Integrand size = 210, antiderivative size = 35 \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=9 \left (-4+\frac {x}{-4+\frac {i \pi +\log \left (-4+4 e^5\right )}{x \log (\log (x))}}\right )^2 \] Output:
3*(x/(ln(-4*exp(5)+4)/ln(ln(x))/x-4)-4)*(3*x/(ln(-4*exp(5)+4)/ln(ln(x))/x- 4)-12)
Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.60 \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=-\frac {9 x^2 \log (\log (x)) \left (-8 i \pi -8 \log \left (4 \left (-1+e^5\right )\right )+x (32+x) \log (\log (x))\right )}{\left (\pi -i \log \left (4 \left (-1+e^5\right )\right )+4 i x \log (\log (x))\right )^2} \] Input:
Integrate[(72*x*(I*Pi + Log[-4 + 4*E^5])^2 + ((-288*x^2 - 18*x^3)*(I*Pi + Log[-4 + 4*E^5]) + 144*x*(I*Pi + Log[-4 + 4*E^5])^2*Log[x])*Log[Log[x]] + (-864*x^2 - 36*x^3)*(I*Pi + Log[-4 + 4*E^5])*Log[x]*Log[Log[x]]^2 + (1152* x^3 + 72*x^4)*Log[x]*Log[Log[x]]^3)/(-((I*Pi + Log[-4 + 4*E^5])^3*Log[x]) + 12*x*(I*Pi + Log[-4 + 4*E^5])^2*Log[x]*Log[Log[x]] - 48*x^2*(I*Pi + Log[ -4 + 4*E^5])*Log[x]*Log[Log[x]]^2 + 64*x^3*Log[x]*Log[Log[x]]^3),x]
Output:
(-9*x^2*Log[Log[x]]*((-8*I)*Pi - 8*Log[4*(-1 + E^5)] + x*(32 + x)*Log[Log[ x]]))/(Pi - I*Log[4*(-1 + E^5)] + (4*I)*x*Log[Log[x]])^2
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.
| \(\displaystyle \int \frac {\left (72 x^4+1152 x^3\right ) \log (x) \log ^3(\log (x))+\left (-36 x^3-864 x^2\right ) \left (\log \left (4 e^5-4\right )+i \pi \right ) \log (x) \log ^2(\log (x))+\left (\left (-18 x^3-288 x^2\right ) \left (\log \left (4 e^5-4\right )+i \pi \right )+144 x \left (\log \left (4 e^5-4\right )+i \pi \right )^2 \log (x)\right ) \log (\log (x))+72 x \left (\log \left (4 e^5-4\right )+i \pi \right )^2}{64 x^3 \log (x) \log ^3(\log (x))-48 x^2 \left (\log \left (4 e^5-4\right )+i \pi \right ) \log (x) \log ^2(\log (x))+12 x \left (\log \left (4 e^5-4\right )+i \pi \right )^2 \log (x) \log (\log (x))-\left (\log \left (4 e^5-4\right )+i \pi \right )^3 \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {18 x \left (-x (x+16) \log (\log (x))+4 i \pi \left (1-\frac {i \log \left (4 \left (e^5-1\right )\right )}{\pi }\right )\right ) \left (2 \log (x) \log (\log (x)) \left (2 x \log (\log (x))-i \pi -\log \left (4 \left (e^5-1\right )\right )\right )-i \pi \left (1-\frac {i \log \left (4 \left (e^5-1\right )\right )}{\pi }\right )\right )}{\log (x) \left (-4 x \log (\log (x))+i \pi \left (1-\frac {i \log \left (4 \left (e^5-1\right )\right )}{\pi }\right )\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 18 \int -\frac {x \left (4 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )-x (x+16) \log (\log (x))\right ) \left (2 \log (x) \log (\log (x)) \left (-2 x \log (\log (x))+\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )+\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )}{\log (x) \left (-4 x \log (\log (x))+\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -18 \int \frac {x \left (4 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )-x (x+16) \log (\log (x))\right ) \left (2 \log (x) \log (\log (x)) \left (-2 x \log (\log (x))+\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )+\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )}{\log (x) \left (-4 x \log (\log (x))+\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )^3}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -18 \int \frac {x \left (4 \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right )-x (x+16) \log (\log (x))\right ) \left (2 \log (x) \log (\log (x)) \left (-2 x \log (\log (x))+\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )+i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )\right )}{\log (x) \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -18 \int \left (\frac {1}{16} (-x-16)-\frac {i x \left (-\pi +i \log \left (-4 \left (1-e^5\right )\right )\right )}{16 \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )}+\frac {(x+16) \left (i \pi +\log \left (-4 \left (1-e^5\right )\right )\right ) \left (-4 x-i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{16 \log (x) \left (4 i x \log (\log (x))+\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )\right )^2}+\frac {x \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2 \left (4 x+i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right ) \log (x)\right )}{16 \log (x) \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -18 \left (\frac {1}{4} \left (\pi -i \log \left (-4 \left (1-e^5\right )\right )\right )^2 \int \frac {x^2}{\log (x) \left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3}dx-\frac {1}{4} \left (\log \left (-4 \left (1-e^5\right )\right )+i \pi \right ) \int \frac {x^2}{\log (x) \left (4 i x \log (\log (x))+\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )\right )^2}dx-\frac {1}{16} \left (\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )^3 \int \frac {x}{\left (i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))\right )^3}dx+\frac {1}{16} \left (\log \left (-4 \left (1-e^5\right )\right )+i \pi \right ) \int \frac {x}{i \pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )-4 x \log (\log (x))}dx-\left (\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )^2 \int \frac {1}{\left (4 i x \log (\log (x))+\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )\right )^2}dx-\frac {1}{16} \left (\log \left (-4 \left (1-e^5\right )\right )+i \pi \right )^2 \int \frac {x}{\left (4 i x \log (\log (x))+\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )\right )^2}dx-4 \left (\log \left (-4 \left (1-e^5\right )\right )+i \pi \right ) \int \frac {x}{\log (x) \left (4 i x \log (\log (x))+\pi \left (1-\frac {i \log \left (4 \left (-1+e^5\right )\right )}{\pi }\right )\right )^2}dx-\frac {1}{32} (x+16)^2\right )\) |
Input:
Int[(72*x*(I*Pi + Log[-4 + 4*E^5])^2 + ((-288*x^2 - 18*x^3)*(I*Pi + Log[-4 + 4*E^5]) + 144*x*(I*Pi + Log[-4 + 4*E^5])^2*Log[x])*Log[Log[x]] + (-864* x^2 - 36*x^3)*(I*Pi + Log[-4 + 4*E^5])*Log[x]*Log[Log[x]]^2 + (1152*x^3 + 72*x^4)*Log[x]*Log[Log[x]]^3)/(-((I*Pi + Log[-4 + 4*E^5])^3*Log[x]) + 12*x *(I*Pi + Log[-4 + 4*E^5])^2*Log[x]*Log[Log[x]] - 48*x^2*(I*Pi + Log[-4 + 4 *E^5])*Log[x]*Log[Log[x]]^2 + 64*x^3*Log[x]*Log[Log[x]]^3),x]
Output:
$Aborted
Time = 3.36 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.11
| method | result | size |
| parallelrisch | \(\frac {144 \ln \left (\ln \left (x \right )\right )^{2} x^{4}+4608 x^{3} \ln \left (\ln \left (x \right )\right )^{2}-1152 x^{2} \ln \left (-4 \,{\mathrm e}^{5}+4\right ) \ln \left (\ln \left (x \right )\right )}{256 x^{2} \ln \left (\ln \left (x \right )\right )^{2}-128 x \ln \left (-4 \,{\mathrm e}^{5}+4\right ) \ln \left (\ln \left (x \right )\right )+16 \ln \left (-4 \,{\mathrm e}^{5}+4\right )^{2}}\) | \(74\) |
