Integrand size = 213, antiderivative size = 29 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=1+\frac {5}{-3+e^2+e^{-4+x}}-\frac {-4+x}{x (5+x)} \] Output:
1-(-4+x)/(5+x)/x+5/(exp(-4+x)+exp(2)-3)
Time = 2.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=\frac {5 e^4}{-3 e^4+e^6+e^x}+\frac {4}{5 x}-\frac {9}{5 (5+x)} \] Input:
Integrate[(-180 - 72*x + 9*x^2 + E^2*(120 + 48*x - 6*x^2) + E^4*(-20 - 8*x + x^2) + E^(-8 + 2*x)*(-20 - 8*x + x^2) + E^(-4 + x)*(120 + 48*x - 131*x^ 2 - 50*x^3 - 5*x^4 + E^2*(-40 - 16*x + 2*x^2)))/(225*x^2 + 90*x^3 + 9*x^4 + E^2*(-150*x^2 - 60*x^3 - 6*x^4) + E^4*(25*x^2 + 10*x^3 + x^4) + E^(-8 + 2*x)*(25*x^2 + 10*x^3 + x^4) + E^(-4 + x)*(-150*x^2 - 60*x^3 - 6*x^4 + E^2 *(50*x^2 + 20*x^3 + 2*x^4))),x]
Output:
(5*E^4)/(-3*E^4 + E^6 + E^x) + 4/(5*x) - 9/(5*(5 + x))
Time = 1.76 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7239, 7293, 2009}
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.
\(\displaystyle \int \frac {9 x^2+e^2 \left (-6 x^2+48 x+120\right )+e^{2 x-8} \left (x^2-8 x-20\right )+e^4 \left (x^2-8 x-20\right )+e^{x-4} \left (-5 x^4-50 x^3-131 x^2+e^2 \left (2 x^2-16 x-40\right )+48 x+120\right )-72 x-180}{9 x^4+90 x^3+225 x^2+e^2 \left (-6 x^4-60 x^3-150 x^2\right )+e^{2 x-8} \left (x^4+10 x^3+25 x^2\right )+e^4 \left (x^4+10 x^3+25 x^2\right )+e^{x-4} \left (-6 x^4-60 x^3-150 x^2+e^2 \left (2 x^4+20 x^3+50 x^2\right )\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{2 x} \left (x^2-8 x-20\right )+2 e^{x+6} \left (x^2-8 x-20\right )+9 e^8 \left (1+\frac {1}{9} e^2 \left (e^2-6\right )\right ) \left (x^2-8 x-20\right )+e^{x+4} \left (-5 x^4-50 x^3-131 x^2+48 x+120\right )}{\left (e^x-3 e^4 \left (1-\frac {e^2}{3}\right )\right )^2 x^2 (x+5)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^2-8 x-20}{x^2 (x+5)^2}+\frac {5 e^4}{3 e^4 \left (1-\frac {e^2}{3}\right )-e^x}+\frac {5 e^8 \left (e^2-3\right )}{\left (e^x-3 e^4 \left (1-\frac {e^2}{3}\right )\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4}{5 x}-\frac {9}{5 (x+5)}+\frac {5 e^4}{e^x-e^4 \left (3-e^2\right )}\) |
Input:
Int[(-180 - 72*x + 9*x^2 + E^2*(120 + 48*x - 6*x^2) + E^4*(-20 - 8*x + x^2 ) + E^(-8 + 2*x)*(-20 - 8*x + x^2) + E^(-4 + x)*(120 + 48*x - 131*x^2 - 50 *x^3 - 5*x^4 + E^2*(-40 - 16*x + 2*x^2)))/(225*x^2 + 90*x^3 + 9*x^4 + E^2* (-150*x^2 - 60*x^3 - 6*x^4) + E^4*(25*x^2 + 10*x^3 + x^4) + E^(-8 + 2*x)*( 25*x^2 + 10*x^3 + x^4) + E^(-4 + x)*(-150*x^2 - 60*x^3 - 6*x^4 + E^2*(50*x ^2 + 20*x^3 + 2*x^4))),x]
Output:
(5*E^4)/(E^x - E^4*(3 - E^2)) + 4/(5*x) - 9/(5*(5 + x))
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {-x +4}{\left (5+x \right ) x}+\frac {5}{{\mathrm e}^{x -4}+{\mathrm e}^{2}-3}\) | \(28\) |
norman | \(\frac {5 x^{2}+\left (-{\mathrm e}^{2}+28\right ) x -x \,{\mathrm e}^{x -4}+4 \,{\mathrm e}^{x -4}+4 \,{\mathrm e}^{2}-12}{x \left (5+x \right ) \left ({\mathrm e}^{x -4}+{\mathrm e}^{2}-3\right )}\) | \(52\) |
parallelrisch | \(\frac {-12-{\mathrm e}^{2} x +5 x^{2}-x \,{\mathrm e}^{x -4}+4 \,{\mathrm e}^{2}+28 x +4 \,{\mathrm e}^{x -4}}{\left ({\mathrm e}^{x -4}+{\mathrm e}^{2}-3\right ) x \left (5+x \right )}\) | \(52\) |
