∫ 1_ 256e−x(256ex −16384x3 +4096x4 −12288x5 +2048x6 −3072x7 +384x8 −320x9 +32x10 −12x11 +x12) dx

Optimal. Leaf size=25 \[ 3+x-e^{-x} \left (2 x+\frac {x^3}{4}\right )^4+\log (4) \]

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Rubi [B] time = 3.44, antiderivative size = 58, normalized size of antiderivative = 2.32, number of steps used = 90, number of rules used = 5, integrand size = 63, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.079, Rules used = {12, 6688, 2196, 2176, 2194} \begin {gather*} -\frac {1}{256} e^{-x} x^{12}-\frac {1}{8} e^{-x} x^{10}-\frac {3}{2} e^{-x} x^8-8 e^{-x} x^6-16 e^{-x} x^4+x \end {gather*}

Int[(256*E^x - 16384*x^3 + 4096*x^4 - 12288*x^5 + 2048*x^6 - 3072*x^7 + 384*x^8 - 320*x^9 + 32*x^10 - 12*x^11

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m

*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x

)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] && !$UseGamma === True

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{256} \int e^{-x} \left (256 e^x-16384 x^3+4096 x^4-12288 x^5+2048 x^6-3072 x^7+384 x^8-320 x^9+32 x^{10}-12 x^{11}+x^{12}\right ) \, dx\\ &=\frac {1}{256} \int \left (256+e^{-x} x^3 \left (8+x^2\right )^3 \left (-32+8 x-12 x^2+x^3\right )\right ) \, dx\\ &=x+\frac {1}{256} \int e^{-x} x^3 \left (8+x^2\right )^3 \left (-32+8 x-12 x^2+x^3\right ) \, dx\\ &=x+\frac {1}{256} \int \left (-16384 e^{-x} x^3+4096 e^{-x} x^4-12288 e^{-x} x^5+2048 e^{-x} x^6-3072 e^{-x} x^7+384 e^{-x} x^8-320 e^{-x} x^9+32 e^{-x} x^{10}-12 e^{-x} x^{11}+e^{-x} x^{12}\right ) \, dx\\ &=x+\frac {1}{256} \int e^{-x} x^{12} \, dx-\frac {3}{64} \int e^{-x} x^{11} \, dx+\frac {1}{8} \int e^{-x} x^{10} \, dx-\frac {5}{4} \int e^{-x} x^9 \, dx+\frac {3}{2} \int e^{-x} x^8 \, dx+8 \int e^{-x} x^6 \, dx-12 \int e^{-x} x^7 \, dx+16 \int e^{-x} x^4 \, dx-48 \int e^{-x} x^5 \, dx-64 \int e^{-x} x^3 \, dx\\ &=x+64 e^{-x} x^3-16 e^{-x} x^4+48 e^{-x} x^5-8 e^{-x} x^6+12 e^{-x} x^7-\frac {3}{2} e^{-x} x^8+\frac {5}{4} e^{-x} x^9-\frac {1}{8} e^{-x} x^{10}+\frac {3}{64} e^{-x} x^{11}-\frac {1}{256} e^{-x} x^{12}+\frac {3}{64} \int e^{-x} x^{11} \, dx-\frac {33}{64} \int e^{-x} x^{10} \, dx+\frac {5}{4} \int e^{-x} x^9 \, dx-\frac {45}{4} \int e^{-x} x^8 \, dx+12 \int e^{-x} x^7 \, dx+48 \int e^{-x} x^5 \, dx+64 \int e^{-x} x^3 \, dx-84 \int e^{-x} x^6 \, dx-192 \int e^{-x} x^2 \, dx-240 \int e^{-x} x^4 \, dx\\ &=x+192 e^{-x} x^2+224 e^{-x} x^4+76 e^{-x} x^6+\frac {39}{4} e^{-x} x^8+\frac {25}{64} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+\frac {33}{64} \int e^{-x} x^{10} \, dx-\frac {165}{32} \int e^{-x} x^9 \, dx+\frac {45}{4} \int e^{-x} x^8 \, dx+84 \int e^{-x} x^6 \, dx-90 \int e^{-x} x^7 \, dx+192 \int e^{-x} x^2 \, dx+240 \int e^{-x} x^4 \, dx-384 \int e^{-x} x \, dx-504 \int e^{-x} x^5 \, dx-960 \int e^{-x} x^3 \, dx\\ &=x+384 e^{-x} x+960 e^{-x} x^3-16 e^{-x} x^4+504 e^{-x} x^5-8 e^{-x} x^6+90 e^{-x} x^7-\frac {3}{2} e^{-x} x^8+\frac {165}{32} e^{-x} x^9-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+\frac {165}{32} \int e^{-x} x^9 \, dx-\frac {1485}{32} \int e^{-x} x^8 \, dx+90 \int e^{-x} x^7 \, dx-384 \int e^{-x} \, dx+384 \int e^{-x} x \, dx+504 \int e^{-x} x^5 \, dx-630 \int e^{-x} x^6 \, dx+960 \int e^{-x} x^3 \, dx-2520 \int e^{-x} x^4 \, dx-2880 \int e^{-x} x^2 \, dx\\ &=384 e^{-x}+x+2880 e^{-x} x^2+2504 e^{-x} x^4+622 e^{-x} x^6+\frac {1437}{32} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+\frac {1485}{32} \int e^{-x} x^8 \, dx-\frac {1485}{4} \int e^{-x} x^7 \, dx+384 \int e^{-x} \, dx+630 \int e^{-x} x^6 \, dx+2520 \int e^{-x} x^4 \, dx+2880 \int e^{-x} x^2 \, dx-3780 \int e^{-x} x^5 \, dx-5760 \int e^{-x} x \, dx-10080 \int e^{-x} x^3 \, dx\\ &=x+5760 e^{-x} x+10080 e^{-x} x^3-16 e^{-x} x^4+3780 e^{-x} x^5-8 e^{-x} x^6+\frac {1485}{4} e^{-x} x^7-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+\frac {1485}{4} \int e^{-x} x^7 \, dx-\frac {10395}{4} \int e^{-x} x^6 \, dx+3780 \int e^{-x} x^5 \, dx-5760 \int e^{-x} \, dx+5760 \int e^{-x} x \, dx+10080 \int e^{-x} x^3 \, dx-18900 \int e^{-x} x^4 \, dx-30240 \int e^{-x} x^2 \, dx\\ &=5760 e^{-x}+x+30240 e^{-x} x^2+18884 e^{-x} x^4+\frac {10363}{4} e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+\frac {10395}{4} \int e^{-x} x^6 \, dx+5760 \int e^{-x} \, dx-\frac {31185}{2} \int e^{-x} x^5 \, dx+18900 \int e^{-x} x^4 \, dx+30240 \int e^{-x} x^2 \, dx-60480 \int e^{-x} x \, dx-75600 \int e^{-x} x^3 \, dx\\ &=x+60480 e^{-x} x+75600 e^{-x} x^3-16 e^{-x} x^4+\frac {31185}{2} e^{-x} x^5-8 e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+\frac {31185}{2} \int e^{-x} x^5 \, dx-60480 \int e^{-x} \, dx+60480 \int e^{-x} x \, dx+75600 \int e^{-x} x^3 \, dx-\frac {155925}{2} \int e^{-x} x^4 \, dx-226800 \int e^{-x} x^2 \, dx\\ &=60480 e^{-x}+x+226800 e^{-x} x^2+\frac {155893}{2} e^{-x} x^4-8 e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+60480 \int e^{-x} \, dx+\frac {155925}{2} \int e^{-x} x^4 \, dx+226800 \int e^{-x} x^2 \, dx-311850 \int e^{-x} x^3 \, dx-453600 \int e^{-x} x \, dx\\ &=x+453600 e^{-x} x+311850 e^{-x} x^3-16 e^{-x} x^4-8 e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+311850 \int e^{-x} x^3 \, dx-453600 \int e^{-x} \, dx+453600 \int e^{-x} x \, dx-935550 \int e^{-x} x^2 \, dx\\ &=453600 e^{-x}+x+935550 e^{-x} x^2-16 e^{-x} x^4-8 e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+453600 \int e^{-x} \, dx+935550 \int e^{-x} x^2 \, dx-1871100 \int e^{-x} x \, dx\\ &=x+1871100 e^{-x} x-16 e^{-x} x^4-8 e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}-1871100 \int e^{-x} \, dx+1871100 \int e^{-x} x \, dx\\ &=1871100 e^{-x}+x-16 e^{-x} x^4-8 e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}+1871100 \int e^{-x} \, dx\\ &=x-16 e^{-x} x^4-8 e^{-x} x^6-\frac {3}{2} e^{-x} x^8-\frac {1}{8} e^{-x} x^{10}-\frac {1}{256} e^{-x} x^{12}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.03, size = 28, normalized size = 1.12 \begin {gather*} x-\frac {1}{256} e^{-x} x^3 \left (8+x^2\right )^3 \left (8 x+x^3\right ) \end {gather*}

