Integrand size = 151, antiderivative size = 29 \[ \int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx=\frac {2 \left (-2+\log \left (\log \left (\frac {\log (x)}{\frac {37}{4}-e^{1-x}}\right )\right )\right )}{5 x} \]
Time = 0.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx=\frac {1}{5} \left (-\frac {4}{x}+\frac {2 \log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x}\right ) \]
Integrate[(-74 + 8*E^(1 - x) + 8*E^(1 - x)*x*Log[x] + (-148 + 16*E^(1 - x) )*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x))] + (74 - 8*E^(1 - x))*Log[x]* Log[(-4*Log[x])/(-37 + 4*E^(1 - x))]*Log[Log[(-4*Log[x])/(-37 + 4*E^(1 - x ))]])/((-185*x^2 + 20*E^(1 - x)*x^2)*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - 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.
| \(\displaystyle \int \frac {8 e^{1-x}+8 e^{1-x} x \log (x)+\left (16 e^{1-x}-148\right ) \log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right ) \log (x)+\left (74-8 e^{1-x}\right ) \log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right ) \log \left (\log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right )\right ) \log (x)-74}{\left (20 e^{1-x} x^2-185 x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (8 e^{1-x}+8 e^{1-x} x \log (x)+\left (16 e^{1-x}-148\right ) \log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right ) \log (x)+\left (74-8 e^{1-x}\right ) \log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right ) \log \left (\log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right )\right ) \log (x)-74\right )}{5 \left (4 e-37 e^x\right ) x^2 \log (x) \log \left (-\frac {4 \log (x)}{4 e^{1-x}-37}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int -\frac {2 e^x \left (-4 e^{1-x} x \log (x)+2 \left (37-4 e^{1-x}\right ) \log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right ) \log (x)-\left (37-4 e^{1-x}\right ) \log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right ) \log \left (\log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right )\right ) \log (x)-4 e^{1-x}+37\right )}{\left (4 e-37 e^x\right ) x^2 \log (x) \log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \int \frac {e^x \left (-4 e^{1-x} x \log (x)+2 \left (37-4 e^{1-x}\right ) \log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right ) \log (x)-\left (37-4 e^{1-x}\right ) \log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right ) \log \left (\log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right )\right ) \log (x)-4 e^{1-x}+37\right )}{\left (4 e-37 e^x\right ) x^2 \log (x) \log \left (\frac {4 \log (x)}{37-4 e^{1-x}}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2}{5} \int \left (\frac {37 e^x}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}-\frac {x \log (x)+2 \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right ) \log (x)-\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right ) \log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right ) \log (x)+1}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{5} \left (-\int \frac {1}{x^2 \log (x) \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}dx+\int \frac {\log \left (\log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )\right )}{x^2}dx-\int \frac {1}{x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}dx+37 \int \frac {e^x}{\left (-4 e+37 e^x\right ) x \log \left (\frac {4 e^x \log (x)}{-4 e+37 e^x}\right )}dx+\frac {2}{x}\right )\) |
