Integrand size = 125, antiderivative size = 21 \[ \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx=e^{-4-7 x-\frac {x^2}{(-4+x)^4}}-x \]
Time = 1.83 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57 \[ \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx=4+e^{-4-\frac {16}{(-4+x)^4}-\frac {8}{(-4+x)^3}-\frac {1}{(-4+x)^2}-7 x}-x \]
Integrate[(1024 - 1280*x + 640*x^2 - 160*x^3 + 20*x^4 - x^5 + E^((-1024 - 768*x + 1407*x^2 - 608*x^3 + 108*x^4 - 7*x^5)/(256 - 256*x + 96*x^2 - 16*x ^3 + x^4))*(7168 - 8952*x + 4482*x^2 - 1120*x^3 + 140*x^4 - 7*x^5))/(-1024 + 1280*x - 640*x^2 + 160*x^3 - 20*x^4 + x^5),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.
\(\displaystyle \int \frac {\left (-7 x^5+140 x^4-1120 x^3+4482 x^2-8952 x+7168\right ) \exp \left (\frac {-7 x^5+108 x^4-608 x^3+1407 x^2-768 x-1024}{x^4-16 x^3+96 x^2-256 x+256}\right )-x^5+20 x^4-160 x^3+640 x^2-1280 x+1024}{x^5-20 x^4+160 x^3-640 x^2+1280 x-1024} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\left (-7 x^5+140 x^4-1120 x^3+4482 x^2-8952 x+7168\right ) \exp \left (\frac {-7 x^5+108 x^4-608 x^3+1407 x^2-768 x-1024}{x^4-16 x^3+96 x^2-256 x+256}\right )-x^5+20 x^4-160 x^3+640 x^2-1280 x+1024}{(x-4)^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (7 x^5-140 x^4+1120 x^3-4482 x^2+8952 x-7168\right ) \exp \left (\frac {-7 x^5+108 x^4-608 x^3+1407 x^2-768 x-1024}{(x-4)^4}\right )}{(4-x)^5}-\frac {x^5}{(x-4)^5}+\frac {20 x^4}{(x-4)^5}-\frac {160 x^3}{(x-4)^5}+\frac {640 x^2}{(x-4)^5}-\frac {1280 x}{(x-4)^5}+\frac {1024}{(x-4)^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -7 \int \exp \left (\frac {-7 x^5+108 x^4-608 x^3+1407 x^2-768 x-1024}{(x-4)^4}\right )dx+64 \int \frac {\exp \left (\frac {-7 x^5+108 x^4-608 x^3+1407 x^2-768 x-1024}{(x-4)^4}\right )}{(x-4)^5}dx+24 \int \frac {\exp \left (\frac {-7 x^5+108 x^4-608 x^3+1407 x^2-768 x-1024}{(x-4)^4}\right )}{(x-4)^4}dx+2 \int \frac {\exp \left (\frac {-7 x^5+108 x^4-608 x^3+1407 x^2-768 x-1024}{(x-4)^4}\right )}{(x-4)^3}dx+\frac {10 x^4}{(4-x)^4}-x+\frac {160}{4-x}-\frac {960}{(4-x)^2}+\frac {2560}{(4-x)^3}-\frac {2560}{(4-x)^4}\) |
Int[(1024 - 1280*x + 640*x^2 - 160*x^3 + 20*x^4 - x^5 + E^((-1024 - 768*x + 1407*x^2 - 608*x^3 + 108*x^4 - 7*x^5)/(256 - 256*x + 96*x^2 - 16*x^3 + x ^4))*(7168 - 8952*x + 4482*x^2 - 1120*x^3 + 140*x^4 - 7*x^5))/(-1024 + 128 0*x - 640*x^2 + 160*x^3 - 20*x^4 + x^5),x]
3.1.5.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 5.67 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81
method | result | size |
risch | \(-x +{\mathrm e}^{-\frac {7 x^{5}-108 x^{4}+608 x^{3}-1407 x^{2}+768 x +1024}{\left (x -4\right )^{4}}}\) | \(38\) |
parallelrisch | \(-x +{\mathrm e}^{-\frac {7 x^{5}-108 x^{4}+608 x^{3}-1407 x^{2}+768 x +1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-40\) | \(54\) |
parts | \(-x +\frac {x^{4} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-256 x \,{\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}+96 x^{2} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-16 x^{3} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}+256 \,{\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}}{\left (x -4\right )^{4}}\) | \(266\) |
norman | \(\frac {x^{4} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-1280 x^{2}+160 x^{3}+3840 x -x^{5}-256 x \,{\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}+96 x^{2} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-16 x^{3} {\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}+256 \,{\mathrm e}^{\frac {-7 x^{5}+108 x^{4}-608 x^{3}+1407 x^{2}-768 x -1024}{x^{4}-16 x^{3}+96 x^{2}-256 x +256}}-4096}{\left (x -4\right )^{4}}\) | \(281\) |
int(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108*x^4-60 8*x^3+1407*x^2-768*x-1024)/(x^4-16*x^3+96*x^2-256*x+256))-x^5+20*x^4-160*x ^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x,method= _RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (20) = 40\).
