Integrand size = 56, antiderivative size = 17 \[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx=e^{\frac {100+e^3}{(1+x)^2}} (5+x) \] Output:
exp((exp(3)+100)/(1+x)^2)*(5+x)
Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx=e^{\frac {100+e^3}{(1+x)^2}} (5+x) \] Input:
Integrate[(E^((100 + E^3)/(1 + 2*x + x^2))*(-999 + E^3*(-10 - 2*x) - 197*x + 3*x^2 + x^3))/(1 + 3*x + 3*x^2 + x^3),x]
Output:
E^((100 + E^3)/(1 + x)^2)*(5 + 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.
\(\displaystyle \int \frac {e^{\frac {100+e^3}{x^2+2 x+1}} \left (x^3+3 x^2-197 x+e^3 (-2 x-10)-999\right )}{x^3+3 x^2+3 x+1} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e^{\frac {100+e^3}{x^2+2 x+1}} \left (x^3+3 x^2-197 x+e^3 (-2 x-10)-999\right )}{(x+1)^3}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {100+e^3}{x^2+2 x+1}} \left (x^3+3 x^2-\left (197+2 e^3\right ) x-10 e^3-999\right )}{(x+1)^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (e^{\frac {100+e^3}{x^2+2 x+1}}-\frac {2 \left (100+e^3\right ) e^{\frac {100+e^3}{x^2+2 x+1}}}{(x+1)^2}-\frac {8 \left (100+e^3\right ) e^{\frac {100+e^3}{x^2+2 x+1}}}{(x+1)^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{\frac {100+e^3}{x^2+2 x+1}}dx-2 \left (100+e^3\right ) \int \frac {e^{\frac {100+e^3}{x^2+2 x+1}}}{(x+1)^2}dx+4 e^{\frac {100+e^3}{x^2+2 x+1}}\) |
Input:
Int[(E^((100 + E^3)/(1 + 2*x + x^2))*(-999 + E^3*(-10 - 2*x) - 197*x + 3*x ^2 + x^3))/(1 + 3*x + 3*x^2 + x^3),x]
Output:
$Aborted
Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
method | result | size |
risch | \({\mathrm e}^{\frac {{\mathrm e}^{3}+100}{\left (1+x \right )^{2}}} \left (5+x \right )\) | \(16\) |
gosper | \(\left (5+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{3}+100}{x^{2}+2 x +1}}\) | \(21\) |
parallelrisch | \(x \,{\mathrm e}^{\frac {{\mathrm e}^{3}+100}{x^{2}+2 x +1}}+5 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+100}{x^{2}+2 x +1}}\) | \(38\) |
norman | \(\frac {x^{3} {\mathrm e}^{\frac {{\mathrm e}^{3}+100}{x^{2}+2 x +1}}+11 x \,{\mathrm e}^{\frac {{\mathrm e}^{3}+100}{x^{2}+2 x +1}}+7 x^{2} {\mathrm e}^{\frac {{\mathrm e}^{3}+100}{x^{2}+2 x +1}}+5 \,{\mathrm e}^{\frac {{\mathrm e}^{3}+100}{x^{2}+2 x +1}}}{\left (1+x \right )^{2}}\) | \(86\) |
derivativedivides | \(\left (1+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{3}+100}{\left (1+x \right )^{2}}}+i \sqrt {{\mathrm e}^{3}+100}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {{\mathrm e}^{3}+100}}{1+x}\right )-\frac {100 i \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {{\mathrm e}^{3}+100}}{1+x}\right )}{\sqrt {{\mathrm e}^{3}+100}}+\frac {400 \,{\mathrm e}^{\frac {{\mathrm e}^{3}}{\left (1+x \right )^{2}}+\frac {100}{\left (1+x \right )^{2}}}}{{\mathrm e}^{3}+100}-\frac {i {\mathrm e}^{3} \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {{\mathrm e}^{3}+100}}{1+x}\right )}{\sqrt {{\mathrm e}^{3}+100}}+\frac {4 \,{\mathrm e}^{3} {\mathrm e}^{\frac {{\mathrm e}^{3}}{\left (1+x \right )^{2}}+\frac {100}{\left (1+x \right )^{2}}}}{{\mathrm e}^{3}+100}\) | \(152\) |
