Integrand size = 138, antiderivative size = 23 \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=\left (3+\frac {1-\frac {x}{e^2}}{25+e^x-x}\right )^2 \] Output:
(3+(1-x/exp(2))/(exp(x)+25-x))^2
Time = 3.57 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.74 \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=-\frac {\left (e^2-x\right ) \left (-6 e^{2+x}+x+e^2 (-151+6 x)\right )}{e^4 \left (25+e^x-x\right )^2} \] Input:
Integrate[(E^4*(152 - 6*x) + 50*x + E^2*(-3800 + 148*x) + E^(2*x)*(-6*E^4 + E^2*(-6 + 6*x)) + E^x*(2*x - 2*x^2 + E^4*(-146 + 6*x) + E^2*(-302 + 160* x - 6*x^2)))/(E^(4 + 3*x) + E^(4 + 2*x)*(75 - 3*x) + E^(4 + x)*(1875 - 150 *x + 3*x^2) + E^4*(15625 - 1875*x + 75*x^2 - x^3)),x]
Output:
-(((E^2 - x)*(-6*E^(2 + x) + x + E^2*(-151 + 6*x)))/(E^4*(25 + E^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.
| \(\displaystyle \int \frac {e^x \left (-2 x^2+e^2 \left (-6 x^2+160 x-302\right )+2 x+e^4 (6 x-146)\right )+e^4 (152-6 x)+50 x+e^2 (148 x-3800)+e^{2 x} \left (e^2 (6 x-6)-6 e^4\right )}{e^{x+4} \left (3 x^2-150 x+1875\right )+e^4 \left (-x^3+75 x^2-1875 x+15625\right )+e^{2 x+4} (75-3 x)+e^{3 x+4}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (-2 x^2+e^2 \left (-6 x^2+160 x-302\right )+2 x+e^4 (6 x-146)\right )+e^4 (152-6 x)+50 x+e^2 (148 x-3800)+e^{2 x} \left (e^2 (6 x-6)-6 e^4\right )}{e^4 \left (-x+e^x+25\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {2 \left (3 e^{2 x} \left (e^2 (1-x)+e^4\right )+2 e^2 (950-37 x)-e^4 (76-3 x)-25 x+e^x \left (x^2-x+e^4 (73-3 x)+e^2 \left (3 x^2-80 x+151\right )\right )\right )}{\left (-x+e^x+25\right )^3}dx}{e^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {3 e^{2 x} \left (e^2 (1-x)+e^4\right )+2 e^2 (950-37 x)-e^4 (76-3 x)-25 x+e^x \left (x^2-x+e^4 (73-3 x)+e^2 \left (3 x^2-80 x+151\right )\right )}{\left (-x+e^x+25\right )^3}dx}{e^4}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2 \int \left (\frac {(x-26) \left (x-e^2\right )^2}{\left (-x+e^x+25\right )^3}+\frac {\left (1-3 e^2\right ) x^2-\left (1-76 e^2-3 e^4\right ) x+e^2 \left (1-77 e^2\right )}{\left (-x+e^x+25\right )^2}+\frac {3 e^2 \left (-x+e^2+1\right )}{-x+e^x+25}\right )dx}{e^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (\int \frac {x^3}{\left (-x+e^x+25\right )^3}dx-2 \left (13+e^2\right ) \int \frac {x^2}{\left (-x+e^x+25\right )^3}dx+\left (1-3 e^2\right ) \int \frac {x^2}{\left (-x+e^x+25\right )^2}dx-26 e^4 \int \frac {1}{\left (-x+e^x+25\right )^3}dx+e^2 \left (1-77 e^2\right ) \int \frac {1}{\left (-x+e^x+25\right )^2}dx+3 e^2 \left (1+e^2\right ) \int \frac {1}{-x+e^x+25}dx+e^2 \left (52+e^2\right ) \int \frac {x}{\left (-x+e^x+25\right )^3}dx-\left (1-76 e^2-3 e^4\right ) \int \frac {x}{\left (-x+e^x+25\right )^2}dx-3 e^2 \int \frac {x}{-x+e^x+25}dx\right )}{e^4}\) |
Input:
Int[(E^4*(152 - 6*x) + 50*x + E^2*(-3800 + 148*x) + E^(2*x)*(-6*E^4 + E^2* (-6 + 6*x)) + E^x*(2*x - 2*x^2 + E^4*(-146 + 6*x) + E^2*(-302 + 160*x - 6* x^2)))/(E^(4 + 3*x) + E^(4 + 2*x)*(75 - 3*x) + E^(4 + x)*(1875 - 150*x + 3 *x^2) + E^4*(15625 - 1875*x + 75*x^2 - x^3)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(23)=46\).
