Integrand size = 75, antiderivative size = 29 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx=\frac {16}{\log ^4\left (\frac {4}{5-\frac {25+x}{5+\frac {1}{1-x \log (x)}}}\right )} \] Output:
16/ln(4/(5-1/(5+1/(-x*ln(x)+1))*(x+25)))^4
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx=\frac {16}{\log ^4\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \] Input:
Integrate[(-1216 - 64*x + (-1600 - 768*x)*Log[x] + 320*x^2*Log[x]^2)/((-30 + 6*x + (25*x - 11*x^2)*Log[x] + 5*x^3*Log[x]^2)*Log[(24 - 20*x*Log[x])/( 5 - x + x^2*Log[x])]^5),x]
Output:
16/Log[(24 - 20*x*Log[x])/(5 - x + x^2*Log[x])]^4
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.
| \(\displaystyle \int \frac {320 x^2 \log ^2(x)-64 x+(-768 x-1600) \log (x)-1216}{\left (5 x^3 \log ^2(x)+\left (25 x-11 x^2\right ) \log (x)+6 x-30\right ) \log ^5\left (\frac {24-20 x \log (x)}{x^2 \log (x)-x+5}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-320 x^2 \log ^2(x)+64 x-(-768 x-1600) \log (x)+1216}{\left (-5 x^3 \log ^2(x)-\left (25 x-11 x^2\right ) \log (x)-6 x+30\right ) \log ^5\left (-\frac {4 (5 x \log (x)-6)}{x^2 \log (x)-x+5}\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {64 \left (-5 x^2 \log ^2(x)+x+12 x \log (x)+25 \log (x)+19\right )}{\left (-5 x^3 \log ^2(x)-\left (25 x-11 x^2\right ) \log (x)-6 x+30\right ) \log ^5\left (-\frac {4 (5 x \log (x)-6)}{x^2 \log (x)-x+5}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 64 \int \frac {-5 x^2 \log ^2(x)+12 x \log (x)+25 \log (x)+x+19}{\left (-5 \log ^2(x) x^3-6 x-\left (25 x-11 x^2\right ) \log (x)+30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 64 \int \left (\frac {5 x^2 \log ^2(x)}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}-\frac {12 x \log (x)}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}-\frac {25 \log (x)}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}-\frac {x}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}-\frac {19}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 64 \left (-19 \int \frac {1}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}dx-\int \frac {x}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}dx-25 \int \frac {\log (x)}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}dx-12 \int \frac {x \log (x)}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}dx+5 \int \frac {x^2 \log ^2(x)}{\left (5 \log ^2(x) x^3-11 \log (x) x^2+25 \log (x) x+6 x-30\right ) \log ^5\left (\frac {4 (6-5 x \log (x))}{\log (x) x^2-x+5}\right )}dx\right )\) |
Input:
Int[(-1216 - 64*x + (-1600 - 768*x)*Log[x] + 320*x^2*Log[x]^2)/((-30 + 6*x + (25*x - 11*x^2)*Log[x] + 5*x^3*Log[x]^2)*Log[(24 - 20*x*Log[x])/(5 - x + x^2*Log[x])]^5),x]
Output:
$Aborted
Time = 13.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
| method | result | size |
| parallelrisch | \(\frac {16}{\ln \left (-\frac {4 \left (5 x \ln \left (x \right )-6\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{4}}\) | \(28\) |
| default | \(\frac {256}{{\left (4 \ln \left (2\right )+2 \ln \left (5\right )+2 i \pi +2 \ln \left (x \ln \left (x \right )-\frac {6}{5}\right )-2 \ln \left (x^{2} \ln \left (x \right )+5-x \right )-i \pi \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )-\frac {6}{5}\right )\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \ln \left (x \right )+5-x}\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )+i \pi \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )-\frac {6}{5}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{x^{2} \ln \left (x \right )+5-x}\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{2}-2 i \pi \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{3}\right )}^{4}}\) | \(238\) |
| risch | \(\frac {256}{{\left (4 \ln \left (2\right )+2 \ln \left (5\right )+2 i \pi +2 \ln \left (x \ln \left (x \right )-\frac {6}{5}\right )-2 \ln \left (x^{2} \ln \left (x \right )+5-x \right )-i \pi \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )-\frac {6}{5}\right )\right ) \operatorname {csgn}\left (\frac {i}{x^{2} \ln \left (x \right )+5-x}\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )+i \pi \,\operatorname {csgn}\left (i \left (x \ln \left (x \right )-\frac {6}{5}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{2}+i \pi \,\operatorname {csgn}\left (\frac {i}{x^{2} \ln \left (x \right )+5-x}\right ) \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{2}-2 i \pi \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{2}+i \pi \operatorname {csgn}\left (\frac {i \left (x \ln \left (x \right )-\frac {6}{5}\right )}{x^{2} \ln \left (x \right )+5-x}\right )^{3}\right )}^{4}}\) | \(238\) |