Input:
int(((x^2-8*x-20)*exp(x-4)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-131*x^2+ 48*x+120)*exp(x-4)+(x^2-8*x-20)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+9*x^2-72 *x-180)/((x^4+10*x^3+25*x^2)*exp(x-4)^2+((2*x^4+20*x^3+50*x^2)*exp(2)-6*x^ 4-60*x^3-150*x^2)*exp(x-4)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4-60*x^3-150 *x^2)*exp(2)+9*x^4+90*x^3+225*x^2),x,method=_RETURNVERBOSE)
Output:
(-x+4)/(5+x)/x+5/(exp(x-4)+exp(2)-3)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=-\frac {5 \, x^{2} - {\left (x - 4\right )} e^{2} - {\left (x - 4\right )} e^{\left (x - 4\right )} + 28 \, x - 12}{3 \, x^{2} - {\left (x^{2} + 5 \, x\right )} e^{2} - {\left (x^{2} + 5 \, x\right )} e^{\left (x - 4\right )} + 15 \, x} \] Input:
integrate(((x^2-8*x-20)*exp(-4+x)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-1 31*x^2+48*x+120)*exp(-4+x)+(x^2-8*x-20)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+ 9*x^2-72*x-180)/((x^4+10*x^3+25*x^2)*exp(-4+x)^2+((2*x^4+20*x^3+50*x^2)*ex p(2)-6*x^4-60*x^3-150*x^2)*exp(-4+x)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4- 60*x^3-150*x^2)*exp(2)+9*x^4+90*x^3+225*x^2),x, algorithm="fricas")
Output:
-(5*x^2 - (x - 4)*e^2 - (x - 4)*e^(x - 4) + 28*x - 12)/(3*x^2 - (x^2 + 5*x )*e^2 - (x^2 + 5*x)*e^(x - 4) + 15*x)
Time = 0.14 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=\frac {4 - x}{x^{2} + 5 x} + \frac {5}{e^{x - 4} - 3 + e^{2}} \] Input:
integrate(((x**2-8*x-20)*exp(-4+x)**2+((2*x**2-16*x-40)*exp(2)-5*x**4-50*x **3-131*x**2+48*x+120)*exp(-4+x)+(x**2-8*x-20)*exp(2)**2+(-6*x**2+48*x+120 )*exp(2)+9*x**2-72*x-180)/((x**4+10*x**3+25*x**2)*exp(-4+x)**2+((2*x**4+20 *x**3+50*x**2)*exp(2)-6*x**4-60*x**3-150*x**2)*exp(-4+x)+(x**4+10*x**3+25* x**2)*exp(2)**2+(-6*x**4-60*x**3-150*x**2)*exp(2)+9*x**4+90*x**3+225*x**2) ,x)
Output:
(4 - x)/(x**2 + 5*x) + 5/(exp(x - 4) - 3 + exp(2))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (27) = 54\).
Time = 0.14 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.34 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=\frac {5 \, x^{2} e^{4} - x {\left (e^{6} - 28 \, e^{4}\right )} - {\left (x - 4\right )} e^{x} + 4 \, e^{6} - 12 \, e^{4}}{x^{2} {\left (e^{6} - 3 \, e^{4}\right )} + 5 \, x {\left (e^{6} - 3 \, e^{4}\right )} + {\left (x^{2} + 5 \, x\right )} e^{x}} \] Input:
integrate(((x^2-8*x-20)*exp(-4+x)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-1 31*x^2+48*x+120)*exp(-4+x)+(x^2-8*x-20)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+ 9*x^2-72*x-180)/((x^4+10*x^3+25*x^2)*exp(-4+x)^2+((2*x^4+20*x^3+50*x^2)*ex p(2)-6*x^4-60*x^3-150*x^2)*exp(-4+x)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4- 60*x^3-150*x^2)*exp(2)+9*x^4+90*x^3+225*x^2),x, algorithm="maxima")
Output:
(5*x^2*e^4 - x*(e^6 - 28*e^4) - (x - 4)*e^x + 4*e^6 - 12*e^4)/(x^2*(e^6 - 3*e^4) + 5*x*(e^6 - 3*e^4) + (x^2 + 5*x)*e^x)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (27) = 54\).
