Integrand size = 138, antiderivative size = 29 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x}{e \left (e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)\right )} \]
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x}{e \left (e^{\frac {\left (-16+x^2\right )^2}{e^4}}-\log (4-x)\right )} \]
Integrate[(E^4*x + E^((256 - 32*x^2 + x^4)/E^4)*(E^4*(-4 + x) - 256*x^2 + 64*x^3 + 16*x^4 - 4*x^5) + E^4*(4 - x)*Log[4 - x])/(E^(5 + (2*(256 - 32*x^ 2 + x^4))/E^4)*(-4 + x) + E^(5 + (256 - 32*x^2 + x^4)/E^4)*(8 - 2*x)*Log[4 - x] + E^5*(-4 + x)*Log[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.
\(\displaystyle \int \frac {e^{\frac {x^4-32 x^2+256}{e^4}} \left (-4 x^5+16 x^4+64 x^3-256 x^2+e^4 (x-4)\right )+e^4 x+e^4 (4-x) \log (4-x)}{e^{\frac {2 \left (x^4-32 x^2+256\right )}{e^4}+5} (x-4)+e^{\frac {x^4-32 x^2+256}{e^4}+5} (8-2 x) \log (4-x)+e^5 (x-4) \log ^2(4-x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {64 x^2}{e^4}-5} \left (-e^{\frac {x^4-32 x^2+256}{e^4}} \left (-4 x^5+16 x^4+64 x^3-256 x^2+e^4 (x-4)\right )-e^4 x-e^4 (4-x) \log (4-x)\right )}{(4-x) \left (e^{\frac {x^4}{e^4}+\frac {256}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{\frac {64 x^2}{e^4}-5} \left (-4 x^4+64 x^2+e^4\right )}{e^{\frac {\left (x^2+16\right )^2}{e^4}}-e^{\frac {64 x^2}{e^4}} \log (4-x)}-\frac {e^{\frac {64 x^2}{e^4}-5} x \left (4 x^4 \log (4-x)-16 x^3 \log (4-x)-64 x^2 \log (4-x)+256 x \log (4-x)-e^4\right )}{(x-4) \left (e^{\frac {x^4}{e^4}+\frac {256}{e^4}}-e^{\frac {32 x^2}{e^4}} \log (4-x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-4 x^4+64 x^2+e^4}{e^5 \left (e^{\frac {\left (x^2-16\right )^2}{e^4}}-\log (4-x)\right )}+\frac {e^{\frac {64 x^2}{e^4}-5} x \left (-4 x^4 \log (4-x)+16 x^3 \log (4-x)+64 x^2 \log (4-x)-256 x \log (4-x)+e^4\right )}{(x-4) \left (e^{\frac {32 x^2}{e^4}} \log (4-x)-e^{\frac {x^4}{e^4}+\frac {256}{e^4}}\right )^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {-4 x^4+64 x^2+e^4}{e^5 \left (e^{\frac {\left (x^2-16\right )^2}{e^4}}-\log (4-x)\right )}+\frac {e^{\frac {64 x^2}{e^4}-5} x \left (-4 x^4 \log (4-x)+16 x^3 \log (4-x)+64 x^2 \log (4-x)-256 x \log (4-x)+e^4\right )}{(x-4) \left (e^{\frac {32 x^2}{e^4}} \log (4-x)-e^{\frac {x^4}{e^4}+\frac {256}{e^4}}\right )^2}\right )dx\) |
Int[(E^4*x + E^((256 - 32*x^2 + x^4)/E^4)*(E^4*(-4 + x) - 256*x^2 + 64*x^3 + 16*x^4 - 4*x^5) + E^4*(4 - x)*Log[4 - x])/(E^(5 + (2*(256 - 32*x^2 + x^ 4))/E^4)*(-4 + x) + E^(5 + (256 - 32*x^2 + x^4)/E^4)*(8 - 2*x)*Log[4 - x] + E^5*(-4 + x)*Log[4 - x]^2),x]
3.2.64.3.1 Defintions of rubi rules used
Time = 1.81 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
risch | \(\frac {{\mathrm e}^{-1} x}{{\mathrm e}^{\left (x -4\right )^{2} \left (4+x \right )^{2} {\mathrm e}^{-4}}-\ln \left (-x +4\right )}\) | \(30\) |
