Integrand size = 177, antiderivative size = 24 \[ \int \frac {3468+2155 x+462 x^2+39 x^3+x^4+e^{-6-6 x} (27+x)+e^{-4-4 x} \left (414+93 x+3 x^2\right )+e^{-2-2 x} \left (2091+876 x+105 x^2+3 x^3\right )+\left (-510 x+180 e^{-4-4 x} x-90 x^2+e^{-2-2 x} \left (930 x+180 x^2\right )\right ) \log (x)}{64 x+e^{-6-6 x} x+48 x^2+12 x^3+x^4+e^{-4-4 x} \left (12 x+3 x^2\right )+e^{-2-2 x} \left (48 x+24 x^2+3 x^3\right )} \, dx=x+3 \left (3+\frac {5}{4+e^{-2-2 x}+x}\right )^2 \log (x) \]
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.29 \[ \int \frac {3468+2155 x+462 x^2+39 x^3+x^4+e^{-6-6 x} (27+x)+e^{-4-4 x} \left (414+93 x+3 x^2\right )+e^{-2-2 x} \left (2091+876 x+105 x^2+3 x^3\right )+\left (-510 x+180 e^{-4-4 x} x-90 x^2+e^{-2-2 x} \left (930 x+180 x^2\right )\right ) \log (x)}{64 x+e^{-6-6 x} x+48 x^2+12 x^3+x^4+e^{-4-4 x} \left (12 x+3 x^2\right )+e^{-2-2 x} \left (48 x+24 x^2+3 x^3\right )} \, dx=\frac {x \left (1+e^{2+2 x} (4+x)\right )^2+3 \left (3+e^{2+2 x} (17+3 x)\right )^2 \log (x)}{\left (1+e^{2+2 x} (4+x)\right )^2} \]
Integrate[(3468 + 2155*x + 462*x^2 + 39*x^3 + x^4 + E^(-6 - 6*x)*(27 + x) + E^(-4 - 4*x)*(414 + 93*x + 3*x^2) + E^(-2 - 2*x)*(2091 + 876*x + 105*x^2 + 3*x^3) + (-510*x + 180*E^(-4 - 4*x)*x - 90*x^2 + E^(-2 - 2*x)*(930*x + 180*x^2))*Log[x])/(64*x + E^(-6 - 6*x)*x + 48*x^2 + 12*x^3 + x^4 + E^(-4 - 4*x)*(12*x + 3*x^2) + E^(-2 - 2*x)*(48*x + 24*x^2 + 3*x^3)),x]
(x*(1 + E^(2 + 2*x)*(4 + x))^2 + 3*(3 + E^(2 + 2*x)*(17 + 3*x))^2*Log[x])/ (1 + E^(2 + 2*x)*(4 + x))^2
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.