Time = 0.76 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35
| method | result | size |
| risch | \(-\frac {\left (6 x \,{\mathrm e}^{4}-6 \,{\mathrm e}^{4+x}-6 x^{2} {\mathrm e}^{2}+6 x \,{\mathrm e}^{2+x}-151 \,{\mathrm e}^{4}+152 \,{\mathrm e}^{2} x -x^{2}\right ) {\mathrm e}^{-4}}{\left (x -{\mathrm e}^{x}-25\right )^{2}}\) | \(54\) |
| parallelrisch | \(-\frac {\left (6 x \,{\mathrm e}^{4}-6 \,{\mathrm e}^{4} {\mathrm e}^{x}-6 x^{2} {\mathrm e}^{2}+6 x \,{\mathrm e}^{2} {\mathrm e}^{x}-151 \,{\mathrm e}^{4}+152 \,{\mathrm e}^{2} x -x^{2}\right ) {\mathrm e}^{-4}}{x^{2}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}-50 x +50 \,{\mathrm e}^{x}+625}\) | \(76\) |
| norman | \(\frac {\left (-2 \left (-25+3 \,{\mathrm e}^{4}-74 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-2} x -\left (6 \,{\mathrm e}^{2}+1\right ) {\mathrm e}^{-2} {\mathrm e}^{2 x}+2 \left (3 \,{\mathrm e}^{4}-150 \,{\mathrm e}^{2}-25\right ) {\mathrm e}^{-2} {\mathrm e}^{x}+2 \left (3 \,{\mathrm e}^{2}+1\right ) {\mathrm e}^{-2} x \,{\mathrm e}^{x}+\left (151 \,{\mathrm e}^{4}-3750 \,{\mathrm e}^{2}-625\right ) {\mathrm e}^{-2}\right ) {\mathrm e}^{-2}}{\left (x -{\mathrm e}^{x}-25\right )^{2}}\) | \(103\) |
Input:
int(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x^2+160 *x-302)*exp(2)-2*x^2+2*x)*exp(x)+(-6*x+152)*exp(2)^2+(148*x-3800)*exp(2)+5 0*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+1875)*exp (2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x,method=_RETURNVERBOSE)
Output:
-(6*x*exp(4)-6*exp(4+x)-6*x^2*exp(2)+6*x*exp(2+x)-151*exp(4)+152*exp(2)*x- x^2)/(x-exp(x)-25)^2*exp(-4)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22 \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=\frac {x^{2} e^{4} - {\left (6 \, x - 151\right )} e^{8} + 2 \, {\left (3 \, x^{2} - 76 \, x\right )} e^{6} - 6 \, {\left (x e^{2} - e^{4}\right )} e^{\left (x + 4\right )}}{{\left (x^{2} - 50 \, x + 625\right )} e^{8} - 2 \, {\left (x - 25\right )} e^{\left (x + 8\right )} + e^{\left (2 \, x + 8\right )}} \] Input:
integrate(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x ^2+160*x-302)*exp(2)-2*x^2+2*x)*exp(x)+(-6*x+152)*exp(2)^2+(148*x-3800)*ex p(2)+50*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+187 5)*exp(2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x, algorithm="fric as")
Output:
(x^2*e^4 - (6*x - 151)*e^8 + 2*(3*x^2 - 76*x)*e^6 - 6*(x*e^2 - e^4)*e^(x + 4))/((x^2 - 50*x + 625)*e^8 - 2*(x - 25)*e^(x + 8) + e^(2*x + 8))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.78 \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=\frac {x^{2} + 6 x^{2} e^{2} - 152 x e^{2} - 6 x e^{4} + \left (- 6 x e^{2} + 6 e^{4}\right ) e^{x} + 151 e^{4}}{x^{2} e^{4} - 50 x e^{4} + \left (- 2 x e^{4} + 50 e^{4}\right ) e^{x} + e^{4} e^{2 x} + 625 e^{4}} \] Input:
