Integrand size = 94, antiderivative size = 25 \[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=\frac {5-\log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{x} \] Output:
(5-ln(ln(4*exp(x^2)/x)+4+x+ln(x)))/x
Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=\frac {5}{x}-\frac {\log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{x} \] Input:
Integrate[(-20 - 6*x - 2*x^2 - 5*Log[(4*E^x^2)/x] - 5*Log[x] + (4 + x + Lo g[(4*E^x^2)/x] + Log[x])*Log[4 + x + Log[(4*E^x^2)/x] + Log[x]])/(4*x^2 + x^3 + x^2*Log[(4*E^x^2)/x] + x^2*Log[x]),x]
Output:
5/x - Log[4 + x + Log[(4*E^x^2)/x] + Log[x]]/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.
| \(\displaystyle \int \frac {-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )+\left (\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4\right ) \log \left (\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4\right )-6 x-5 \log (x)-20}{x^3+4 x^2+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {5 \log \left (\frac {4 e^{x^2}}{x}\right )}{x^2 \left (\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4\right )}+\frac {\log \left (\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4\right )}{x^2}-\frac {5 \log (x)}{x^2 \left (\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4\right )}-\frac {6}{x \left (\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4\right )}-\frac {20}{x^2 \left (\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4\right )}-\frac {2}{\log \left (\frac {4 e^{x^2}}{x}\right )+x+\log (x)+4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {1}{x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)+4}dx-\int \frac {1}{x \left (x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)+4\right )}dx+\int \frac {\log \left (x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)+4\right )}{x^2}dx+\frac {5}{x}\) |
Input:
Int[(-20 - 6*x - 2*x^2 - 5*Log[(4*E^x^2)/x] - 5*Log[x] + (4 + x + Log[(4*E ^x^2)/x] + Log[x])*Log[4 + x + Log[(4*E^x^2)/x] + Log[x]])/(4*x^2 + x^3 + x^2*Log[(4*E^x^2)/x] + x^2*Log[x]),x]
Output:
$Aborted
Time = 1.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04
| method | result | size |
| parallelrisch | \(\frac {10-2 \ln \left (\ln \left (\frac {4 \,{\mathrm e}^{x^{2}}}{x}\right )+4+x +\ln \left (x \right )\right )}{2 x}\) | \(26\) |
| default | \(-\frac {\ln \left (\ln \left (\frac {4 \,{\mathrm e}^{x^{2}}}{x}\right )+4+x +\ln \left (x \right )\right )}{x}+\frac {5}{x}\) | \(28\) |
| risch | \(-\frac {\ln \left (2 \ln \left (2\right )+\ln \left ({\mathrm e}^{x^{2}}\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{x^{2}}}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x^{2}}}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i {\mathrm e}^{x^{2}}}{x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{x^{2}}\right )\right )}{2}+4+x \right )}{x}+\frac {5}{x}\) | \(83\) |
Input:
int(((ln(4*exp(x^2)/x)+4+x+ln(x))*ln(ln(4*exp(x^2)/x)+4+x+ln(x))-5*ln(4*ex p(x^2)/x)-5*ln(x)-2*x^2-6*x-20)/(x^2*ln(4*exp(x^2)/x)+x^2*ln(x)+x^3+4*x^2) ,x,method=_RETURNVERBOSE)
Output:
1/2*(10-2*ln(ln(4*exp(x^2)/x)+4+x+ln(x)))/x
Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=-\frac {\log \left (x^{2} + x + 2 \, \log \left (2\right ) + 4\right ) - 5}{x} \] Input:
integrate(((log(4*exp(x^2)/x)+4+x+log(x))*log(log(4*exp(x^2)/x)+4+x+log(x) )-5*log(4*exp(x^2)/x)-5*log(x)-2*x^2-6*x-20)/(x^2*log(4*exp(x^2)/x)+x^2*lo g(x)+x^3+4*x^2),x, algorithm="fricas")
Output:
-(log(x^2 + x + 2*log(2) + 4) - 5)/x
Timed out. \[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=\text {Timed out} \] Input:
integrate(((ln(4*exp(x**2)/x)+4+x+ln(x))*ln(ln(4*exp(x**2)/x)+4+x+ln(x))-5 *ln(4*exp(x**2)/x)-5*ln(x)-2*x**2-6*x-20)/(x**2*ln(4*exp(x**2)/x)+x**2*ln( x)+x**3+4*x**2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (23) = 46\).