| \(\displaystyle \int \frac {x^4+39 x^3+462 x^2+e^{-4 x-4} \left (3 x^2+93 x+414\right )+\left (-90 x^2+e^{-2 x-2} \left (180 x^2+930 x\right )+180 e^{-4 x-4} x-510 x\right ) \log (x)+e^{-2 x-2} \left (3 x^3+105 x^2+876 x+2091\right )+2155 x+e^{-6 x-6} (x+27)+3468}{x^4+12 x^3+48 x^2+e^{-4 x-4} \left (3 x^2+12 x\right )+e^{-2 x-2} \left (3 x^3+24 x^2+48 x\right )+e^{-6 x-6} x+64 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{6 x+6} \left (x^4+39 x^3+462 x^2+e^{-4 x-4} \left (3 x^2+93 x+414\right )+\left (-90 x^2+e^{-2 x-2} \left (180 x^2+930 x\right )+180 e^{-4 x-4} x-510 x\right ) \log (x)+e^{-2 x-2} \left (3 x^3+105 x^2+876 x+2091\right )+2155 x+e^{-6 x-6} (x+27)+3468\right )}{x \left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {15 e^{6 x+6} (x+4) \left (12 x^2 \log (x)+34 x \log (x)-5\right )}{x \left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^2}-\frac {15 e^{4 x+4} \left (24 x^2 \log (x)+6 x+82 x \log (x)+19\right )}{x}+\frac {15 e^{6 x+6} (x+4) \left (24 x^2 \log (x)+6 x+82 x \log (x)+19\right )}{x \left (e^{2 x+2} x+4 e^{2 x+2}+1\right )}+\frac {x+27}{x}-\frac {150 e^{6 x+6} (2 x+9) \log (x)}{\left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^3}+\frac {90 e^{2 x+2} (2 x \log (x)+1)}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x+90 e^{2 x+2} \log (x)+\frac {45}{2} e^{4 x+4} \log (x)-\frac {15}{2} e^{4 x+4} (12 x+41) \log (x)+27 \log (x)-1350 \log (x) \int \frac {e^{6 x+6}}{\left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^3}dx-300 \log (x) \int \frac {e^{6 x+6} x}{\left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^3}dx+2040 \log (x) \int \frac {e^{6 x+6}}{\left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^2}dx-75 \int \frac {e^{6 x+6}}{\left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^2}dx-300 \int \frac {e^{6 x+6}}{x \left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^2}dx+1230 \log (x) \int \frac {e^{6 x+6} x}{\left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^2}dx+180 \log (x) \int \frac {e^{6 x+6} x^2}{\left (e^{2 x+2} x+4 e^{2 x+2}+1\right )^2}dx+4920 \log (x) \int \frac {e^{6 x+6}}{e^{2 x+2} x+4 e^{2 x+2}+1}dx+645 \int \frac {e^{6 x+6}}{e^{2 x+2} x+4 e^{2 x+2}+1}dx+1140 \int \frac {e^{6 x+6}}{x \left (e^{2 x+2} x+4 e^{2 x+2}+1\right )}dx+2670 \log (x) \int \frac {e^{6 x+6} x}{e^{2 x+2} x+4 e^{2 x+2}+1}dx+90 \int \frac {e^{6 x+6} x}{e^{2 