Integrand size = 156, antiderivative size = 24 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=2+x-5 \left (x^2-\frac {x}{1+e^{-1+x}+x}\right )^2 \] Output:
2+x-5*(x^2-x/(1+x+exp(-1+x)))^2
Time = 4.42 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=x-5 x^4-\frac {5 e^2 x^2}{\left (e+e^x+e x\right )^2}+\frac {10 e x^3}{e+e^x+e x} \] Input:
Integrate[(1 - 7*x + 33*x^2 + 31*x^3 - 40*x^4 - 60*x^5 - 20*x^6 + E^(-3 + 3*x)*(1 - 20*x^3) + E^(-2 + 2*x)*(3 + 3*x + 30*x^2 - 70*x^3 - 60*x^4) + E^ (-1 + x)*(3 - 4*x + 73*x^2 - 20*x^3 - 130*x^4 - 60*x^5))/(1 + E^(-3 + 3*x) + 3*x + 3*x^2 + x^3 + E^(-2 + 2*x)*(3 + 3*x) + E^(-1 + x)*(3 + 6*x + 3*x^ 2)),x]
Output:
x - 5*x^4 - (5*E^2*x^2)/(E + E^x + E*x)^2 + (10*E*x^3)/(E + E^x + E*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.
\(\displaystyle \int \frac {-20 x^6-60 x^5-40 x^4+31 x^3+e^{3 x-3} \left (1-20 x^3\right )+33 x^2+e^{2 x-2} \left (-60 x^4-70 x^3+30 x^2+3 x+3\right )+e^{x-1} \left (-60 x^5-130 x^4-20 x^3+73 x^2-4 x+3\right )-7 x+1}{x^3+3 x^2+e^{x-1} \left (3 x^2+6 x+3\right )+3 x+e^{3 x-3}+e^{2 x-2} (3 x+3)+1} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^3 \left (-20 x^6-60 x^5-40 x^4+31 x^3+e^{3 x-3} \left (1-20 x^3\right )+33 x^2+e^{2 x-2} \left (-60 x^4-70 x^3+30 x^2+3 x+3\right )+e^{x-1} \left (-60 x^5-130 x^4-20 x^3+73 x^2-4 x+3\right )-7 x+1\right )}{\left (e x+e^x+e\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^3 \int \frac {-20 x^6-60 x^5-40 x^4+31 x^3+33 x^2-7 x+e^{3 x-3} \left (1-20 x^3\right )+e^{2 x-2} \left (-60 x^4-70 x^3+30 x^2+3 x+3\right )+e^{x-1} \left (-60 x^5-130 x^4-20 x^3+73 x^2-4 x+3\right )+1}{\left (e x+e^x+e\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle e^3 \int \left (-\frac {10 x^3}{\left (e x+e^x+e\right )^3}-\frac {10 (x-3) x^2}{e^2 \left (e x+e^x+e\right )}+\frac {10 \left (x^3+x-1\right ) x}{e \left (e x+e^x+e\right )^2}+\frac {1-20 x^3}{e^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^3 \left (\frac {10 \int \frac {x^4}{\left (e x+e^x+e\right )^2}dx}{e}-10 \int \frac {x^3}{\left (e x+e^x+e\right )^3}dx-\frac {10 \int \frac {x^3}{e x+e^x+e}dx}{e^2}+\frac {10 \int \frac {x^2}{\left (e x+e^x+e\right )^2}dx}{e}+\frac {30 \int \frac {x^2}{e x+e^x+e}dx}{e^2}-\frac {10 \int \frac {x}{\left (e x+e^x+e\right )^2}dx}{e}-\frac {5 x^4}{e^3}+\frac {x}{e^3}\right )\) |
Input:
Int[(1 - 7*x + 33*x^2 + 31*x^3 - 40*x^4 - 60*x^5 - 20*x^6 + E^(-3 + 3*x)*( 1 - 20*x^3) + E^(-2 + 2*x)*(3 + 3*x + 30*x^2 - 70*x^3 - 60*x^4) + E^(-1 + x)*(3 - 4*x + 73*x^2 - 20*x^3 - 130*x^4 - 60*x^5))/(1 + E^(-3 + 3*x) + 3*x + 3*x^2 + x^3 + E^(-2 + 2*x)*(3 + 3*x) + E^(-1 + x)*(3 + 6*x + 3*x^2)),x]
Output:
$Aborted
Time = 1.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62
method | result | size |
risch | \(-5 x^{4}+x +\frac {5 x^{2} \left (2 x^{2}+2 x \,{\mathrm e}^{-1+x}+2 x -1\right )}{\left (1+x +{\mathrm e}^{-1+x}\right )^{2}}\) | \(39\) |
norman | \(\frac {x \,{\mathrm e}^{-2+2 x}+3 \,{\mathrm e}^{-2+2 x}+6 \,{\mathrm e}^{-1+x}+7 x +8 x \,{\mathrm e}^{-1+x}+11 x^{3}+5 x^{4}-10 x^{5}-5 x^{6}+2 \,{\mathrm e}^{-1+x} x^{2}+10 \,{\mathrm e}^{-1+x} x^{3}-10 \,{\mathrm e}^{-1+x} x^{4}-10 \,{\mathrm e}^{-1+x} x^{5}-5 \,{\mathrm e}^{-2+2 x} x^{4}+3}{\left (1+x +{\mathrm e}^{-1+x}\right )^{2}}\) | \(112\) |
parallelrisch | \(-\frac {5 x^{6}+10 \,{\mathrm e}^{-1+x} x^{5}+5 \,{\mathrm e}^{-2+2 x} x^{4}+10 x^{5}+10 \,{\mathrm e}^{-1+x} x^{4}-5 x^{4}-10 \,{\mathrm e}^{-1+x} x^{3}-11 x^{3}-2 \,{\mathrm e}^{-1+x} x^{2}-x \,{\mathrm e}^{-2+2 x}+3 x^{2}-2 x \,{\mathrm e}^{-1+x}-x}{x^{2}+2 x \,{\mathrm e}^{-1+x}+{\mathrm e}^{-2+2 x}+2 x +2 \,{\mathrm e}^{-1+x}+1}\) | \(124\) |
Input:
int(((-20*x^3+1)*exp(-1+x)^3+(-60*x^4-70*x^3+30*x^2+3*x+3)*exp(-1+x)^2+(-6 0*x^5-130*x^4-20*x^3+73*x^2-4*x+3)*exp(-1+x)-20*x^6-60*x^5-40*x^4+31*x^3+3 3*x^2-7*x+1)/(exp(-1+x)^3+(3*x+3)*exp(-1+x)^2+(3*x^2+6*x+3)*exp(-1+x)+x^3+ 3*x^2+3*x+1),x,method=_RETURNVERBOSE)
Output:
-5*x^4+x+5*x^2*(2*x^2+2*x*exp(-1+x)+2*x-1)/(1+x+exp(-1+x))^2
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.25 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=-\frac {5 \, x^{6} + 10 \, x^{5} - 5 \, x^{4} - 11 \, x^{3} + 3 \, x^{2} + {\left (5 \, x^{4} - x\right )} e^{\left (2 \, x - 2\right )} + 2 \, {\left (5 \, x^{5} + 5 \, x^{4} - 5 \, x^{3} - x^{2} - x\right )} e^{\left (x - 1\right )} - x}{x^{2} + 2 \, {\left (x + 1\right )} e^{\left (x - 1\right )} + 2 \, x + e^{\left (2 \, x - 2\right )} + 1} \] Input:
integrate(((-20*x^3+1)*exp(-1+x)^3+(-60*x^4-70*x^3+30*x^2+3*x+3)*exp(-1+x) ^2+(-60*x^5-130*x^4-20*x^3+73*x^2-4*x+3)*exp(-1+x)-20*x^6-60*x^5-40*x^4+31 *x^3+33*x^2-7*x+1)/(exp(-1+x)^3+(3*x+3)*exp(-1+x)^2+(3*x^2+6*x+3)*exp(-1+x )+x^3+3*x^2+3*x+1),x, algorithm="fricas")
Output:
-(5*x^6 + 10*x^5 - 5*x^4 - 11*x^3 + 3*x^2 + (5*x^4 - x)*e^(2*x - 2) + 2*(5 *x^5 + 5*x^4 - 5*x^3 - x^2 - x)*e^(x - 1) - x)/(x^2 + 2*(x + 1)*e^(x - 1) + 2*x + e^(2*x - 2) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.33 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=- 5 x^{4} + x + \frac {10 x^{4} + 10 x^{3} e^{x - 1} + 10 x^{3} - 5 x^{2}}{x^{2} + 2 x + \left (2 x + 2\right ) e^{x - 1} + e^{2 x - 2} + 1} \] Input:
integrate(((-20*x**3+1)*exp(-1+x)**3+(-60*x**4-70*x**3+30*x**2+3*x+3)*exp( -1+x)**2+(-60*x**5-130*x**4-20*x**3+73*x**2-4*x+3)*exp(-1+x)-20*x**6-60*x* *5-40*x**4+31*x**3+33*x**2-7*x+1)/(exp(-1+x)**3+(3*x+3)*exp(-1+x)**2+(3*x* *2+6*x+3)*exp(-1+x)+x**3+3*x**2+3*x+1),x)
Output:
-5*x**4 + x + (10*x**4 + 10*x**3*exp(x - 1) + 10*x**3 - 5*x**2)/(x**2 + 2* x + (2*x + 2)*exp(x - 1) + exp(2*x - 2) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 5.25 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=-\frac {5 \, x^{6} e^{2} + 10 \, x^{5} e^{2} - 5 \, x^{4} e^{2} - 11 \, x^{3} e^{2} + 3 \, x^{2} e^{2} - x e^{2} + {\left (5 \, x^{4} - x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (5 \, x^{5} e + 5 \, x^{4} e - 5 \, x^{3} e - x^{2} e - x e\right )} e^{x}}{x^{2} e^{2} + 2 \, x e^{2} + 2 \, {\left (x e + e\right )} e^{x} + e^{2} + e^{\left (2 \, x\right )}} \] Input:
integrate(((-20*x^3+1)*exp(-1+x)^3+(-60*x^4-70*x^3+30*x^2+3*x+3)*exp(-1+x) ^2+(-60*x^5-130*x^4-20*x^3+73*x^2-4*x+3)*exp(-1+x)-20*x^6-60*x^5-40*x^4+31 *x^3+33*x^2-7*x+1)/(exp(-1+x)^3+(3*x+3)*exp(-1+x)^2+(3*x^2+6*x+3)*exp(-1+x )+x^3+3*x^2+3*x+1),x, algorithm="maxima")
Output:
-(5*x^6*e^2 + 10*x^5*e^2 - 5*x^4*e^2 - 11*x^3*e^2 + 3*x^2*e^2 - x*e^2 + (5 *x^4 - x)*e^(2*x) + 2*(5*x^5*e + 5*x^4*e - 5*x^3*e - x^2*e - x*e)*e^x)/(x^ 2*e^2 + 2*x*e^2 + 2*(x*e + e)*e^x + e^2 + e^(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (23) = 46\).
Time = 0.13 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.62 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=-\frac {5 \, x^{6} e^{2} + 10 \, x^{5} e^{2} + 10 \, x^{5} e^{\left (x + 1\right )} - 5 \, x^{4} e^{2} + 5 \, x^{4} e^{\left (2 \, x\right )} + 10 \, x^{4} e^{\left (x + 1\right )} - 11 \, x^{3} e^{2} - 10 \, x^{3} e^{\left (x + 1\right )} + 3 \, x^{2} e^{2} - 2 \, x^{2} e^{\left (x + 1\right )} - x e^{2} - x e^{\left (2 \, x\right )} - 2 \, x e^{\left (x + 1\right )}}{x^{2} e^{2} + 2 \, x e^{2} + 2 \, x e^{\left (x + 1\right )} + e^{2} + e^{\left (2 \, x\right )} + 2 \, e^{\left (x + 1\right )}} \] Input:
integrate(((-20*x^3+1)*exp(-1+x)^3+(-60*x^4-70*x^3+30*x^2+3*x+3)*exp(-1+x) ^2+(-60*x^5-130*x^4-20*x^3+73*x^2-4*x+3)*exp(-1+x)-20*x^6-60*x^5-40*x^4+31 *x^3+33*x^2-7*x+1)/(exp(-1+x)^3+(3*x+3)*exp(-1+x)^2+(3*x^2+6*x+3)*exp(-1+x )+x^3+3*x^2+3*x+1),x, algorithm="giac")
Output:
-(5*x^6*e^2 + 10*x^5*e^2 + 10*x^5*e^(x + 1) - 5*x^4*e^2 + 5*x^4*e^(2*x) + 10*x^4*e^(x + 1) - 11*x^3*e^2 - 10*x^3*e^(x + 1) + 3*x^2*e^2 - 2*x^2*e^(x + 1) - x*e^2 - x*e^(2*x) - 2*x*e^(x + 1))/(x^2*e^2 + 2*x*e^2 + 2*x*e^(x + 1) + e^2 + e^(2*x) + 2*e^(x + 1))
Time = 3.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.54 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=-\frac {x\,\left (3\,x-{\mathrm {e}}^{2\,x-2}+5\,x^3\,{\mathrm {e}}^{2\,x-2}-11\,x^2-5\,x^3+10\,x^4+5\,x^5-1\right )-x\,{\mathrm {e}}^{x-1}\,\left (-10\,x^4-10\,x^3+10\,x^2+2\,x+2\right )}{{\left (x+{\mathrm {e}}^{x-1}+1\right )}^2} \] Input:
int(-(7*x + exp(3*x - 3)*(20*x^3 - 1) + exp(x - 1)*(4*x - 73*x^2 + 20*x^3 + 130*x^4 + 60*x^5 - 3) - exp(2*x - 2)*(3*x + 30*x^2 - 70*x^3 - 60*x^4 + 3 ) - 33*x^2 - 31*x^3 + 40*x^4 + 60*x^5 + 20*x^6 - 1)/(3*x + exp(3*x - 3) + exp(x - 1)*(6*x + 3*x^2 + 3) + exp(2*x - 2)*(3*x + 3) + 3*x^2 + x^3 + 1),x )
Output:
-(x*(3*x - exp(2*x - 2) + 5*x^3*exp(2*x - 2) - 11*x^2 - 5*x^3 + 10*x^4 + 5 *x^5 - 1) - x*exp(x - 1)*(2*x + 10*x^2 - 10*x^3 - 10*x^4 + 2))/(x + exp(x - 1) + 1)^2
Time = 0.17 (sec) , antiderivative size = 156, normalized size of antiderivative = 6.50 \[ \int \frac {1-7 x+33 x^2+31 x^3-40 x^4-60 x^5-20 x^6+e^{-3+3 x} \left (1-20 x^3\right )+e^{-2+2 x} \left (3+3 x+30 x^2-70 x^3-60 x^4\right )+e^{-1+x} \left (3-4 x+73 x^2-20 x^3-130 x^4-60 x^5\right )}{1+e^{-3+3 x}+3 x+3 x^2+x^3+e^{-2+2 x} (3+3 x)+e^{-1+x} \left (3+6 x+3 x^2\right )} \, dx=\frac {-5 e^{2 x} x^{4}+e^{2 x} x -e^{2 x}-10 e^{x} e \,x^{5}-10 e^{x} e \,x^{4}+10 e^{x} e \,x^{3}+2 e^{x} e \,x^{2}-2 e^{x} e -5 e^{2} x^{6}-10 e^{2} x^{5}+5 e^{2} x^{4}+11 e^{2} x^{3}-4 e^{2} x^{2}-e^{2} x -e^{2}}{e^{2 x}+2 e^{x} e x +2 e^{x} e +e^{2} x^{2}+2 e^{2} x +e^{2}} \] Input:
int(((-20*x^3+1)*exp(-1+x)^3+(-60*x^4-70*x^3+30*x^2+3*x+3)*exp(-1+x)^2+(-6 0*x^5-130*x^4-20*x^3+73*x^2-4*x+3)*exp(-1+x)-20*x^6-60*x^5-40*x^4+31*x^3+3 3*x^2-7*x+1)/(exp(-1+x)^3+(3*x+3)*exp(-1+x)^2+(3*x^2+6*x+3)*exp(-1+x)+x^3+ 3*x^2+3*x+1),x)
Output:
( - 5*e**(2*x)*x**4 + e**(2*x)*x - e**(2*x) - 10*e**x*e*x**5 - 10*e**x*e*x **4 + 10*e**x*e*x**3 + 2*e**x*e*x**2 - 2*e**x*e - 5*e**2*x**6 - 10*e**2*x* *5 + 5*e**2*x**4 + 11*e**2*x**3 - 4*e**2*x**2 - e**2*x - e**2)/(e**(2*x) + 2*e**x*e*x + 2*e**x*e + e**2*x**2 + 2*e**2*x + e**2)