Integrand size = 106, antiderivative size = 24 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=-e^{\frac {256}{81}+x+\frac {\log (2)}{-x^5+\log (x)}}+x \] Output:
x-exp(256/81+x+ln(2)/(ln(x)-x^5))
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=-2^{\frac {1}{-x^5+\log (x)}} e^{\frac {256}{81}+x}+x \] Input:
Integrate[(x^11 - 2*x^6*Log[x] + x*Log[x]^2 + E^((-256*x^5 - 81*x^6 + 81*L og[2] + (256 + 81*x)*Log[x])/(-81*x^5 + 81*Log[x]))*(-x^11 + (1 - 5*x^5)*L og[2] + 2*x^6*Log[x] - x*Log[x]^2))/(x^11 - 2*x^6*Log[x] + x*Log[x]^2),x]
Output:
-(2^(-x^5 + Log[x])^(-1)*E^(256/81 + x)) + x
Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(24)=48\).
Time = 5.57 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {7292, 7293, 2009}
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.
\(\displaystyle \int \frac {\left (-x^{11}+2 x^6 \log (x)+\left (1-5 x^5\right ) \log (2)-x \log ^2(x)\right ) \exp \left (\frac {-81 x^6-256 x^5+(81 x+256) \log (x)+81 \log (2)}{81 \log (x)-81 x^5}\right )+x^{11}-2 x^6 \log (x)+x \log ^2(x)}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-x^{11}+2 x^6 \log (x)+\left (1-5 x^5\right ) \log (2)-x \log ^2(x)\right ) \exp \left (\frac {-81 x^6-256 x^5+(81 x+256) \log (x)+81 \log (2)}{81 \log (x)-81 x^5}\right )+x^{11}-2 x^6 \log (x)+x \log ^2(x)}{x \left (x^5-\log (x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2^{\frac {1}{\log (x)-x^5}} e^{\frac {x^5 (81 x+256)}{81 \left (x^5-\log (x)\right )}} \left (-x^{11}+2 x^6 \log (x)-x^5 \log (32)-x \log ^2(x)+\log (2)\right ) x^{\frac {-81 x^5-81 x+81 \log (x)-256}{81 \left (x^5-\log (x)\right )}}}{\left (x^5-\log (x)\right )^2}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x-\frac {2^{\frac {1}{\log (x)-x^5}} e^{\frac {x^5 (81 x+256)}{81 \left (x^5-\log (x)\right )}} x^{-\frac {81 x^5+81 x-81 \log (x)+256}{81 \left (x^5-\log (x)\right )}} \left (\log (2)-x^5 \log (32)\right )}{\left (\frac {1}{x}-5 x^4\right ) \log (2)}\) |
Input:
Int[(x^11 - 2*x^6*Log[x] + x*Log[x]^2 + E^((-256*x^5 - 81*x^6 + 81*Log[2] + (256 + 81*x)*Log[x])/(-81*x^5 + 81*Log[x]))*(-x^11 + (1 - 5*x^5)*Log[2] + 2*x^6*Log[x] - x*Log[x]^2))/(x^11 - 2*x^6*Log[x] + x*Log[x]^2),x]
Output:
x - (2^(-x^5 + Log[x])^(-1)*E^((x^5*(256 + 81*x))/(81*(x^5 - Log[x])))*(Lo g[2] - x^5*Log[32]))/(x^((256 + 81*x + 81*x^5 - 81*Log[x])/(81*(x^5 - Log[ x])))*(x^(-1) - 5*x^4)*Log[2])
Time = 1.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
\[x -2^{\frac {1}{\ln \left (x \right )-x^{5}}} {\mathrm e}^{x +\frac {256}{81}}\]
Input:
int(((-x*ln(x)^2+2*x^6*ln(x)+(-5*x^5+1)*ln(2)-x^11)*exp(((81*x+256)*ln(x)+ 81*ln(2)-81*x^6-256*x^5)/(81*ln(x)-81*x^5))+x*ln(x)^2-2*x^6*ln(x)+x^11)/(x *ln(x)^2-2*x^6*ln(x)+x^11),x)
Output:
x-2^(1/(ln(x)-x^5))*exp(x+256/81)
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\left (\frac {81 \, x^{6} + 256 \, x^{5} - {\left (81 \, x + 256\right )} \log \left (x\right ) - 81 \, \log \left (2\right )}{81 \, {\left (x^{5} - \log \left (x\right )\right )}}\right )} \] Input:
integrate(((-x*log(x)^2+2*x^6*log(x)+(-5*x^5+1)*log(2)-x^11)*exp(((81*x+25 6)*log(x)+81*log(2)-81*x^6-256*x^5)/(81*log(x)-81*x^5))+x*log(x)^2-2*x^6*l og(x)+x^11)/(x*log(x)^2-2*x^6*log(x)+x^11),x, algorithm="fricas")
Output:
x - e^(1/81*(81*x^6 + 256*x^5 - (81*x + 256)*log(x) - 81*log(2))/(x^5 - lo g(x)))
Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\frac {- 81 x^{6} - 256 x^{5} + \left (81 x + 256\right ) \log {\left (x \right )} + 81 \log {\left (2 \right )}}{- 81 x^{5} + 81 \log {\left (x \right )}}} \] Input:
integrate(((-x*ln(x)**2+2*x**6*ln(x)+(-5*x**5+1)*ln(2)-x**11)*exp(((81*x+2 56)*ln(x)+81*ln(2)-81*x**6-256*x**5)/(81*ln(x)-81*x**5))+x*ln(x)**2-2*x**6 *ln(x)+x**11)/(x*ln(x)**2-2*x**6*ln(x)+x**11),x)
Output:
x - exp((-81*x**6 - 256*x**5 + (81*x + 256)*log(x) + 81*log(2))/(-81*x**5 + 81*log(x)))
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\left (x - \frac {\log \left (2\right )}{x^{5} - \log \left (x\right )} + \frac {256}{81}\right )} \] Input:
integrate(((-x*log(x)^2+2*x^6*log(x)+(-5*x^5+1)*log(2)-x^11)*exp(((81*x+25 6)*log(x)+81*log(2)-81*x^6-256*x^5)/(81*log(x)-81*x^5))+x*log(x)^2-2*x^6*l og(x)+x^11)/(x*log(x)^2-2*x^6*log(x)+x^11),x, algorithm="maxima")
Output:
x - e^(x - log(2)/(x^5 - log(x)) + 256/81)
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x - e^{\left (\frac {81 \, x^{6} + 256 \, x^{5} - 81 \, x \log \left (x\right ) - 81 \, \log \left (2\right ) - 256 \, \log \left (x\right )}{81 \, {\left (x^{5} - \log \left (x\right )\right )}}\right )} \] Input:
integrate(((-x*log(x)^2+2*x^6*log(x)+(-5*x^5+1)*log(2)-x^11)*exp(((81*x+25 6)*log(x)+81*log(2)-81*x^6-256*x^5)/(81*log(x)-81*x^5))+x*log(x)^2-2*x^6*l og(x)+x^11)/(x*log(x)^2-2*x^6*log(x)+x^11),x, algorithm="giac")
Output:
x - e^(1/81*(81*x^6 + 256*x^5 - 81*x*log(x) - 81*log(2) - 256*log(x))/(x^5 - log(x)))
Time = 3.01 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.71 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=x-2^{\frac {81}{81\,\ln \left (x\right )-81\,x^5}}\,x^{\frac {256}{81\,\ln \left (x\right )-81\,x^5}}\,x^{\frac {81\,x}{81\,\ln \left (x\right )-81\,x^5}}\,{\mathrm {e}}^{-\frac {81\,x^6}{81\,\ln \left (x\right )-81\,x^5}}\,{\mathrm {e}}^{-\frac {256\,x^5}{81\,\ln \left (x\right )-81\,x^5}} \] Input:
int((x*log(x)^2 - 2*x^6*log(x) - exp((81*log(2) + log(x)*(81*x + 256) - 25 6*x^5 - 81*x^6)/(81*log(x) - 81*x^5))*(x*log(x)^2 - 2*x^6*log(x) + log(2)* (5*x^5 - 1) + x^11) + x^11)/(x*log(x)^2 - 2*x^6*log(x) + x^11),x)
Output:
x - 2^(81/(81*log(x) - 81*x^5))*x^(256/(81*log(x) - 81*x^5))*x^((81*x)/(81 *log(x) - 81*x^5))*exp(-(81*x^6)/(81*log(x) - 81*x^5))*exp(-(256*x^5)/(81* log(x) - 81*x^5))
Time = 0.20 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.46 \[ \int \frac {x^{11}-2 x^6 \log (x)+x \log ^2(x)+e^{\frac {-256 x^5-81 x^6+81 \log (2)+(256+81 x) \log (x)}{-81 x^5+81 \log (x)}} \left (-x^{11}+\left (1-5 x^5\right ) \log (2)+2 x^6 \log (x)-x \log ^2(x)\right )}{x^{11}-2 x^6 \log (x)+x \log ^2(x)} \, dx=\frac {e^{\frac {13 x^{5}}{81 \,\mathrm {log}\left (x \right )-81 x^{5}}} x -e^{\frac {81 \,\mathrm {log}\left (x \right ) x +13 \,\mathrm {log}\left (x \right )+81 \,\mathrm {log}\left (2\right )-81 x^{6}}{81 \,\mathrm {log}\left (x \right )-81 x^{5}}} e^{3}}{e^{\frac {13 x^{5}}{81 \,\mathrm {log}\left (x \right )-81 x^{5}}}} \] Input:
int(((-x*log(x)^2+2*x^6*log(x)+(-5*x^5+1)*log(2)-x^11)*exp(((81*x+256)*log (x)+81*log(2)-81*x^6-256*x^5)/(81*log(x)-81*x^5))+x*log(x)^2-2*x^6*log(x)+ x^11)/(x*log(x)^2-2*x^6*log(x)+x^11),x)
Output:
(e**((13*x**5)/(81*log(x) - 81*x**5))*x - e**((81*log(x)*x + 13*log(x) + 8 1*log(2) - 81*x**6)/(81*log(x) - 81*x**5))*e**3)/e**((13*x**5)/(81*log(x) - 81*x**5))