Integrand size = 227, antiderivative size = 19 \[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=16 x \left (4+e^{e^{-x} (4+e)}+x\right )^4 \] Output:
16*x*(4+exp((exp(1)+4)/exp(x))+x)^4
Time = 5.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=16 x \left (4+e^{e^{-x} (4+e)}+x\right )^4 \] Input:
Integrate[(E^((4*(4 + E))/E^x)*(16*E^x - 256*x - 64*E*x) + E^x*(4096 + 819 2*x + 4608*x^2 + 1024*x^3 + 80*x^4) + E^((3*(4 + E))/E^x)*(-3072*x - 768*x ^2 + E^x*(256 + 128*x) + E*(-768*x - 192*x^2)) + E^((2*(4 + E))/E^x)*(-122 88*x - 6144*x^2 - 768*x^3 + E^x*(1536 + 1536*x + 288*x^2) + E*(-3072*x - 1 536*x^2 - 192*x^3)) + E^((4 + E)/E^x)*(-16384*x - 12288*x^2 - 3072*x^3 - 2 56*x^4 + E^x*(4096 + 6144*x + 2304*x^2 + 256*x^3) + E*(-4096*x - 3072*x^2 - 768*x^3 - 64*x^4)))/E^x,x]
Output:
16*x*(4 + E^((4 + E)/E^x) + x)^4
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.
| \(\displaystyle \int e^{-x} \left (e^{3 (4+e) e^{-x}} \left (-768 x^2+e \left (-192 x^2-768 x\right )-3072 x+e^x (128 x+256)\right )+e^{2 (4+e) e^{-x}} \left (-768 x^3-6144 x^2+e^x \left (288 x^2+1536 x+1536\right )+e \left (-192 x^3-1536 x^2-3072 x\right )-12288 x\right )+e^x \left (80 x^4+1024 x^3+4608 x^2+8192 x+4096\right )+e^{(4+e) e^{-x}} \left (-256 x^4-3072 x^3-12288 x^2+e^x \left (256 x^3+2304 x^2+6144 x+4096\right )+e \left (-64 x^4-768 x^3-3072 x^2-4096 x\right )-16384 x\right )+e^{4 (4+e) e^{-x}} \left (-64 e x-256 x+16 e^x\right )\right ) \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 16 e^{-x} \left (x+e^{(4+e) e^{-x}}+4\right )^3 \left (-16 \left (1+\frac {e}{4}\right ) e^{(4+e) e^{-x}} x+e^{e^{-x} \left (e^x x+e+4\right )}+e^x (5 x+4)\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 16 \int e^{-x} \left (x+e^{e^{-x} (4+e)}+4\right )^3 \left (-4 e^{e^{-x} (4+e)} (4+e) x+e^{e^{-x} \left (e^x x+e+4\right )}+e^x (5 x+4)\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 16 \int \left (4 (-4-e) e^{e^{-x} (4+e)-x} x \left (x+e^{e^{1-x}+4 e^{-x}}+4\right )^3+\left (5 x+e^{e^{1-x}+4 e^{-x}}+4\right ) \left (x+e^{e^{1-x}+4 e^{-x}}+4\right )^3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 16 \left (-4 (4+e) \int e^{e^{-x} (4+e)-x} x^4dx+16 \int e^{e^{-x} (4+e)} x^3dx-48 (4+e) \int e^{e^{-x} (4+e)-x} x^3dx-12 (4+e) \int e^{2 e^{-x} (4+e)-x} x^3dx+144 \int e^{e^{-x} (4+e)} x^2dx+18 \int e^{2 e^{-x} (4+e)} x^2dx-192 (4+e) \int e^{e^{-x} (4+e)-x} x^2dx-96 (4+e) \int e^{2 e^{-x} (4+e)-x} x^2dx-12 (4+e) \int e^{3 e^{-x} (4+e)-x} x^2dx+384 \int e^{e^{-x} (4+e)} xdx+96 \int e^{2 e^{-x} (4+e)} xdx+8 \int e^{3 e^{-x} (4+e)} xdx-256 (4+e) \int e^{e^{-x} (4+e)-x} xdx-192 (4+e) \int e^{2 e^{-x} (4+e)-x} xdx-48 (4+e) \int e^{3 e^{-x} (4+e)-x} xdx-4 (4+e) \int e^{4 e^{-x} (4+e)-x} xdx-256 \operatorname {ExpIntegralEi}\left (e^{-x} (4+e)\right )-96 \operatorname {ExpIntegralEi}\left (2 e^{-x} (4+e)\right )-16 \operatorname {ExpIntegralEi}\left (3 e^{-x} (4+e)\right )-\operatorname {ExpIntegralEi}\left (4 e^{-x} (4+e)\right )+x (x+4)^4\right )\) |
Input:
Int[(E^((4*(4 + E))/E^x)*(16*E^x - 256*x - 64*E*x) + E^x*(4096 + 8192*x + 4608*x^2 + 1024*x^3 + 80*x^4) + E^((3*(4 + E))/E^x)*(-3072*x - 768*x^2 + E ^x*(256 + 128*x) + E*(-768*x - 192*x^2)) + E^((2*(4 + E))/E^x)*(-12288*x - 6144*x^2 - 768*x^3 + E^x*(1536 + 1536*x + 288*x^2) + E*(-3072*x - 1536*x^ 2 - 192*x^3)) + E^((4 + E)/E^x)*(-16384*x - 12288*x^2 - 3072*x^3 - 256*x^4 + E^x*(4096 + 6144*x + 2304*x^2 + 256*x^3) + E*(-4096*x - 3072*x^2 - 768* x^3 - 64*x^4)))/E^x,x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(18)=36\).