Int[(-74 + 8*E^(1 - x) + 8*E^(1 - x)*x*Log[x] + (-148 + 16*E^(1 - x))*Log[ x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x))] + (74 - 8*E^(1 - x))*Log[x]*Log[(- 4*Log[x])/(-37 + 4*E^(1 - x))]*Log[Log[(-4*Log[x])/(-37 + 4*E^(1 - x))]])/ ((-185*x^2 + 20*E^(1 - x)*x^2)*Log[x]*Log[(-4*Log[x])/(-37 + 4*E^(1 - x))] ),x]
3.26.35.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 15.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
| method | result | size |
| parallelrisch | \(\frac {-1184+592 \ln \left (\ln \left (-\frac {4 \ln \left (x \right )}{4 \,{\mathrm e}^{1-x}-37}\right )\right )}{1480 x}\) | \(28\) |
| risch | \(\frac {2 \ln \left (i \pi +\ln \left (\ln \left (x \right )\right )-\ln \left ({\mathrm e}^{1-x}-\frac {37}{4}\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )+\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )\right )}{2}+i \pi \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )^{2} \left (\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{{\mathrm e}^{1-x}-\frac {37}{4}}\right )-1\right )\right )}{5 x}-\frac {4}{5 x}\) | \(150\) |
int(((-8*exp(1-x)+74)*ln(x)*ln(-4*ln(x)/(4*exp(1-x)-37))*ln(ln(-4*ln(x)/(4 *exp(1-x)-37)))+(16*exp(1-x)-148)*ln(x)*ln(-4*ln(x)/(4*exp(1-x)-37))+8*x*e xp(1-x)*ln(x)+8*exp(1-x)-74)/(20*x^2*exp(1-x)-185*x^2)/ln(x)/ln(-4*ln(x)/( 4*exp(1-x)-37)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx=\frac {2 \, {\left (\log \left (\log \left (-\frac {4 \, \log \left (x\right )}{4 \, e^{\left (-x + 1\right )} - 37}\right )\right ) - 2\right )}}{5 \, x} \]
integrate(((-8*exp(1-x)+74)*log(x)*log(-4*log(x)/(4*exp(1-x)-37))*log(log( -4*log(x)/(4*exp(1-x)-37)))+(16*exp(1-x)-148)*log(x)*log(-4*log(x)/(4*exp( 1-x)-37))+8*x*exp(1-x)*log(x)+8*exp(1-x)-74)/(20*x^2*exp(1-x)-185*x^2)/log (x)/log(-4*log(x)/(4*exp(1-x)-37)),x, algorithm=\
Time = 18.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx=\frac {2 \log {\left (\log {\left (- \frac {4 \log {\left (x \right )}}{4 e^{1 - x} - 37} \right )} \right )}}{5 x} - \frac {4}{5 x} \]
integrate(((-8*exp(1-x)+74)*ln(x)*ln(-4*ln(x)/(4*exp(1-x)-37))*ln(ln(-4*ln (x)/(4*exp(1-x)-37)))+(16*exp(1-x)-148)*ln(x)*ln(-4*ln(x)/(4*exp(1-x)-37)) +8*x*exp(1-x)*ln(x)+8*exp(1-x)-74)/(20*x**2*exp(1-x)-185*x**2)/ln(x)/ln(-4 *ln(x)/(4*exp(1-x)-37)),x)
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx=\frac {2 \, {\left (\log \left (x + 2 \, \log \left (2\right ) - \log \left (-4 \, e + 37 \, e^{x}\right ) + \log \left (\log \left (x\right )\right )\right ) - 2\right )}}{5 \, x} \]
integrate(((-8*exp(1-x)+74)*log(x)*log(-4*log(x)/(4*exp(1-x)-37))*log(log( -4*log(x)/(4*exp(1-x)-37)))+(16*exp(1-x)-148)*log(x)*log(-4*log(x)/(4*exp( 1-x)-37))+8*x*exp(1-x)*log(x)+8*exp(1-x)-74)/(20*x^2*exp(1-x)-185*x^2)/log (x)/log(-4*log(x)/(4*exp(1-x)-37)),x, algorithm=\