parallelrisch | \(-\frac {x \,{\mathrm e}^{-1}}{\ln \left (-x +4\right )-{\mathrm e}^{\left (x^{4}-32 x^{2}+256\right ) {\mathrm e}^{-4}}}\) | \(35\) |
int(((-x+4)*exp(4)*ln(-x+4)+((x-4)*exp(4)-4*x^5+16*x^4+64*x^3-256*x^2)*exp ((x^4-32*x^2+256)/exp(4))+x*exp(4))/((x-4)*exp(1)*exp(4)*ln(-x+4)^2+(-2*x+ 8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*ln(-x+4)+(x-4)*exp(1)*exp(4) *exp((x^4-32*x^2+256)/exp(4))^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=-\frac {x e^{4}}{e^{5} \log \left (-x + 4\right ) - e^{\left ({\left (x^{4} - 32 \, x^{2} + 5 \, e^{4} + 256\right )} e^{\left (-4\right )}\right )}} \]
integrate(((-x+4)*exp(4)*log(-x+4)+((x-4)*exp(4)-4*x^5+16*x^4+64*x^3-256*x ^2)*exp((x^4-32*x^2+256)/exp(4))+x*exp(4))/((x-4)*exp(1)*exp(4)*log(-x+4)^ 2+(-2*x+8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*log(-x+4)+(x-4)*exp( 1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))^2),x, algorithm=\
Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x}{e e^{\frac {x^{4} - 32 x^{2} + 256}{e^{4}}} - e \log {\left (4 - x \right )}} \]
integrate(((-x+4)*exp(4)*ln(-x+4)+((x-4)*exp(4)-4*x**5+16*x**4+64*x**3-256 *x**2)*exp((x**4-32*x**2+256)/exp(4))+x*exp(4))/((x-4)*exp(1)*exp(4)*ln(-x +4)**2+(-2*x+8)*exp(1)*exp(4)*exp((x**4-32*x**2+256)/exp(4))*ln(-x+4)+(x-4 )*exp(1)*exp(4)*exp((x**4-32*x**2+256)/exp(4))**2),x)
Time = 0.43 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=-\frac {x e^{\left (32 \, x^{2} e^{\left (-4\right )}\right )}}{e^{\left (32 \, x^{2} e^{\left (-4\right )} + 1\right )} \log \left (-x + 4\right ) - e^{\left (x^{4} e^{\left (-4\right )} + 256 \, e^{\left (-4\right )} + 1\right )}} \]
integrate(((-x+4)*exp(4)*log(-x+4)+((x-4)*exp(4)-4*x^5+16*x^4+64*x^3-256*x ^2)*exp((x^4-32*x^2+256)/exp(4))+x*exp(4))/((x-4)*exp(1)*exp(4)*log(-x+4)^ 2+(-2*x+8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*log(-x+4)+(x-4)*exp( 1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 926 vs. \(2 (26) = 52\).
Time = 1.20 (sec) , antiderivative size = 926, normalized size of antiderivative = 31.93 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\text {Too large to display} \]
integrate(((-x+4)*exp(4)*log(-x+4)+((x-4)*exp(4)-4*x^5+16*x^4+64*x^3-256*x ^2)*exp((x^4-32*x^2+256)/exp(4))+x*exp(4))/((x-4)*exp(1)*exp(4)*log(-x+4)^ 2+(-2*x+8)*exp(1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))*log(-x+4)+(x-4)*exp( 1)*exp(4)*exp((x^4-32*x^2+256)/exp(4))^2),x, algorithm=\
