Integrand size = 70, antiderivative size = 29 \[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=e^{-1-e^x} \left (-\frac {15 x}{16}+\frac {e^x}{x+e^2 x}\right ) \] Output:
(-15/16*x+exp(x)/(x+exp(2)*x))/exp(1+exp(x))
Time = 0.35 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=\frac {e^{-1-e^x} \left (16 e^x-15 x^2-15 e^2 x^2\right )}{16 \left (1+e^2\right ) x} \] Input:
Integrate[(E^(-1 - E^x)*(-16*E^(2*x)*x - 15*x^2 - 15*E^2*x^2 + E^x*(-16 + 16*x + 15*x^3 + 15*E^2*x^3)))/(16*x^2 + 16*E^2*x^2),x]
Output:
(E^(-1 - E^x)*(16*E^x - 15*x^2 - 15*E^2*x^2))/(16*(1 + E^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.
\(\displaystyle \int \frac {e^{-e^x-1} \left (e^x \left (15 e^2 x^3+15 x^3+16 x-16\right )-15 e^2 x^2-15 x^2-16 e^{2 x} x\right )}{16 e^2 x^2+16 x^2} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{-e^x-1} \left (e^x \left (15 e^2 x^3+15 x^3+16 x-16\right )-15 e^2 x^2-15 x^2-16 e^{2 x} x\right )}{\left (16+16 e^2\right ) x^2}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {e^{-e^x-1} \left (e^x \left (15 e^2 x^3+15 x^3+16 x-16\right )+\left (-15-15 e^2\right ) x^2-16 e^{2 x} x\right )}{\left (16+16 e^2\right ) x^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {e^{-1-e^x} \left (15 \left (1+e^2\right ) x^2+16 e^{2 x} x+e^x \left (-15 e^2 x^3-15 x^3-16 x+16\right )\right )}{x^2}dx}{16 \left (1+e^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {e^{-1-e^x} \left (15 \left (1+e^2\right ) x^2+16 e^{2 x} x+e^x \left (-15 e^2 x^3-15 x^3-16 x+16\right )\right )}{x^2}dx}{16 \left (1+e^2\right )}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {\int \left (\frac {e^{x-e^x-1} \left (-15 \left (1+e^2\right ) x^3-16 x+16\right )}{x^2}+\frac {16 e^{2 x-e^x-1}}{x}+15 e^{-1-e^x} \left (1+e^2\right )\right )dx}{16 \left (1+e^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {16 \int \frac {e^{x-e^x-1}}{x^2}dx-16 \int \frac {e^{x-e^x-1}}{x}dx+16 \int \frac {e^{2 x-e^x-1}}{x}dx-15 \left (1+e^2\right ) \int e^{x-e^x-1} xdx+\frac {15 \left (1+e^2\right ) \operatorname {ExpIntegralEi}\left (-e^x\right )}{e}}{16 \left (1+e^2\right )}\) |
Input:
Int[(E^(-1 - E^x)*(-16*E^(2*x)*x - 15*x^2 - 15*E^2*x^2 + E^x*(-16 + 16*x + 15*x^3 + 15*E^2*x^3)))/(16*x^2 + 16*E^2*x^2),x]
Output:
$Aborted
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
norman | \(\frac {\left (\frac {{\mathrm e}^{x}}{{\mathrm e}^{2}+1}-\frac {15 x^{2}}{16}\right ) {\mathrm e}^{-{\mathrm e}^{x}-1}}{x}\) | \(27\) |
risch | \(-\frac {\left (15 x^{2} {\mathrm e}^{2}+15 x^{2}-16 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{x}-1}}{16 x \left ({\mathrm e}^{2}+1\right )}\) | \(36\) |
parallelrisch | \(\frac {\left (-15 x^{2} {\mathrm e}^{2}-15 x^{2}+16 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-{\mathrm e}^{x}-1}}{16 \left ({\mathrm e}^{2}+1\right ) x}\) | \(36\) |
Input:
int((-16*x*exp(x)^2+(15*x^3*exp(2)+15*x^3+16*x-16)*exp(x)-15*x^2*exp(2)-15 *x^2)/(16*x^2*exp(2)+16*x^2)/exp(exp(x)+1),x,method=_RETURNVERBOSE)