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.00 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=\frac {5 \, {\left (x - 4\right )}^{2} - {\left (x - 4\right )} e^{2} - {\left (x - 4\right )} e^{\left (x - 4\right )} + 68 \, x - 92}{{\left (x - 4\right )}^{2} e^{2} + {\left (x - 4\right )}^{2} e^{\left (x - 4\right )} - 3 \, {\left (x - 4\right )}^{2} + 13 \, {\left (x - 4\right )} e^{2} + 13 \, {\left (x - 4\right )} e^{\left (x - 4\right )} - 39 \, x + 36 \, e^{2} + 36 \, e^{\left (x - 4\right )} + 48} \] Input:
integrate(((x^2-8*x-20)*exp(-4+x)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-1 31*x^2+48*x+120)*exp(-4+x)+(x^2-8*x-20)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+ 9*x^2-72*x-180)/((x^4+10*x^3+25*x^2)*exp(-4+x)^2+((2*x^4+20*x^3+50*x^2)*ex p(2)-6*x^4-60*x^3-150*x^2)*exp(-4+x)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4- 60*x^3-150*x^2)*exp(2)+9*x^4+90*x^3+225*x^2),x, algorithm="giac")
Output:
(5*(x - 4)^2 - (x - 4)*e^2 - (x - 4)*e^(x - 4) + 68*x - 92)/((x - 4)^2*e^2 + (x - 4)^2*e^(x - 4) - 3*(x - 4)^2 + 13*(x - 4)*e^2 + 13*(x - 4)*e^(x - 4) - 39*x + 36*e^2 + 36*e^(x - 4) + 48)
Time = 0.37 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=\frac {5}{{\mathrm {e}}^{x-4}+{\mathrm {e}}^2-3}-\frac {x-4}{x^2+5\,x} \] Input:
int(-(72*x + exp(4)*(8*x - x^2 + 20) - exp(2)*(48*x - 6*x^2 + 120) + exp(2 *x - 8)*(8*x - x^2 + 20) + exp(x - 4)*(exp(2)*(16*x - 2*x^2 + 40) - 48*x + 131*x^2 + 50*x^3 + 5*x^4 - 120) - 9*x^2 + 180)/(exp(4)*(25*x^2 + 10*x^3 + x^4) - exp(x - 4)*(150*x^2 - exp(2)*(50*x^2 + 20*x^3 + 2*x^4) + 60*x^3 + 6*x^4) + exp(2*x - 8)*(25*x^2 + 10*x^3 + x^4) - exp(2)*(150*x^2 + 60*x^3 + 6*x^4) + 225*x^2 + 90*x^3 + 9*x^4),x)
Output:
5/(exp(x - 4) + exp(2) - 3) - (x - 4)/(5*x + x^2)
Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.86 \[ \int \frac {-180-72 x+9 x^2+e^2 \left (120+48 x-6 x^2\right )+e^4 \left (-20-8 x+x^2\right )+e^{-8+2 x} \left (-20-8 x+x^2\right )+e^{-4+x} \left (120+48 x-131 x^2-50 x^3-5 x^4+e^2 \left (-40-16 x+2 x^2\right )\right )}{225 x^2+90 x^3+9 x^4+e^2 \left (-150 x^2-60 x^3-6 x^4\right )+e^4 \left (25 x^2+10 x^3+x^4\right )+e^{-8+2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+x} \left (-150 x^2-60 x^3-6 x^4+e^2 \left (50 x^2+20 x^3+2 x^4\right )\right )} \, dx=\frac {e^{x} x^{2}+20 e^{x}+e^{6} x^{2}+20 e^{6}+22 e^{4} x^{2}+125 e^{4} x -60 e^{4}}{5 x \left (e^{x} x +5 e^{x}+e^{6} x +5 e^{6}-3 e^{4} x -15 e^{4}\right )} \] Input:
int(((x^2-8*x-20)*exp(-4+x)^2+((2*x^2-16*x-40)*exp(2)-5*x^4-50*x^3-131*x^2 +48*x+120)*exp(-4+x)+(x^2-8*x-20)*exp(2)^2+(-6*x^2+48*x+120)*exp(2)+9*x^2- 72*x-180)/((x^4+10*x^3+25*x^2)*exp(-4+x)^2+((2*x^4+20*x^3+50*x^2)*exp(2)-6 *x^4-60*x^3-150*x^2)*exp(-4+x)+(x^4+10*x^3+25*x^2)*exp(2)^2+(-6*x^4-60*x^3 -150*x^2)*exp(2)+9*x^4+90*x^3+225*x^2),x)
Output:
(e**x*x**2 + 20*e**x + e**6*x**2 + 20*e**6 + 22*e**4*x**2 + 125*e**4*x - 6 0*e**4)/(5*x*(e**x*x + 5*e**x + e**6*x + 5*e**6 - 3*e**4*x - 15*e**4))