Integrand size = 115, antiderivative size = 21 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=\frac {-4+\frac {3}{e^5}+x}{2-11 e^{-x}+x} \] Output:
(3/exp(5)+x-4)/(2+exp(-x)-12/exp(x)+x)
Time = 3.55 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=\frac {11 e^5+3 e^x-6 e^{5+x}}{e^5 \left (-11+e^x (2+x)\right )} \] Input:
Integrate[(E^(2*x)*(-3 + 6*E^5 + (3 + E^5*(-3 + x))/E^x) + E^x*(-36 + E^5* (36 - 12*x)))/(144*E^5 + E^x*(-24*E^(5 - x) + E^5*(-48 - 24*x)) + E^(2*x)* (E^(5 - 2*x) + E^(5 - x)*(4 + 2*x) + E^5*(4 + 4*x + x^2))),x]
Output:
(11*E^5 + 3*E^x - 6*E^(5 + x))/(E^5*(-11 + E^x*(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.
\(\displaystyle \int \frac {e^{2 x} \left (e^{-x} \left (e^5 (x-3)+3\right )+6 e^5-3\right )+e^x \left (e^5 (36-12 x)-36\right )}{e^{2 x} \left (e^5 \left (x^2+4 x+4\right )+e^{5-x} (2 x+4)+e^{5-2 x}\right )+e^x \left (e^5 (-24 x-48)-24 e^{5-x}\right )+144 e^5} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{x-5} \left (-11 e^5 (x-3)-3 \left (1-2 e^5\right ) e^x-33\right )}{\left (11-e^x (x+2)\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 \left (2 e^5-1\right ) e^{x-5}}{(x+2) \left (e^x x+2 e^x-11\right )}-\frac {11 e^{x-5} (x+3) \left (e^5 x-4 e^5+3\right )}{(x+2) \left (e^x x+2 e^x-11\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -33 \left (1-e^5\right ) \int \frac {e^{x-5}}{\left (e^x x+2 e^x-11\right )^2}dx-11 \int \frac {e^x x}{\left (e^x x+2 e^x-11\right )^2}dx-33 \left (1-2 e^5\right ) \int \frac {e^{x-5}}{(x+2) \left (e^x x+2 e^x-11\right )^2}dx-3 \left (1-2 e^5\right ) \int \frac {e^{x-5}}{(x+2) \left (e^x x+2 e^x-11\right )}dx\) |
Input:
Int[(E^(2*x)*(-3 + 6*E^5 + (3 + E^5*(-3 + x))/E^x) + E^x*(-36 + E^5*(36 - 12*x)))/(144*E^5 + E^x*(-24*E^(5 - x) + E^5*(-48 - 24*x)) + E^(2*x)*(E^(5 - 2*x) + E^(5 - x)*(4 + 2*x) + E^5*(4 + 4*x + x^2))),x]
Output:
$Aborted
Time = 0.68 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52
method | result | size |
risch | \(-\frac {\left (6 \,{\mathrm e}^{5+x}-11 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-5}}{{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-11}\) | \(32\) |
norman | \(\frac {\left (11 \,{\mathrm e}^{3 x}-3 \left (2 \,{\mathrm e}^{5}-1\right ) {\mathrm e}^{-5} {\mathrm e}^{4 x}\right ) {\mathrm e}^{-3 x}}{{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-11}\) | \(41\) |
parallelrisch | \(\frac {\left ({\mathrm e}^{x} {\mathrm e}^{5} x -4 \,{\mathrm e}^{5} {\mathrm e}^{x}+3 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-5}}{{\mathrm e}^{-x} {\mathrm e}^{x}+{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-12}\) | \(42\) |
Input:
int(((((-3+x)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp(5)-36 )*exp(x))/((exp(5)*exp(-x)^2+(4+2*x)*exp(5)*exp(-x)+(x^2+4*x+4)*exp(5))*ex p(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x,method= _RETURNVERBOSE)
Output:
-(6*exp(5+x)-11*exp(5)-3*exp(x))*exp(-5)/(exp(x)*x+2*exp(x)-11)