\[ \int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx=\int { -\frac {2 \, {\left ({\left (4 \, e^{\left (-x + 1\right )} - 37\right )} \log \left (x\right ) \log \left (-\frac {4 \, \log \left (x\right )}{4 \, e^{\left (-x + 1\right )} - 37}\right ) \log \left (\log \left (-\frac {4 \, \log \left (x\right )}{4 \, e^{\left (-x + 1\right )} - 37}\right )\right ) - 4 \, x e^{\left (-x + 1\right )} \log \left (x\right ) - 2 \, {\left (4 \, e^{\left (-x + 1\right )} - 37\right )} \log \left (x\right ) \log \left (-\frac {4 \, \log \left (x\right )}{4 \, e^{\left (-x + 1\right )} - 37}\right ) - 4 \, e^{\left (-x + 1\right )} + 37\right )}}{5 \, {\left (4 \, x^{2} e^{\left (-x + 1\right )} - 37 \, x^{2}\right )} \log \left (x\right ) \log \left (-\frac {4 \, \log \left (x\right )}{4 \, e^{\left (-x + 1\right )} - 37}\right )} \,d x } \]
integrate(((-8*exp(1-x)+74)*log(x)*log(-4*log(x)/(4*exp(1-x)-37))*log(log( -4*log(x)/(4*exp(1-x)-37)))+(16*exp(1-x)-148)*log(x)*log(-4*log(x)/(4*exp( 1-x)-37))+8*x*exp(1-x)*log(x)+8*exp(1-x)-74)/(20*x^2*exp(1-x)-185*x^2)/log (x)/log(-4*log(x)/(4*exp(1-x)-37)),x, algorithm=\
integrate(-2/5*((4*e^(-x + 1) - 37)*log(x)*log(-4*log(x)/(4*e^(-x + 1) - 3 7))*log(log(-4*log(x)/(4*e^(-x + 1) - 37))) - 4*x*e^(-x + 1)*log(x) - 2*(4 *e^(-x + 1) - 37)*log(x)*log(-4*log(x)/(4*e^(-x + 1) - 37)) - 4*e^(-x + 1) + 37)/((4*x^2*e^(-x + 1) - 37*x^2)*log(x)*log(-4*log(x)/(4*e^(-x + 1) - 3 7))), x)
Timed out. \[ \int \frac {-74+8 e^{1-x}+8 e^{1-x} x \log (x)+\left (-148+16 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )+\left (74-8 e^{1-x}\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right ) \log \left (\log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )\right )}{\left (-185 x^2+20 e^{1-x} x^2\right ) \log (x) \log \left (-\frac {4 \log (x)}{-37+4 e^{1-x}}\right )} \, dx=\int \frac {8\,{\mathrm {e}}^{1-x}+\ln \left (-\frac {4\,\ln \left (x\right )}{4\,{\mathrm {e}}^{1-x}-37}\right )\,\ln \left (x\right )\,\left (16\,{\mathrm {e}}^{1-x}-148\right )+8\,x\,{\mathrm {e}}^{1-x}\,\ln \left (x\right )-\ln \left (-\frac {4\,\ln \left (x\right )}{4\,{\mathrm {e}}^{1-x}-37}\right )\,\ln \left (\ln \left (-\frac {4\,\ln \left (x\right )}{4\,{\mathrm {e}}^{1-x}-37}\right )\right )\,\ln \left (x\right )\,\left (8\,{\mathrm {e}}^{1-x}-74\right )-74}{\ln \left (-\frac {4\,\ln \left (x\right )}{4\,{\mathrm {e}}^{1-x}-37}\right )\,\ln \left (x\right )\,\left (20\,x^2\,{\mathrm {e}}^{1-x}-185\,x^2\right )} \,d x \]
int((8*exp(1 - x) + log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(16*exp(1 - x) - 148) + 8*x*exp(1 - x)*log(x) - log(-(4*log(x))/(4*exp(1 - x) - 37)) *log(log(-(4*log(x))/(4*exp(1 - x) - 37)))*log(x)*(8*exp(1 - x) - 74) - 74 )/(log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(20*x^2*exp(1 - x) - 185*x^ 2)),x)
int((8*exp(1 - x) + log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(16*exp(1 - x) - 148) + 8*x*exp(1 - x)*log(x) - log(-(4*log(x))/(4*exp(1 - x) - 37)) *log(log(-(4*log(x))/(4*exp(1 - x) - 37)))*log(x)*(8*exp(1 - x) - 74) - 74 )/(log(-(4*log(x))/(4*exp(1 - x) - 37))*log(x)*(20*x^2*exp(1 - x) - 185*x^ 2)), x)