Output:
(1/(exp(2)+1)*exp(x)-15/16*x^2)/x/exp(exp(x)+1)
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=-\frac {{\left (15 \, x^{2} e^{2} + 15 \, x^{2} - 16 \, e^{x}\right )} e^{\left (-e^{x} - 1\right )}}{16 \, {\left (x e^{2} + x\right )}} \] Input:
integrate((-16*x*exp(x)^2+(15*x^3*exp(2)+15*x^3+16*x-16)*exp(x)-15*x^2*exp (2)-15*x^2)/(16*x^2*exp(2)+16*x^2)/exp(1+exp(x)),x, algorithm="fricas")
Output:
-1/16*(15*x^2*e^2 + 15*x^2 - 16*e^x)*e^(-e^x - 1)/(x*e^2 + x)
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=\frac {\left (- 15 x^{2} e^{2} - 15 x^{2} + 16 e^{x}\right ) e^{- e^{x} - 1}}{16 x + 16 x e^{2}} \] Input:
integrate((-16*x*exp(x)**2+(15*x**3*exp(2)+15*x**3+16*x-16)*exp(x)-15*x**2 *exp(2)-15*x**2)/(16*x**2*exp(2)+16*x**2)/exp(1+exp(x)),x)
Output:
(-15*x**2*exp(2) - 15*x**2 + 16*exp(x))*exp(-exp(x) - 1)/(16*x + 16*x*exp( 2))
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=-\frac {{\left (15 \, x^{2} {\left (e^{2} + 1\right )} - 16 \, e^{x}\right )} e^{\left (-e^{x}\right )}}{16 \, x {\left (e^{3} + e\right )}} \] Input:
integrate((-16*x*exp(x)^2+(15*x^3*exp(2)+15*x^3+16*x-16)*exp(x)-15*x^2*exp (2)-15*x^2)/(16*x^2*exp(2)+16*x^2)/exp(1+exp(x)),x, algorithm="maxima")
Output:
-1/16*(15*x^2*(e^2 + 1) - 16*e^x)*e^(-e^x)/(x*(e^3 + e))
\[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=\int { -\frac {{\left (15 \, x^{2} e^{2} + 15 \, x^{2} + 16 \, x e^{\left (2 \, x\right )} - {\left (15 \, x^{3} e^{2} + 15 \, x^{3} + 16 \, x - 16\right )} e^{x}\right )} e^{\left (-e^{x} - 1\right )}}{16 \, {\left (x^{2} e^{2} + x^{2}\right )}} \,d x } \] Input:
integrate((-16*x*exp(x)^2+(15*x^3*exp(2)+15*x^3+16*x-16)*exp(x)-15*x^2*exp (2)-15*x^2)/(16*x^2*exp(2)+16*x^2)/exp(1+exp(x)),x, algorithm="giac")
Output:
integrate(-1/16*(15*x^2*e^2 + 15*x^2 + 16*x*e^(2*x) - (15*x^3*e^2 + 15*x^3 + 16*x - 16)*e^x)*e^(-e^x - 1)/(x^2*e^2 + x^2), x)
Time = 4.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=\frac {{\mathrm {e}}^{x-{\mathrm {e}}^x-1}-\frac {x^2\,{\mathrm {e}}^{-{\mathrm {e}}^x-1}\,\left (15\,{\mathrm {e}}^2+15\right )}{16}}{x\,\left ({\mathrm {e}}^2+1\right )} \] Input:
int(-(exp(- exp(x) - 1)*(16*x*exp(2*x) - exp(x)*(16*x + 15*x^3*exp(2) + 15 *x^3 - 16) + 15*x^2*exp(2) + 15*x^2))/(16*x^2*exp(2) + 16*x^2),x)
Output:
(exp(x - exp(x) - 1) - (x^2*exp(- exp(x) - 1)*(15*exp(2) + 15))/16)/(x*(ex p(2) + 1))
Time = 0.16 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-1-e^x} \left (-16 e^{2 x} x-15 x^2-15 e^2 x^2+e^x \left (-16+16 x+15 x^3+15 e^2 x^3\right )\right )}{16 x^2+16 e^2 x^2} \, dx=\frac {16 e^{x}-15 e^{2} x^{2}-15 x^{2}}{16 e^{e^{x}} e x \left (e^{2}+1\right )} \] Input:
int((-16*x*exp(x)^2+(15*x^3*exp(2)+15*x^3+16*x-16)*exp(x)-15*x^2*exp(2)-15 *x^2)/(16*x^2*exp(2)+16*x^2)/exp(1+exp(x)),x)
Output:
(16*e**x - 15*e**2*x**2 - 15*x**2)/(16*e**(e**x)*e*x*(e**2 + 1))