Input:
int((320*x^2*ln(x)^2+(-768*x-1600)*ln(x)-64*x-1216)/(5*x^3*ln(x)^2+(-11*x^ 2+25*x)*ln(x)+6*x-30)/ln((-20*x*ln(x)+24)/(x^2*ln(x)+5-x))^5,x,method=_RET URNVERBOSE)
Output:
16/ln(-4*(5*x*ln(x)-6)/(x^2*ln(x)+5-x))^4
Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx=\frac {16}{\log \left (-\frac {4 \, {\left (5 \, x \log \left (x\right ) - 6\right )}}{x^{2} \log \left (x\right ) - x + 5}\right )^{4}} \] Input:
integrate((320*x^2*log(x)^2+(-768*x-1600)*log(x)-64*x-1216)/(5*x^3*log(x)^ 2+(-11*x^2+25*x)*log(x)+6*x-30)/log((-20*x*log(x)+24)/(x^2*log(x)+5-x))^5, x, algorithm="fricas")
Output:
16/log(-4*(5*x*log(x) - 6)/(x^2*log(x) - x + 5))^4
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx=\frac {16}{\log {\left (\frac {- 20 x \log {\left (x \right )} + 24}{x^{2} \log {\left (x \right )} - x + 5} \right )}^{4}} \] Input:
integrate((320*x**2*ln(x)**2+(-768*x-1600)*ln(x)-64*x-1216)/(5*x**3*ln(x)* *2+(-11*x**2+25*x)*ln(x)+6*x-30)/ln((-20*x*ln(x)+24)/(x**2*ln(x)+5-x))**5, x)
Output:
16/log((-20*x*log(x) + 24)/(x**2*log(x) - x + 5))**4
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 332, normalized size of antiderivative = 11.45 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx=\frac {16}{\pi ^{4} - 8 i \, \pi ^{3} \log \left (2\right ) - 24 \, \pi ^{2} \log \left (2\right )^{2} + 32 i \, \pi \log \left (2\right )^{3} + 16 \, \log \left (2\right )^{4} - 4 \, {\left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (5 \, x \log \left (x\right ) - 6\right )\right )} \log \left (x^{2} \log \left (x\right ) - x + 5\right )^{3} + \log \left (x^{2} \log \left (x\right ) - x + 5\right )^{4} - 4 \, {\left (-i \, \pi - 2 \, \log \left (2\right )\right )} \log \left (5 \, x \log \left (x\right ) - 6\right )^{3} + \log \left (5 \, x \log \left (x\right ) - 6\right )^{4} - 6 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (2\right ) - 4 \, \log \left (2\right )^{2} + 2 \, {\left (-i \, \pi - 2 \, \log \left (2\right )\right )} \log \left (5 \, x \log \left (x\right ) - 6\right ) - \log \left (5 \, x \log \left (x\right ) - 6\right )^{2}\right )} \log \left (x^{2} \log \left (x\right ) - x + 5\right )^{2} - 6 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (2\right ) - 4 \, \log \left (2\right )^{2}\right )} \log \left (5 \, x \log \left (x\right ) - 6\right )^{2} - 4 \, {\left (-i \, \pi ^{3} - 6 \, \pi ^{2} \log \left (2\right ) + 12 i \, \pi \log \left (2\right )^{2} + 8 \, \log \left (2\right )^{3} + 3 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )} \log \left (5 \, x \log \left (x\right ) - 6\right )^{2} + \log \left (5 \, x \log \left (x\right ) - 6\right )^{3} - 3 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (2\right ) - 4 \, \log \left (2\right )^{2}\right )} \log \left (5 \, x \log \left (x\right ) - 6\right )\right )} \log \left (x^{2} \log \left (x\right ) - x + 5\right ) - 4 \, {\left (i \, \pi ^{3} + 6 \, \pi ^{2} \log \left (2\right ) - 12 i \, \pi \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3}\right )} \log \left (5 \, x \log \left (x\right ) - 6\right )} \] Input:
integrate((320*x^2*log(x)^2+(-768*x-1600)*log(x)-64*x-1216)/(5*x^3*log(x)^ 2+(-11*x^2+25*x)*log(x)+6*x-30)/log((-20*x*log(x)+24)/(x^2*log(x)+5-x))^5, x, algorithm="maxima")
Output:
16/(pi^4 - 8*I*pi^3*log(2) - 24*pi^2*log(2)^2 + 32*I*pi*log(2)^3 + 16*log( 2)^4 - 4*(I*pi + 2*log(2) + log(5*x*log(x) - 6))*log(x^2*log(x) - x + 5)^3 + log(x^2*log(x) - x + 5)^4 - 4*(-I*pi - 2*log(2))*log(5*x*log(x) - 6)^3 + log(5*x*log(x) - 6)^4 - 6*(pi^2 - 4*I*pi*log(2) - 4*log(2)^2 + 2*(-I*pi - 2*log(2))*log(5*x*log(x) - 6) - log(5*x*log(x) - 6)^2)*log(x^2*log(x) - x + 5)^2 - 6*(pi^2 - 4*I*pi*log(2) - 4*log(2)^2)*log(5*x*log(x) - 6)^2 - 4 *(-I*pi^3 - 6*pi^2*log(2) + 12*I*pi*log(2)^2 + 8*log(2)^3 + 3*(I*pi + 2*lo g(2))*log(5*x*log(x) - 6)^2 + log(5*x*log(x) - 6)^3 - 3*(pi^2 - 4*I*pi*log (2) - 4*log(2)^2)*log(5*x*log(x) - 6))*log(x^2*log(x) - x + 5) - 4*(I*pi^3 + 6*pi^2*log(2) - 12*I*pi*log(2)^2 - 8*log(2)^3)*log(5*x*log(x) - 6))
Leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (27) = 54\).
