Integrand size = 91, antiderivative size = 29 \[ \int \frac {e (2-x)+48 x-32 x^2+(16-16 x) \log (4)+(-16+16 x) \log (-3+3 x)+e^{e^x} \left (-32+16 x+e^x \left (-16 x+16 x^2+(-16+16 x) \log (4)\right )+e^x (16-16 x) \log (-3+3 x)\right )}{-16+16 x} \, dx=\left (\frac {e}{16}-e^{e^x}+x\right ) (-x-\log (4)+\log (3 (-1+x))) \]
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(29)=58\).
Time = 2.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {e (2-x)+48 x-32 x^2+(16-16 x) \log (4)+(-16+16 x) \log (-3+3 x)+e^{e^x} \left (-32+16 x+e^x \left (-16 x+16 x^2+(-16+16 x) \log (4)\right )+e^x (16-16 x) \log (-3+3 x)\right )}{-16+16 x} \, dx=-\frac {e x}{16}+e^{e^x} x-x^2+e^{e^x} \log (4)-x \log (4)+\frac {1}{16} e \log (1-x)+\left (-e^{e^x}+x\right ) \log (3 (-1+x)) \]
Integrate[(E*(2 - x) + 48*x - 32*x^2 + (16 - 16*x)*Log[4] + (-16 + 16*x)*L og[-3 + 3*x] + E^E^x*(-32 + 16*x + E^x*(-16*x + 16*x^2 + (-16 + 16*x)*Log[ 4]) + E^x*(16 - 16*x)*Log[-3 + 3*x]))/(-16 + 16*x),x]
-1/16*(E*x) + E^E^x*x - x^2 + E^E^x*Log[4] - x*Log[4] + (E*Log[1 - x])/16 + (-E^E^x + x)*Log[3*(-1 + 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.
\(\displaystyle \int \frac {-32 x^2+e^{e^x} \left (e^x \left (16 x^2-16 x+(16 x-16) \log (4)\right )+16 x+e^x (16-16 x) \log (3 x-3)-32\right )+48 x+e (2-x)+(16 x-16) \log (3 x-3)+(16-16 x) \log (4)}{16 x-16} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {32 x^2-16 e^{e^x} x+32 e^{e^x}-16 x \log (3 (x-1))-48 x \left (1-\frac {e}{48}-\frac {2 \log (2)}{3}\right )+16 \log (3 (x-1))-2 e \left (1+\frac {8 \log (4)}{e}\right )}{16 (1-x)}+e^{x+e^x} (x-\log (3 x-3)+\log (4))\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {e^{e^x}}{1-x}dx+\int \frac {e^{e^x}}{x-1}dx+\int e^{x+e^x} xdx+\operatorname {ExpIntegralEi}\left (e^x\right )-x^2-x+\frac {1}{16} x (16-e-16 \log (4))-(1-x) \log (-3 (1-x))+\frac {1}{16} (16+e) \log (1-x)-e^{e^x} \log (3 x-3)+e^{e^x} \log (4)\) |
Int[(E*(2 - x) + 48*x - 32*x^2 + (16 - 16*x)*Log[4] + (-16 + 16*x)*Log[-3 + 3*x] + E^E^x*(-32 + 16*x + E^x*(-16*x + 16*x^2 + (-16 + 16*x)*Log[4]) + E^x*(16 - 16*x)*Log[-3 + 3*x]))/(-16 + 16*x),x]
3.22.77.3.1 Defintions of rubi rules used
Time = 0.71 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76
method | result | size |
risch | \(\ln \left (-3+3 x \right ) x -\frac {x \,{\mathrm e}}{16}-2 x \ln \left (2\right )-x^{2}+\frac {{\mathrm e} \ln \left (-1+x \right )}{16}+\left (x +2 \ln \left (2\right )-\ln \left (-3+3 x \right )\right ) {\mathrm e}^{{\mathrm e}^{x}}\) | \(51\) |