x+2} x+4 e^{2 x+2}+1}dx+360 \log (x) \int \frac {e^{6 x+6} x^2}{e^{2 x+2} x+4 e^{2 x+2}+1}dx+1350 \int \frac {\int \frac {e^{6 x+6}}{\left (e^{2 x+2} (x+4)+1\right )^3}dx}{x}dx+300 \int \frac {\int \frac {e^{6 x+6} x}{\left (e^{2 x+2} (x+4)+1\right )^3}dx}{x}dx-2040 \int \frac {\int \frac {e^{6 x+6}}{\left (e^{2 x+2} (x+4)+1\right )^2}dx}{x}dx-1230 \int \frac {\int \frac {e^{6 x+6} x}{\left (e^{2 x+2} (x+4)+1\right )^2}dx}{x}dx-180 \int \frac {\int \frac {e^{6 x+6} x^2}{\left (e^{2 x+2} (x+4)+1\right )^2}dx}{x}dx-4920 \int \frac {\int \frac {e^{6 x+6}}{e^{2 x+2} (x+4)+1}dx}{x}dx-2670 \int \frac {\int \frac {e^{6 x+6} x}{e^{2 x+2} (x+4)+1}dx}{x}dx-360 \int \frac {\int \frac {e^{6 x+6} x^2}{e^{2 x+2} (x+4)+1}dx}{x}dx\) |
Int[(3468 + 2155*x + 462*x^2 + 39*x^3 + x^4 + E^(-6 - 6*x)*(27 + x) + E^(- 4 - 4*x)*(414 + 93*x + 3*x^2) + E^(-2 - 2*x)*(2091 + 876*x + 105*x^2 + 3*x ^3) + (-510*x + 180*E^(-4 - 4*x)*x - 90*x^2 + E^(-2 - 2*x)*(930*x + 180*x^ 2))*Log[x])/(64*x + E^(-6 - 6*x)*x + 48*x^2 + 12*x^3 + x^4 + E^(-4 - 4*x)* (12*x + 3*x^2) + E^(-2 - 2*x)*(48*x + 24*x^2 + 3*x^3)),x]
3.8.91.3.1 Defintions of rubi rules used
Time = 0.47 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
| method | result | size |
| risch | \(\frac {15 \left (6 \,{\mathrm e}^{-2-2 x}+6 x +29\right ) \ln \left (x \right )}{\left ({\mathrm e}^{-2-2 x}+4+x \right )^{2}}+x +27 \ln \left (x \right )\) | \(35\) |
| parallelrisch | \(\frac {-128-32 x +108 \ln \left (x \right ) {\mathrm e}^{-2-2 x} x +2 x \,{\mathrm e}^{-4-4 x}+612 x \ln \left (x \right )-64 \,{\mathrm e}^{-2-2 x}+4 \,{\mathrm e}^{-2-2 x} x^{2}+2 x^{3}+8 x^{2}+1734 \ln \left (x \right )+54 x^{2} \ln \left (x \right )+612 \ln \left (x \right ) {\mathrm e}^{-2-2 x}+54 \ln \left (x \right ) {\mathrm e}^{-4-4 x}-8 \,{\mathrm e}^{-4-4 x}}{2 \,{\mathrm e}^{-4-4 x}+4 \,{\mathrm e}^{-2-2 x} x +2 x^{2}+16 \,{\mathrm e}^{-2-2 x}+16 x +32}\) | \(154\) |
int(((180*x*exp(-1-x)^4+(180*x^2+930*x)*exp(-1-x)^2-90*x^2-510*x)*ln(x)+(x +27)*exp(-1-x)^6+(3*x^2+93*x+414)*exp(-1-x)^4+(3*x^3+105*x^2+876*x+2091)*e xp(-1-x)^2+x^4+39*x^3+462*x^2+2155*x+3468)/(x*exp(-1-x)^6+(3*x^2+12*x)*exp (-1-x)^4+(3*x^3+24*x^2+48*x)*exp(-1-x)^2+x^4+12*x^3+48*x^2+64*x),x,method= _RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.08 \[ \int \frac {3468+2155 x+462 x^2+39 x^3+x^4+e^{-6-6 x} (27+x)+e^{-4-4 x} \left (414+93 x+3 x^2\right )+e^{-2-2 