Integrand size = 80, antiderivative size = 24 \[ \int \frac {2 e^6 (17+8 x-\log (x))}{\left (18 x+4 x^2-x \log (x)\right ) \left (54 x+12 x^2-3 x \log (x)+\frac {2 e^6 \left (-18 x-4 x^2+x \log (x)\right )}{18 x+4 x^2-x \log (x)}\right )} \, dx=\log \left (3-\frac {e^6}{x \left (9+2 x-\frac {\log (x)}{2}\right )}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
Time = 0.48 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {2 e^6 (17+8 x-\log (x))}{\left (18 x+4 x^2-x \log (x)\right ) \left (54 x+12 x^2-3 x \log (x)+\frac {2 e^6 \left (-18 x-4 x^2+x \log (x)\right )}{18 x+4 x^2-x \log (x)}\right )} \, dx=2 e^6 \left (-\frac {\log (x)}{2 e^6}-\frac {\log (18+4 x-\log (x))}{2 e^6}+\frac {\log \left (2 e^6-54 x-12 x^2+3 x \log (x)\right )}{2 e^6}\right ) \]
Integrate[(2*E^6*(17 + 8*x - Log[x]))/((18*x + 4*x^2 - x*Log[x])*(54*x + 1 2*x^2 - 3*x*Log[x] + (2*E^6*(-18*x - 4*x^2 + x*Log[x]))/(18*x + 4*x^2 - x* Log[x]))),x]
2*E^6*(-1/2*Log[x]/E^6 - Log[18 + 4*x - Log[x]]/(2*E^6) + Log[2*E^6 - 54*x - 12*x^2 + 3*x*Log[x]]/(2*E^6))
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.
\(\displaystyle \int \frac {2 e^6 (8 x-\log (x)+17)}{\left (4 x^2+18 x-x \log (x)\right ) \left (12 x^2+\frac {2 e^6 \left (-4 x^2-18 x+x \log (x)\right )}{4 x^2+18 x-x \log (x)}+54 x-3 x \log (x)\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 e^6 \int -\frac {8 x-\log (x)+17}{\left (4 x^2-\log (x) x+18 x\right ) \left (-12 x^2+3 \log (x) x-54 x+2 e^6\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 e^6 \int \frac {8 x-\log (x)+17}{\left (4 x^2-\log (x) x+18 x\right ) \left (-12 x^2+3 \log (x) x-54 x+2 e^6\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 e^6 \int \left (\frac {4 x-1}{2 e^6 x (4 x-\log (x)+18)}+\frac {12 x^2-3 x+2 e^6}{2 e^6 x \left (-12 x^2+3 \log (x) x-54 x+2 e^6\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 e^6 \left (-\int \frac {1}{x \left (12 x^2-3 \log (x) x+54 x-2 e^6\right )}dx-\frac {6 \int \frac {x}{12 x^2-3 \log (x) x+54 x-2 e^6}dx}{e^6}-\frac {3 \int \frac {1}{-12 x^2+3 \log (x) x-54 x+2 e^6}dx}{2 e^6}+\frac {\log (4 x-\log (x)+18)}{2 e^6}\right )\) |
Int[(2*E^6*(17 + 8*x - Log[x]))/((18*x + 4*x^2 - x*Log[x])*(54*x + 12*x^2 - 3*x*Log[x] + (2*E^6*(-18*x - 4*x^2 + x*Log[x]))/(18*x + 4*x^2 - x*Log[x] ))),x]
3.27.21.3.1 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 = 1.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\ln \left ({\mathrm e}^{-\ln \left (-\frac {x \left (\ln \left (x \right )-4 x -18\right )}{2}\right )+6}-3\right )\) | \(20\) |
default | \(2 \,{\mathrm e}^{6} \left (-\frac {{\mathrm e}^{-6} \ln \left (x \right )}{2}-\frac {{\mathrm e}^{-6} \ln \left (\ln \left (x \right )-4 x -18\right )}{2}+\frac {{\mathrm e}^{-6} \ln \left (3 x \ln \left (x \right )-12 x^{2}+2 \,{\mathrm e}^{6}-54 x \right )}{2}\right )\) | \(53\) |
risch | \(-\ln \left (\ln \left (x \right )-4 x -18\right )-\ln \left (2\right )+\ln \left (-\frac {\ln \left (x \right )}{4}+x +\frac {9}{2}\right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right ) \left (\operatorname {csgn}\left (i x \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )+\operatorname {csgn}\left (i x \right )\right ) \left (\operatorname {csgn}\left (i x \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )-\operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )\right )}{2}-6+\ln \left (\frac {2 \,{\mathrm e}^{6} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i x \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )^{3}}{2}} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i x \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )^{2} \operatorname {csgn}\left (i x \right )}{2}} {\mathrm e}^{\frac {i \pi \operatorname {csgn}\left (i x \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )^{2} \operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )}{2}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i x \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (\frac {\ln \left (x \right )}{4}-x -\frac {9}{2}\right )\right )}{2}}}{x \left (-\frac {\ln \left (x \right )}{4}+x +\frac {9}{2}\right )}-3\right )\) | \(230\) |
int((-ln(x)+8*x+17)*exp(-ln(-1/2*x*ln(x)+2*x^2+9*x)+6)/((x*ln(x)-4*x^2-18* x)*exp(-ln(-1/2*x*ln(x)+2*x^2+9*x)+6)-3*x*ln(x)+12*x^2+54*x),x,method=_RET URNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {2 e^6 (17+8 x-\log (x))}{\left (18 x+4 x^2-x \log (x)\right ) \left (54 x+12 x^2-3 x \log (x)+\frac {2 e^6 \left (-18 x-4 x^2+x \log (x)\right )}{18 x+4 x^2-x \log (x)}\right )} \, dx=-\log \left (-4 \, x + \log \left (x\right ) - 18\right ) + \log \left (-\frac {12 \, x^{2} - 3 \, x \log \left (x\right ) + 54 \, x - 2 \, e^{6}}{x}\right ) \]
integrate((-log(x)+8*x+17)*exp(-log(-1/2*x*log(x)+2*x^2+9*x)+6)/((x*log(x) -4*x^2-18*x)*exp(-log(-1/2*x*log(x)+2*x^2+9*x)+6)-3*x*log(x)+12*x^2+54*x), x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (17) = 34\).