Integrate[(256*E^x - 16384*x^3 + 4096*x^4 - 12288*x^5 + 2048*x^6 - 3072*x^7 + 384*x^8 - 320*x^9 + 32*x^10 - 12

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fricas [A] time = 0.58, size = 35, normalized size = 1.40 \begin {gather*} -\frac {1}{256} \, {\left (x^{12} + 32 \, x^{10} + 384 \, x^{8} + 2048 \, x^{6} + 4096 \, x^{4} - 256 \, x e^{x}\right )} e^{\left (-x\right )} \end {gather*}

integrate(1/256*(256*exp(x)+x^12-12*x^11+32*x^10-320*x^9+384*x^8-3072*x^7+2048*x^6-12288*x^5+4096*x^4-16384*x^

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giac [A] time = 0.37, size = 32, normalized size = 1.28 \begin {gather*} -\frac {1}{256} \, {\left (x^{12} + 32 \, x^{10} + 384 \, x^{8} + 2048 \, x^{6} + 4096 \, x^{4}\right )} e^{\left (-x\right )} + x \end {gather*}

integrate(1/256*(256*exp(x)+x^12-12*x^11+32*x^10-320*x^9+384*x^8-3072*x^7+2048*x^6-12288*x^5+4096*x^4-16384*x^

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method	result	size

risch	$x +\frac {\left (-x^{12}-32 x^{10}-384 x^{8}-2048 x^{6}-4096 x^{4}\right ) {\mathrm e}^{-x}}{256}$	$35$
norman	$\left ({\mathrm e}^{x} x -16 x^{4}-8 x^{6}-\frac {3 x^{8}}{2}-\frac {x^{10}}{8}-\frac {x^{12}}{256}\right ) {\mathrm e}^{-x}$	$36$
default	$x -16 x^{4} {\mathrm e}^{-x}-\frac {{\mathrm e}^{-x} x^{12}}{256}-\frac {{\mathrm e}^{-x} x^{10}}{8}-\frac {3 \,{\mathrm e}^{-x} x^{8}}{2}-8 \,{\mathrm e}^{-x} x^{6}$	$48$

int(1/256*(256*exp(x)+x^12-12*x^11+32*x^10-320*x^9+384*x^8-3072*x^7+2048*x^6-12288*x^5+4096*x^4-16384*x^3)/exp

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maxima [B] time = 0.41, size = 417, normalized size = 16.68 \begin {gather*} -\frac {1}{256} \, {\left (x^{12} + 12 \, x^{11} + 132 \, x^{10} + 1320 \, x^{9} + 11880 \, x^{8} + 95040 \, x^{7} + 665280 \, x^{6} + 3991680 \, x^{5} + 19958400 \, x^{4} + 79833600 \, x^{3} + 239500800 \, x^{2} + 479001600 \, x + 479001600\right )} e^{\left (-x\right )} + \frac {3}{64} \, {\left (x^{11} + 11 \, x^{10} + 110 \, x^{9} + 990 \, x^{8} + 7920 \, x^{7} + 55440 \, x^{6} + 332640 \, x^{5} + 1663200 \, x^{4} + 6652800 \, x^{3} + 19958400 \, x^{2} + 39916800 \, x + 39916800\right )} e^{\left (-x\right )} - \frac {1}{8} \, {\left (x^{10} + 10 \, x^{9} + 90 \, x^{8} + 720 \, x^{7} + 5040 \, x^{6} + 30240 \, x^{5} + 151200 \, x^{4} + 604800 \, x^{3} + 1814400 \, x^{2} + 3628800 \, x + 3628800\right )} e^{\left (-x\right )} + \frac {5}{4} \, {\left (x^{9} + 9 \, x^{8} + 72 \, x^{7} + 504 \, x^{6} + 3024 \, x^{5} + 15120 \, x^{4} + 60480 \, x^{3} + 181440 \, x^{2} + 362880 \, x + 362880\right )} e^{\left (-x\right )} - \frac {3}{2} \, {\left (x^{8} + 8 \, x^{7} + 56 \, x^{6} + 336 \, x^{5} + 1680 \, x^{4} + 6720 \, x^{3} + 20160 \, x^{2} + 40320 \, x + 40320\right )} e^{\left (-x\right )} + 12 \, {\left (x^{7} + 7 \, x^{6} + 42 \, x^{5} + 210 \, x^{4} + 840 \, x^{3} + 2520 \, x^{2} + 5040 \, x + 5040\right )} e^{\left (-x\right )} - 8 \, {\left (x^{6} + 6 \, x^{5} + 30 \, x^{4} + 120 \, x^{3} + 360 \, x^{2} + 720 \, x + 720\right )} e^{\left (-x\right )} + 48 \, {\left (x^{5} + 5 \, x^{4} + 20 \, x^{3} + 60 \, x^{2} + 120 \, x + 120\right )} e^{\left (-x\right )} - 16 \, {\left (x^{4} + 4 \, x^{3} + 12 \, x^{2} + 24 \, x + 24\right )} e^{\left (-x\right )} + 64 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} + x \end {gather*}