x} \left (2091+876 x+105 x^2+3 x^3\right )+\left (-510 x+180 e^{-4-4 x} x-90 x^2+e^{-2-2 x} \left (930 x+180 x^2\right )\right ) \log (x)}{64 x+e^{-6-6 x} x+48 x^2+12 x^3+x^4+e^{-4-4 x} \left (12 x+3 x^2\right )+e^{-2-2 x} \left (48 x+24 x^2+3 x^3\right )} \, dx=\frac {x^{3} + 8 \, x^{2} + 2 \, {\left (x^{2} + 4 \, x\right )} e^{\left (-2 \, x - 2\right )} + x e^{\left (-4 \, x - 4\right )} + 3 \, {\left (9 \, x^{2} + 6 \, {\left (3 \, x + 17\right )} e^{\left (-2 \, x - 2\right )} + 102 \, x + 9 \, e^{\left (-4 \, x - 4\right )} + 289\right )} \log \left (x\right ) + 16 \, x}{x^{2} + 2 \, {\left (x + 4\right )} e^{\left (-2 \, x - 2\right )} + 8 \, x + e^{\left (-4 \, x - 4\right )} + 16} \]
integrate(((180*x*exp(-1-x)^4+(180*x^2+930*x)*exp(-1-x)^2-90*x^2-510*x)*lo g(x)+(x+27)*exp(-1-x)^6+(3*x^2+93*x+414)*exp(-1-x)^4+(3*x^3+105*x^2+876*x+ 2091)*exp(-1-x)^2+x^4+39*x^3+462*x^2+2155*x+3468)/(x*exp(-1-x)^6+(3*x^2+12 *x)*exp(-1-x)^4+(3*x^3+24*x^2+48*x)*exp(-1-x)^2+x^4+12*x^3+48*x^2+64*x),x, algorithm=\
(x^3 + 8*x^2 + 2*(x^2 + 4*x)*e^(-2*x - 2) + x*e^(-4*x - 4) + 3*(9*x^2 + 6* (3*x + 17)*e^(-2*x - 2) + 102*x + 9*e^(-4*x - 4) + 289)*log(x) + 16*x)/(x^ 2 + 2*(x + 4)*e^(-2*x - 2) + 8*x + e^(-4*x - 4) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.54 \[ \int \frac {3468+2155 x+462 x^2+39 x^3+x^4+e^{-6-6 x} (27+x)+e^{-4-4 x} \left (414+93 x+3 x^2\right )+e^{-2-2 x} \left (2091+876 x+105 x^2+3 x^3\right )+\left (-510 x+180 e^{-4-4 x} x-90 x^2+e^{-2-2 x} \left (930 x+180 x^2\right )\right ) \log (x)}{64 x+e^{-6-6 x} x+48 x^2+12 x^3+x^4+e^{-4-4 x} \left (12 x+3 x^2\right )+e^{-2-2 x} \left (48 x+24 x^2+3 x^3\right )} \, dx=x + \frac {90 x \log {\left (x \right )} + 90 e^{- 2 x - 2} \log {\left (x \right )} + 435 \log {\left (x \right )}}{x^{2} + 8 x + \left (2 x + 8\right ) e^{- 2 x - 2} + e^{- 4 x - 4} + 16} + 27 \log {\left (x \right )} \]
integrate(((180*x*exp(-1-x)**4+(180*x**2+930*x)*exp(-1-x)**2-90*x**2-510*x )*ln(x)+(x+27)*exp(-1-x)**6+(3*x**2+93*x+414)*exp(-1-x)**4+(3*x**3+105*x** 2+876*x+2091)*exp(-1-x)**2+x**4+39*x**3+462*x**2+2155*x+3468)/(x*exp(-1-x) **6+(3*x**2+12*x)*exp(-1-x)**4+(3*x**3+24*x**2+48*x)*exp(-1-x)**2+x**4+12* x**3+48*x**2+64*x),x)
x + (90*x*log(x) + 90*exp(-2*x - 2)*log(x) + 435*log(x))/(x**2 + 8*x + (2* x + 8)*exp(-2*x - 2) + exp(-4*x - 4) + 16) + 27*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (23) = 46\).