Time = 0.30 (sec) , antiderivative size = 311, normalized size of antiderivative = 12.44 \[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=\frac {{\left (3 \, \log \left (2\right ) + 1\right )} \log \left (x\right )}{\log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 4} + \frac {{\left (4 \, \log \left (2\right )^{2} - 7 \, \log \left (2\right ) - 25\right )} \arctan \left (\frac {2 \, x + 1}{\sqrt {8 \, \log \left (2\right ) + 15}}\right )}{{\left (\log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 4\right )} \sqrt {8 \, \log \left (2\right ) + 15}} - \frac {4 \, \arctan \left (\frac {2 \, x + 1}{\sqrt {8 \, \log \left (2\right ) + 15}}\right )}{\sqrt {8 \, \log \left (2\right ) + 15}} + \frac {5 \, {\left (4 \, \log \left (2\right ) + 7\right )} \arctan \left (\frac {2 \, x + 1}{\sqrt {8 \, \log \left (2\right ) + 15}}\right )}{{\left (\log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 4\right )} \sqrt {8 \, \log \left (2\right ) + 15}} - \frac {5 \, \log \left (x^{2} + x + 2 \, \log \left (2\right ) + 4\right )}{2 \, {\left (\log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 4\right )}} + \frac {3 \, \log \left (x^{2} + x + 2 \, \log \left (2\right ) + 4\right )}{2 \, {\left (\log \left (2\right ) + 2\right )}} + \frac {5 \, \log \left (x\right )}{\log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 4} - \frac {3 \, \log \left (x\right )}{\log \left (2\right ) + 2} + \frac {3 \, \arctan \left (\frac {2 \, x + 1}{\sqrt {8 \, \log \left (2\right ) + 15}}\right )}{\sqrt {8 \, \log \left (2\right ) + 15} {\left (\log \left (2\right ) + 2\right )}} + \frac {10 \, \log \left (2\right )^{2} - {\left (x {\left (3 \, \log \left (2\right ) + 1\right )} + 2 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 8\right )} \log \left (x^{2} + x + 2 \, \log \left (2\right ) + 4\right ) + 20 \, \log \left (2\right )}{2 \, {\left (\log \left (2\right )^{2} + 4 \, \log \left (2\right ) + 4\right )} x} + \frac {10}{x {\left (\log \left (2\right ) + 2\right )}} \] Input:
integrate(((log(4*exp(x^2)/x)+4+x+log(x))*log(log(4*exp(x^2)/x)+4+x+log(x) )-5*log(4*exp(x^2)/x)-5*log(x)-2*x^2-6*x-20)/(x^2*log(4*exp(x^2)/x)+x^2*lo g(x)+x^3+4*x^2),x, algorithm="maxima")
Output:
(3*log(2) + 1)*log(x)/(log(2)^2 + 4*log(2) + 4) + (4*log(2)^2 - 7*log(2) - 25)*arctan((2*x + 1)/sqrt(8*log(2) + 15))/((log(2)^2 + 4*log(2) + 4)*sqrt (8*log(2) + 15)) - 4*arctan((2*x + 1)/sqrt(8*log(2) + 15))/sqrt(8*log(2) + 15) + 5*(4*log(2) + 7)*arctan((2*x + 1)/sqrt(8*log(2) + 15))/((log(2)^2 + 4*log(2) + 4)*sqrt(8*log(2) + 15)) - 5/2*log(x^2 + x + 2*log(2) + 4)/(log (2)^2 + 4*log(2) + 4) + 3/2*log(x^2 + x + 2*log(2) + 4)/(log(2) + 2) + 5*l og(x)/(log(2)^2 + 4*log(2) + 4) - 3*log(x)/(log(2) + 2) + 3*arctan((2*x + 1)/sqrt(8*log(2) + 15))/(sqrt(8*log(2) + 15)*(log(2) + 2)) + 1/2*(10*log(2 )^2 - (x*(3*log(2) + 1) + 2*log(2)^2 + 8*log(2) + 8)*log(x^2 + x + 2*log(2 ) + 4) + 20*log(2))/((log(2)^2 + 4*log(2) + 4)*x) + 10/(x*(log(2) + 2))
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=-\frac {\log \left (x^{2} + x + 2 \, \log \left (2\right ) + 4\right )}{x} + \frac {5}{x} \] Input:
integrate(((log(4*exp(x^2)/x)+4+x+log(x))*log(log(4*exp(x^2)/x)+4+x+log(x) )-5*log(4*exp(x^2)/x)-5*log(x)-2*x^2-6*x-20)/(x^2*log(4*exp(x^2)/x)+x^2*lo g(x)+x^3+4*x^2),x, algorithm="giac")
Output:
-log(x^2 + x + 2*log(2) + 4)/x + 5/x
Time = 2.85 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=-\frac {\ln \left (x+\ln \left (\frac {4}{x}\right )+\ln \left (x\right )+x^2+4\right )-5}{x} \] Input:
int(-(6*x + 5*log((4*exp(x^2))/x) + 5*log(x) - log(x + log((4*exp(x^2))/x) + log(x) + 4)*(x + log((4*exp(x^2))/x) + log(x) + 4) + 2*x^2 + 20)/(x^2*l og((4*exp(x^2))/x) + x^2*log(x) + 4*x^2 + x^3),x)
Output:
-(log(x + log(4/x) + log(x) + x^2 + 4) - 5)/x
\[ \int \frac {-20-6 x-2 x^2-5 \log \left (\frac {4 e^{x^2}}{x}\right )-5 \log (x)+\left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right ) \log \left (4+x+\log \left (\frac {4 e^{x^2}}{x}\right )+\log (x)\right )}{4 x^2+x^3+x^2 \log \left (\frac {4 e^{x^2}}{x}\right )+x^2 \log (x)} \, dx=\int \frac {\left (\mathrm {log}\left (\frac {4 \,{\mathrm e}^{x^{2}}}{x}\right )+4+x +\mathrm {log}\left (x \right )\right ) \mathrm {log}\left (\mathrm {log}\left (\frac {4 \,{\mathrm e}^{x^{2}}}{x}\right )+4+x +\mathrm {log}\left (x \right )\right )-5 \,\mathrm {log}\left (\frac {4 \,{\mathrm e}^{x^{2}}}{x}\right )-5 \,\mathrm {log}\left (x \right )-2 x^{2}-6 x -20}{x^{2} \mathrm {log}\left (\frac {4 \,{\mathrm e}^{x^{2}}}{x}\right )+\mathrm {log}\left (x \right ) x^{2}+x^{3}+4 x^{2}}d x \] Input:
int(((log(4*exp(x^2)/x)+4+x+log(x))*log(log(4*exp(x^2)/x)+4+x+log(x))-5*lo g(4*exp(x^2)/x)-5*log(x)-2*x^2-6*x-20)/(x^2*log(4*exp(x^2)/x)+x^2*log(x)+x ^3+4*x^2),x)
Output:
int(((log(4*exp(x^2)/x)+4+x+log(x))*log(log(4*exp(x^2)/x)+4+x+log(x))-5*lo g(4*exp(x^2)/x)-5*log(x)-2*x^2-6*x-20)/(x^2*log(4*exp(x^2)/x)+x^2*log(x)+x ^3+4*x^2),x)