\(\int \frac {(15-17 x-4 x^2) \log (\frac {9-24 x+16 x^2}{x^4})+(-120+92 x-8 x^2+(60-40 x) \log (x)) \log (10-x-5 \log (x))+(30 x-43 x^2+4 x^3+(-15 x+20 x^2) \log (x)) \log ^2(10-x-5 \log (x))}{(30 x-43 x^2+4 x^3+(-15 x+20 x^2) \log (x)) \log ^2(10-x-5 \log (x))} \, dx\) [1343]

Optimal result

Integrand size = 136, antiderivative size = 32 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=x+\frac {\log \left (\frac {\left (-4+\frac {3}{x}\right )^2}{x^2}\right )}{\log (-x+5 (2-\log (x)))} \] Output:

x+ln((3/x-4)^2/x^2)/ln(-5*ln(x)+10-x)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=x+\frac {\log \left (\frac {(-3+4 x)^2}{x^4}\right )}{\log (10-x-5 \log (x))} \] Input:

Integrate[((15 - 17*x - 4*x^2)*Log[(9 - 24*x + 16*x^2)/x^4] + (-120 + 92*x 
 - 8*x^2 + (60 - 40*x)*Log[x])*Log[10 - x - 5*Log[x]] + (30*x - 43*x^2 + 4 
*x^3 + (-15*x + 20*x^2)*Log[x])*Log[10 - x - 5*Log[x]]^2)/((30*x - 43*x^2 
+ 4*x^3 + (-15*x + 20*x^2)*Log[x])*Log[10 - x - 5*Log[x]]^2),x]
 

Output:

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

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.

\(\displaystyle \int \frac {\left (-8 x^2+92 x+(60-40 x) \log (x)-120\right ) \log (-x-5 \log (x)+10)+\left (-4 x^2-17 x+15\right ) \log \left (\frac {16 x^2-24 x+9}{x^4}\right )+\left (4 x^3-43 x^2+\left (20 x^2-15 x\right ) \log (x)+30 x\right ) \log ^2(-x-5 \log (x)+10)}{\left (4 x^3-43 x^2+\left (20 x^2-15 x\right ) \log (x)+30 x\right ) \log ^2(-x-5 \log (x)+10)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-8 x^2+92 x+(60-40 x) \log (x)-120\right ) \log (-x-5 \log (x)+10)+\left (-4 x^2-17 x+15\right ) \log \left (\frac {16 x^2-24 x+9}{x^4}\right )+\left (4 x^3-43 x^2+\left (20 x^2-15 x\right ) \log (x)+30 x\right ) \log ^2(-x-5 \log (x)+10)}{(3-4 x) x (-x-5 \log (x)+10) \log ^2(-x-5 \log (x)+10)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {(x+5) \log \left (\frac {(4 x-3)^2}{x^4}\right )}{x (x+5 \log (x)-10) \log ^2(-x-5 \log (x)+10)}-\frac {4 (2 x-3)}{x (4 x-3) \log (-x-5 \log (x)+10)}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \frac {\log \left (\frac {(4 x-3)^2}{x^4}\right )}{(x+5 \log (x)-10) \log ^2(-x-5 \log (x)+10)}dx-5 \int \frac {\log \left (\frac {(4 x-3)^2}{x^4}\right )}{x (x+5 \log (x)-10) \log ^2(-x-5 \log (x)+10)}dx-4 \int \frac {1}{x \log (-x-5 \log (x)+10)}dx+8 \int \frac {1}{(4 x-3) \log (-x-5 \log (x)+10)}dx+x\)

Input:

Int[((15 - 17*x - 4*x^2)*Log[(9 - 24*x + 16*x^2)/x^4] + (-120 + 92*x - 8*x 
^2 + (60 - 40*x)*Log[x])*Log[10 - x - 5*Log[x]] + (30*x - 43*x^2 + 4*x^3 + 
 (-15*x + 20*x^2)*Log[x])*Log[10 - x - 5*Log[x]]^2)/((30*x - 43*x^2 + 4*x^ 
3 + (-15*x + 20*x^2)*Log[x])*Log[10 - x - 5*Log[x]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 5.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81

