Integrand size = 111, antiderivative size = 33 \[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=\frac {1}{3} \log \left (-x+\frac {x \log \left (\frac {25}{\log ^2(2 x)}\right )}{(3+x) (-2+3 \log (2))}\right ) \] Output:
1/3*ln(ln(25/ln(2*x)^2)*x/(3*ln(2)-2)/(3+x)-x)
\[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=\int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx \] Input:
Integrate[(-6 - 2*x + (18 + 12*x + 2*x^2 + (-27 - 18*x - 3*x^2)*Log[2])*Lo g[2*x] + 3*Log[2*x]*Log[25/Log[2*x]^2])/((54*x + 36*x^2 + 6*x^3 + (-81*x - 54*x^2 - 9*x^3)*Log[2])*Log[2*x] + (9*x + 3*x^2)*Log[2*x]*Log[25/Log[2*x] ^2]),x]
Output:
Integrate[(-6 - 2*x + (18 + 12*x + 2*x^2 + (-27 - 18*x - 3*x^2)*Log[2])*Lo g[2*x] + 3*Log[2*x]*Log[25/Log[2*x]^2])/((54*x + 36*x^2 + 6*x^3 + (-81*x - 54*x^2 - 9*x^3)*Log[2])*Log[2*x] + (9*x + 3*x^2)*Log[2*x]*Log[25/Log[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.
| \(\displaystyle \int \frac {\left (2 x^2+\left (-3 x^2-18 x-27\right ) \log (2)+12 x+18\right ) \log (2 x)-2 x+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )-6}{\left (3 x^2+9 x\right ) \log \left (\frac {25}{\log ^2(2 x)}\right ) \log (2 x)+\left (6 x^3+36 x^2+\left (-9 x^3-54 x^2-81 x\right ) \log (2)+54 x\right ) \log (2 x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (2 x^2+\left (-3 x^2-18 x-27\right ) \log (2)+12 x+18\right ) \log (2 x)-2 x+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )-6}{3 x (x+3) \log (2 x) \left (\log \left (\frac {25}{\log ^2(2 x)}\right )+2 x \left (1-\frac {\log (8)}{2}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {2 x-\left (2 x^2+12 x-3 \left (x^2+6 x+9\right ) \log (2)+18\right ) \log (2 x)-3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )+6}{x (x+3) \log (2 x) \left ((2-\log (8)) x+\log \left (\frac {25}{\log ^2(2 x)}\right )-\log (512)+6\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {2 x-\left (2 x^2+12 x-3 \left (x^2+6 x+9\right ) \log (2)+18\right ) \log (2 x)-3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )+6}{x (x+3) \log (2 x) \left ((2-\log (8)) x+\log \left (\frac {25}{\log ^2(2 x)}\right )-\log (512)+6\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {1}{3} \int \frac {2 x-\left (2 x^2+12 x-3 \left (x^2+6 x+9\right ) \log (2)+18\right ) \log (2 x)-3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )+6}{x (x+3) \log (2 x) \left ((2-\log (8)) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (\frac {-2 \left (1-\frac {\log (8)}{2}\right ) \log (2 x) x^2-6 \left (1-\frac {\log (512)}{6}\right ) \log (2 x) x+2 x+6}{x (x+3) \log (2 x) \left (2 \left (1-\frac {\log (8)}{2}\right ) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}-\frac {3}{x (x+3)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (-(2-\log (8)) \int \frac {1}{-2 \left (1-\frac {\log (8)}{2}\right ) x-\log \left (\frac {25}{\log ^2(2 x)}\right )-6 \left (1-\frac {\log (512)}{6}\right )}dx-\frac {2}{3} \int \frac {1}{\log (2 x) \left (-2 \left (1-\frac {\log (8)}{2}\right ) x-\log \left (\frac {25}{\log ^2(2 x)}\right )-6 \left (1-\frac {\log (512)}{6}\right )\right )}dx-(6-\log (512)) \int \frac {1}{(-x-3) \left (2 \left (1-\frac {\log (8)}{2}\right ) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx-3 (2-\log (8)) \int \frac {1}{(x+3) \left (2 \left (1-\frac {\log (8)}{2}\right ) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx-\frac {2}{3} \int \frac {1}{\log (2 x) \left (2 \left (1-\frac {\log (8)}{2}\right ) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx-2 \int \frac {1}{(-x-3) \log (2 x) \left (2 \left (1-\frac {\log (8)}{2}\right ) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx-2 \int \frac {1}{x \log (2 x) \left (2 \left (1-\frac {\log (8)}{2}\right ) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx-2 \int \frac {1}{(x+3) \log (2 x) \left (2 \left (1-\frac {\log (8)}{2}\right ) x+\log \left (\frac {25}{\log ^2(2 x)}\right )+6 \left (1-\frac {\log (512)}{6}\right )\right )}dx+\log (x)-\log (x+3)\right )\) |
