Integrand size = 188, antiderivative size = 32 \[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\frac {4}{\left (-e^{\frac {2 \left (-x+3 \left (3+(-x+\log (\log (x)))^2\right )\right )}{x}}+x\right )^2} \] Output:
4/(x-exp((3*(ln(ln(x))-x)^2+9-x)/x)^2)^2
Time = 0.13 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.22 \[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\frac {4 e^4 \log ^{24}(x)}{\left (e^{\frac {6 \left (3+x^2+\log ^2(\log (x))\right )}{x}}-e^2 x \log ^{12}(x)\right )^2} \] Input:
Integrate[(8*x^2*Log[x] + E^((2*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[L og[x]]^2))/x)*(96*x + (144 - 48*x^2)*Log[x] - 96*Log[Log[x]] + 48*Log[x]*L og[Log[x]]^2))/(E^((6*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2)) /x)*x^2*Log[x] - 3*E^((4*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^ 2))/x)*x^3*Log[x] + 3*E^((2*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x ]]^2))/x)*x^4*Log[x] - x^5*Log[x]),x]
Output:
(4*E^4*Log[x]^24)/(E^((6*(3 + x^2 + Log[Log[x]]^2))/x) - E^2*x*Log[x]^12)^ 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.
| \(\displaystyle \int \frac {\left (\left (144-48 x^2\right ) \log (x)+96 x+48 \log (x) \log ^2(\log (x))-96 \log (\log (x))\right ) \exp \left (\frac {2 \left (3 x^2-x+3 \log ^2(\log (x))-6 x \log (\log (x))+9\right )}{x}\right )+8 x^2 \log (x)}{x^2 \log (x) \exp \left (\frac {6 \left (3 x^2-x+3 \log ^2(\log (x))-6 x \log (\log (x))+9\right )}{x}\right )+3 x^4 \log (x) \exp \left (\frac {2 \left (3 x^2-x+3 \log ^2(\log (x))-6 x \log (\log (x))+9\right )}{x}\right )-3 x^3 \log (x) \exp \left (\frac {4 \left (3 x^2-x+3 \log ^2(\log (x))-6 x \log (\log (x))+9\right )}{x}\right )+x^5 (-\log (x))} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^6 \log ^{35}(x) \left (\left (\left (144-48 x^2\right ) \log (x)+96 x+48 \log (x) \log ^2(\log (x))-96 \log (\log (x))\right ) \exp \left (\frac {2 \left (3 x^2-x+3 \log ^2(\log (x))-6 x \log (\log (x))+9\right )}{x}\right )+8 x^2 \log (x)\right )}{x^2 \left (e^{6 x+\frac {18}{x}+\frac {6 \log ^2(\log (x))}{x}}-e^2 x \log ^{12}(x)\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^6 \int \frac {8 \log ^{35}(x) \left (\log (x) x^2+\frac {6 e^{\frac {2 \left (3 x^2-x+3 \log ^2(\log (x))+9\right )}{x}} \left (\log (x) \log ^2(\log (x))-2 \log (\log (x))+2 x+\left (3-x^2\right ) \log (x)\right )}{\log ^{12}(x)}\right )}{x^2 \left (e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}-e^2 x \log ^{12}(x)\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 8 e^6 \int \frac {\log ^{35}(x) \left (\log (x) x^2+\frac {6 e^{\frac {2 \left (3 x^2-x+3 \log ^2(\log (x))+9\right )}{x}} \left (\log (x) \log ^2(\log (x))-2 \log (\log (x))+2 x+\left (3-x^2\right ) \log (x)\right )}{\log ^{12}(x)}\right )}{x^2 \left (e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}-e^2 x \log ^{12}(x)\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 8 e^6 \int \left (\frac {\log ^{35}(x) \left (6 \log (x) x^2-\log (x) x-12 x-6 \log (x) \log ^2(\log (x))-18 \log (x)+12 \log (\log (x))\right )}{x \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^3}-\frac {6 \log ^{23}(x) \left (\log (x) x^2-2 x-\log (x) \log ^2(\log (x))-3 \log (x)+2 \log (\log (x))\right )}{e^2 x^2 \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 8 e^6 \left (\frac {18 \int \frac {\log ^{24}(x)}{x^2 \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^2}dx}{e^2}+\frac {6 \int \frac {\log ^{24}(x) \log ^2(\log (x))}{x^2 \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^2}dx}{e^2}-\frac {12 \int \frac {\log ^{23}(x) \log (\log (x))}{x^2 \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^2}dx}{e^2}-\int \frac {\log ^{36}(x)}{\left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^3}dx-18 \int \frac {\log ^{36}(x)}{x \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^3}dx+6 \int \frac {x \log ^{36}(x)}{\left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^3}dx-6 \int \frac {\log ^{36}(x) \log ^2(\log (x))}{x \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^3}dx-12 \int \frac {\log ^{35}(x)}{\left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^3}dx+12 \int \frac {\log ^{35}(x) \log (\log (x))}{x \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^3}dx-\frac {6 \int \frac {\log ^{24}(x)}{\left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^2}dx}{e^2}+\frac {12 \int \frac {\log ^{23}(x)}{x \left (e^2 x \log ^{12}(x)-e^{\frac {6 \log ^2(\log (x))}{x}+6 x+\frac {18}{x}}\right )^2}dx}{e^2}\right )\) |
Input:
Int[(8*x^2*Log[x] + E^((2*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]] ^2))/x)*(96*x + (144 - 48*x^2)*Log[x] - 96*Log[Log[x]] + 48*Log[x]*Log[Log [x]]^2))/(E^((6*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x)*x^ 2*Log[x] - 3*E^((4*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2))/x) *x^3*Log[x] + 3*E^((2*(9 - x + 3*x^2 - 6*x*Log[Log[x]] + 3*Log[Log[x]]^2)) /x)*x^4*Log[x] - x^5*Log[x]),x]
Output:
$Aborted
Time = 59.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12
| method | result | size |
| risch | \(\frac {4}{\left (-\frac {{\mathrm e}^{\frac {6 \ln \left (\ln \left (x \right )\right )^{2}+6 x^{2}-2 x +18}{x}}}{\ln \left (x \right )^{12}}+x \right )^{2}}\) | \(36\) |
| parallelrisch | \(\frac {4}{{\mathrm e}^{\frac {12 \ln \left (\ln \left (x \right )\right )^{2}-24 x \ln \left (\ln \left (x \right )\right )+12 x^{2}-4 x +36}{x}}-2 \,{\mathrm e}^{\frac {6 \ln \left (\ln \left (x \right )\right )^{2}-12 x \ln \left (\ln \left (x \right )\right )+6 x^{2}-2 x +18}{x}} x +x^{2}}\) | \(72\) |
Input:
int(((48*ln(x)*ln(ln(x))^2-96*ln(ln(x))+(-48*x^2+144)*ln(x)+96*x)*exp((3*l n(ln(x))^2-6*x*ln(ln(x))+3*x^2-x+9)/x)^2+8*x^2*ln(x))/(x^2*ln(x)*exp((3*ln (ln(x))^2-6*x*ln(ln(x))+3*x^2-x+9)/x)^6-3*x^3*ln(x)*exp((3*ln(ln(x))^2-6*x *ln(ln(x))+3*x^2-x+9)/x)^4+3*x^4*ln(x)*exp((3*ln(ln(x))^2-6*x*ln(ln(x))+3* x^2-x+9)/x)^2-x^5*ln(x)),x,method=_RETURNVERBOSE)
Output:
4/(-1/ln(x)^12*exp(2*(3*ln(ln(x))^2+3*x^2-x+9)/x)+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (30) = 60\).