Time = 0.44 (sec) , antiderivative size = 603, normalized size of antiderivative = 20.79 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx =\text {Too large to display} \] Input:
integrate((320*x^2*log(x)^2+(-768*x-1600)*log(x)-64*x-1216)/(5*x^3*log(x)^ 2+(-11*x^2+25*x)*log(x)+6*x-30)/log((-20*x*log(x)+24)/(x^2*log(x)+5-x))^5, x, algorithm="giac")
Output:
16*(5*x^2*log(x)^2 - 12*x*log(x) - x - 25*log(x) - 19)/(5*x^2*log(x^2*log( x) - x + 5)^4*log(x)^2 - 20*x^2*log(x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24)*log(x)^2 + 30*x^2*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 24)^ 2*log(x)^2 - 20*x^2*log(x^2*log(x) - x + 5)*log(-20*x*log(x) + 24)^3*log(x )^2 + 5*x^2*log(-20*x*log(x) + 24)^4*log(x)^2 - 12*x*log(x^2*log(x) - x + 5)^4*log(x) + 48*x*log(x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24)*log(x) - 72*x*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 24)^2*log(x) + 48*x*l og(x^2*log(x) - x + 5)*log(-20*x*log(x) + 24)^3*log(x) - 12*x*log(-20*x*lo g(x) + 24)^4*log(x) - x*log(x^2*log(x) - x + 5)^4 + 4*x*log(x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24) - 6*x*log(x^2*log(x) - x + 5)^2*log(-20*x*l og(x) + 24)^2 + 4*x*log(x^2*log(x) - x + 5)*log(-20*x*log(x) + 24)^3 - x*l og(-20*x*log(x) + 24)^4 - 25*log(x^2*log(x) - x + 5)^4*log(x) + 100*log(x^ 2*log(x) - x + 5)^3*log(-20*x*log(x) + 24)*log(x) - 150*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 24)^2*log(x) + 100*log(x^2*log(x) - x + 5)*log( -20*x*log(x) + 24)^3*log(x) - 25*log(-20*x*log(x) + 24)^4*log(x) - 19*log( x^2*log(x) - x + 5)^4 + 76*log(x^2*log(x) - x + 5)^3*log(-20*x*log(x) + 24 ) - 114*log(x^2*log(x) - x + 5)^2*log(-20*x*log(x) + 24)^2 + 76*log(x^2*lo g(x) - x + 5)*log(-20*x*log(x) + 24)^3 - 19*log(-20*x*log(x) + 24)^4)
Time = 2.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx=\frac {16}{{\ln \left (-\frac {20\,x\,\ln \left (x\right )-24}{x^2\,\ln \left (x\right )-x+5}\right )}^4} \] Input:
int(-(64*x + log(x)*(768*x + 1600) - 320*x^2*log(x)^2 + 1216)/(log(-(20*x* log(x) - 24)/(x^2*log(x) - x + 5))^5*(6*x + 5*x^3*log(x)^2 + log(x)*(25*x - 11*x^2) - 30)),x)
Output:
16/log(-(20*x*log(x) - 24)/(x^2*log(x) - x + 5))^4
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {-1216-64 x+(-1600-768 x) \log (x)+320 x^2 \log ^2(x)}{\left (-30+6 x+\left (25 x-11 x^2\right ) \log (x)+5 x^3 \log ^2(x)\right ) \log ^5\left (\frac {24-20 x \log (x)}{5-x+x^2 \log (x)}\right )} \, dx=\frac {16}{\mathrm {log}\left (\frac {-20 \,\mathrm {log}\left (x \right ) x +24}{\mathrm {log}\left (x \right ) x^{2}-x +5}\right )^{4}} \] Input:
int((320*x^2*log(x)^2+(-768*x-1600)*log(x)-64*x-1216)/(5*x^3*log(x)^2+(-11 *x^2+25*x)*log(x)+6*x-30)/log((-20*x*log(x)+24)/(x^2*log(x)+5-x))^5,x)
Output:
16/log(( - 20*log(x)*x + 24)/(log(x)*x**2 - x + 5))**4