Time = 0.88 (sec) , antiderivative size = 104, normalized size of antiderivative = 5.47
| method | result | size |
| risch | \(16 x^{5}+16 \,{\mathrm e}^{4 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x +256 x^{4}+1536 x^{3}+4096 x^{2}+4096 x +64 \left (4+x \right ) x \,{\mathrm e}^{3 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}}+96 \left (x^{2}+8 x +16\right ) x \,{\mathrm e}^{2 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}}+64 \left (x^{3}+12 x^{2}+48 x +64\right ) x \,{\mathrm e}^{\left ({\mathrm e}+4\right ) {\mathrm e}^{-x}}\) | \(104\) |
| parallelrisch | \(64 \,{\mathrm e}^{\left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x^{4}+16 x^{5}+96 \,{\mathrm e}^{2 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x^{3}+768 \,{\mathrm e}^{\left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x^{3}+256 x^{4}+64 \,{\mathrm e}^{3 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x^{2}+768 \,{\mathrm e}^{2 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x^{2}+3072 \,{\mathrm e}^{\left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x^{2}+1536 x^{3}+256 \,{\mathrm e}^{3 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x +1536 \,{\mathrm e}^{2 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x +4096 \,{\mathrm e}^{\left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x +16 \,{\mathrm e}^{4 \left ({\mathrm e}+4\right ) {\mathrm e}^{-x}} x +4096 x^{2}+4096 x\) | \(179\) |
Input:
int(((16*exp(x)-64*x*exp(1)-256*x)*exp((exp(1)+4)/exp(x))^4+((128*x+256)*e xp(x)+(-192*x^2-768*x)*exp(1)-768*x^2-3072*x)*exp((exp(1)+4)/exp(x))^3+((2 88*x^2+1536*x+1536)*exp(x)+(-192*x^3-1536*x^2-3072*x)*exp(1)-768*x^3-6144* x^2-12288*x)*exp((exp(1)+4)/exp(x))^2+((256*x^3+2304*x^2+6144*x+4096)*exp( x)+(-64*x^4-768*x^3-3072*x^2-4096*x)*exp(1)-256*x^4-3072*x^3-12288*x^2-163 84*x)*exp((exp(1)+4)/exp(x))+(80*x^4+1024*x^3+4608*x^2+8192*x+4096)*exp(x) )/exp(x),x,method=_RETURNVERBOSE)
Output:
16*x^5+16*exp(4*(exp(1)+4)*exp(-x))*x+256*x^4+1536*x^3+4096*x^2+4096*x+64* (4+x)*x*exp(3*(exp(1)+4)*exp(-x))+96*(x^2+8*x+16)*x*exp(2*(exp(1)+4)*exp(- x))+64*(x^3+12*x^2+48*x+64)*x*exp((exp(1)+4)*exp(-x))
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.89 \[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=16 \, x^{5} + 256 \, x^{4} + 1536 \, x^{3} + 4096 \, x^{2} + 16 \, x e^{\left (4 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 64 \, {\left (x^{2} + 4 \, x\right )} e^{\left (3 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 96 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{\left (2 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 64 \, {\left (x^{4} + 12 \, x^{3} + 48 \, x^{2} + 64 \, x\right )} e^{\left ({\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4096 \, x \] Input:
integrate(((16*exp(x)-64*exp(1)*x-256*x)*exp((exp(1)+4)/exp(x))^4+((128*x+ 256)*exp(x)+(-192*x^2-768*x)*exp(1)-768*x^2-3072*x)*exp((exp(1)+4)/exp(x)) ^3+((288*x^2+1536*x+1536)*exp(x)+(-192*x^3-1536*x^2-3072*x)*exp(1)-768*x^3 -6144*x^2-12288*x)*exp((exp(1)+4)/exp(x))^2+((256*x^3+2304*x^2+6144*x+4096 )*exp(x)+(-64*x^4-768*x^3-3072*x^2-4096*x)*exp(1)-256*x^4-3072*x^3-12288*x ^2-16384*x)*exp((exp(1)+4)/exp(x))+(80*x^4+1024*x^3+4608*x^2+8192*x+4096)* exp(x))/exp(x),x, algorithm="fricas")
Output:
16*x^5 + 256*x^4 + 1536*x^3 + 4096*x^2 + 16*x*e^(4*(e + 4)*e^(-x)) + 64*(x ^2 + 4*x)*e^(3*(e + 4)*e^(-x)) + 96*(x^3 + 8*x^2 + 16*x)*e^(2*(e + 4)*e^(- x)) + 64*(x^4 + 12*x^3 + 48*x^2 + 64*x)*e^((e + 4)*e^(-x)) + 4096*x
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (17) = 34\).
Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.89 \[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=16 x^{5} + 256 x^{4} + 1536 x^{3} + 4096 x^{2} + 16 x e^{4 \left (e + 4\right ) e^{- x}} + 4096 x + \left (64 x^{2} + 256 x\right ) e^{3 \left (e + 4\right ) e^{- x}} + \left (96 x^{3} + 768 x^{2} + 1536 x\right ) e^{2 \left (e + 4\right ) e^{- x}} + \left (64 x^{4} + 768 x^{3} + 3072 x^{2} + 4096 x\right ) e^{\left (e + 4\right ) e^{- x}} \] Input:
integrate(((16*exp(x)-64*exp(1)*x-256*x)*exp((exp(1)+4)/exp(x))**4+((128*x +256)*exp(x)+(-192*x**2-768*x)*exp(1)-768*x**2-3072*x)*exp((exp(1)+4)/exp( x))**3+((288*x**2+1536*x+1536)*exp(x)+(-192*x**3-1536*x**2-3072*x)*exp(1)- 768*x**3-6144*x**2-12288*x)*exp((exp(1)+4)/exp(x))**2+((256*x**3+2304*x**2 +6144*x+4096)*exp(x)+(-64*x**4-768*x**3-3072*x**2-4096*x)*exp(1)-256*x**4- 3072*x**3-12288*x**2-16384*x)*exp((exp(1)+4)/exp(x))+(80*x**4+1024*x**3+46 08*x**2+8192*x+4096)*exp(x))/exp(x),x)
Output:
16*x**5 + 256*x**4 + 1536*x**3 + 4096*x**2 + 16*x*exp(4*(E + 4)*exp(-x)) + 4096*x + (64*x**2 + 256*x)*exp(3*(E + 4)*exp(-x)) + (96*x**3 + 768*x**2 + 1536*x)*exp(2*(E + 4)*exp(-x)) + (64*x**4 + 768*x**3 + 3072*x**2 + 4096*x )*exp((E + 4)*exp(-x))
\[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=\int { -16 \, {\left ({\left (4 \, x e + 16 \, x - e^{x}\right )} e^{\left (4 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4 \, {\left (12 \, x^{2} + 3 \, {\left (x^{2} + 4 \, x\right )} e - 2 \, {\left (x + 2\right )} e^{x} + 48 \, x\right )} e^{\left (3 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 6 \, {\left (8 \, x^{3} + 64 \, x^{2} + 2 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e - {\left (3 \, x^{2} + 16 \, x + 16\right )} e^{x} + 128 \, x\right )} e^{\left (2 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4 \, {\left (4 \, x^{4} + 48 \, x^{3} + 192 \, x^{2} + {\left (x^{4} + 12 \, x^{3} + 48 \, x^{2} + 64 \, x\right )} e - 4 \, {\left (x^{3} + 9 \, x^{2} + 24 \, x + 16\right )} e^{x} + 256 \, x\right )} e^{\left ({\left (e + 4\right )} e^{\left (-x\right )}\right )} - {\left (5 \, x^{4} + 64 \, x^{3} + 288 \, x^{2} + 512 \, x + 256\right )} e^{x}\right )} e^{\left (-x\right )} \,d x } \] Input:
integrate(((16*exp(x)-64*exp(1)*x-256*x)*exp((exp(1)+4)/exp(x))^4+((128*x+ 256)*exp(x)+(-192*x^2-768*x)*exp(1)-768*x^2-3072*x)*exp((exp(1)+4)/exp(x)) ^3+((288*x^2+1536*x+1536)*exp(x)+(-192*x^3-1536*x^2-3072*x)*exp(1)-768*x^3 -6144*x^2-12288*x)*exp((exp(1)+4)/exp(x))^2+((256*x^3+2304*x^2+6144*x+4096 )*exp(x)+(-64*x^4-768*x^3-3072*x^2-4096*x)*exp(1)-256*x^4-3072*x^3-12288*x ^2-16384*x)*exp((exp(1)+4)/exp(x))+(80*x^4+1024*x^3+4608*x^2+8192*x+4096)* exp(x))/exp(x),x, algorithm="maxima")
