Integrand size = 57, antiderivative size = 21 \[ \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (-2+x)\right ) x}{\left (-3 e^4+3 e^x\right ) \left (e^8+e^{2 x}-2 e^{4+x}\right )^4} \, dx=\frac {e^{-x} x^2}{3 \left (-1+e^{-4+x}\right )^8} \]
Time = 1.94 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (-2+x)\right ) x}{\left (-3 e^4+3 e^x\right ) \left (e^8+e^{2 x}-2 e^{4+x}\right )^4} \, dx=\frac {e^{32-x} x^2}{3 \left (e^4-e^x\right )^8} \]
Integrate[(E^(32 - x)*(E^x*(2 - 9*x) + E^4*(-2 + x))*x)/((-3*E^4 + 3*E^x)* (E^8 + E^(2*x) - 2*E^(4 + x))^4),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.
\(\displaystyle \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (x-2)\right ) x}{\left (3 e^x-3 e^4\right ) \left (e^{2 x}-2 e^{x+4}+e^8\right )^4} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{32-x} \left (-e^x (2-9 x)-e^4 (x-2)\right ) x}{3 \left (e^4-e^x\right )^9}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -\frac {e^{32-x} \left (e^x (2-9 x)-e^4 (2-x)\right ) x}{\left (e^4-e^x\right )^9}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{3} \int \frac {e^{32-x} \left (e^x (2-9 x)-e^4 (2-x)\right ) x}{\left (e^4-e^x\right )^9}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{3} \int \left (\frac {e^{32-x} x (9 x-2)}{\left (-e^4+e^x\right )^8}-\frac {8 e^{36-x} x^2}{\left (e^4-e^x\right )^9}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (8 \int \frac {e^{36-x} x^2}{\left (e^4-e^x\right )^9}dx-9 \int \frac {e^{32-x} x^2}{\left (-e^4+e^x\right )^8}dx+2 \int \frac {e^{32-x} x}{\left (-e^4+e^x\right )^8}dx\right )\) |
Int[(E^(32 - x)*(E^x*(2 - 9*x) + E^4*(-2 + x))*x)/((-3*E^4 + 3*E^x)*(E^8 + E^(2*x) - 2*E^(4 + x))^4),x]
3.5.74.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 1.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81
method | result | size |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{-x +32}}{3 \left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{4+x}+{\mathrm e}^{8}\right )^{4}}\) | \(38\) |
int(((-9*x+2)*exp(x)+(-2+x)*exp(2)^2)/(3*exp(x)-3*exp(2)^2)/exp(4*ln((exp( x)^2-2*exp(2)^2*exp(x)+exp(2)^4)/exp(2)^4)+x-ln(x)),x,method=_RETURNVERBOS E)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.62 \[ \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (-2+x)\right ) x}{\left (-3 e^4+3 e^x\right ) \left (e^8+e^{2 x}-2 e^{4+x}\right )^4} \, dx=\frac {x^{2} e^{68}}{3 \, {\left (e^{\left (9 \, x + 36\right )} - 8 \, e^{\left (8 \, x + 40\right )} + 28 \, e^{\left (7 \, x + 44\right )} - 56 \, e^{\left (6 \, x + 48\right )} + 70 \, e^{\left (5 \, x + 52\right )} - 56 \, e^{\left (4 \, x + 56\right )} + 28 \, e^{\left (3 \, x + 60\right )} - 8 \, e^{\left (2 \, x + 64\right )} + e^{\left (x + 68\right )}\right )}} \]
integrate(((-9*x+2)*exp(x)+(-2+x)*exp(2)^2)/(3*exp(x)-3*exp(2)^2)/exp(4*lo g((exp(x)^2-2*exp(2)^2*exp(x)+exp(2)^4)/exp(2)^4)+x-log(x)),x, algorithm=\
