Integrand size = 91, antiderivative size = 28 \[ \int \frac {e^{\frac {36+12 \log \left (\frac {4}{x}\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (-36+36 e^4 x+\left (-12+12 e^4 x\right ) \log \left (\frac {4}{x}\right )+\left (-48-12 \log \left (\frac {4}{x}\right )\right ) \log \left (e^{-e^4 x} x\right )\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx=e^{\frac {12 \left (3+\log \left (\frac {4}{x}\right )\right )}{x \log \left (e^{-e^4 x} x\right )}} \]
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {36+12 \log \left (\frac {4}{x}\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (-36+36 e^4 x+\left (-12+12 e^4 x\right ) \log \left (\frac {4}{x}\right )+\left (-48-12 \log \left (\frac {4}{x}\right )\right ) \log \left (e^{-e^4 x} x\right )\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx=e^{\frac {12 \left (3+\log \left (\frac {4}{x}\right )\right )}{x \log \left (e^{-e^4 x} x\right )}} \]
Integrate[(E^((36 + 12*Log[4/x])/(x*Log[x/E^(E^4*x)]))*(-36 + 36*E^4*x + ( -12 + 12*E^4*x)*Log[4/x] + (-48 - 12*Log[4/x])*Log[x/E^(E^4*x)]))/(x^2*Log [x/E^(E^4*x)]^2),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.
\(\displaystyle \int \frac {e^{\frac {12 \log \left (\frac {4}{x}\right )+36}{x \log \left (e^{-e^4 x} x\right )}} \left (36 e^4 x+\left (12 e^4 x-12\right ) \log \left (\frac {4}{x}\right )+\left (-12 \log \left (\frac {4}{x}\right )-48\right ) \log \left (e^{-e^4 x} x\right )-36\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (36 e^4 x+\left (12 e^4 x-12\right ) \log \left (\frac {4}{x}\right )+\left (-12 \log \left (\frac {4}{x}\right )-48\right ) \log \left (e^{-e^4 x} x\right )-36\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {12 \left (e^4 x-1\right ) e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (\log \left (\frac {4}{x}\right )+3\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )}-\frac {12 e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (\log \left (\frac {4}{x}\right )+4\right )}{x^2 \log \left (e^{-e^4 x} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -36 \int \frac {e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}}}{x^2 \log ^2\left (e^{-e^4 x} x\right )}dx-12 \int \frac {e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}} \log \left (\frac {4}{x}\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )}dx-48 \int \frac {e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}}}{x^2 \log \left (e^{-e^4 x} x\right )}dx-12 \int \frac {e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}} \log \left (\frac {4}{x}\right )}{x^2 \log \left (e^{-e^4 x} x\right )}dx+36 \int \frac {e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}+4}}{x \log ^2\left (e^{-e^4 x} x\right )}dx+12 \int \frac {e^{\frac {12 \left (\log \left (\frac {4}{x}\right )+3\right )}{x \log \left (e^{-e^4 x} x\right )}+4} \log \left (\frac {4}{x}\right )}{x \log ^2\left (e^{-e^4 x} x\right )}dx\) |
Int[(E^((36 + 12*Log[4/x])/(x*Log[x/E^(E^4*x)]))*(-36 + 36*E^4*x + (-12 + 12*E^4*x)*Log[4/x] + (-48 - 12*Log[4/x])*Log[x/E^(E^4*x)]))/(x^2*Log[x/E^( E^4*x)]^2),x]
3.18.15.3.1 Defintions of rubi rules used
Time = 77.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {12 \ln \left (\frac {4}{x}\right )+36}{x \ln \left (x \,{\mathrm e}^{-x \,{\mathrm e}^{4}}\right )}}\) | \(27\) |
risch | \({\mathrm e}^{\frac {48 \ln \left (2\right )-24 \ln \left (x \right )+72}{x \left (-i \pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-x \,{\mathrm e}^{4}}\right )^{3}+i \pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-x \,{\mathrm e}^{4}}\right )^{2} \operatorname {csgn}\left (i x \right )+i \pi \operatorname {csgn}\left (i x \,{\mathrm e}^{-x \,{\mathrm e}^{4}}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-x \,{\mathrm e}^{4}}\right )-i \pi \,\operatorname {csgn}\left (i x \,{\mathrm e}^{-x \,{\mathrm e}^{4}}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i {\mathrm e}^{-x \,{\mathrm e}^{4}}\right )+2 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{x \,{\mathrm e}^{4}}\right )\right )}}\) | \(128\) |
