log(x)^2 - 8*x*lo 
g(-2*x*log(x) + 2*log(x))*log(x)^2 + 8*x*log(x - 2)*log(x)^2 - 4*x*log(x)^ 
3 - 3*x*log(-2*x*log(x) + 2*log(x))^2 + 6*x*log(-2*x*log(x) + 2*log(x))*lo 
g(x - 2) - 3*x*log(x - 2)^2 - 6*x*log(-2*x*log(x) + 2*log(x))*log(x) + 2*l 
og(-2*x*log(x) + 2*log(x))^2*log(x) + 6*x*log(x - 2)*log(x) - 4*log(-2*x*l 
og(x) + 2*log(x))*log(x - 2)*log(x) + 2*log(x - 2)^2*log(x) - 3*x*log(x)^2 
 + 4*log(-2*x*log(x) + 2*log(x))*log(x)^2 - 4*log(x - 2)*log(x)^2 + 2*log( 
x)^3 + 2*log(-2*x*log(x) + 2*log(x))^2 - 4*log(-2*x*log(x) + 2*log(x))*log 
(x - 2) + 2*log(x - 2)^2 + 4*log(-2*x*log(x) + 2*log(x))*log(x) - 4*log(x 
- 2)*log(x) + 2*log(x)^2)
 

3.21.49.9 Mupad [B] (verification not implemented)

Time = 14.01 (sec) , antiderivative size = 843, normalized size of antiderivative = 35.12 \[ \int \frac {-36 x+54 x^2-18 x^3+\left (-36 x+72 x^2-18 x^3\right ) \log (x)+\left (36 x-54 x^2+18 x^3\right ) \log (x) \log \left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )}{\left (2-3 x+x^2\right ) \log (x) \log ^3\left (\frac {\left (2 x-2 x^2\right ) \log (x)}{-2+x}\right )} \, dx =\text {Too large to display} \]

input

int(-(36*x - 54*x^2 + 18*x^3 + log(x)*(36*x - 72*x^2 + 18*x^3) - log((log( 
x)*(2*x - 2*x^2))/(x - 2))*log(x)*(36*x - 54*x^2 + 18*x^3))/(log((log(x)*( 
2*x - 2*x^2))/(x - 2))^3*log(x)*(x^2 - 3*x + 2)),x)
 

output

27*x + ((9*x^2*log(x)*(x^2 - 3*x + 2))/(2*log(x) - 3*x + x^2*log(x) - 4*x* 
log(x) + x^2 + 2) - (9*x*log((log(x)*(2*x - 2*x^2))/(x - 2))*log(x)*(x^2 - 
 3*x + 2)*(4*x + 8*x*log(x)^2 - 24*x^2*log(x) + 26*x^3*log(x) - 12*x^4*log 
(x) + 2*x^5*log(x) - 26*x^2*log(x)^2 + 32*x^3*log(x)^2 - 15*x^4*log(x)^2 + 
 2*x^5*log(x)^2 + 8*x*log(x) - 12*x^2 + 13*x^3 - 6*x^4 + x^5))/(2*log(x) - 
 3*x + x^2*log(x) - 4*x*log(x) + x^2 + 2)^3)/log((log(x)*(2*x - 2*x^2))/(x 
 - 2)) + (9*x^2 - (9*x^2*log((log(x)*(2*x - 2*x^2))/(x - 2))*log(x)*(x^2 - 
 3*x + 2))/(2*log(x) - 3*x + x^2*log(x) - 4*x*log(x) + x^2 + 2))/log((log( 
x)*(2*x - 2*x^2))/(x - 2))^2 + (3672*x - 7452*x^2 + 6336*x^3 - 1944*x^4 + 
162*x^5 - 648)/(108*x^2 - 48*x - 112*x^3 + 54*x^4 - 12*x^5 + x^6 + 8) + 18 
*x^2 + (9*(384*x^3 - 4288*x^4 + 22528*x^5 - 72992*x^6 + 161528*x^7 - 25626 
8*x^8 + 298436*x^9 - 257452*x^10 + 164354*x^11 - 76855*x^12 + 25783*x^13 - 
 5986*x^14 + 904*x^15 - 79*x^16 + 3*x^17))/((x^2 - 4*x + 2)^3*(log(x)^2*(x 
^2 - 4*x + 2)^2 + (x^2 - 3*x + 2)^2 + 2*log(x)*(x^2 - 3*x + 2)*(x^2 - 4*x 
+ 2))*(4*x - 14*x^2 + 20*x^3 - 9*x^4 + x^5)) - (9*(256*x^3 - 3328*x^4 + 20 
416*x^5 - 77824*x^6 + 204976*x^7 - 393296*x^8 + 565812*x^9 - 619816*x^10 + 
 520484*x^11 - 335036*x^12 + 164269*x^13 - 60526*x^14 + 16379*x^15 - 3136* 
x^16 + 399*x^17 - 30*x^18 + x^19))/((x^2 - 4*x + 2)^3*(log(x)^3*(x^2 - 4*x 
 + 2)^3 + (x^2 - 3*x + 2)^3 + 3*log(x)^2*(x^2 - 3*x + 2)*(x^2 - 4*x + 2)^2 
 + 3*log(x)*(x^2 - 3*x + 2)^2*(x^2 - 4*x + 2))*(4*x - 14*x^2 + 20*x^3 -...