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {2 e^6 (17+8 x-\log (x))}{\left (18 x+4 x^2-x \log (x)\right ) \left (54 x+12 x^2-3 x \log (x)+\frac {2 e^6 \left (-18 x-4 x^2+x \log (x)\right )}{18 x+4 x^2-x \log (x)}\right )} \, dx=- \log {\left (\log {\left (x \right )} + \frac {- 12 x^{2} - 54 x}{3 x} \right )} + \log {\left (\log {\left (x \right )} + \frac {- 12 x^{2} - 54 x + 2 e^{6}}{3 x} \right )} \]
integrate((-ln(x)+8*x+17)*exp(-ln(-1/2*x*ln(x)+2*x**2+9*x)+6)/((x*ln(x)-4* x**2-18*x)*exp(-ln(-1/2*x*ln(x)+2*x**2+9*x)+6)-3*x*ln(x)+12*x**2+54*x),x)
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {2 e^6 (17+8 x-\log (x))}{\left (18 x+4 x^2-x \log (x)\right ) \left (54 x+12 x^2-3 x \log (x)+\frac {2 e^6 \left (-18 x-4 x^2+x \log (x)\right )}{18 x+4 x^2-x \log (x)}\right )} \, dx=-{\left (e^{\left (-6\right )} \log \left (-4 \, x + \log \left (x\right ) - 18\right ) - e^{\left (-6\right )} \log \left (-\frac {12 \, x^{2} - 3 \, x \log \left (x\right ) + 54 \, x - 2 \, e^{6}}{3 \, x}\right )\right )} e^{6} \]
integrate((-log(x)+8*x+17)*exp(-log(-1/2*x*log(x)+2*x^2+9*x)+6)/((x*log(x) -4*x^2-18*x)*exp(-log(-1/2*x*log(x)+2*x^2+9*x)+6)-3*x*log(x)+12*x^2+54*x), x, algorithm=\
-(e^(-6)*log(-4*x + log(x) - 18) - e^(-6)*log(-1/3*(12*x^2 - 3*x*log(x) + 54*x - 2*e^6)/x))*e^6
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {2 e^6 (17+8 x-\log (x))}{\left (18 x+4 x^2-x \log (x)\right ) \left (54 x+12 x^2-3 x \log (x)+\frac {2 e^6 \left (-18 x-4 x^2+x \log (x)\right )}{18 x+4 x^2-x \log (x)}\right )} \, dx=\log \left (-12 \, x^{2} + 3 \, x \log \left (x\right ) - 54 \, x + 2 \, e^{6}\right ) - \log \left (4 \, x - \log \left (x\right ) + 18\right ) - \log \left (x\right ) \]
integrate((-log(x)+8*x+17)*exp(-log(-1/2*x*log(x)+2*x^2+9*x)+6)/((x*log(x) -4*x^2-18*x)*exp(-log(-1/2*x*log(x)+2*x^2+9*x)+6)-3*x*log(x)+12*x^2+54*x), x, algorithm=\
Time = 14.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {2 e^6 (17+8 x-\log (x))}{\left (18 x+4 x^2-x \log (x)\right ) \left (54 x+12 x^2-3 x \log (x)+\frac {2 e^6 \left (-18 x-4 x^2+x \log (x)\right )}{18 x+4 x^2-x \log (x)}\right )} \, dx=\ln \left (54\,x-2\,{\mathrm {e}}^6-3\,x\,\ln \left (x\right )+12\,x^2\right )-\ln \left (x\right )-\ln \left (4\,x-\ln \left (x\right )+18\right ) \]