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.16 \[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\frac {4}{x^{2} - 2 \, x e^{\left (\frac {2 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \left (x\right )\right ) + 3 \, \log \left (\log \left (x\right )\right )^{2} - x + 9\right )}}{x}\right )} + e^{\left (\frac {4 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \left (x\right )\right ) + 3 \, \log \left (\log \left (x\right )\right )^{2} - x + 9\right )}}{x}\right )}} \] Input:
integrate(((48*log(x)*log(log(x))^2-96*log(log(x))+(-48*x^2+144)*log(x)+96 *x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2+8*x^2*log(x))/(x^ 2*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^6-3*x^3*log(x) *exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^4+3*x^4*log(x)*exp((3* log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2-x^5*log(x)),x, algorithm="fr icas")
Output:
4/(x^2 - 2*x*e^(2*(3*x^2 - 6*x*log(log(x)) + 3*log(log(x))^2 - x + 9)/x) + e^(4*(3*x^2 - 6*x*log(log(x)) + 3*log(log(x))^2 - x + 9)/x))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\frac {4}{x^{2} - 2 x e^{\frac {2 \cdot \left (3 x^{2} - 6 x \log {\left (\log {\left (x \right )} \right )} - x + 3 \log {\left (\log {\left (x \right )} \right )}^{2} + 9\right )}{x}} + e^{\frac {4 \cdot \left (3 x^{2} - 6 x \log {\left (\log {\left (x \right )} \right )} - x + 3 \log {\left (\log {\left (x \right )} \right )}^{2} + 9\right )}{x}}} \] Input:
integrate(((48*ln(x)*ln(ln(x))**2-96*ln(ln(x))+(-48*x**2+144)*ln(x)+96*x)* exp((3*ln(ln(x))**2-6*x*ln(ln(x))+3*x**2-x+9)/x)**2+8*x**2*ln(x))/(x**2*ln (x)*exp((3*ln(ln(x))**2-6*x*ln(ln(x))+3*x**2-x+9)/x)**6-3*x**3*ln(x)*exp(( 3*ln(ln(x))**2-6*x*ln(ln(x))+3*x**2-x+9)/x)**4+3*x**4*ln(x)*exp((3*ln(ln(x ))**2-6*x*ln(ln(x))+3*x**2-x+9)/x)**2-x**5*ln(x)),x)
Output:
4/(x**2 - 2*x*exp(2*(3*x**2 - 6*x*log(log(x)) - x + 3*log(log(x))**2 + 9)/ x) + exp(4*(3*x**2 - 6*x*log(log(x)) - x + 3*log(log(x))**2 + 9)/x))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (30) = 60\).
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.16 \[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\frac {4 \, e^{4} \log \left (x\right )^{24}}{x^{2} e^{4} \log \left (x\right )^{24} - 2 \, x e^{\left (6 \, x + \frac {6 \, \log \left (\log \left (x\right )\right )^{2}}{x} + \frac {18}{x} + 2\right )} \log \left (x\right )^{12} + e^{\left (12 \, x + \frac {12 \, \log \left (\log \left (x\right )\right )^{2}}{x} + \frac {36}{x}\right )}} \] Input:
integrate(((48*log(x)*log(log(x))^2-96*log(log(x))+(-48*x^2+144)*log(x)+96 *x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2+8*x^2*log(x))/(x^ 2*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^6-3*x^3*log(x) *exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^4+3*x^4*log(x)*exp((3* log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2-x^5*log(x)),x, algorithm="ma xima")
Output:
4*e^4*log(x)^24/(x^2*e^4*log(x)^24 - 2*x*e^(6*x + 6*log(log(x))^2/x + 18/x + 2)*log(x)^12 + e^(12*x + 12*log(log(x))^2/x + 36/x))