Time = 0.34 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.17 \[ \int \frac {3468+2155 x+462 x^2+39 x^3+x^4+e^{-6-6 x} (27+x)+e^{-4-4 x} \left (414+93 x+3 x^2\right )+e^{-2-2 x} \left (2091+876 x+105 x^2+3 x^3\right )+\left (-510 x+180 e^{-4-4 x} x-90 x^2+e^{-2-2 x} \left (930 x+180 x^2\right )\right ) \log (x)}{64 x+e^{-6-6 x} x+48 x^2+12 x^3+x^4+e^{-4-4 x} \left (12 x+3 x^2\right )+e^{-2-2 x} \left (48 x+24 x^2+3 x^3\right )} \, dx=\frac {{\left (x^{3} e^{4} + 8 \, x^{2} e^{4} + 16 \, x e^{4} + 3 \, {\left (9 \, x^{2} e^{4} + 102 \, x e^{4} + 289 \, e^{4}\right )} \log \left (x\right )\right )} e^{\left (4 \, x\right )} + 2 \, {\left (x^{2} e^{2} + 4 \, x e^{2} + 9 \, {\left (3 \, x e^{2} + 17 \, e^{2}\right )} \log \left (x\right )\right )} e^{\left (2 \, x\right )} + x + 27 \, \log \left (x\right )}{{\left (x^{2} e^{4} + 8 \, x e^{4} + 16 \, e^{4}\right )} e^{\left (4 \, x\right )} + 2 \, {\left (x e^{2} + 4 \, e^{2}\right )} e^{\left (2 \, x\right )} + 1} \]
integrate(((180*x*exp(-1-x)^4+(180*x^2+930*x)*exp(-1-x)^2-90*x^2-510*x)*lo g(x)+(x+27)*exp(-1-x)^6+(3*x^2+93*x+414)*exp(-1-x)^4+(3*x^3+105*x^2+876*x+ 2091)*exp(-1-x)^2+x^4+39*x^3+462*x^2+2155*x+3468)/(x*exp(-1-x)^6+(3*x^2+12 *x)*exp(-1-x)^4+(3*x^3+24*x^2+48*x)*exp(-1-x)^2+x^4+12*x^3+48*x^2+64*x),x, algorithm=\
((x^3*e^4 + 8*x^2*e^4 + 16*x*e^4 + 3*(9*x^2*e^4 + 102*x*e^4 + 289*e^4)*log (x))*e^(4*x) + 2*(x^2*e^2 + 4*x*e^2 + 9*(3*x*e^2 + 17*e^2)*log(x))*e^(2*x) + x + 27*log(x))/((x^2*e^4 + 8*x*e^4 + 16*e^4)*e^(4*x) + 2*(x*e^2 + 4*e^2 )*e^(2*x) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (23) = 46\).
Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 5.67 \[ \int \frac {3468+2155 x+462 x^2+39 x^3+x^4+e^{-6-6 x} (27+x)+e^{-4-4 x} \left (414+93 x+3 x^2\right )+e^{-2-2 x} \left (2091+876 x+105 x^2+3 x^3\right )+\left (-510 x+180 e^{-4-4 x} x-90 x^2+e^{-2-2 x} \left (930 x+180 x^2\right )\right ) \log (x)}{64 x+e^{-6-6 x} x+48 x^2+12 x^3+x^4+e^{-4-4 x} \left (12 x+3 x^2\right )+e^{-2-2 x} \left (48 x+24 x^2+3 x^3\right )} \, dx=\frac {x^{3} e^{4} + 27 \, x^{2} e^{4} \log \left (x\right ) + 8 \, x^{2} e^{4} + 2 \, x^{2} e^{\left (-2 \, x + 2\right )} + 306 \, x e^{4} \log \left (x\right ) + 54 \, x e^{\left (-2 \, x + 2\right )} \log \left (x\right ) + 16 \, x e^{4} + x e^{\left (-4 \, x\right )} + 8 \, x e^{\left (-2 \, x + 2\right )} + 867 \, e^{4} \log \left (x\right ) + 27 \, e^{\left (-4 \, x\right )} \log \left (x\right ) + 306 \, e^{\left (-2 \, x + 2\right )} \log \left (x\right )}{x^{2} e^{4} + 8 \, x e^{4} + 2 \, x e^{\left (-2 \, x + 2\right )} + 16 \, e^{4} + e^{\left (-4 \, x\right )} + 8 \, e^{\left (-2 \, x + 2\right )}} \]