-(4*x^5*e^(x^4*e^(-4) + 4)*log(-x + 4)^2 - 4*x^5*e^(x^4*e^(-4) + (x^4 - 32 *x^2)*e^(-4) + 256*e^(-4) + 4)*log(-x + 4) - 16*x^4*e^(x^4*e^(-4) + 4)*log (-x + 4)^2 + 16*x^4*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4) + 4 )*log(-x + 4) - 64*x^3*e^(x^4*e^(-4) + 4)*log(-x + 4)^2 + 64*x^3*e^(x^4*e^ (-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4) + 4)*log(-x + 4) + 256*x^2*e^(x^ 4*e^(-4) + 4)*log(-x + 4)^2 - 256*x^2*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4 ) + 256*e^(-4) + 4)*log(-x + 4) - x*e^(x^4*e^(-4) + 8)*log(-x + 4) + x*e^( 2*(x^4 - 16*x^2)*e^(-4) + 256*e^(-4) + 8))/(4*x^4*e^(x^4*e^(-4) + 5)*log(- x + 4)^3 - 4*x^4*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4) + 5)*l og(-x + 4)^2 - 4*x^4*e^(2*(x^4 - 16*x^2)*e^(-4) + 256*e^(-4) + 5)*log(-x + 4)^2 - 16*x^3*e^(x^4*e^(-4) + 5)*log(-x + 4)^3 + 4*x^4*e^(2*(x^4 - 16*x^2 )*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 512*e^(-4) + 5)*log(-x + 4) + 16*x^3*e^ (x^4*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 256*e^(-4) + 5)*log(-x + 4)^2 + 16*x ^3*e^(2*(x^4 - 16*x^2)*e^(-4) + 256*e^(-4) + 5)*log(-x + 4)^2 - 64*x^2*e^( x^4*e^(-4) + 5)*log(-x + 4)^3 - 16*x^3*e^(2*(x^4 - 16*x^2)*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 512*e^(-4) + 5)*log(-x + 4) + 64*x^2*e^(x^4*e^(-4) + (x^ 4 - 32*x^2)*e^(-4) + 256*e^(-4) + 5)*log(-x + 4)^2 + 64*x^2*e^(2*(x^4 - 16 *x^2)*e^(-4) + 256*e^(-4) + 5)*log(-x + 4)^2 + 256*x*e^(x^4*e^(-4) + 5)*lo g(-x + 4)^3 - 64*x^2*e^(2*(x^4 - 16*x^2)*e^(-4) + (x^4 - 32*x^2)*e^(-4) + 512*e^(-4) + 5)*log(-x + 4) - 256*x*e^(x^4*e^(-4) + (x^4 - 32*x^2)*e^(-...
Time = 9.85 (sec) , antiderivative size = 158, normalized size of antiderivative = 5.45 \[ \int \frac {e^4 x+e^{\frac {256-32 x^2+x^4}{e^4}} \left (e^4 (-4+x)-256 x^2+64 x^3+16 x^4-4 x^5\right )+e^4 (4-x) \log (4-x)}{e^{5+\frac {2 \left (256-32 x^2+x^4\right )}{e^4}} (-4+x)+e^{5+\frac {256-32 x^2+x^4}{e^4}} (8-2 x) \log (4-x)+e^5 (-4+x) \log ^2(4-x)} \, dx=\frac {x\,{\mathrm {e}}^{-1}\,{\left (4\,{\mathrm {e}}^4-x\,{\mathrm {e}}^4\right )}^2-{\mathrm {e}}^{-5}\,\ln \left (4-x\right )\,{\left (4\,{\mathrm {e}}^4-x\,{\mathrm {e}}^4\right )}^2\,\left (4\,x^5-16\,x^4-64\,x^3+256\,x^2\right )}{\left (\ln \left (4-x\right )-{\mathrm {e}}^{{\mathrm {e}}^{-4}\,x^4-32\,{\mathrm {e}}^{-4}\,x^2+256\,{\mathrm {e}}^{-4}}\right )\,\left (x-4\right )\,\left (4\,{\mathrm {e}}^8-x\,{\mathrm {e}}^8+512\,x^2\,{\mathrm {e}}^4\,\ln \left (4-x\right )-32\,x^4\,{\mathrm {e}}^4\,\ln \left (4-x\right )+4\,x^5\,{\mathrm {e}}^4\,\ln \left (4-x\right )-1024\,x\,{\mathrm {e}}^4\,\ln \left (4-x\right )\right )} \]
int((x*exp(4) + exp(exp(-4)*(x^4 - 32*x^2 + 256))*(exp(4)*(x - 4) - 256*x^ 2 + 64*x^3 + 16*x^4 - 4*x^5) - exp(4)*log(4 - x)*(x - 4))/(exp(5)*exp(2*ex p(-4)*(x^4 - 32*x^2 + 256))*(x - 4) + exp(5)*log(4 - x)^2*(x - 4) - exp(5) *exp(exp(-4)*(x^4 - 32*x^2 + 256))*log(4 - x)*(2*x - 8)),x)
(x*exp(-1)*(4*exp(4) - x*exp(4))^2 - exp(-5)*log(4 - x)*(4*exp(4) - x*exp( 4))^2*(256*x^2 - 64*x^3 - 16*x^4 + 4*x^5))/((log(4 - x) - exp(256*exp(-4) - 32*x^2*exp(-4) + x^4*exp(-4)))*(x - 4)*(4*exp(8) - x*exp(8) + 512*x^2*ex p(4)*log(4 - x) - 32*x^4*exp(4)*log(4 - x) + 4*x^5*exp(4)*log(4 - x) - 102 4*x*exp(4)*log(4 - x)))