integrate(1/256*(256*exp(x)+x^12-12*x^11+32*x^10-320*x^9+384*x^8-3072*x^7+2048*x^6-12288*x^5+4096*x^4-16384*x^

-1/256*(x^12 + 12*x^11 + 132*x^10 + 1320*x^9 + 11880*x^8 + 95040*x^7 + 665280*x^6 + 3991680*x^5 + 19958400*x^4

+ 79833600*x^3 + 239500800*x^2 + 479001600*x + 479001600)*e^(-x) + 3/64*(x^11 + 11*x^10 + 110*x^9 + 990*x^8 +

7920*x^7 + 55440*x^6 + 332640*x^5 + 1663200*x^4 + 6652800*x^3 + 19958400*x^2 + 39916800*x + 39916800)*e^(-x)

- 1/8*(x^10 + 10*x^9 + 90*x^8 + 720*x^7 + 5040*x^6 + 30240*x^5 + 151200*x^4 + 604800*x^3 + 1814400*x^2 + 36288

00*x + 3628800)*e^(-x) + 5/4*(x^9 + 9*x^8 + 72*x^7 + 504*x^6 + 3024*x^5 + 15120*x^4 + 60480*x^3 + 181440*x^2 +

362880*x + 362880)*e^(-x) - 3/2*(x^8 + 8*x^7 + 56*x^6 + 336*x^5 + 1680*x^4 + 6720*x^3 + 20160*x^2 + 40320*x +

40320)*e^(-x) + 12*(x^7 + 7*x^6 + 42*x^5 + 210*x^4 + 840*x^3 + 2520*x^2 + 5040*x + 5040)*e^(-x) - 8*(x^6 + 6*

x^5 + 30*x^4 + 120*x^3 + 360*x^2 + 720*x + 720)*e^(-x) + 48*(x^5 + 5*x^4 + 20*x^3 + 60*x^2 + 120*x + 120)*e^(-

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mupad [B] time = 0.12, size = 35, normalized size = 1.40 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-x}\,\left (4096\,x^3-256\,{\mathrm {e}}^x+2048\,x^5+384\,x^7+32\,x^9+x^{11}\right )}{256} \end {gather*}

int(exp(-x)*(exp(x) - 64*x^3 + 16*x^4 - 48*x^5 + 8*x^6 - 12*x^7 + (3*x^8)/2 - (5*x^9)/4 + x^10/8 - (3*x^11)/64

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sympy [A] time = 0.14, size = 31, normalized size = 1.24 \begin {gather*} x + \frac {\left (- x^{12} - 32 x^{10} - 384 x^{8} - 2048 x^{6} - 4096 x^{4}\right ) e^{- x}}{256} \end {gather*}

integrate(1/256*(256*exp(x)+x**12-12*x**11+32*x**10-320*x**9+384*x**8-3072*x**7+2048*x**6-12288*x**5+4096*x**4

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3.10.33 \(\int \frac {1}{256} e^{-x} (256 e^x-16384 x^3+4096 x^4-12288 x^5+2048 x^6-3072 x^7+384 x^8-320 x^9+32 x^{10}-12 x^{11}+x^{12}) \, dx\)