method	result	size
parallelrisch	\(\frac {9000 x \ln \left (-5 \ln \left (x \right )+10-x \right )+3375 \ln \left (-5 \ln \left (x \right )+10-x \right )+9000 \ln \left (\frac {16 x^{2}-24 x +9}{x^{4}}\right )}{9000 \ln \left (-5 \ln \left (x \right )+10-x \right )}\)	\(58\)
risch	\(x +\frac {-i \pi \operatorname {csgn}\left (\frac {i \left (x -\frac {3}{4}\right )^{2}}{x^{4}}\right )^{3}-i \pi \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )-8 \ln \left (x \right )+8 \ln \left (2\right )+4 \ln \left (x -\frac {3}{4}\right )-i \pi \operatorname {csgn}\left (i \left (x -\frac {3}{4}\right )^{2}\right )^{3}+i \pi \,\operatorname {csgn}\left (i \left (x -\frac {3}{4}\right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (x -\frac {3}{4}\right )^{2}}{x^{4}}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (x -\frac {3}{4}\right )\right )^{2} \operatorname {csgn}\left (i \left (x -\frac {3}{4}\right )^{2}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{x^{4}}\right ) \operatorname {csgn}\left (\frac {i \left (x -\frac {3}{4}\right )^{2}}{x^{4}}\right )^{2}+2 i \pi \,\operatorname {csgn}\left (i \left (x -\frac {3}{4}\right )\right ) \operatorname {csgn}\left (i \left (x -\frac {3}{4}\right )^{2}\right )^{2}-2 i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+i \pi \operatorname {csgn}\left (i x^{4}\right )^{3}-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+i \pi \operatorname {csgn}\left (i x^{3}\right )^{3}-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-i \pi \,\operatorname {csgn}\left (\frac {i}{x^{4}}\right ) \operatorname {csgn}\left (i \left (x -\frac {3}{4}\right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (x -\frac {3}{4}\right )^{2}}{x^{4}}\right )-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right )^{2}}{2 \ln \left (-5 \ln \left (x \right )+10-x \right )}\)	\(390\)

Input:

int((((20*x^2-15*x)*ln(x)+4*x^3-43*x^2+30*x)*ln(-5*ln(x)+10-x)^2+((-40*x+6 
0)*ln(x)-8*x^2+92*x-120)*ln(-5*ln(x)+10-x)+(-4*x^2-17*x+15)*ln((16*x^2-24* 
x+9)/x^4))/((20*x^2-15*x)*ln(x)+4*x^3-43*x^2+30*x)/ln(-5*ln(x)+10-x)^2,x,m 
ethod=_RETURNVERBOSE)
 

Output:

1/9000*(9000*x*ln(-5*ln(x)+10-x)+3375*ln(-5*ln(x)+10-x)+9000*ln((16*x^2-24 
*x+9)/x^4))/ln(-5*ln(x)+10-x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=\frac {x \log \left (-x - 5 \, \log \left (x\right ) + 10\right ) + \log \left (\frac {16 \, x^{2} - 24 \, x + 9}{x^{4}}\right )}{\log \left (-x - 5 \, \log \left (x\right ) + 10\right )} \] Input:

integrate((((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)*log(-5*log(x)+10-x)^2+ 
((-40*x+60)*log(x)-8*x^2+92*x-120)*log(-5*log(x)+10-x)+(-4*x^2-17*x+15)*lo 
g((16*x^2-24*x+9)/x^4))/((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)/log(-5*lo 
g(x)+10-x)^2,x, algorithm="fricas")
 

Output:

(x*log(-x - 5*log(x) + 10) + log((16*x^2 - 24*x + 9)/x^4))/log(-x - 5*log( 
x) + 10)
 

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=x + \frac {\log {\left (\frac {16 x^{2} - 24 x + 9}{x^{4}} \right )}}{\log {\left (- x - 5 \log {\left (x \right )} + 10 \right )}} \] Input:

integrate((((20*x**2-15*x)*ln(x)+4*x**3-43*x**2+30*x)*ln(-5*ln(x)+10-x)**2 
+((-40*x+60)*ln(x)-8*x**2+92*x-120)*ln(-5*ln(x)+10-x)+(-4*x**2-17*x+15)*ln 
((16*x**2-24*x+9)/x**4))/((20*x**2-15*x)*ln(x)+4*x**3-43*x**2+30*x)/ln(-5* 
ln(x)+10-x)**2,x)
 