1/3*x^2*e^68/(e^(9*x + 36) - 8*e^(8*x + 40) + 28*e^(7*x + 44) - 56*e^(6*x + 48) + 70*e^(5*x + 52) - 56*e^(4*x + 56) + 28*e^(3*x + 60) - 8*e^(2*x + 6 4) + e^(x + 68))
Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 8.81 \[ \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (-2+x)\right ) x}{\left (-3 e^4+3 e^x\right ) \left (e^8+e^{2 x}-2 e^{4+x}\right )^4} \, dx=\frac {x^{2} e^{- x}}{3} + \frac {- x^{2} e^{7 x} + 8 x^{2} e^{4} e^{6 x} - 28 x^{2} e^{8} e^{5 x} + 56 x^{2} e^{12} e^{4 x} - 70 x^{2} e^{16} e^{3 x} + 56 x^{2} e^{20} e^{2 x} - 28 x^{2} e^{24} e^{x} + 8 x^{2} e^{28}}{3 e^{8 x} - 24 e^{4} e^{7 x} + 84 e^{8} e^{6 x} - 168 e^{12} e^{5 x} + 210 e^{16} e^{4 x} - 168 e^{20} e^{3 x} + 84 e^{24} e^{2 x} - 24 e^{28} e^{x} + 3 e^{32}} \]
integrate(((-9*x+2)*exp(x)+(-2+x)*exp(2)**2)/(3*exp(x)-3*exp(2)**2)/exp(4* ln((exp(x)**2-2*exp(2)**2*exp(x)+exp(2)**4)/exp(2)**4)+x-ln(x)),x)
x**2*exp(-x)/3 + (-x**2*exp(7*x) + 8*x**2*exp(4)*exp(6*x) - 28*x**2*exp(8) *exp(5*x) + 56*x**2*exp(12)*exp(4*x) - 70*x**2*exp(16)*exp(3*x) + 56*x**2* exp(20)*exp(2*x) - 28*x**2*exp(24)*exp(x) + 8*x**2*exp(28))/(3*exp(8*x) - 24*exp(4)*exp(7*x) + 84*exp(8)*exp(6*x) - 168*exp(12)*exp(5*x) + 210*exp(1 6)*exp(4*x) - 168*exp(20)*exp(3*x) + 84*exp(24)*exp(2*x) - 24*exp(28)*exp( x) + 3*exp(32))
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.52 \[ \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (-2+x)\right ) x}{\left (-3 e^4+3 e^x\right ) \left (e^8+e^{2 x}-2 e^{4+x}\right )^4} \, dx=\frac {x^{2} e^{\left (-x + 32\right )}}{3 \, {\left (e^{32} + e^{\left (8 \, x\right )} - 8 \, e^{\left (7 \, x + 4\right )} + 28 \, e^{\left (6 \, x + 8\right )} - 56 \, e^{\left (5 \, x + 12\right )} + 70 \, e^{\left (4 \, x + 16\right )} - 56 \, e^{\left (3 \, x + 20\right )} + 28 \, e^{\left (2 \, x + 24\right )} - 8 \, e^{\left (x + 28\right )}\right )}} \]
integrate(((-9*x+2)*exp(x)+(-2+x)*exp(2)^2)/(3*exp(x)-3*exp(2)^2)/exp(4*lo g((exp(x)^2-2*exp(2)^2*exp(x)+exp(2)^4)/exp(2)^4)+x-log(x)),x, algorithm=\
1/3*x^2*e^(-x + 32)/(e^32 + e^(8*x) - 8*e^(7*x + 4) + 28*e^(6*x + 8) - 56* e^(5*x + 12) + 70*e^(4*x + 16) - 56*e^(3*x + 20) + 28*e^(2*x + 24) - 8*e^( x + 28))
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.52 \[ \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (-2+x)\right ) x}{\left (-3 e^4+3 e^x\right ) \left (e^8+e^{2 x}-2 e^{4+x}\right )^4} \, dx=\frac {x^{2} e^{32}}{3 \, {\left (e^{\left (9 \, x\right )} - 8 \, e^{\left (8 \, x + 4\right )} + 28 \, e^{\left (7 \, x + 8\right )} - 56 \, e^{\left (6 \, x + 12\right )} + 70 \, e^{\left (5 \, x + 16\right )} - 56 \, e^{\left (4 \, x + 20\right )} + 28 \, e^{\left (3 \, x + 24\right )} - 8 \, e^{\left (2 \, x + 28\right )} + e^{\left (x + 32\right )}\right )}} \]
integrate(((-9*x+2)*exp(x)+(-2+x)*exp(2)^2)/(3*exp(x)-3*exp(2)^2)/exp(4*lo g((exp(x)^2-2*exp(2)^2*exp(x)+exp(2)^4)/exp(2)^4)+x-log(x)),x, algorithm=\