parallelrisch | \(1-\frac {x \,{\mathrm e}}{16}+\frac {\ln \left (-3+3 x \right ) {\mathrm e}}{16}-2 x \ln \left (2\right )+2 \ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{x}}-x^{2}+\ln \left (-3+3 x \right ) x +x \,{\mathrm e}^{{\mathrm e}^{x}}-{\mathrm e}^{{\mathrm e}^{x}} \ln \left (-3+3 x \right )-\frac {{\mathrm e}}{8}-4 \ln \left (2\right )\) | \(67\) |
int((((-16*x+16)*exp(x)*ln(-3+3*x)+(2*(16*x-16)*ln(2)+16*x^2-16*x)*exp(x)+ 16*x-32)*exp(exp(x))+(16*x-16)*ln(-3+3*x)+2*(-16*x+16)*ln(2)+(2-x)*exp(1)- 32*x^2+48*x)/(16*x-16),x,method=_RETURNVERBOSE)
ln(-3+3*x)*x-1/16*x*exp(1)-2*x*ln(2)-x^2+1/16*exp(1)*ln(-1+x)+(x+2*ln(2)-l n(-3+3*x))*exp(exp(x))
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {e (2-x)+48 x-32 x^2+(16-16 x) \log (4)+(-16+16 x) \log (-3+3 x)+e^{e^x} \left (-32+16 x+e^x \left (-16 x+16 x^2+(-16+16 x) \log (4)\right )+e^x (16-16 x) \log (-3+3 x)\right )}{-16+16 x} \, dx=-x^{2} - \frac {1}{16} \, x e + {\left (x + 2 \, \log \left (2\right ) - \log \left (3 \, x - 3\right )\right )} e^{\left (e^{x}\right )} - 2 \, x \log \left (2\right ) + \frac {1}{16} \, {\left (16 \, x + e\right )} \log \left (3 \, x - 3\right ) \]
integrate((((-16*x+16)*exp(x)*log(-3+3*x)+(2*(16*x-16)*log(2)+16*x^2-16*x) *exp(x)+16*x-32)*exp(exp(x))+(16*x-16)*log(-3+3*x)+2*(-16*x+16)*log(2)+(2- x)*exp(1)-32*x^2+48*x)/(16*x-16),x, algorithm=\
-x^2 - 1/16*x*e + (x + 2*log(2) - log(3*x - 3))*e^(e^x) - 2*x*log(2) + 1/1 6*(16*x + e)*log(3*x - 3)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 17.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {e (2-x)+48 x-32 x^2+(16-16 x) \log (4)+(-16+16 x) \log (-3+3 x)+e^{e^x} \left (-32+16 x+e^x \left (-16 x+16 x^2+(-16+16 x) \log (4)\right )+e^x (16-16 x) \log (-3+3 x)\right )}{-16+16 x} \, dx=- x^{2} + x \log {\left (3 x - 3 \right )} - x \left (\frac {e}{16} + 2 \log {\left (2 \right )}\right ) + \left (x - \log {\left (3 x - 3 \right )} + 2 \log {\left (2 \right )}\right ) e^{e^{x}} + \frac {e \log {\left (x - 1 \right )}}{16} \]
integrate((((-16*x+16)*exp(x)*ln(-3+3*x)+(2*(16*x-16)*ln(2)+16*x**2-16*x)* exp(x)+16*x-32)*exp(exp(x))+(16*x-16)*ln(-3+3*x)+2*(-16*x+16)*ln(2)+(2-x)* exp(1)-32*x**2+48*x)/(16*x-16),x)
-x**2 + x*log(3*x - 3) - x*(E/16 + 2*log(2)) + (x - log(3*x - 3) + 2*log(2 ))*exp(exp(x)) + E*log(x - 1)/16
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (27) = 54\).