| risch | \(\frac {9 x^{2}}{16}+18 x -\frac {9 x \left (-8 x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (1-{\mathrm e}^{5}\right )-16 x^{2} \ln \left (\ln \left (x \right )\right ) \ln \left (2\right )+x \ln \left (1-{\mathrm e}^{5}\right )^{2}+4 x \ln \left (2\right ) \ln \left (1-{\mathrm e}^{5}\right )-128 x \ln \left (\ln \left (x \right )\right ) \ln \left (1-{\mathrm e}^{5}\right )+4 x \ln \left (2\right )^{2}-256 x \ln \left (\ln \left (x \right )\right ) \ln \left (2\right )+32 \ln \left (1-{\mathrm e}^{5}\right )^{2}+128 \ln \left (2\right ) \ln \left (1-{\mathrm e}^{5}\right )+128 \ln \left (2\right )^{2}\right )}{16 {\left (-4 x \ln \left (\ln \left (x \right )\right )+\ln \left (1-{\mathrm e}^{5}\right )+2 \ln \left (2\right )\right )}^{2}}\) | \(138\) |
| default | \(\text {Expression too large to display}\) | \(1972\) |
| parts | \(\text {Expression too large to display}\) | \(1972\) |
Input:
int(((72*x^4+1152*x^3)*ln(x)*ln(ln(x))^3+(-36*x^3-864*x^2)*ln(-4*exp(5)+4) *ln(x)*ln(ln(x))^2+(144*x*ln(-4*exp(5)+4)^2*ln(x)+(-18*x^3-288*x^2)*ln(-4* exp(5)+4))*ln(ln(x))+72*x*ln(-4*exp(5)+4)^2)/(64*x^3*ln(x)*ln(ln(x))^3-48* x^2*ln(-4*exp(5)+4)*ln(x)*ln(ln(x))^2+12*x*ln(-4*exp(5)+4)^2*ln(x)*ln(ln(x ))-ln(-4*exp(5)+4)^3*ln(x)),x,method=_RETURNVERBOSE)
Output:
1/16*(144*ln(ln(x))^2*x^4+4608*x^3*ln(ln(x))^2-1152*x^2*ln(-4*exp(5)+4)*ln (ln(x)))/(16*x^2*ln(ln(x))^2-8*x*ln(-4*exp(5)+4)*ln(ln(x))+ln(-4*exp(5)+4) ^2)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (28) = 56\).
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=-\frac {9 \, {\left (8 \, x^{2} \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right ) - {\left (x^{4} + 32 \, x^{3}\right )} \log \left (\log \left (x\right )\right )^{2}\right )}}{16 \, x^{2} \log \left (\log \left (x\right )\right )^{2} - 8 \, x \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right ) + \log \left (-4 \, e^{5} + 4\right )^{2}} \] Input:
integrate(((72*x^4+1152*x^3)*log(x)*log(log(x))^3+(-36*x^3-864*x^2)*log(-4 *exp(5)+4)*log(x)*log(log(x))^2+(144*x*log(-4*exp(5)+4)^2*log(x)+(-18*x^3- 288*x^2)*log(-4*exp(5)+4))*log(log(x))+72*x*log(-4*exp(5)+4)^2)/(64*x^3*lo g(x)*log(log(x))^3-48*x^2*log(-4*exp(5)+4)*log(x)*log(log(x))^2+12*x*log(- 4*exp(5)+4)^2*log(x)*log(log(x))-log(-4*exp(5)+4)^3*log(x)),x, algorithm=" fricas")
Output:
-9*(8*x^2*log(-4*e^5 + 4)*log(log(x)) - (x^4 + 32*x^3)*log(log(x))^2)/(16* x^2*log(log(x))^2 - 8*x*log(-4*e^5 + 4)*log(log(x)) + log(-4*e^5 + 4)^2)
Timed out. \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=\text {Timed out} \] Input:
integrate(((72*x**4+1152*x**3)*ln(x)*ln(ln(x))**3+(-36*x**3-864*x**2)*ln(- 4*exp(5)+4)*ln(x)*ln(ln(x))**2+(144*x*ln(-4*exp(5)+4)**2*ln(x)+(-18*x**3-2 88*x**2)*ln(-4*exp(5)+4))*ln(ln(x))+72*x*ln(-4*exp(5)+4)**2)/(64*x**3*ln(x )*ln(ln(x))**3-48*x**2*ln(-4*exp(5)+4)*ln(x)*ln(ln(x))**2+12*x*ln(-4*exp(5 )+4)**2*ln(x)*ln(ln(x))-ln(-4*exp(5)+4)**3*ln(x)),x)
Output:
Timed out
