Integrand size = 88, antiderivative size = 23 \[ \int \frac {e^{\frac {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx=e^{x+\frac {5 \left (-3-\frac {4}{x}+x\right ) \log \left (14+e^x\right )}{x}} \]
Time = 5.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx=e^x \left (14+e^x\right )^{5-\frac {5 (4+3 x)}{x^2}} \]
Integrate[(E^((x^3 + (-20 - 15*x + 5*x^2)*Log[14 + E^x])/x^2)*(14*x^3 + E^ x*(-20*x - 15*x^2 + 6*x^3) + (560 + 210*x + E^x*(40 + 15*x))*Log[14 + E^x] ))/(14*x^3 + E^x*x^3),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.
\(\displaystyle \int \frac {e^{\frac {x^3+\left (5 x^2-15 x-20\right ) \log \left (e^x+14\right )}{x^2}} \left (14 x^3+e^x \left (6 x^3-15 x^2-20 x\right )+\left (210 x+e^x (15 x+40)+560\right ) \log \left (e^x+14\right )\right )}{e^x x^3+14 x^3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {\left (5 x^2-15 x-20\right ) \log \left (e^x+14\right )}{x^2}+x} \left (14 x^3+e^x \left (6 x^3-15 x^2-20 x\right )+\left (210 x+e^x (15 x+40)+560\right ) \log \left (e^x+14\right )\right )}{e^x x^3+14 x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{\frac {\left (5 x^2-15 x-20\right ) \log \left (e^x+14\right )}{x^2}+x} \left (6 x^3-15 x^2-20 x+15 x \log \left (e^x+14\right )+40 \log \left (e^x+14\right )\right )}{x^3}-\frac {70 \left (x^2-3 x-4\right ) e^{\frac {\left (5 x^2-15 x-20\right ) \log \left (e^x+14\right )}{x^2}+x}}{\left (e^x+14\right ) x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 \int e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}}dx-70 \int \frac {e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}}}{14+e^x}dx-20 \int \frac {e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}}}{x^2}dx+280 \int \frac {e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}}}{\left (14+e^x\right ) x^2}dx-15 \int \frac {e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}}}{x}dx+210 \int \frac {e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}}}{\left (14+e^x\right ) x}dx+15 \int \frac {e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}} \log \left (14+e^x\right )}{x^2}dx+40 \int \frac {e^{x+\frac {\left (5 x^2-15 x-20\right ) \log \left (14+e^x\right )}{x^2}} \log \left (14+e^x\right )}{x^3}dx\) |
Int[(E^((x^3 + (-20 - 15*x + 5*x^2)*Log[14 + E^x])/x^2)*(14*x^3 + E^x*(-20 *x - 15*x^2 + 6*x^3) + (560 + 210*x + E^x*(40 + 15*x))*Log[14 + E^x]))/(14 *x^3 + E^x*x^3),x]
3.10.41.3.1 Defintions of rubi rules used
Time = 0.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {\left (5 x^{2}-15 x -20\right ) \ln \left ({\mathrm e}^{x}+14\right )+x^{3}}{x^{2}}}\) | \(26\) |
risch | \(\left ({\mathrm e}^{x}+14\right )^{5} \left ({\mathrm e}^{x}+14\right )^{-\frac {15}{x}} \left ({\mathrm e}^{x}+14\right )^{-\frac {20}{x^{2}}} {\mathrm e}^{x}\) | \(30\) |
int((((15*x+40)*exp(x)+210*x+560)*ln(exp(x)+14)+(6*x^3-15*x^2-20*x)*exp(x) +14*x^3)*exp(((5*x^2-15*x-20)*ln(exp(x)+14)+x^3)/x^2)/(exp(x)*x^3+14*x^3), x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx=e^{\left (\frac {x^{3} + 5 \, {\left (x^{2} - 3 \, x - 4\right )} \log \left (e^{x} + 14\right )}{x^{2}}\right )} \]
integrate((((15*x+40)*exp(x)+210*x+560)*log(exp(x)+14)+(6*x^3-15*x^2-20*x) *exp(x)+14*x^3)*exp(((5*x^2-15*x-20)*log(exp(x)+14)+x^3)/x^2)/(exp(x)*x^3+ 14*x^3),x, algorithm=\
Timed out. \[ \int \frac {e^{\frac {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx=\text {Timed out} \]
integrate((((15*x+40)*exp(x)+210*x+560)*ln(exp(x)+14)+(6*x**3-15*x**2-20*x )*exp(x)+14*x**3)*exp(((5*x**2-15*x-20)*ln(exp(x)+14)+x**3)/x**2)/(exp(x)* x**3+14*x**3),x)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {e^{\frac {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx={\left (e^{\left (6 \, x\right )} + 70 \, e^{\left (5 \, x\right )} + 1960 \, e^{\left (4 \, x\right )} + 27440 \, e^{\left (3 \, x\right )} + 192080 \, e^{\left (2 \, x\right )} + 537824 \, e^{x}\right )} e^{\left (-\frac {15 \, \log \left (e^{x} + 14\right )}{x} - \frac {20 \, \log \left (e^{x} + 14\right )}{x^{2}}\right )} \]
integrate((((15*x+40)*exp(x)+210*x+560)*log(exp(x)+14)+(6*x^3-15*x^2-20*x) *exp(x)+14*x^3)*exp(((5*x^2-15*x-20)*log(exp(x)+14)+x^3)/x^2)/(exp(x)*x^3+ 14*x^3),x, algorithm=\
(e^(6*x) + 70*e^(5*x) + 1960*e^(4*x) + 27440*e^(3*x) + 192080*e^(2*x) + 53 7824*e^x)*e^(-15*log(e^x + 14)/x - 20*log(e^x + 14)/x^2)
Time = 0.77 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {e^{\frac {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx=e^{\left (x - \frac {15 \, \log \left (e^{x} + 14\right )}{x} - \frac {20 \, \log \left (e^{x} + 14\right )}{x^{2}} + 5 \, \log \left (e^{x} + 14\right )\right )} \]
integrate((((15*x+40)*exp(x)+210*x+560)*log(exp(x)+14)+(6*x^3-15*x^2-20*x) *exp(x)+14*x^3)*exp(((5*x^2-15*x-20)*log(exp(x)+14)+x^3)/x^2)/(exp(x)*x^3+ 14*x^3),x, algorithm=\
Time = 9.71 (sec) , antiderivative size = 178, normalized size of antiderivative = 7.74 \[ \int \frac {e^{\frac {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx=\frac {192080\,{\mathrm {e}}^{2\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {27440\,{\mathrm {e}}^{3\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {1960\,{\mathrm {e}}^{4\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {70\,{\mathrm {e}}^{5\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {{\mathrm {e}}^{6\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {537824\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}} \]
int((exp(-(log(exp(x) + 14)*(15*x - 5*x^2 + 20) - x^3)/x^2)*(log(exp(x) + 14)*(210*x + exp(x)*(15*x + 40) + 560) + 14*x^3 - exp(x)*(20*x + 15*x^2 - 6*x^3)))/(x^3*exp(x) + 14*x^3),x)
(192080*exp(2*x))/((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2)) + (27440*e xp(3*x))/((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2)) + (1960*exp(4*x))/( (exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2)) + (70*exp(5*x))/((exp(x) + 14 )^(15/x)*(exp(x) + 14)^(20/x^2)) + exp(6*x)/((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2)) + (537824*exp(x))/((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/ x^2))