\[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\int { -\frac {8 \, {\left (x^{2} \log \left (x\right ) + 6 \, {\left (\log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} - {\left (x^{2} - 3\right )} \log \left (x\right ) + 2 \, x - 2 \, \log \left (\log \left (x\right )\right )\right )} e^{\left (\frac {2 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \left (x\right )\right ) + 3 \, \log \left (\log \left (x\right )\right )^{2} - x + 9\right )}}{x}\right )}\right )}}{x^{5} \log \left (x\right ) - 3 \, x^{4} e^{\left (\frac {2 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \left (x\right )\right ) + 3 \, \log \left (\log \left (x\right )\right )^{2} - x + 9\right )}}{x}\right )} \log \left (x\right ) + 3 \, x^{3} e^{\left (\frac {4 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \left (x\right )\right ) + 3 \, \log \left (\log \left (x\right )\right )^{2} - x + 9\right )}}{x}\right )} \log \left (x\right ) - x^{2} e^{\left (\frac {6 \, {\left (3 \, x^{2} - 6 \, x \log \left (\log \left (x\right )\right ) + 3 \, \log \left (\log \left (x\right )\right )^{2} - x + 9\right )}}{x}\right )} \log \left (x\right )} \,d x } \] Input:
integrate(((48*log(x)*log(log(x))^2-96*log(log(x))+(-48*x^2+144)*log(x)+96 *x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2+8*x^2*log(x))/(x^ 2*log(x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^6-3*x^3*log(x) *exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^4+3*x^4*log(x)*exp((3* log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2-x^5*log(x)),x, algorithm="gi ac")
Output:
undef
Time = 3.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\frac {4}{x^2+\frac {{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{\frac {12\,{\ln \left (\ln \left (x\right )\right )}^2}{x}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{36/x}}{{\ln \left (x\right )}^{24}}-\frac {2\,x\,{\mathrm {e}}^{6\,x}\,{\mathrm {e}}^{\frac {6\,{\ln \left (\ln \left (x\right )\right )}^2}{x}}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{18/x}}{{\ln \left (x\right )}^{12}}} \] Input:
int(-(8*x^2*log(x) + exp((2*(3*log(log(x))^2 - 6*x*log(log(x)) - x + 3*x^2 + 9))/x)*(96*x - 96*log(log(x)) - log(x)*(48*x^2 - 144) + 48*log(log(x))^ 2*log(x)))/(x^5*log(x) - 3*x^4*exp((2*(3*log(log(x))^2 - 6*x*log(log(x)) - x + 3*x^2 + 9))/x)*log(x) + 3*x^3*exp((4*(3*log(log(x))^2 - 6*x*log(log(x )) - x + 3*x^2 + 9))/x)*log(x) - x^2*exp((6*(3*log(log(x))^2 - 6*x*log(log (x)) - x + 3*x^2 + 9))/x)*log(x)),x)
Output:
4/(x^2 + (exp(12*x)*exp((12*log(log(x))^2)/x)*exp(-4)*exp(36/x))/log(x)^24 - (2*x*exp(6*x)*exp((6*log(log(x))^2)/x)*exp(-2)*exp(18/x))/log(x)^12)
Time = 11.92 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.28 \[ \int \frac {8 x^2 \log (x)+e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} \left (96 x+\left (144-48 x^2\right ) \log (x)-96 \log (\log (x))+48 \log (x) \log ^2(\log (x))\right )}{e^{\frac {6 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^2 \log (x)-3 e^{\frac {4 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^3 \log (x)+3 e^{\frac {2 \left (9-x+3 x^2-6 x \log (\log (x))+3 \log ^2(\log (x))\right )}{x}} x^4 \log (x)-x^5 \log (x)} \, dx=\frac {4 \mathrm {log}\left (x \right )^{24} e^{4}}{e^{\frac {12 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}+12 x^{2}+36}{x}}-2 e^{\frac {6 \mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2}+6 x^{2}+18}{x}} \mathrm {log}\left (x \right )^{12} e^{2} x +\mathrm {log}\left (x \right )^{24} e^{4} x^{2}} \] Input:
int(((48*log(x)*log(log(x))^2-96*log(log(x))+(-48*x^2+144)*log(x)+96*x)*ex p((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2+8*x^2*log(x))/(x^2*log( x)*exp((3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^6-3*x^3*log(x)*exp(( 3*log(log(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^4+3*x^4*log(x)*exp((3*log(lo g(x))^2-6*x*log(log(x))+3*x^2-x+9)/x)^2-x^5*log(x)),x)
Output:
(4*log(x)**24*e**4)/(e**((12*log(log(x))**2 + 12*x**2 + 36)/x) - 2*e**((6* log(log(x))**2 + 6*x**2 + 18)/x)*log(x)**12*e**2*x + log(x)**24*e**4*x**2)