integrate(((-6*exp(2)**2+(6*x-6)*exp(2))*exp(x)**2+((6*x-146)*exp(2)**2+(- 6*x**2+160*x-302)*exp(2)-2*x**2+2*x)*exp(x)+(-6*x+152)*exp(2)**2+(148*x-38 00)*exp(2)+50*x)/(exp(2)**2*exp(x)**3+(-3*x+75)*exp(2)**2*exp(x)**2+(3*x** 2-150*x+1875)*exp(2)**2*exp(x)+(-x**3+75*x**2-1875*x+15625)*exp(2)**2),x)
Output:
(x**2 + 6*x**2*exp(2) - 152*x*exp(2) - 6*x*exp(4) + (-6*x*exp(2) + 6*exp(4 ))*exp(x) + 151*exp(4))/(x**2*exp(4) - 50*x*exp(4) + (-2*x*exp(4) + 50*exp (4))*exp(x) + exp(4)*exp(2*x) + 625*exp(4))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.39 \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=\frac {x^{2} {\left (6 \, e^{2} + 1\right )} - 2 \, x {\left (3 \, e^{4} + 76 \, e^{2}\right )} - 6 \, {\left (x e^{2} - e^{4}\right )} e^{x} + 151 \, e^{4}}{x^{2} e^{4} - 50 \, x e^{4} - 2 \, {\left (x e^{4} - 25 \, e^{4}\right )} e^{x} + 625 \, e^{4} + e^{\left (2 \, x + 4\right )}} \] Input:
integrate(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x ^2+160*x-302)*exp(2)-2*x^2+2*x)*exp(x)+(-6*x+152)*exp(2)^2+(148*x-3800)*ex p(2)+50*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+187 5)*exp(2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x, algorithm="maxi ma")
Output:
(x^2*(6*e^2 + 1) - 2*x*(3*e^4 + 76*e^2) - 6*(x*e^2 - e^4)*e^x + 151*e^4)/( x^2*e^4 - 50*x*e^4 - 2*(x*e^4 - 25*e^4)*e^x + 625*e^4 + e^(2*x + 4))
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (20) = 40\).
Time = 0.14 (sec) , antiderivative size = 116, normalized size of antiderivative = 5.04 \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=\frac {6 \, {\left (x + 4\right )}^{2} e^{6} + {\left (x + 4\right )}^{2} e^{4} - 6 \, {\left (x + 4\right )} e^{8} - 200 \, {\left (x + 4\right )} e^{6} - 8 \, {\left (x + 4\right )} e^{4} - 6 \, {\left (x + 4\right )} e^{\left (x + 6\right )} + 175 \, e^{8} + 704 \, e^{6} + 16 \, e^{4} + 6 \, e^{\left (x + 8\right )} + 24 \, e^{\left (x + 6\right )}}{{\left (x + 4\right )}^{2} e^{8} - 58 \, {\left (x + 4\right )} e^{8} - 2 \, {\left (x + 4\right )} e^{\left (x + 8\right )} + 841 \, e^{8} + e^{\left (2 \, x + 8\right )} + 58 \, e^{\left (x + 8\right )}} \] Input:
integrate(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x ^2+160*x-302)*exp(2)-2*x^2+2*x)*exp(x)+(-6*x+152)*exp(2)^2+(148*x-3800)*ex p(2)+50*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+187 5)*exp(2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x, algorithm="giac ")
