Integrand size = 111, antiderivative size = 24 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=1+x+\frac {x}{e^4}-e^5 x-\frac {4}{3+e^x+x} \] Output:
x/exp(4)+1-x*exp(5)-4/(exp(x)+3+x)+x
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(24)=48\).
Time = 0.77 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=\frac {e^x x+e^{4+x} x-e^{9+x} x+x (3+x)-e^9 x (3+x)+e^4 \left (-4+3 x+x^2\right )}{e^4 \left (3+e^x+x\right )} \] Input:
Integrate[(9 + E^(2*x)*(1 + E^4 - E^9) + 6*x + x^2 + E^9*(-9 - 6*x - x^2) + E^4*(13 + 6*x + x^2) + E^x*(6 + E^9*(-6 - 2*x) + 2*x + E^4*(10 + 2*x)))/ (E^(4 + 2*x) + E^(4 + x)*(6 + 2*x) + E^4*(9 + 6*x + x^2)),x]
Output:
(E^x*x + E^(4 + x)*x - E^(9 + x)*x + x*(3 + x) - E^9*x*(3 + x) + E^4*(-4 + 3*x + x^2))/(E^4*(3 + E^x + 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.
\(\displaystyle \int \frac {x^2+e^9 \left (-x^2-6 x-9\right )+e^4 \left (x^2+6 x+13\right )+6 x+e^x \left (e^9 (-2 x-6)+2 x+e^4 (2 x+10)+6\right )+\left (1+e^4-e^9\right ) e^{2 x}+9}{e^4 \left (x^2+6 x+9\right )+e^{x+4} (2 x+6)+e^{2 x+4}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x^2+e^9 \left (-x^2-6 x-9\right )+e^4 \left (x^2+6 x+13\right )+6 x+e^x \left (e^9 (-2 x-6)+2 x+e^4 (2 x+10)+6\right )+\left (1+e^4-e^9\right ) e^{2 x}+9}{e^4 \left (x+e^x+3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {x^2+6 x-e^9 \left (x^2+6 x+9\right )+e^4 \left (x^2+6 x+13\right )+2 e^x \left (x-e^9 (x+3)+e^4 (x+5)+3\right )+e^{2 x} \left (1+e^4-e^9\right )+9}{\left (x+e^x+3\right )^2}dx}{e^4}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\int \left (-\frac {4 e^4 (x+2)}{\left (x+e^x+3\right )^2}+\frac {4 e^4}{x+e^x+3}-e^9+e^4+1\right )dx}{e^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-8 e^4 \int \frac {1}{\left (x+e^x+3\right )^2}dx-4 e^4 \int \frac {x}{\left (x+e^x+3\right )^2}dx+4 e^4 \int \frac {1}{x+e^x+3}dx+\left (1+e^4-e^9\right ) x}{e^4}\) |
Input:
Int[(9 + E^(2*x)*(1 + E^4 - E^9) + 6*x + x^2 + E^9*(-9 - 6*x - x^2) + E^4* (13 + 6*x + x^2) + E^x*(6 + E^9*(-6 - 2*x) + 2*x + E^4*(10 + 2*x)))/(E^(4 + 2*x) + E^(4 + x)*(6 + 2*x) + E^4*(9 + 6*x + x^2)),x]
Output:
$Aborted
Time = 0.49 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17
method | result | size |
risch | \({\mathrm e}^{-4} {\mathrm e}^{4} x -{\mathrm e}^{-4} {\mathrm e}^{9} x +{\mathrm e}^{-4} x -\frac {4}{{\mathrm e}^{x}+3+x}\) | \(28\) |
parallelrisch | \(-\frac {\left ({\mathrm e}^{5} {\mathrm e}^{4} x^{2}+{\mathrm e}^{5} {\mathrm e}^{4} {\mathrm e}^{x} x +3 x \,{\mathrm e}^{4} {\mathrm e}^{5}-x^{2} {\mathrm e}^{4}-x \,{\mathrm e}^{4} {\mathrm e}^{x}-3 x \,{\mathrm e}^{4}-x^{2}-{\mathrm e}^{x} x +4 \,{\mathrm e}^{4}-3 x \right ) {\mathrm e}^{-4}}{{\mathrm e}^{x}+3+x}\) | \(74\) |
norman | \(\frac {3 \,{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) {\mathrm e}^{x}-{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) x^{2}-{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) x \,{\mathrm e}^{x}+{\mathrm e}^{-4} \left (9 \,{\mathrm e}^{4} {\mathrm e}^{5}-13 \,{\mathrm e}^{4}-9\right )}{{\mathrm e}^{x}+3+x}\) | \(86\) |
Input:
int(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x+10)*e xp(4)+2*x+6)*exp(x)+(-x^2-6*x-9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x^2+6*x +9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x,method=_R ETURNVERBOSE)
Output:
exp(-4)*exp(4)*x-exp(-4)*exp(9)*x+exp(-4)*x-4/(exp(x)+3+x)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.79 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=-\frac {{\left (x^{2} + 3 \, x\right )} e^{13} - {\left (x^{2} + 3 \, x - 4\right )} e^{8} - {\left (x^{2} + 3 \, x\right )} e^{4} + {\left (x e^{9} - x e^{4} - x\right )} e^{\left (x + 4\right )}}{{\left (x + 3\right )} e^{8} + e^{\left (x + 8\right )}} \] Input:
integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x +10)*exp(4)+2*x+6)*exp(x)+(-x^2-6*x-9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x ^2+6*x+9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x, al gorithm="fricas")
Output:
-((x^2 + 3*x)*e^13 - (x^2 + 3*x - 4)*e^8 - (x^2 + 3*x)*e^4 + (x*e^9 - x*e^ 4 - x)*e^(x + 4))/((x + 3)*e^8 + e^(x + 8))
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=\frac {x \left (- e^{9} + 1 + e^{4}\right )}{e^{4}} - \frac {4}{x + e^{x} + 3} \] Input:
integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)**2+((-2*x-6)*exp(4)*exp(5)+(2* x+10)*exp(4)+2*x+6)*exp(x)+(-x**2-6*x-9)*exp(4)*exp(5)+(x**2+6*x+13)*exp(4 )+x**2+6*x+9)/(exp(4)*exp(x)**2+(2*x+6)*exp(4)*exp(x)+(x**2+6*x+9)*exp(4)) ,x)
Output:
x*(-exp(9) + 1 + exp(4))*exp(-4) - 4/(x + exp(x) + 3)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (21) = 42\).
Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=-\frac {x^{2} {\left (e^{9} - e^{4} - 1\right )} + x {\left (e^{9} - e^{4} - 1\right )} e^{x} + 3 \, x {\left (e^{9} - e^{4} - 1\right )} + 4 \, e^{4}}{x e^{4} + 3 \, e^{4} + e^{\left (x + 4\right )}} \] Input:
integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x +10)*exp(4)+2*x+6)*exp(x)+(-x^2-6*x-9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x ^2+6*x+9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x, al gorithm="maxima")
Output:
-(x^2*(e^9 - e^4 - 1) + x*(e^9 - e^4 - 1)*e^x + 3*x*(e^9 - e^4 - 1) + 4*e^ 4)/(x*e^4 + 3*e^4 + e^(x + 4))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (21) = 42\).
Time = 0.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=-\frac {x^{2} e^{9} - x^{2} e^{4} - x^{2} + 3 \, x e^{9} - 3 \, x e^{4} + x e^{\left (x + 9\right )} - x e^{\left (x + 4\right )} - x e^{x} - 3 \, x + 4 \, e^{4}}{x e^{4} + 3 \, e^{4} + e^{\left (x + 4\right )}} \] Input:
integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x +10)*exp(4)+2*x+6)*exp(x)+(-x^2-6*x-9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x ^2+6*x+9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x, al gorithm="giac")
Output:
-(x^2*e^9 - x^2*e^4 - x^2 + 3*x*e^9 - 3*x*e^4 + x*e^(x + 9) - x*e^(x + 4) - x*e^x - 3*x + 4*e^4)/(x*e^4 + 3*e^4 + e^(x + 4))
Time = 2.56 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=x+x\,{\mathrm {e}}^{-4}-x\,{\mathrm {e}}^5-\frac {4\,{\mathrm {e}}^4}{{\mathrm {e}}^{x+4}+3\,{\mathrm {e}}^4+x\,{\mathrm {e}}^4} \] Input:
int((6*x + exp(2*x)*(exp(4) - exp(9) + 1) + exp(x)*(2*x + exp(4)*(2*x + 10 ) - exp(9)*(2*x + 6) + 6) + exp(4)*(6*x + x^2 + 13) - exp(9)*(6*x + x^2 + 9) + x^2 + 9)/(exp(2*x)*exp(4) + exp(4)*(6*x + x^2 + 9) + exp(4)*exp(x)*(2 *x + 6)),x)
Output:
x + x*exp(-4) - x*exp(5) - (4*exp(4))/(exp(x + 4) + 3*exp(4) + x*exp(4))
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.54 \[ \int \frac {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx=\frac {-e^{x} e^{9} x +3 e^{x} e^{9}+e^{x} e^{4} x -3 e^{x} e^{4}+e^{x} x -3 e^{x}-e^{9} x^{2}+9 e^{9}+e^{4} x^{2}-13 e^{4}+x^{2}-9}{e^{4} \left (e^{x}+x +3\right )} \] Input:
int(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x+10)*e xp(4)+2*x+6)*exp(x)+(-x^2-6*x-9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x^2+6*x +9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x)
Output:
( - e**x*e**9*x + 3*e**x*e**9 + e**x*e**4*x - 3*e**x*e**4 + e**x*x - 3*e** x - e**9*x**2 + 9*e**9 + e**4*x**2 - 13*e**4 + x**2 - 9)/(e**4*(e**x + x + 3))