int(((-12*ln(4/x)-48)*ln(x/exp(x*exp(4)))+(12*x*exp(4)-12)*ln(4/x)+36*x*ex p(4)-36)*exp((12*ln(4/x)+36)/x/ln(x/exp(x*exp(4))))/x^2/ln(x/exp(x*exp(4)) )^2,x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {36+12 \log \left (\frac {4}{x}\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (-36+36 e^4 x+\left (-12+12 e^4 x\right ) \log \left (\frac {4}{x}\right )+\left (-48-12 \log \left (\frac {4}{x}\right )\right ) \log \left (e^{-e^4 x} x\right )\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx=e^{\left (-\frac {12 \, {\left (\log \left (\frac {4}{x}\right ) + 3\right )}}{x^{2} e^{4} - 2 \, x \log \left (2\right ) + x \log \left (\frac {4}{x}\right )}\right )} \]
integrate(((-12*log(4/x)-48)*log(x/exp(x*exp(4)))+(12*x*exp(4)-12)*log(4/x )+36*x*exp(4)-36)*exp((12*log(4/x)+36)/x/log(x/exp(x*exp(4))))/x^2/log(x/e xp(x*exp(4)))^2,x, algorithm=\
Timed out. \[ \int \frac {e^{\frac {36+12 \log \left (\frac {4}{x}\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (-36+36 e^4 x+\left (-12+12 e^4 x\right ) \log \left (\frac {4}{x}\right )+\left (-48-12 \log \left (\frac {4}{x}\right )\right ) \log \left (e^{-e^4 x} x\right )\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx=\text {Timed out} \]
integrate(((-12*ln(4/x)-48)*ln(x/exp(x*exp(4)))+(12*x*exp(4)-12)*ln(4/x)+3 6*x*exp(4)-36)*exp((12*ln(4/x)+36)/x/ln(x/exp(x*exp(4))))/x**2/ln(x/exp(x* exp(4)))**2,x)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
Time = 0.45 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93 \[ \int \frac {e^{\frac {36+12 \log \left (\frac {4}{x}\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (-36+36 e^4 x+\left (-12+12 e^4 x\right ) \log \left (\frac {4}{x}\right )+\left (-48-12 \log \left (\frac {4}{x}\right )\right ) \log \left (e^{-e^4 x} x\right )\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx=e^{\left (-\frac {24 \, e^{4} \log \left (2\right )}{x e^{4} \log \left (x\right ) - \log \left (x\right )^{2}} - \frac {36 \, e^{4}}{x e^{4} \log \left (x\right ) - \log \left (x\right )^{2}} + \frac {12 \, e^{4}}{x e^{4} - \log \left (x\right )} - \frac {12}{x} + \frac {24 \, \log \left (2\right )}{x \log \left (x\right )} + \frac {36}{x \log \left (x\right )}\right )} \]
integrate(((-12*log(4/x)-48)*log(x/exp(x*exp(4)))+(12*x*exp(4)-12)*log(4/x )+36*x*exp(4)-36)*exp((12*log(4/x)+36)/x/log(x/exp(x*exp(4))))/x^2/log(x/e xp(x*exp(4)))^2,x, algorithm=\
e^(-24*e^4*log(2)/(x*e^4*log(x) - log(x)^2) - 36*e^4/(x*e^4*log(x) - log(x )^2) + 12*e^4/(x*e^4 - log(x)) - 12/x + 24*log(2)/(x*log(x)) + 36/(x*log(x )))
Time = 0.64 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {36+12 \log \left (\frac {4}{x}\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (-36+36 e^4 x+\left (-12+12 e^4 x\right ) \log \left (\frac {4}{x}\right )+\left (-48-12 \log \left (\frac {4}{x}\right )\right ) \log \left (e^{-e^4 x} x\right )\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx=e^{\left (-\frac {12 \, {\left (2 \, \log \left (2\right ) - \log \left (x\right ) + 3\right )}}{x^{2} e^{4} - x \log \left (x\right )}\right )} \]
integrate(((-12*log(4/x)-48)*log(x/exp(x*exp(4)))+(12*x*exp(4)-12)*log(4/x )+36*x*exp(4)-36)*exp((12*log(4/x)+36)/x/log(x/exp(x*exp(4))))/x^2/log(x/e xp(x*exp(4)))^2,x, algorithm=\
Time = 9.73 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {e^{\frac {36+12 \log \left (\frac {4}{x}\right )}{x \log \left (e^{-e^4 x} x\right )}} \left (-36+36 e^4 x+\left (-12+12 e^4 x\right ) \log \left (\frac {4}{x}\right )+\left (-48-12 \log \left (\frac {4}{x}\right )\right ) \log \left (e^{-e^4 x} x\right )\right )}{x^2 \log ^2\left (e^{-e^4 x} x\right )} \, dx=\frac {{\mathrm {e}}^{-\frac {36}{x^2\,{\mathrm {e}}^4-x\,\ln \left (x\right )}}}{2^{\frac {24}{x^2\,{\mathrm {e}}^4-x\,\ln \left (x\right )}}\,{\left (\frac {1}{x}\right )}^{\frac {12}{x^2\,{\mathrm {e}}^4-x\,\ln \left (x\right )}}} \]
int((exp((12*log(4/x) + 36)/(x*log(x*exp(-x*exp(4)))))*(36*x*exp(4) + log( 4/x)*(12*x*exp(4) - 12) - log(x*exp(-x*exp(4)))*(12*log(4/x) + 48) - 36))/ (x^2*log(x*exp(-x*exp(4)))^2),x)