Integrand size = 181, antiderivative size = 27 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-1+\frac {x^2}{-5+\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )}\right ) \]
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=-\log \left (5-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right )+\log \left (5+x^2-\log \left (x-e^{\sqrt [4]{e}} x (1+x)\right )\right ) \]
Integrate[(-11*x + E^E^(1/4)*(11*x + 12*x^2) + (2*x + E^E^(1/4)*(-2*x - 2* x^2))*Log[x + E^E^(1/4)*(-x - x^2)])/(-25 - 5*x^2 + E^E^(1/4)*(25 + 25*x + 5*x^2 + 5*x^3) + (10 + x^2 + E^E^(1/4)*(-10 - 10*x - x^2 - x^3))*Log[x + E^E^(1/4)*(-x - x^2)] + (-1 + E^E^(1/4)*(1 + x))*Log[x + E^E^(1/4)*(-x - x ^2)]^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.
| \(\displaystyle \int \frac {e^{\sqrt [4]{e}} \left (12 x^2+11 x\right )+\left (e^{\sqrt [4]{e}} \left (-2 x^2-2 x\right )+2 x\right ) \log \left (e^{\sqrt [4]{e}} \left (-x^2-x\right )+x\right )-11 x}{-5 x^2+\left (e^{\sqrt [4]{e}} (x+1)-1\right ) \log ^2\left (e^{\sqrt [4]{e}} \left (-x^2-x\right )+x\right )+e^{\sqrt [4]{e}} \left (5 x^3+5 x^2+25 x+25\right )+\left (x^2+e^{\sqrt [4]{e}} \left (-x^3-x^2-10 x-10\right )+10\right ) \log \left (e^{\sqrt [4]{e}} \left (-x^2-x\right )+x\right )-25} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{\sqrt [4]{e}} \left (12 x^2+11 x\right )-\left (e^{\sqrt [4]{e}} \left (-2 x^2-2 x\right )+2 x\right ) \log \left (e^{\sqrt [4]{e}} \left (-x^2-x\right )+x\right )+11 x}{\left (-e^{\sqrt [4]{e}} x-e^{\sqrt [4]{e}}+1\right ) \left (5-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )\right ) \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 x \log \left (x \left (-e^{\sqrt [4]{e}} x-e^{\sqrt [4]{e}}+1\right )\right )}{\left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right ) \left (\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )-5\right )}-\frac {x \left (12 e^{\sqrt [4]{e}} x+11 e^{\sqrt [4]{e}}-11\right )}{\left (e^{\sqrt [4]{e}} x+e^{\sqrt [4]{e}}-1\right ) \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right ) \left (\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -11 \int \frac {1}{x \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right )}dx-e^{\sqrt [4]{e}} \int \frac {1}{\left (e^{\sqrt [4]{e}} x+e^{\sqrt [4]{e}}-1\right ) \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right )}dx+2 \int \frac {\log \left (x \left (-e^{\sqrt [4]{e}} x-e^{\sqrt [4]{e}}+1\right )\right )}{x \left (x^2-\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )+5\right )}dx-11 \int \frac {1}{x \left (\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )-5\right )}dx-e^{\sqrt [4]{e}} \int \frac {1}{\left (e^{\sqrt [4]{e}} x+e^{\sqrt [4]{e}}-1\right ) \left (\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )-5\right )}dx+2 \int \frac {\log \left (x \left (-e^{\sqrt [4]{e}} x-e^{\sqrt [4]{e}}+1\right )\right )}{x \left (\log \left (x-e^{\sqrt [4]{e}} x (x+1)\right )-5\right )}dx\) |
Int[(-11*x + E^E^(1/4)*(11*x + 12*x^2) + (2*x + E^E^(1/4)*(-2*x - 2*x^2))* Log[x + E^E^(1/4)*(-x - x^2)])/(-25 - 5*x^2 + E^E^(1/4)*(25 + 25*x + 5*x^2 + 5*x^3) + (10 + x^2 + E^E^(1/4)*(-10 - 10*x - x^2 - x^3))*Log[x + E^E^(1 /4)*(-x - x^2)] + (-1 + E^E^(1/4)*(1 + x))*Log[x + E^E^(1/4)*(-x - x^2)]^2 ),x]
3.12.33.3.1 Defintions of rubi rules used