integrate(((180*x*exp(-1-x)^4+(180*x^2+930*x)*exp(-1-x)^2-90*x^2-510*x)*lo g(x)+(x+27)*exp(-1-x)^6+(3*x^2+93*x+414)*exp(-1-x)^4+(3*x^3+105*x^2+876*x+ 2091)*exp(-1-x)^2+x^4+39*x^3+462*x^2+2155*x+3468)/(x*exp(-1-x)^6+(3*x^2+12 *x)*exp(-1-x)^4+(3*x^3+24*x^2+48*x)*exp(-1-x)^2+x^4+12*x^3+48*x^2+64*x),x, algorithm=\
(x^3*e^4 + 27*x^2*e^4*log(x) + 8*x^2*e^4 + 2*x^2*e^(-2*x + 2) + 306*x*e^4* log(x) + 54*x*e^(-2*x + 2)*log(x) + 16*x*e^4 + x*e^(-4*x) + 8*x*e^(-2*x + 2) + 867*e^4*log(x) + 27*e^(-4*x)*log(x) + 306*e^(-2*x + 2)*log(x))/(x^2*e ^4 + 8*x*e^4 + 2*x*e^(-2*x + 2) + 16*e^4 + e^(-4*x) + 8*e^(-2*x + 2))
Timed out. \[ \int \frac {3468+2155 x+462 x^2+39 x^3+x^4+e^{-6-6 x} (27+x)+e^{-4-4 x} \left (414+93 x+3 x^2\right )+e^{-2-2 x} \left (2091+876 x+105 x^2+3 x^3\right )+\left (-510 x+180 e^{-4-4 x} x-90 x^2+e^{-2-2 x} \left (930 x+180 x^2\right )\right ) \log (x)}{64 x+e^{-6-6 x} x+48 x^2+12 x^3+x^4+e^{-4-4 x} \left (12 x+3 x^2\right )+e^{-2-2 x} \left (48 x+24 x^2+3 x^3\right )} \, dx=\int \frac {2155\,x+{\mathrm {e}}^{-4\,x-4}\,\left (3\,x^2+93\,x+414\right )-\ln \left (x\right )\,\left (510\,x-{\mathrm {e}}^{-2\,x-2}\,\left (180\,x^2+930\,x\right )-180\,x\,{\mathrm {e}}^{-4\,x-4}+90\,x^2\right )+{\mathrm {e}}^{-2\,x-2}\,\left (3\,x^3+105\,x^2+876\,x+2091\right )+{\mathrm {e}}^{-6\,x-6}\,\left (x+27\right )+462\,x^2+39\,x^3+x^4+3468}{64\,x+{\mathrm {e}}^{-4\,x-4}\,\left (3\,x^2+12\,x\right )+x\,{\mathrm {e}}^{-6\,x-6}+{\mathrm {e}}^{-2\,x-2}\,\left (3\,x^3+24\,x^2+48\,x\right )+48\,x^2+12\,x^3+x^4} \,d x \]
int((2155*x + exp(- 4*x - 4)*(93*x + 3*x^2 + 414) - log(x)*(510*x - exp(- 2*x - 2)*(930*x + 180*x^2) - 180*x*exp(- 4*x - 4) + 90*x^2) + exp(- 2*x - 2)*(876*x + 105*x^2 + 3*x^3 + 2091) + exp(- 6*x - 6)*(x + 27) + 462*x^2 + 39*x^3 + x^4 + 3468)/(64*x + exp(- 4*x - 4)*(12*x + 3*x^2) + x*exp(- 6*x - 6) + exp(- 2*x - 2)*(48*x + 24*x^2 + 3*x^3) + 48*x^2 + 12*x^3 + x^4),x)
int((2155*x + exp(- 4*x - 4)*(93*x + 3*x^2 + 414) - log(x)*(510*x - exp(- 2*x - 2)*(930*x + 180*x^2) - 180*x*exp(- 4*x - 4) + 90*x^2) + exp(- 2*x - 2)*(876*x + 105*x^2 + 3*x^3 + 2091) + exp(- 6*x - 6)*(x + 27) + 462*x^2 + 39*x^3 + x^4 + 3468)/(64*x + exp(- 4*x - 4)*(12*x + 3*x^2) + x*exp(- 6*x - 6) + exp(- 2*x - 2)*(48*x + 24*x^2 + 3*x^3) + 48*x^2 + 12*x^3 + x^4), x)