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=-\frac {3 \, {\left (2 \, e^{5} - 1\right )} e^{x} - 11 \, e^{5}}{{\left (x + 2\right )} e^{\left (x + 5\right )} - 11 \, e^{5}} \] Input:
integrate(((((-3+x)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp (5)-36)*exp(x))/((exp(5)*exp(-x)^2+(4+2*x)*exp(5)*exp(-x)+(x^2+4*x+4)*exp( 5))*exp(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x, algorithm="fricas")
Output:
-(3*(2*e^5 - 1)*e^x - 11*e^5)/((x + 2)*e^(x + 5) - 11*e^5)
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=\frac {- x e^{5} - 3 + 4 e^{5}}{- x e^{5} - 2 e^{5} + 11 e^{5} e^{- x}} \] Input:
integrate(((((-3+x)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)**2+((-12*x+36)*ex p(5)-36)*exp(x))/((exp(5)*exp(-x)**2+(4+2*x)*exp(5)*exp(-x)+(x**2+4*x+4)*e xp(5))*exp(x)**2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)) ,x)
Output:
(-x*exp(5) - 3 + 4*exp(5))/(-x*exp(5) - 2*exp(5) + 11*exp(5)*exp(-x))
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=-\frac {3 \, {\left (2 \, e^{5} - 1\right )} e^{x} - 11 \, e^{5}}{{\left (x e^{5} + 2 \, e^{5}\right )} e^{x} - 11 \, e^{5}} \] Input:
integrate(((((-3+x)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp (5)-36)*exp(x))/((exp(5)*exp(-x)^2+(4+2*x)*exp(5)*exp(-x)+(x^2+4*x+4)*exp( 5))*exp(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x, algorithm="maxima")
Output:
-(3*(2*e^5 - 1)*e^x - 11*e^5)/((x*e^5 + 2*e^5)*e^x - 11*e^5)
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=\frac {{\left (x - 5\right )} e^{5} + e^{5} + 3}{{\left (x - 5\right )} e^{5} + 7 \, e^{5} - 11 \, e^{\left (-x + 5\right )}} \] Input:
integrate(((((-3+x)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp (5)-36)*exp(x))/((exp(5)*exp(-x)^2+(4+2*x)*exp(5)*exp(-x)+(x^2+4*x+4)*exp( 5))*exp(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x, algorithm="giac")
Output:
((x - 5)*e^5 + e^5 + 3)/((x - 5)*e^5 + 7*e^5 - 11*e^(-x + 5))
Time = 2.65 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=\frac {{\mathrm {e}}^{x-5}\,\left (x\,{\mathrm {e}}^5-4\,{\mathrm {e}}^5+3\right )}{2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x-11} \] Input:
int((exp(2*x)*(6*exp(5) + exp(-x)*(exp(5)*(x - 3) + 3) - 3) - exp(x)*(exp( 5)*(12*x - 36) + 36))/(144*exp(5) + exp(2*x)*(exp(-2*x)*exp(5) + exp(5)*(4 *x + x^2 + 4) + exp(-x)*exp(5)*(2*x + 4)) - exp(x)*(24*exp(-x)*exp(5) + ex p(5)*(24*x + 48))),x)
Output:
(exp(x - 5)*(x*exp(5) - 4*exp(5) + 3))/(2*exp(x) + x*exp(x) - 11)
Time = 0.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx=\frac {e^{x} \left (e^{5} x -4 e^{5}+3\right )}{e^{5} \left (e^{x} x +2 e^{x}-11\right )} \] Input:
int(((((-3+x)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp(5)-36 )*exp(x))/((exp(5)*exp(-x)^2+(4+2*x)*exp(5)*exp(-x)+(x^2+4*x+4)*exp(5))*ex p(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x)
Output:
(e**x*(e**5*x - 4*e**5 + 3))/(e**5*(e**x*x + 2*e**x - 11))