Time = 5.57 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74
| method | result | size |
| norman | \(-\ln \left (\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )+\ln \left (x^{2}-\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )+5\right )\) | \(47\) |
| risch | \(\ln \left (-x^{2}+\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )-\ln \left (\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right )\) | \(47\) |
| parallelrisch | \(\left (-\ln \left (\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )-5\right ) {\mathrm e}^{2 \,{\mathrm e}^{\frac {1}{4}}}+\ln \left (x^{2}-\ln \left (\left (-x^{2}-x \right ) {\mathrm e}^{{\mathrm e}^{\frac {1}{4}}}+x \right )+5\right ) {\mathrm e}^{2 \,{\mathrm e}^{\frac {1}{4}}}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{\frac {1}{4}}}\) | \(64\) |
int((((-2*x^2-2*x)*exp(exp(1/4))+2*x)*ln((-x^2-x)*exp(exp(1/4))+x)+(12*x^2 +11*x)*exp(exp(1/4))-11*x)/(((1+x)*exp(exp(1/4))-1)*ln((-x^2-x)*exp(exp(1/ 4))+x)^2+((-x^3-x^2-10*x-10)*exp(exp(1/4))+x^2+10)*ln((-x^2-x)*exp(exp(1/4 ))+x)+(5*x^3+5*x^2+25*x+25)*exp(exp(1/4))-5*x^2-25),x,method=_RETURNVERBOS E)
Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-x^{2} + \log \left (-{\left (x^{2} + x\right )} e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) - \log \left (\log \left (-{\left (x^{2} + x\right )} e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) \]
integrate((((-2*x^2-2*x)*exp(exp(1/4))+2*x)*log((-x^2-x)*exp(exp(1/4))+x)+ (12*x^2+11*x)*exp(exp(1/4))-11*x)/(((1+x)*exp(exp(1/4))-1)*log((-x^2-x)*ex p(exp(1/4))+x)^2+((-x^3-x^2-10*x-10)*exp(exp(1/4))+x^2+10)*log((-x^2-x)*ex p(exp(1/4))+x)+(5*x^3+5*x^2+25*x+25)*exp(exp(1/4))-5*x^2-25),x, algorithm= \
Exception generated. \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\text {Exception raised: PolynomialError} \]
integrate((((-2*x**2-2*x)*exp(exp(1/4))+2*x)*ln((-x**2-x)*exp(exp(1/4))+x) +(12*x**2+11*x)*exp(exp(1/4))-11*x)/(((1+x)*exp(exp(1/4))-1)*ln((-x**2-x)* exp(exp(1/4))+x)**2+((-x**3-x**2-10*x-10)*exp(exp(1/4))+x**2+10)*ln((-x**2 -x)*exp(exp(1/4))+x)+(5*x**3+5*x**2+25*x+25)*exp(exp(1/4))-5*x**2-25),x)
Exception raised: PolynomialError >> 1/(x**2*exp(exp(1/4)) - x + x*exp(exp (1/4))) contains an element of the set of generators.
Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-x^{2} + \log \left (-x e^{\left (e^{\frac {1}{4}}\right )} - e^{\left (e^{\frac {1}{4}}\right )} + 1\right ) + \log \left (x\right ) - 5\right ) - \log \left (\log \left (-x e^{\left (e^{\frac {1}{4}}\right )} - e^{\left (e^{\frac {1}{4}}\right )} + 1\right ) + \log \left (x\right ) - 5\right ) \]
integrate((((-2*x^2-2*x)*exp(exp(1/4))+2*x)*log((-x^2-x)*exp(exp(1/4))+x)+ (12*x^2+11*x)*exp(exp(1/4))-11*x)/(((1+x)*exp(exp(1/4))-1)*log((-x^2-x)*ex p(exp(1/4))+x)^2+((-x^3-x^2-10*x-10)*exp(exp(1/4))+x^2+10)*log((-x^2-x)*ex p(exp(1/4))+x)+(5*x^3+5*x^2+25*x+25)*exp(exp(1/4))-5*x^2-25),x, algorithm= \
log(-x^2 + log(-x*e^(e^(1/4)) - e^(e^(1/4)) + 1) + log(x) - 5) - log(log(- x*e^(e^(1/4)) - e^(e^(1/4)) + 1) + log(x) - 5)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.43 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\log \left (-x^{2} + \log \left (-x^{2} e^{\left (e^{\frac {1}{4}}\right )} - x e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) - \log \left (\log \left (-x^{2} e^{\left (e^{\frac {1}{4}}\right )} - x e^{\left (e^{\frac {1}{4}}\right )} + x\right ) - 5\right ) \]