1/3*x^2*e^32/(e^(9*x) - 8*e^(8*x + 4) + 28*e^(7*x + 8) - 56*e^(6*x + 12) + 70*e^(5*x + 16) - 56*e^(4*x + 20) + 28*e^(3*x + 24) - 8*e^(2*x + 28) + e^ (x + 32))
Time = 14.56 (sec) , antiderivative size = 371, normalized size of antiderivative = 17.67 \[ \int \frac {e^{32-x} \left (e^x (2-9 x)+e^4 (-2+x)\right ) x}{\left (-3 e^4+3 e^x\right ) \left (e^8+e^{2 x}-2 e^{4+x}\right )^4} \, dx=\frac {x^2\,{\mathrm {e}}^{-x}}{3}+\frac {x^2}{3\,\left ({\mathrm {e}}^4-{\mathrm {e}}^x\right )}+\frac {x^2\,{\mathrm {e}}^{28}}{3\,\left ({\mathrm {e}}^{8\,x}-8\,{\mathrm {e}}^{x+28}+{\mathrm {e}}^{32}-8\,{\mathrm {e}}^{7\,x+4}+28\,{\mathrm {e}}^{6\,x+8}-56\,{\mathrm {e}}^{5\,x+12}+70\,{\mathrm {e}}^{4\,x+16}-56\,{\mathrm {e}}^{3\,x+20}+28\,{\mathrm {e}}^{2\,x+24}\right )}+\frac {x^2\,{\mathrm {e}}^{20}}{3\,\left ({\mathrm {e}}^{6\,x}-6\,{\mathrm {e}}^{x+20}+{\mathrm {e}}^{24}-6\,{\mathrm {e}}^{5\,x+4}+15\,{\mathrm {e}}^{4\,x+8}-20\,{\mathrm {e}}^{3\,x+12}+15\,{\mathrm {e}}^{2\,x+16}\right )}+\frac {x^2\,{\mathrm {e}}^{12}}{3\,\left ({\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{x+12}+{\mathrm {e}}^{16}-4\,{\mathrm {e}}^{3\,x+4}+6\,{\mathrm {e}}^{2\,x+8}\right )}+\frac {x^2\,{\mathrm {e}}^4}{3\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^{x+4}+{\mathrm {e}}^8\right )}-\frac {x^2\,{\mathrm {e}}^{24}}{3\,\left ({\mathrm {e}}^{7\,x}+7\,{\mathrm {e}}^{x+24}-{\mathrm {e}}^{28}-7\,{\mathrm {e}}^{6\,x+4}+21\,{\mathrm {e}}^{5\,x+8}-35\,{\mathrm {e}}^{4\,x+12}+35\,{\mathrm {e}}^{3\,x+16}-21\,{\mathrm {e}}^{2\,x+20}\right )}-\frac {x^2\,{\mathrm {e}}^{16}}{3\,\left ({\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^{x+16}-{\mathrm {e}}^{20}-5\,{\mathrm {e}}^{4\,x+4}+10\,{\mathrm {e}}^{3\,x+8}-10\,{\mathrm {e}}^{2\,x+12}\right )}-\frac {x^2\,{\mathrm {e}}^8}{3\,\left ({\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^{x+8}-{\mathrm {e}}^{12}-3\,{\mathrm {e}}^{2\,x+4}\right )} \]
int((exp(log(x) - 4*log(exp(-8)*(exp(2*x) + exp(8) - 2*exp(4)*exp(x))) - x )*(exp(x)*(9*x - 2) - exp(4)*(x - 2)))/(3*exp(4) - 3*exp(x)),x)
(x^2*exp(-x))/3 + x^2/(3*(exp(4) - exp(x))) + (x^2*exp(28))/(3*(exp(8*x) - 8*exp(x + 28) + exp(32) - 8*exp(7*x + 4) + 28*exp(6*x + 8) - 56*exp(5*x + 12) + 70*exp(4*x + 16) - 56*exp(3*x + 20) + 28*exp(2*x + 24))) + (x^2*exp (20))/(3*(exp(6*x) - 6*exp(x + 20) + exp(24) - 6*exp(5*x + 4) + 15*exp(4*x + 8) - 20*exp(3*x + 12) + 15*exp(2*x + 16))) + (x^2*exp(12))/(3*(exp(4*x) - 4*exp(x + 12) + exp(16) - 4*exp(3*x + 4) + 6*exp(2*x + 8))) + (x^2*exp( 4))/(3*(exp(2*x) - 2*exp(x + 4) + exp(8))) - (x^2*exp(24))/(3*(exp(7*x) + 7*exp(x + 24) - exp(28) - 7*exp(6*x + 4) + 21*exp(5*x + 8) - 35*exp(4*x + 12) + 35*exp(3*x + 16) - 21*exp(2*x + 20))) - (x^2*exp(16))/(3*(exp(5*x) + 5*exp(x + 16) - exp(20) - 5*exp(4*x + 4) + 10*exp(3*x + 8) - 10*exp(2*x + 12))) - (x^2*exp(8))/(3*(exp(3*x) + 3*exp(x + 8) - exp(12) - 3*exp(2*x + 4)))