Integrand size = 117, antiderivative size = 29 \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx=e^{\frac {(1+x) \left (\frac {1}{x^2}-3 \left (x+x^2\right )\right )}{5 \left (e^x-x\right )}} \] Output:
exp(1/5*(1+x)/(exp(x)-x)*(1/x^2-3*x^2-3*x))
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx=e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 \left (e^x-x\right ) x^2}} \] Input:
Integrate[(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*E^x*x^2 - 5*x^3))*(3*x + 2*x^2 + 6*x^5 + 6*x^6 + E^x*(-2 - 2*x - x^2 - 3*x^3 - 9*x^4 - 3*x^5 + 3*x^ 6)))/(5*E^(2*x)*x^3 - 10*E^x*x^4 + 5*x^5),x]
Output:
E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*(E^x - x)*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.
\(\displaystyle \int \frac {\left (6 x^6+6 x^5+2 x^2+e^x \left (3 x^6-3 x^5-9 x^4-3 x^3-x^2-2 x-2\right )+3 x\right ) \exp \left (\frac {-3 x^5-6 x^4-3 x^3+x+1}{5 e^x x^2-5 x^3}\right )}{5 x^5-10 e^x x^4+5 e^{2 x} x^3} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (6 x^6+6 x^5+2 x^2+e^x \left (3 x^6-3 x^5-9 x^4-3 x^3-x^2-2 x-2\right )+3 x\right ) \exp \left (\frac {-3 x^5-6 x^4-3 x^3+x+1}{5 \left (e^x-x\right ) x^2}\right )}{5 \left (e^x-x\right )^2 x^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{5} \int \frac {\exp \left (\frac {-3 x^5-6 x^4-3 x^3+x+1}{5 \left (e^x-x\right ) x^2}\right ) \left (6 x^6+6 x^5+2 x^2+3 x-e^x \left (-3 x^6+3 x^5+9 x^4+3 x^3+x^2+2 x+2\right )\right )}{\left (e^x-x\right )^2 x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{5} \int \left (\frac {\exp \left (\frac {-3 x^5-6 x^4-3 x^3+x+1}{5 \left (e^x-x\right ) x^2}\right ) \left (3 x^6-3 x^5-9 x^4-3 x^3-x^2-2 x-2\right )}{\left (e^x-x\right ) x^3}+\frac {\exp \left (\frac {-3 x^5-6 x^4-3 x^3+x+1}{5 \left (e^x-x\right ) x^2}\right ) \left (3 x^6+3 x^5-3 x^4-3 x^3-x^2+1\right )}{\left (e^x-x\right )^2 x^2}\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{5} \int \left (\frac {\exp \left (-\frac {3 x^5+6 x^4+3 x^3-x-1}{5 \left (e^x-x\right ) x^2}\right ) \left (3 x^6-3 x^5-9 x^4-3 x^3-x^2-2 x-2\right )}{\left (e^x-x\right ) x^3}+\frac {\exp \left (-\frac {3 x^5+6 x^4+3 x^3-x-1}{5 \left (e^x-x\right ) x^2}\right ) \left (3 x^6+3 x^5-3 x^4-3 x^3-x^2+1\right )}{\left (e^x-x\right )^2 x^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {1}{5} \int \left (\frac {\exp \left (-\frac {3 x^5+6 x^4+3 x^3-x-1}{5 \left (e^x-x\right ) x^2}\right ) \left (3 x^6-3 x^5-9 x^4-3 x^3-x^2-2 x-2\right )}{\left (e^x-x\right ) x^3}+\frac {\exp \left (-\frac {3 x^5+6 x^4+3 x^3-x-1}{5 \left (e^x-x\right ) x^2}\right ) \left (3 x^6+3 x^5-3 x^4-3 x^3-x^2+1\right )}{\left (e^x-x\right )^2 x^2}\right )dx\) |
Input:
Int[(E^((1 + x - 3*x^3 - 6*x^4 - 3*x^5)/(5*E^x*x^2 - 5*x^3))*(3*x + 2*x^2 + 6*x^5 + 6*x^6 + E^x*(-2 - 2*x - x^2 - 3*x^3 - 9*x^4 - 3*x^5 + 3*x^6)))/( 5*E^(2*x)*x^3 - 10*E^x*x^4 + 5*x^5),x]
Output:
$Aborted
Time = 3.67 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03
method | result | size |
risch | \({\mathrm e}^{-\frac {\left (1+x \right ) \left (3 x^{4}+3 x^{3}-1\right )}{5 x^{2} \left ({\mathrm e}^{x}-x \right )}}\) | \(30\) |