integrate((((-2*x^2-2*x)*exp(exp(1/4))+2*x)*log((-x^2-x)*exp(exp(1/4))+x)+ (12*x^2+11*x)*exp(exp(1/4))-11*x)/(((1+x)*exp(exp(1/4))-1)*log((-x^2-x)*ex p(exp(1/4))+x)^2+((-x^3-x^2-10*x-10)*exp(exp(1/4))+x^2+10)*log((-x^2-x)*ex p(exp(1/4))+x)+(5*x^3+5*x^2+25*x+25)*exp(exp(1/4))-5*x^2-25),x, algorithm= \
log(-x^2 + log(-x^2*e^(e^(1/4)) - x*e^(e^(1/4)) + x) - 5) - log(log(-x^2*e ^(e^(1/4)) - x*e^(e^(1/4)) + x) - 5)
Time = 15.10 (sec) , antiderivative size = 250, normalized size of antiderivative = 9.26 \[ \int \frac {-11 x+e^{\sqrt [4]{e}} \left (11 x+12 x^2\right )+\left (2 x+e^{\sqrt [4]{e}} \left (-2 x-2 x^2\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )}{-25-5 x^2+e^{\sqrt [4]{e}} \left (25+25 x+5 x^2+5 x^3\right )+\left (10+x^2+e^{\sqrt [4]{e}} \left (-10-10 x-x^2-x^3\right )\right ) \log \left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )+\left (-1+e^{\sqrt [4]{e}} (1+x)\right ) \log ^2\left (x+e^{\sqrt [4]{e}} \left (-x-x^2\right )\right )} \, dx=\ln \left (\frac {{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}}{2}-x-x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}+x^2+x^3-\frac {1}{2}\right )+\ln \left (\left (x^2-\ln \left (-x\,\left ({\mathrm {e}}^{{\mathrm {e}}^{1/4}}+x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-1\right )\right )+5\right )\,\left ({\mathrm {e}}^{{\mathrm {e}}^{1/4}}+2\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-1\right )\right )-\ln \left (20\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-2\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-10\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}+20\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}+2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )+4\,x^2\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-20\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-20\,x^2+4\,x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-4\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )-4\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\,\ln \left (x-x^2\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}-x\,{\mathrm {e}}^{{\mathrm {e}}^{1/4}}\right )+10\right )-\ln \left (x-\frac {{\mathrm {e}}^{-{\mathrm {e}}^{1/4}}}{2}+\frac {1}{2}\right ) \]
int((log(x - exp(exp(1/4))*(x + x^2))*(2*x - exp(exp(1/4))*(2*x + 2*x^2)) - 11*x + exp(exp(1/4))*(11*x + 12*x^2))/(log(x - exp(exp(1/4))*(x + x^2))^ 2*(exp(exp(1/4))*(x + 1) - 1) + log(x - exp(exp(1/4))*(x + x^2))*(x^2 - ex p(exp(1/4))*(10*x + x^2 + x^3 + 10) + 10) + exp(exp(1/4))*(25*x + 5*x^2 + 5*x^3 + 25) - 5*x^2 - 25),x)
log(exp(-exp(1/4))/2 - x - x^2*exp(-exp(1/4)) + x^2 + x^3 - 1/2) + log((x^ 2 - log(-x*(exp(exp(1/4)) + x*exp(exp(1/4)) - 1)) + 5)*(exp(exp(1/4)) + 2* x*exp(exp(1/4)) - 1)) - log(20*x^2*exp(exp(1/4)) - 2*log(x - x^2*exp(exp(1 /4)) - x*exp(exp(1/4))) - 10*exp(exp(1/4)) + 20*x^3*exp(exp(1/4)) + 2*exp( exp(1/4))*log(x - x^2*exp(exp(1/4)) - x*exp(exp(1/4))) + 4*x^2*log(x - x^2 *exp(exp(1/4)) - x*exp(exp(1/4))) - 20*x*exp(exp(1/4)) - 20*x^2 + 4*x*exp( exp(1/4))*log(x - x^2*exp(exp(1/4)) - x*exp(exp(1/4))) - 4*x^2*exp(exp(1/4 ))*log(x - x^2*exp(exp(1/4)) - x*exp(exp(1/4))) - 4*x^3*exp(exp(1/4))*log( x - x^2*exp(exp(1/4)) - x*exp(exp(1/4))) + 10) - log(x - exp(-exp(1/4))/2 + 1/2)