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 188, normalized size of antiderivative = 5.37 \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=-\frac {9 \, {\left (8 \, {\left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} x^{2} \log \left (\log \left (x\right )\right ) - {\left (x^{4} + 32 \, x^{3}\right )} \log \left (\log \left (x\right )\right )^{2}\right )}}{16 \, x^{2} \log \left (\log \left (x\right )\right )^{2} - 8 \, {\left (i \, \pi + 2 \, \log \left (2\right ) + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} x \log \left (\log \left (x\right )\right ) - \pi ^{2} + 2 i \, \pi {\left (2 \, \log \left (2\right ) + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} + 4 \, {\left (\log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) + \log \left (e - 1\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + \log \left (e^{4} + e^{3} + e^{2} + e + 1\right )^{2} + 2 \, \log \left (e^{4} + e^{3} + e^{2} + e + 1\right ) \log \left (e - 1\right ) + \log \left (e - 1\right )^{2}} \] Input:
integrate(((72*x^4+1152*x^3)*log(x)*log(log(x))^3+(-36*x^3-864*x^2)*log(-4 *exp(5)+4)*log(x)*log(log(x))^2+(144*x*log(-4*exp(5)+4)^2*log(x)+(-18*x^3- 288*x^2)*log(-4*exp(5)+4))*log(log(x))+72*x*log(-4*exp(5)+4)^2)/(64*x^3*lo g(x)*log(log(x))^3-48*x^2*log(-4*exp(5)+4)*log(x)*log(log(x))^2+12*x*log(- 4*exp(5)+4)^2*log(x)*log(log(x))-log(-4*exp(5)+4)^3*log(x)),x, algorithm=" maxima")
Output:
-9*(8*(I*pi + 2*log(2) + log(e^4 + e^3 + e^2 + e + 1) + log(e - 1))*x^2*lo g(log(x)) - (x^4 + 32*x^3)*log(log(x))^2)/(16*x^2*log(log(x))^2 - 8*(I*pi + 2*log(2) + log(e^4 + e^3 + e^2 + e + 1) + log(e - 1))*x*log(log(x)) - pi ^2 + 2*I*pi*(2*log(2) + log(e^4 + e^3 + e^2 + e + 1) + log(e - 1)) + 4*(lo g(e^4 + e^3 + e^2 + e + 1) + log(e - 1))*log(2) + 4*log(2)^2 + log(e^4 + e ^3 + e^2 + e + 1)^2 + 2*log(e^4 + e^3 + e^2 + e + 1)*log(e - 1) + log(e - 1)^2)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (28) = 56\).
Time = 0.17 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.06 \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=\frac {9 \, {\left (x^{4} \log \left (\log \left (x\right )\right )^{2} + 32 \, x^{3} \log \left (\log \left (x\right )\right )^{2} - 8 \, x^{2} \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right )\right )}}{16 \, x^{2} \log \left (\log \left (x\right )\right )^{2} - 8 \, x \log \left (-4 \, e^{5} + 4\right ) \log \left (\log \left (x\right )\right ) + \log \left (-4 \, e^{5} + 4\right )^{2}} \] Input:
integrate(((72*x^4+1152*x^3)*log(x)*log(log(x))^3+(-36*x^3-864*x^2)*log(-4 *exp(5)+4)*log(x)*log(log(x))^2+(144*x*log(-4*exp(5)+4)^2*log(x)+(-18*x^3- 288*x^2)*log(-4*exp(5)+4))*log(log(x))+72*x*log(-4*exp(5)+4)^2)/(64*x^3*lo g(x)*log(log(x))^3-48*x^2*log(-4*exp(5)+4)*log(x)*log(log(x))^2+12*x*log(- 4*exp(5)+4)^2*log(x)*log(log(x))-log(-4*exp(5)+4)^3*log(x)),x, algorithm=" giac")
Output:
9*(x^4*log(log(x))^2 + 32*x^3*log(log(x))^2 - 8*x^2*log(-4*e^5 + 4)*log(lo g(x)))/(16*x^2*log(log(x))^2 - 8*x*log(-4*e^5 + 4)*log(log(x)) + log(-4*e^ 5 + 4)^2)
Time = 2.52 (sec) , antiderivative size = 3724, normalized size of antiderivative = 106.40 \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=\text {Too large to display} \] Input:
int((log(log(x))*(log(4 - 4*exp(5))*(288*x^2 + 18*x^3) - 144*x*log(4 - 4*e xp(5))^2*log(x)) - 72*x*log(4 - 4*exp(5))^2 - log(log(x))^3*log(x)*(1152*x ^3 + 72*x^4) + log(log(x))^2*log(4 - 4*exp(5))*log(x)*(864*x^2 + 36*x^3))/ (log(4 - 4*exp(5))^3*log(x) - 64*x^3*log(log(x))^3*log(x) + 48*x^2*log(log (x))^2*log(4 - 4*exp(5))*log(x) - 12*x*log(log(x))*log(4 - 4*exp(5))^2*log (x)),x)