default | \(\left (1+x \right ) {\mathrm e}^{\frac {{\mathrm e}^{3}+100}{\left (1+x \right )^{2}}}+i \sqrt {{\mathrm e}^{3}+100}\, \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {{\mathrm e}^{3}+100}}{1+x}\right )-\frac {100 i \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {{\mathrm e}^{3}+100}}{1+x}\right )}{\sqrt {{\mathrm e}^{3}+100}}+\frac {400 \,{\mathrm e}^{\frac {{\mathrm e}^{3}}{\left (1+x \right )^{2}}+\frac {100}{\left (1+x \right )^{2}}}}{{\mathrm e}^{3}+100}-\frac {i {\mathrm e}^{3} \sqrt {\pi }\, \operatorname {erf}\left (\frac {i \sqrt {{\mathrm e}^{3}+100}}{1+x}\right )}{\sqrt {{\mathrm e}^{3}+100}}+\frac {4 \,{\mathrm e}^{3} {\mathrm e}^{\frac {{\mathrm e}^{3}}{\left (1+x \right )^{2}}+\frac {100}{\left (1+x \right )^{2}}}}{{\mathrm e}^{3}+100}\) | \(152\) |
Input:
int(((-2*x-10)*exp(3)+x^3+3*x^2-197*x-999)*exp((exp(3)+100)/(x^2+2*x+1))/( x^3+3*x^2+3*x+1),x,method=_RETURNVERBOSE)
Output:
exp((exp(3)+100)/(1+x)^2)*(5+x)
Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx={\left (x + 5\right )} e^{\left (\frac {e^{3} + 100}{x^{2} + 2 \, x + 1}\right )} \] Input:
integrate(((-2*x-10)*exp(3)+x^3+3*x^2-197*x-999)*exp((exp(3)+100)/(x^2+2*x +1))/(x^3+3*x^2+3*x+1),x, algorithm="fricas")
Output:
(x + 5)*e^((e^3 + 100)/(x^2 + 2*x + 1))
Time = 0.18 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx=\left (x + 5\right ) e^{\frac {e^{3} + 100}{x^{2} + 2 x + 1}} \] Input:
integrate(((-2*x-10)*exp(3)+x**3+3*x**2-197*x-999)*exp((exp(3)+100)/(x**2+ 2*x+1))/(x**3+3*x**2+3*x+1),x)
Output:
(x + 5)*exp((exp(3) + 100)/(x**2 + 2*x + 1))
\[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx=\int { \frac {{\left (x^{3} + 3 \, x^{2} - 2 \, {\left (x + 5\right )} e^{3} - 197 \, x - 999\right )} e^{\left (\frac {e^{3} + 100}{x^{2} + 2 \, x + 1}\right )}}{x^{3} + 3 \, x^{2} + 3 \, x + 1} \,d x } \] Input:
integrate(((-2*x-10)*exp(3)+x^3+3*x^2-197*x-999)*exp((exp(3)+100)/(x^2+2*x +1))/(x^3+3*x^2+3*x+1),x, algorithm="maxima")
Output:
999/2*e^(e^3/(x^2 + 2*x + 1) + 100/(x^2 + 2*x + 1))/(e^3 + 100) + integrat e((x^3 + 3*x^2 - x*(2*e^3 + 197) - 10*e^3)*e^(e^3/(x^2 + 2*x + 1) + 100/(x ^2 + 2*x + 1))/(x^3 + 3*x^2 + 3*x + 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 79, normalized size of antiderivative = 4.65 \[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx=x e^{\left (-\frac {x^{2} e^{3} + 100 \, x^{2} + 2 \, x e^{3} + 200 \, x}{x^{2} + 2 \, x + 1} + e^{3} + 100\right )} + 5 \, e^{\left (-\frac {x^{2} e^{3} + 100 \, x^{2} + 2 \, x e^{3} + 200 \, x}{x^{2} + 2 \, x + 1} + e^{3} + 100\right )} \] Input:
integrate(((-2*x-10)*exp(3)+x^3+3*x^2-197*x-999)*exp((exp(3)+100)/(x^2+2*x +1))/(x^3+3*x^2+3*x+1),x, algorithm="giac")
Output:
x*e^(-(x^2*e^3 + 100*x^2 + 2*x*e^3 + 200*x)/(x^2 + 2*x + 1) + e^3 + 100) + 5*e^(-(x^2*e^3 + 100*x^2 + 2*x*e^3 + 200*x)/(x^2 + 2*x + 1) + e^3 + 100)
Time = 2.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx={\mathrm {e}}^{\frac {100}{x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^3}{x^2+2\,x+1}}\,\left (x+5\right ) \] Input:
int(-(exp((exp(3) + 100)/(2*x + x^2 + 1))*(197*x - 3*x^2 - x^3 + exp(3)*(2 *x + 10) + 999))/(3*x + 3*x^2 + x^3 + 1),x)
Output:
exp(100/(2*x + x^2 + 1))*exp(exp(3)/(2*x + x^2 + 1))*(x + 5)
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\frac {100+e^3}{1+2 x+x^2}} \left (-999+e^3 (-10-2 x)-197 x+3 x^2+x^3\right )}{1+3 x+3 x^2+x^3} \, dx=e^{\frac {e^{3}+100}{x^{2}+2 x +1}} \left (x +5\right ) \] Input:
int(((-2*x-10)*exp(3)+x^3+3*x^2-197*x-999)*exp((exp(3)+100)/(x^2+2*x+1))/( x^3+3*x^2+3*x+1),x)
Output:
e**((e**3 + 100)/(x**2 + 2*x + 1))*(x + 5)