Output:
(6*(x + 4)^2*e^6 + (x + 4)^2*e^4 - 6*(x + 4)*e^8 - 200*(x + 4)*e^6 - 8*(x + 4)*e^4 - 6*(x + 4)*e^(x + 6) + 175*e^8 + 704*e^6 + 16*e^4 + 6*e^(x + 8) + 24*e^(x + 6))/((x + 4)^2*e^8 - 58*(x + 4)*e^8 - 2*(x + 4)*e^(x + 8) + 84 1*e^8 + e^(2*x + 8) + 58*e^(x + 8))
Timed out. \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=\int \frac {50\,x-{\mathrm {e}}^{2\,x}\,\left (6\,{\mathrm {e}}^4-{\mathrm {e}}^2\,\left (6\,x-6\right )\right )+{\mathrm {e}}^x\,\left (2\,x-{\mathrm {e}}^2\,\left (6\,x^2-160\,x+302\right )-2\,x^2+{\mathrm {e}}^4\,\left (6\,x-146\right )\right )-{\mathrm {e}}^4\,\left (6\,x-152\right )+{\mathrm {e}}^2\,\left (148\,x-3800\right )}{{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^4-{\mathrm {e}}^4\,\left (x^3-75\,x^2+1875\,x-15625\right )+{\mathrm {e}}^4\,{\mathrm {e}}^x\,\left (3\,x^2-150\,x+1875\right )-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^4\,\left (3\,x-75\right )} \,d x \] Input:
int((50*x - exp(2*x)*(6*exp(4) - exp(2)*(6*x - 6)) + exp(x)*(2*x - exp(2)* (6*x^2 - 160*x + 302) - 2*x^2 + exp(4)*(6*x - 146)) - exp(4)*(6*x - 152) + exp(2)*(148*x - 3800))/(exp(3*x)*exp(4) - exp(4)*(1875*x - 75*x^2 + x^3 - 15625) + exp(4)*exp(x)*(3*x^2 - 150*x + 1875) - exp(2*x)*exp(4)*(3*x - 75 )),x)
Output:
int((50*x - exp(2*x)*(6*exp(4) - exp(2)*(6*x - 6)) + exp(x)*(2*x - exp(2)* (6*x^2 - 160*x + 302) - 2*x^2 + exp(4)*(6*x - 146)) - exp(4)*(6*x - 152) + exp(2)*(148*x - 3800))/(exp(3*x)*exp(4) - exp(4)*(1875*x - 75*x^2 + x^3 - 15625) + exp(4)*exp(x)*(3*x^2 - 150*x + 1875) - exp(2*x)*exp(4)*(3*x - 75 )), x)
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91 \[ \int \frac {e^4 (152-6 x)+50 x+e^2 (-3800+148 x)+e^{2 x} \left (-6 e^4+e^2 (-6+6 x)\right )+e^x \left (2 x-2 x^2+e^4 (-146+6 x)+e^2 \left (-302+160 x-6 x^2\right )\right )}{e^{4+3 x}+e^{4+2 x} (75-3 x)+e^{4+x} \left (1875-150 x+3 x^2\right )+e^4 \left (15625-1875 x+75 x^2-x^3\right )} \, dx=\frac {-3 e^{2 x} e^{2}+6 e^{x} e^{4}-150 e^{x} e^{2}-6 e^{4} x +151 e^{4}+3 e^{2} x^{2}-2 e^{2} x -1875 e^{2}+x^{2}}{e^{4} \left (e^{2 x}-2 e^{x} x +50 e^{x}+x^{2}-50 x +625\right )} \] Input:
int(((-6*exp(2)^2+(6*x-6)*exp(2))*exp(x)^2+((6*x-146)*exp(2)^2+(-6*x^2+160 *x-302)*exp(2)-2*x^2+2*x)*exp(x)+(-6*x+152)*exp(2)^2+(148*x-3800)*exp(2)+5 0*x)/(exp(2)^2*exp(x)^3+(-3*x+75)*exp(2)^2*exp(x)^2+(3*x^2-150*x+1875)*exp (2)^2*exp(x)+(-x^3+75*x^2-1875*x+15625)*exp(2)^2),x)
Output:
( - 3*e**(2*x)*e**2 + 6*e**x*e**4 - 150*e**x*e**2 - 6*e**4*x + 151*e**4 + 3*e**2*x**2 - 2*e**2*x - 1875*e**2 + x**2)/(e**4*(e**(2*x) - 2*e**x*x + 50 *e**x + x**2 - 50*x + 625))