Time = 0.31 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {e (2-x)+48 x-32 x^2+(16-16 x) \log (4)+(-16+16 x) \log (-3+3 x)+e^{e^x} \left (-32+16 x+e^x \left (-16 x+16 x^2+(-16+16 x) \log (4)\right )+e^x (16-16 x) \log (-3+3 x)\right )}{-16+16 x} \, dx=-x^{2} + x {\left (\log \left (3\right ) - 1\right )} - \frac {1}{16} \, {\left (x + \log \left (x - 1\right )\right )} e + {\left (x - \log \left (3\right ) + 2 \, \log \left (2\right ) - \log \left (x - 1\right )\right )} e^{\left (e^{x}\right )} - 2 \, {\left (x + \log \left (x - 1\right )\right )} \log \left (2\right ) + {\left (x - 1\right )} \log \left (x - 1\right ) + \frac {1}{8} \, e \log \left (x - 1\right ) + 2 \, \log \left (2\right ) \log \left (x - 1\right ) + x + \log \left (x - 1\right ) \]
integrate((((-16*x+16)*exp(x)*log(-3+3*x)+(2*(16*x-16)*log(2)+16*x^2-16*x) *exp(x)+16*x-32)*exp(exp(x))+(16*x-16)*log(-3+3*x)+2*(-16*x+16)*log(2)+(2- x)*exp(1)-32*x^2+48*x)/(16*x-16),x, algorithm=\
-x^2 + x*(log(3) - 1) - 1/16*(x + log(x - 1))*e + (x - log(3) + 2*log(2) - log(x - 1))*e^(e^x) - 2*(x + log(x - 1))*log(2) + (x - 1)*log(x - 1) + 1/ 8*e*log(x - 1) + 2*log(2)*log(x - 1) + x + log(x - 1)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.69 \[ \int \frac {e (2-x)+48 x-32 x^2+(16-16 x) \log (4)+(-16+16 x) \log (-3+3 x)+e^{e^x} \left (-32+16 x+e^x \left (-16 x+16 x^2+(-16+16 x) \log (4)\right )+e^x (16-16 x) \log (-3+3 x)\right )}{-16+16 x} \, dx=-\frac {1}{16} \, {\left (16 \, x^{2} e^{x} + 32 \, x e^{x} \log \left (2\right ) - 16 \, x e^{x} \log \left (3 \, x - 3\right ) - 16 \, x e^{\left (x + e^{x}\right )} + x e^{\left (x + 1\right )} - 32 \, e^{\left (x + e^{x}\right )} \log \left (2\right ) + 16 \, e^{\left (x + e^{x}\right )} \log \left (3 \, x - 3\right ) - e^{\left (x + 1\right )} \log \left (x - 1\right )\right )} e^{\left (-x\right )} \]
integrate((((-16*x+16)*exp(x)*log(-3+3*x)+(2*(16*x-16)*log(2)+16*x^2-16*x) *exp(x)+16*x-32)*exp(exp(x))+(16*x-16)*log(-3+3*x)+2*(-16*x+16)*log(2)+(2- x)*exp(1)-32*x^2+48*x)/(16*x-16),x, algorithm=\
-1/16*(16*x^2*e^x + 32*x*e^x*log(2) - 16*x*e^x*log(3*x - 3) - 16*x*e^(x + e^x) + x*e^(x + 1) - 32*e^(x + e^x)*log(2) + 16*e^(x + e^x)*log(3*x - 3) - e^(x + 1)*log(x - 1))*e^(-x)
Timed out. \[ \int \frac {e (2-x)+48 x-32 x^2+(16-16 x) \log (4)+(-16+16 x) \log (-3+3 x)+e^{e^x} \left (-32+16 x+e^x \left (-16 x+16 x^2+(-16+16 x) \log (4)\right )+e^x (16-16 x) \log (-3+3 x)\right )}{-16+16 x} \, dx=\int \frac {48\,x-2\,\ln \left (2\right )\,\left (16\,x-16\right )+\ln \left (3\,x-3\right )\,\left (16\,x-16\right )-\mathrm {e}\,\left (x-2\right )+{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (16\,x+{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (16\,x-16\right )-16\,x+16\,x^2\right )-{\mathrm {e}}^x\,\ln \left (3\,x-3\right )\,\left (16\,x-16\right )-32\right )-32\,x^2}{16\,x-16} \,d x \]
int((48*x - 2*log(2)*(16*x - 16) + log(3*x - 3)*(16*x - 16) - exp(1)*(x - 2) + exp(exp(x))*(16*x + exp(x)*(2*log(2)*(16*x - 16) - 16*x + 16*x^2) - e xp(x)*log(3*x - 3)*(16*x - 16) - 32) - 32*x^2)/(16*x - 16),x)