Output:

x + log((16*x**2 - 24*x + 9)/x**4)/log(-x - 5*log(x) + 10)
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.19 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=\frac {x \log \left (-x - 5 \, \log \left (x\right ) + 10\right ) + 2 \, \log \left (4 \, x - 3\right ) - 4 \, \log \left (x\right )}{\log \left (-x - 5 \, \log \left (x\right ) + 10\right )} \] Input:

integrate((((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)*log(-5*log(x)+10-x)^2+ 
((-40*x+60)*log(x)-8*x^2+92*x-120)*log(-5*log(x)+10-x)+(-4*x^2-17*x+15)*lo 
g((16*x^2-24*x+9)/x^4))/((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)/log(-5*lo 
g(x)+10-x)^2,x, algorithm="maxima")
 

Output:

(x*log(-x - 5*log(x) + 10) + 2*log(4*x - 3) - 4*log(x))/log(-x - 5*log(x) 
+ 10)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=x + \frac {\log \left (16 \, x^{2} - 24 \, x + 9\right )}{\log \left (-x - 5 \, \log \left (x\right ) + 10\right )} - \frac {4 \, \log \left (x\right )}{\log \left (-x - 5 \, \log \left (x\right ) + 10\right )} \] Input:

integrate((((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)*log(-5*log(x)+10-x)^2+ 
((-40*x+60)*log(x)-8*x^2+92*x-120)*log(-5*log(x)+10-x)+(-4*x^2-17*x+15)*lo 
g((16*x^2-24*x+9)/x^4))/((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)/log(-5*lo 
g(x)+10-x)^2,x, algorithm="giac")
 

Output:

x + log(16*x^2 - 24*x + 9)/log(-x - 5*log(x) + 10) - 4*log(x)/log(-x - 5*l 
og(x) + 10)
 

Mupad [B] (verification not implemented)

Time = 3.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=x+\frac {\ln \left (\frac {16\,x^2-24\,x+9}{x^4}\right )}{\ln \left (10-5\,\ln \left (x\right )-x\right )} \] Input:

int(-(log(10 - 5*log(x) - x)*(log(x)*(40*x - 60) - 92*x + 8*x^2 + 120) + l 
og((16*x^2 - 24*x + 9)/x^4)*(17*x + 4*x^2 - 15) - log(10 - 5*log(x) - x)^2 
*(30*x - log(x)*(15*x - 20*x^2) - 43*x^2 + 4*x^3))/(log(10 - 5*log(x) - x) 
^2*(30*x - log(x)*(15*x - 20*x^2) - 43*x^2 + 4*x^3)),x)
 

Output:

x + log((16*x^2 - 24*x + 9)/x^4)/log(10 - 5*log(x) - x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {\left (15-17 x-4 x^2\right ) \log \left (\frac {9-24 x+16 x^2}{x^4}\right )+\left (-120+92 x-8 x^2+(60-40 x) \log (x)\right ) \log (10-x-5 \log (x))+\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))}{\left (30 x-43 x^2+4 x^3+\left (-15 x+20 x^2\right ) \log (x)\right ) \log ^2(10-x-5 \log (x))} \, dx=\frac {\mathrm {log}\left (-5 \,\mathrm {log}\left (x \right )-x +10\right ) x +\mathrm {log}\left (\frac {16 x^{2}-24 x +9}{x^{4}}\right )}{\mathrm {log}\left (-5 \,\mathrm {log}\left (x \right )-x +10\right )} \] Input:

int((((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)*log(-5*log(x)+10-x)^2+((-40* 
x+60)*log(x)-8*x^2+92*x-120)*log(-5*log(x)+10-x)+(-4*x^2-17*x+15)*log((16* 
x^2-24*x+9)/x^4))/((20*x^2-15*x)*log(x)+4*x^3-43*x^2+30*x)/log(-5*log(x)+1 
0-x)^2,x)
 

Output:

(log( - 5*log(x) - x + 10)*x + log((16*x**2 - 24*x + 9)/x**4))/log( - 5*lo 
g(x) - x + 10)