parallelrisch | \({\mathrm e}^{-\frac {3 x^{5}+6 x^{4}+3 x^{3}-x -1}{5 x^{2} \left ({\mathrm e}^{x}-x \right )}}\) | \(35\) |
Input:
int(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3*x)*exp ((-3*x^5-6*x^4-3*x^3+x+1)/(5*exp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*exp(x)* x^4+5*x^5),x,method=_RETURNVERBOSE)
Output:
exp(-1/5*(1+x)*(3*x^4+3*x^3-1)/x^2/(exp(x)-x))
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx=e^{\left (\frac {3 \, x^{5} + 6 \, x^{4} + 3 \, x^{3} - x - 1}{5 \, {\left (x^{3} - x^{2} e^{x}\right )}}\right )} \] Input:
integrate(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3* x)*exp((-3*x^5-6*x^4-3*x^3+x+1)/(5*exp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*e xp(x)*x^4+5*x^5),x, algorithm="fricas")
Output:
e^(1/5*(3*x^5 + 6*x^4 + 3*x^3 - x - 1)/(x^3 - x^2*e^x))
Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx=e^{\frac {- 3 x^{5} - 6 x^{4} - 3 x^{3} + x + 1}{- 5 x^{3} + 5 x^{2} e^{x}}} \] Input:
integrate(((3*x**6-3*x**5-9*x**4-3*x**3-x**2-2*x-2)*exp(x)+6*x**6+6*x**5+2 *x**2+3*x)*exp((-3*x**5-6*x**4-3*x**3+x+1)/(5*exp(x)*x**2-5*x**3))/(5*exp( x)**2*x**3-10*exp(x)*x**4+5*x**5),x)
Output:
exp((-3*x**5 - 6*x**4 - 3*x**3 + x + 1)/(-5*x**3 + 5*x**2*exp(x)))
Exception generated. \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3* x)*exp((-3*x^5-6*x^4-3*x^3+x+1)/(5*exp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*e xp(x)*x^4+5*x^5),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3* x)*exp((-3*x^5-6*x^4-3*x^3+x+1)/(5*exp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*e xp(x)*x^4+5*x^5),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{6075,[1,36]%%%}+%%%{6075,[1,35]%%%}+%%%{-66825,[1,34]%%%}+ %%%{-1154
Time = 2.48 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66 \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx={\mathrm {e}}^{\frac {3\,x}{5\,x-5\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {1}{5\,x\,{\mathrm {e}}^x-5\,x^2}}\,{\mathrm {e}}^{\frac {3\,x^3}{5\,x-5\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {6\,x^2}{5\,x-5\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {1}{5\,x^2\,{\mathrm {e}}^x-5\,x^3}} \] Input:
int((exp(-(3*x^3 - x + 6*x^4 + 3*x^5 - 1)/(5*x^2*exp(x) - 5*x^3))*(3*x - e xp(x)*(2*x + x^2 + 3*x^3 + 9*x^4 + 3*x^5 - 3*x^6 + 2) + 2*x^2 + 6*x^5 + 6* x^6))/(5*x^3*exp(2*x) - 10*x^4*exp(x) + 5*x^5),x)
Output:
exp((3*x)/(5*x - 5*exp(x)))*exp(1/(5*x*exp(x) - 5*x^2))*exp((3*x^3)/(5*x - 5*exp(x)))*exp((6*x^2)/(5*x - 5*exp(x)))*exp(1/(5*x^2*exp(x) - 5*x^3))
Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {1+x-3 x^3-6 x^4-3 x^5}{5 e^x x^2-5 x^3}} \left (3 x+2 x^2+6 x^5+6 x^6+e^x \left (-2-2 x-x^2-3 x^3-9 x^4-3 x^5+3 x^6\right )\right )}{5 e^{2 x} x^3-10 e^x x^4+5 x^5} \, dx=\frac {e^{\frac {x +1}{5 e^{x} x^{2}-5 x^{3}}}}{e^{\frac {3 x^{3}+6 x^{2}+3 x}{5 e^{x}-5 x}}} \] Input:
int(((3*x^6-3*x^5-9*x^4-3*x^3-x^2-2*x-2)*exp(x)+6*x^6+6*x^5+2*x^2+3*x)*exp ((-3*x^5-6*x^4-3*x^3+x+1)/(5*exp(x)*x^2-5*x^3))/(5*exp(x)^2*x^3-10*exp(x)* x^4+5*x^5),x)
Output:
e**((x + 1)/(5*e**x*x**2 - 5*x**3))/e**((3*x**3 + 6*x**2 + 3*x)/(5*e**x - 5*x))