Input:
Int[(-6 - 2*x + (18 + 12*x + 2*x^2 + (-27 - 18*x - 3*x^2)*Log[2])*Log[2*x] + 3*Log[2*x]*Log[25/Log[2*x]^2])/((54*x + 36*x^2 + 6*x^3 + (-81*x - 54*x^ 2 - 9*x^3)*Log[2])*Log[2*x] + (9*x + 3*x^2)*Log[2*x]*Log[25/Log[2*x]^2]),x ]
Output:
$Aborted
Time = 4.49 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(\frac {\ln \left (x \right )}{3}-\frac {\ln \left (3+x \right )}{3}+\frac {\ln \left (3 x \ln \left (2\right )+9 \ln \left (2\right )-2 \ln \left (5\right )-2 x -\ln \left (\frac {1}{\left (\ln \left (2\right )+\ln \left (x \right )\right )^{2}}\right )-6\right )}{3}\) | \(43\) |
| parallelrisch | \(\frac {\ln \left (2 x \right )}{3}+\frac {\ln \left (\frac {3 x \ln \left (2\right )+9 \ln \left (2\right )-2 x -\ln \left (\frac {25}{\ln \left (2 x \right )^{2}}\right )-6}{3 \ln \left (2\right )-2}\right )}{3}-\frac {\ln \left (3+x \right )}{3}\) | \(51\) |
| risch | \(-\frac {\ln \left (3+x \right )}{3}+\frac {\ln \left (x \right )}{3}+\frac {\ln \left (-3-\frac {i \pi \operatorname {csgn}\left (i \ln \left (2 x \right )\right )^{2} \operatorname {csgn}\left (i \ln \left (2 x \right )^{2}\right )}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \ln \left (2 x \right )\right ) \operatorname {csgn}\left (i \ln \left (2 x \right )^{2}\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i \ln \left (2 x \right )^{2}\right )^{3}}{4}+\frac {3 x \ln \left (2\right )}{2}+\frac {9 \ln \left (2\right )}{2}-\ln \left (5\right )-x +\ln \left (\ln \left (2 x \right )\right )\right )}{3}\) | \(102\) |
Input:
int((3*ln(2*x)*ln(25/ln(2*x)^2)+((-3*x^2-18*x-27)*ln(2)+2*x^2+12*x+18)*ln( 2*x)-2*x-6)/((3*x^2+9*x)*ln(2*x)*ln(25/ln(2*x)^2)+((-9*x^3-54*x^2-81*x)*ln (2)+6*x^3+36*x^2+54*x)*ln(2*x)),x,method=_RETURNVERBOSE)
Output:
1/3*ln(x)-1/3*ln(3+x)+1/3*ln(3*x*ln(2)+9*ln(2)-2*ln(5)-2*x-ln(1/(ln(2)+ln( x))^2)-6)
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=\frac {1}{3} \, \log \left (-3 \, {\left (x + 3\right )} \log \left (2\right ) + 2 \, x + \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) + 6\right ) - \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{3} \, \log \left (x\right ) \] Input:
integrate((3*log(2*x)*log(25/log(2*x)^2)+((-3*x^2-18*x-27)*log(2)+2*x^2+12 *x+18)*log(2*x)-2*x-6)/((3*x^2+9*x)*log(2*x)*log(25/log(2*x)^2)+((-9*x^3-5 4*x^2-81*x)*log(2)+6*x^3+36*x^2+54*x)*log(2*x)),x, algorithm="fricas")
Output:
1/3*log(-3*(x + 3)*log(2) + 2*x + log(25/log(2*x)^2) + 6) - 1/3*log(x + 3) + 1/3*log(x)
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=\frac {\log {\left (x \right )}}{3} - \frac {\log {\left (x + 3 \right )}}{3} + \frac {\log {\left (- 3 x \log {\left (2 \right )} + 2 x + \log {\left (\frac {25}{\log {\left (2 x \right )}^{2}} \right )} - 9 \log {\left (2 \right )} + 6 \right )}}{3} \] Input:
integrate((3*ln(2*x)*ln(25/ln(2*x)**2)+((-3*x**2-18*x-27)*ln(2)+2*x**2+12* x+18)*ln(2*x)-2*x-6)/((3*x**2+9*x)*ln(2*x)*ln(25/ln(2*x)**2)+((-9*x**3-54* x**2-81*x)*ln(2)+6*x**3+36*x**2+54*x)*ln(2*x)),x)
Output:
log(x)/3 - log(x + 3)/3 + log(-3*x*log(2) + 2*x + log(25/log(2*x)**2) - 9* log(2) + 6)/3
Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=\frac {1}{3} \, \log \left (\frac {1}{2} \, x {\left (3 \, \log \left (2\right ) - 2\right )} - \log \left (5\right ) + \frac {9}{2} \, \log \left (2\right ) + \log \left (\log \left (2\right ) + \log \left (x\right )\right ) - 3\right ) - \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{3} \, \log \left (x\right ) \] Input:
integrate((3*log(2*x)*log(25/log(2*x)^2)+((-3*x^2-18*x-27)*log(2)+2*x^2+12 *x+18)*log(2*x)-2*x-6)/((3*x^2+9*x)*log(2*x)*log(25/log(2*x)^2)+((-9*x^3-5 4*x^2-81*x)*log(2)+6*x^3+36*x^2+54*x)*log(2*x)),x, algorithm="maxima")
Output:
1/3*log(1/2*x*(3*log(2) - 2) - log(5) + 9/2*log(2) + log(log(2) + log(x)) - 3) - 1/3*log(x + 3) + 1/3*log(x)
\[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=\int { \frac {{\left (2 \, x^{2} - 3 \, {\left (x^{2} + 6 \, x + 9\right )} \log \left (2\right ) + 12 \, x + 18\right )} \log \left (2 \, x\right ) + 3 \, \log \left (2 \, x\right ) \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) - 2 \, x - 6}{3 \, {\left ({\left (x^{2} + 3 \, x\right )} \log \left (2 \, x\right ) \log \left (\frac {25}{\log \left (2 \, x\right )^{2}}\right ) + {\left (2 \, x^{3} + 12 \, x^{2} - 3 \, {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \left (2\right ) + 18 \, x\right )} \log \left (2 \, x\right )\right )}} \,d x } \] Input:
integrate((3*log(2*x)*log(25/log(2*x)^2)+((-3*x^2-18*x-27)*log(2)+2*x^2+12 *x+18)*log(2*x)-2*x-6)/((3*x^2+9*x)*log(2*x)*log(25/log(2*x)^2)+((-9*x^3-5 4*x^2-81*x)*log(2)+6*x^3+36*x^2+54*x)*log(2*x)),x, algorithm="giac")
Output:
integrate(1/3*((2*x^2 - 3*(x^2 + 6*x + 9)*log(2) + 12*x + 18)*log(2*x) + 3 *log(2*x)*log(25/log(2*x)^2) - 2*x - 6)/((x^2 + 3*x)*log(2*x)*log(25/log(2 *x)^2) + (2*x^3 + 12*x^2 - 3*(x^3 + 6*x^2 + 9*x)*log(2) + 18*x)*log(2*x)), x)
Timed out. \[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=-\int \frac {2\,x-3\,\ln \left (2\,x\right )\,\ln \left (\frac {25}{{\ln \left (2\,x\right )}^2}\right )-\ln \left (2\,x\right )\,\left (12\,x-\ln \left (2\right )\,\left (3\,x^2+18\,x+27\right )+2\,x^2+18\right )+6}{\ln \left (2\,x\right )\,\left (54\,x-\ln \left (2\right )\,\left (9\,x^3+54\,x^2+81\,x\right )+36\,x^2+6\,x^3\right )+\ln \left (2\,x\right )\,\ln \left (\frac {25}{{\ln \left (2\,x\right )}^2}\right )\,\left (3\,x^2+9\,x\right )} \,d x \] Input:
int(-(2*x - 3*log(2*x)*log(25/log(2*x)^2) - log(2*x)*(12*x - log(2)*(18*x + 3*x^2 + 27) + 2*x^2 + 18) + 6)/(log(2*x)*(54*x - log(2)*(81*x + 54*x^2 + 9*x^3) + 36*x^2 + 6*x^3) + log(2*x)*log(25/log(2*x)^2)*(9*x + 3*x^2)),x)
Output:
-int((2*x - 3*log(2*x)*log(25/log(2*x)^2) - log(2*x)*(12*x - log(2)*(18*x + 3*x^2 + 27) + 2*x^2 + 18) + 6)/(log(2*x)*(54*x - log(2)*(81*x + 54*x^2 + 9*x^3) + 36*x^2 + 6*x^3) + log(2*x)*log(25/log(2*x)^2)*(9*x + 3*x^2)), x)
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6-2 x+\left (18+12 x+2 x^2+\left (-27-18 x-3 x^2\right ) \log (2)\right ) \log (2 x)+3 \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )}{\left (54 x+36 x^2+6 x^3+\left (-81 x-54 x^2-9 x^3\right ) \log (2)\right ) \log (2 x)+\left (9 x+3 x^2\right ) \log (2 x) \log \left (\frac {25}{\log ^2(2 x)}\right )} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {25}{\mathrm {log}\left (2 x \right )^{2}}\right )-3 \,\mathrm {log}\left (2\right ) x -9 \,\mathrm {log}\left (2\right )+2 x +6\right )}{3}-\frac {\mathrm {log}\left (x +3\right )}{3}+\frac {\mathrm {log}\left (x \right )}{3} \] Input:
int((3*log(2*x)*log(25/log(2*x)^2)+((-3*x^2-18*x-27)*log(2)+2*x^2+12*x+18) *log(2*x)-2*x-6)/((3*x^2+9*x)*log(2*x)*log(25/log(2*x)^2)+((-9*x^3-54*x^2- 81*x)*log(2)+6*x^3+36*x^2+54*x)*log(2*x)),x)
Output:
(log(log(25/log(2*x)**2) - 3*log(2)*x - 9*log(2) + 2*x + 6) - log(x + 3) + log(x))/3