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.48 \[ \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx=-x + e^{\left (-\frac {7 \, x^{5} - 108 \, x^{4} + 608 \, x^{3} - 1407 \, x^{2} + 768 \, x + 1024}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256}\right )} \]
integrate(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108* x^4-608*x^3+1407*x^2-768*x-1024)/(x^4-16*x^3+96*x^2-256*x+256))-x^5+20*x^4 -160*x^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x, algorithm=\
-x + e^(-(7*x^5 - 108*x^4 + 608*x^3 - 1407*x^2 + 768*x + 1024)/(x^4 - 16*x ^3 + 96*x^2 - 256*x + 256))
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx=- x + e^{\frac {- 7 x^{5} + 108 x^{4} - 608 x^{3} + 1407 x^{2} - 768 x - 1024}{x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 256}} \]
integrate(((-7*x**5+140*x**4-1120*x**3+4482*x**2-8952*x+7168)*exp((-7*x**5 +108*x**4-608*x**3+1407*x**2-768*x-1024)/(x**4-16*x**3+96*x**2-256*x+256)) -x**5+20*x**4-160*x**3+640*x**2-1280*x+1024)/(x**5-20*x**4+160*x**3-640*x* *2+1280*x-1024),x)
-x + exp((-7*x**5 + 108*x**4 - 608*x**3 + 1407*x**2 - 768*x - 1024)/(x**4 - 16*x**3 + 96*x**2 - 256*x + 256))
Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (20) = 40\).
Time = 0.42 (sec) , antiderivative size = 249, normalized size of antiderivative = 11.86 \[ \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx=-x + \frac {32 \, {\left (15 \, x^{3} - 150 \, x^{2} + 520 \, x - 616\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} - \frac {320 \, {\left (3 \, x^{3} - 27 \, x^{2} + 88 \, x - 100\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} + \frac {160 \, {\left (x^{3} - 6 \, x^{2} + 16 \, x - 16\right )}}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256} - \frac {320 \, {\left (3 \, x^{2} - 8 \, x + 8\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} + \frac {1280 \, {\left (x - 1\right )}}{3 \, {\left (x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256\right )}} - \frac {256}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256} + e^{\left (-7 \, x - \frac {16}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256} - \frac {8}{x^{3} - 12 \, x^{2} + 48 \, x - 64} - \frac {1}{x^{2} - 8 \, x + 16} - 4\right )} \]
integrate(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108* x^4-608*x^3+1407*x^2-768*x-1024)/(x^4-16*x^3+96*x^2-256*x+256))-x^5+20*x^4 -160*x^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x, algorithm=\
-x + 32/3*(15*x^3 - 150*x^2 + 520*x - 616)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) - 320/3*(3*x^3 - 27*x^2 + 88*x - 100)/(x^4 - 16*x^3 + 96*x^2 - 256* x + 256) + 160*(x^3 - 6*x^2 + 16*x - 16)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) - 320/3*(3*x^2 - 8*x + 8)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) + 128 0/3*(x - 1)/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) - 256/(x^4 - 16*x^3 + 96 *x^2 - 256*x + 256) + e^(-7*x - 16/(x^4 - 16*x^3 + 96*x^2 - 256*x + 256) - 8/(x^3 - 12*x^2 + 48*x - 64) - 1/(x^2 - 8*x + 16) - 4)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 0.69 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.76 \[ \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx=-{\left (x e^{4} - e^{\left (-\frac {7 \, x^{5} - 112 \, x^{4} + 672 \, x^{3} - 1791 \, x^{2} + 1792 \, x}{x^{4} - 16 \, x^{3} + 96 \, x^{2} - 256 \, x + 256}\right )}\right )} e^{\left (-4\right )} \]
integrate(((-7*x^5+140*x^4-1120*x^3+4482*x^2-8952*x+7168)*exp((-7*x^5+108* x^4-608*x^3+1407*x^2-768*x-1024)/(x^4-16*x^3+96*x^2-256*x+256))-x^5+20*x^4 -160*x^3+640*x^2-1280*x+1024)/(x^5-20*x^4+160*x^3-640*x^2+1280*x-1024),x, algorithm=\
-(x*e^4 - e^(-(7*x^5 - 112*x^4 + 672*x^3 - 1791*x^2 + 1792*x)/(x^4 - 16*x^ 3 + 96*x^2 - 256*x + 256)))*e^(-4)
Time = 10.83 (sec) , antiderivative size = 156, normalized size of antiderivative = 7.43 \[ \int \frac {1024-1280 x+640 x^2-160 x^3+20 x^4-x^5+e^{\frac {-1024-768 x+1407 x^2-608 x^3+108 x^4-7 x^5}{256-256 x+96 x^2-16 x^3+x^4}} \left (7168-8952 x+4482 x^2-1120 x^3+140 x^4-7 x^5\right )}{-1024+1280 x-640 x^2+160 x^3-20 x^4+x^5} \, dx={\mathrm {e}}^{-\frac {768\,x}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{-\frac {7\,x^5}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{\frac {108\,x^4}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{-\frac {608\,x^3}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{\frac {1407\,x^2}{x^4-16\,x^3+96\,x^2-256\,x+256}}\,{\mathrm {e}}^{-\frac {1024}{x^4-16\,x^3+96\,x^2-256\,x+256}}-x \]
int(-(1280*x + exp(-(768*x - 1407*x^2 + 608*x^3 - 108*x^4 + 7*x^5 + 1024)/ (96*x^2 - 256*x - 16*x^3 + x^4 + 256))*(8952*x - 4482*x^2 + 1120*x^3 - 140 *x^4 + 7*x^5 - 7168) - 640*x^2 + 160*x^3 - 20*x^4 + x^5 - 1024)/(1280*x - 640*x^2 + 160*x^3 - 20*x^4 + x^5 - 1024),x)
exp(-(768*x)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp(-(7*x^5)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp((108*x^4)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp(-(608*x^3)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp((1407*x ^2)/(96*x^2 - 256*x - 16*x^3 + x^4 + 256))*exp(-1024/(96*x^2 - 256*x - 16* x^3 + x^4 + 256)) - x