Output:
x*((39*log(4 - 4*exp(5)))/32 + 18) + (159*log(4 - 4*exp(5))^2*log(x))/128 - ((9*log(4 - 4*exp(5))^2*(64*x + 16*log(4 - 4*exp(5))*log(x) + x*log(4 - 4*exp(5))*log(x)))/(256*x*(4*x + log(4 - 4*exp(5))*log(x))) - (9*log(log(x ))*log(4 - 4*exp(5))*(64*x + 16*log(4 - 4*exp(5))*log(x) + 4*x^2 + 3*x*log (4 - 4*exp(5))*log(x)))/(64*(4*x + log(4 - 4*exp(5))*log(x))) + (9*x^2*log (log(x))^2*log(4 - 4*exp(5))*log(x))/(16*(4*x + log(4 - 4*exp(5))*log(x))) )/(log(4 - 4*exp(5))^2/(16*x^2) + log(log(x))^2 - (log(log(x))*log(4 - 4*e xp(5)))/(2*x)) - log(x)^2*(((135*x^2*log(4 - 4*exp(5))^5)/8 + (297*x^3*log (4 - 4*exp(5))^4)/8 + 27*x^4*log(4 - 4*exp(5))^3 + (135*x*log(4 - 4*exp(5) )^6)/64 + (27*log(4 - 4*exp(5))^7)/256)/(160*x^2*log(4 - 4*exp(5))^3 + 640 *x^3*log(4 - 4*exp(5))^2 + 20*x*log(4 - 4*exp(5))^4 + 1280*x^4*log(4 - 4*e xp(5)) + 1024*x^5 + log(4 - 4*exp(5))^5) - (27*log(4 - 4*exp(5))^2)/256) + (8040*x^2*log(4 - 4*exp(5))^5 + 20976*x^3*log(4 - 4*exp(5))^4 + 18432*x^4 *log(4 - 4*exp(5))^3 + 1239*x*log(4 - 4*exp(5))^6 + (279*log(4 - 4*exp(5)) ^7)/4)/(20480*x^2*log(4 - 4*exp(5))^3 + 81920*x^3*log(4 - 4*exp(5))^2 + 25 60*x*log(4 - 4*exp(5))^4 + 163840*x^4*log(4 - 4*exp(5)) + 131072*x^5 + 128 *log(4 - 4*exp(5))^5) + ((9*log(4 - 4*exp(5))*(4*log(4 - 4*exp(5))^3*log(x )^3 + 256*x^3 + 16*x^4 + 48*x*log(4 - 4*exp(5))^2*log(x)^2 - x^2*log(4 - 4 *exp(5))^2*log(x) + 4*x^2*log(4 - 4*exp(5))^2*log(x)^2 + 192*x^2*log(4 - 4 *exp(5))*log(x) + 16*x^3*log(4 - 4*exp(5))*log(x)))/(16*(4*x + log(4 - ...
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {72 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2+\left (\left (-288 x^2-18 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right )+144 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x)\right ) \log (\log (x))+\left (-864 x^2-36 x^3\right ) \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+\left (1152 x^3+72 x^4\right ) \log (x) \log ^3(\log (x))}{-\left (i \pi +\log \left (-4+4 e^5\right )\right )^3 \log (x)+12 x \left (i \pi +\log \left (-4+4 e^5\right )\right )^2 \log (x) \log (\log (x))-48 x^2 \left (i \pi +\log \left (-4+4 e^5\right )\right ) \log (x) \log ^2(\log (x))+64 x^3 \log (x) \log ^3(\log (x))} \, dx=\frac {9 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2} \left (\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x^{2}+32 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x -8 \,\mathrm {log}\left (-4 e^{5}+4\right )\right )}{16 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x^{2}-8 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (-4 e^{5}+4\right ) x +\mathrm {log}\left (-4 e^{5}+4\right )^{2}} \] Input:
int(((72*x^4+1152*x^3)*log(x)*log(log(x))^3+(-36*x^3-864*x^2)*log(-4*exp(5 )+4)*log(x)*log(log(x))^2+(144*x*log(-4*exp(5)+4)^2*log(x)+(-18*x^3-288*x^ 2)*log(-4*exp(5)+4))*log(log(x))+72*x*log(-4*exp(5)+4)^2)/(64*x^3*log(x)*l og(log(x))^3-48*x^2*log(-4*exp(5)+4)*log(x)*log(log(x))^2+12*x*log(-4*exp( 5)+4)^2*log(x)*log(log(x))-log(-4*exp(5)+4)^3*log(x)),x)
Output:
(9*log(log(x))*x**2*(log(log(x))*x**2 + 32*log(log(x))*x - 8*log( - 4*e**5 + 4)))/(16*log(log(x))**2*x**2 - 8*log(log(x))*log( - 4*e**5 + 4)*x + log ( - 4*e**5 + 4)**2)