Output:
16*x^5 + 256*x^4 + 1536*x^3 + 4096*x^2 + 4096*x - 4096*Ei((e + 4)*e^(-x)) + 16*integrate(-(4*x*(e + 4) - e^x)*e^(-x + 16*e^(-x) + 4*e^(-x + 1)), x) + 16*integrate(-4*(3*x^2*(e + 4) + 12*x*(e + 4) - 2*(x + 2)*e^x)*e^(-x + 1 2*e^(-x) + 3*e^(-x + 1)), x) + 16*integrate(-6*(2*x^3*(e + 4) + 16*x^2*(e + 4) + 32*x*(e + 4) - (3*x^2 + 16*x + 16)*e^x)*e^(-x + 8*e^(-x) + 2*e^(-x + 1)), x) + 16*integrate(-4*(x^4*(e + 4) + 12*x^3*(e + 4) + 48*x^2*(e + 4) + 64*x*(e + 4) - 4*(x^3 + 9*x^2 + 24*x)*e^x)*e^(-x + 4*e^(-x) + e^(-x + 1 )), x)
\[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=\int { -16 \, {\left ({\left (4 \, x e + 16 \, x - e^{x}\right )} e^{\left (4 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4 \, {\left (12 \, x^{2} + 3 \, {\left (x^{2} + 4 \, x\right )} e - 2 \, {\left (x + 2\right )} e^{x} + 48 \, x\right )} e^{\left (3 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 6 \, {\left (8 \, x^{3} + 64 \, x^{2} + 2 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e - {\left (3 \, x^{2} + 16 \, x + 16\right )} e^{x} + 128 \, x\right )} e^{\left (2 \, {\left (e + 4\right )} e^{\left (-x\right )}\right )} + 4 \, {\left (4 \, x^{4} + 48 \, x^{3} + 192 \, x^{2} + {\left (x^{4} + 12 \, x^{3} + 48 \, x^{2} + 64 \, x\right )} e - 4 \, {\left (x^{3} + 9 \, x^{2} + 24 \, x + 16\right )} e^{x} + 256 \, x\right )} e^{\left ({\left (e + 4\right )} e^{\left (-x\right )}\right )} - {\left (5 \, x^{4} + 64 \, x^{3} + 288 \, x^{2} + 512 \, x + 256\right )} e^{x}\right )} e^{\left (-x\right )} \,d x } \] Input:
integrate(((16*exp(x)-64*exp(1)*x-256*x)*exp((exp(1)+4)/exp(x))^4+((128*x+ 256)*exp(x)+(-192*x^2-768*x)*exp(1)-768*x^2-3072*x)*exp((exp(1)+4)/exp(x)) ^3+((288*x^2+1536*x+1536)*exp(x)+(-192*x^3-1536*x^2-3072*x)*exp(1)-768*x^3 -6144*x^2-12288*x)*exp((exp(1)+4)/exp(x))^2+((256*x^3+2304*x^2+6144*x+4096 )*exp(x)+(-64*x^4-768*x^3-3072*x^2-4096*x)*exp(1)-256*x^4-3072*x^3-12288*x ^2-16384*x)*exp((exp(1)+4)/exp(x))+(80*x^4+1024*x^3+4608*x^2+8192*x+4096)* exp(x))/exp(x),x, algorithm="giac")
Output:
integrate(-16*((4*x*e + 16*x - e^x)*e^(4*(e + 4)*e^(-x)) + 4*(12*x^2 + 3*( x^2 + 4*x)*e - 2*(x + 2)*e^x + 48*x)*e^(3*(e + 4)*e^(-x)) + 6*(8*x^3 + 64* x^2 + 2*(x^3 + 8*x^2 + 16*x)*e - (3*x^2 + 16*x + 16)*e^x + 128*x)*e^(2*(e + 4)*e^(-x)) + 4*(4*x^4 + 48*x^3 + 192*x^2 + (x^4 + 12*x^3 + 48*x^2 + 64*x )*e - 4*(x^3 + 9*x^2 + 24*x + 16)*e^x + 256*x)*e^((e + 4)*e^(-x)) - (5*x^4 + 64*x^3 + 288*x^2 + 512*x + 256)*e^x)*e^(-x), x)
Time = 2.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=16\,x\,{\left (x+{\mathrm {e}}^{4\,{\mathrm {e}}^{-x}+{\mathrm {e}}^{-x}\,\mathrm {e}}+4\right )}^4 \] Input:
int(-exp(-x)*(exp(3*exp(-x)*(exp(1) + 4))*(3072*x + exp(1)*(768*x + 192*x^ 2) - exp(x)*(128*x + 256) + 768*x^2) - exp(x)*(8192*x + 4608*x^2 + 1024*x^ 3 + 80*x^4 + 4096) + exp(exp(-x)*(exp(1) + 4))*(16384*x + exp(1)*(4096*x + 3072*x^2 + 768*x^3 + 64*x^4) + 12288*x^2 + 3072*x^3 + 256*x^4 - exp(x)*(6 144*x + 2304*x^2 + 256*x^3 + 4096)) + exp(4*exp(-x)*(exp(1) + 4))*(256*x - 16*exp(x) + 64*x*exp(1)) + exp(2*exp(-x)*(exp(1) + 4))*(12288*x + exp(1)* (3072*x + 1536*x^2 + 192*x^3) - exp(x)*(1536*x + 288*x^2 + 1536) + 6144*x^ 2 + 768*x^3)),x)
Output:
16*x*(x + exp(4*exp(-x) + exp(-x)*exp(1)) + 4)^4
Time = 0.19 (sec) , antiderivative size = 173, normalized size of antiderivative = 9.11 \[ \int e^{-x} \left (e^{4 e^{-x} (4+e)} \left (16 e^x-256 x-64 e x\right )+e^x \left (4096+8192 x+4608 x^2+1024 x^3+80 x^4\right )+e^{3 e^{-x} (4+e)} \left (-3072 x-768 x^2+e^x (256+128 x)+e \left (-768 x-192 x^2\right )\right )+e^{2 e^{-x} (4+e)} \left (-12288 x-6144 x^2-768 x^3+e^x \left (1536+1536 x+288 x^2\right )+e \left (-3072 x-1536 x^2-192 x^3\right )\right )+e^{e^{-x} (4+e)} \left (-16384 x-12288 x^2-3072 x^3-256 x^4+e^x \left (4096+6144 x+2304 x^2+256 x^3\right )+e \left (-4096 x-3072 x^2-768 x^3-64 x^4\right )\right )\right ) \, dx=16 x \left (e^{\frac {4 e +16}{e^{x}}}+4 e^{\frac {3 e +12}{e^{x}}} x +16 e^{\frac {3 e +12}{e^{x}}}+6 e^{\frac {2 e +8}{e^{x}}} x^{2}+48 e^{\frac {2 e +8}{e^{x}}} x +96 e^{\frac {2 e +8}{e^{x}}}+4 e^{\frac {e +4}{e^{x}}} x^{3}+48 e^{\frac {e +4}{e^{x}}} x^{2}+192 e^{\frac {e +4}{e^{x}}} x +256 e^{\frac {e +4}{e^{x}}}+x^{4}+16 x^{3}+96 x^{2}+256 x +256\right ) \] Input:
int(((16*exp(x)-64*exp(1)*x-256*x)*exp((exp(1)+4)/exp(x))^4+((128*x+256)*e xp(x)+(-192*x^2-768*x)*exp(1)-768*x^2-3072*x)*exp((exp(1)+4)/exp(x))^3+((2 88*x^2+1536*x+1536)*exp(x)+(-192*x^3-1536*x^2-3072*x)*exp(1)-768*x^3-6144* x^2-12288*x)*exp((exp(1)+4)/exp(x))^2+((256*x^3+2304*x^2+6144*x+4096)*exp( x)+(-64*x^4-768*x^3-3072*x^2-4096*x)*exp(1)-256*x^4-3072*x^3-12288*x^2-163 84*x)*exp((exp(1)+4)/exp(x))+(80*x^4+1024*x^3+4608*x^2+8192*x+4096)*exp(x) )/exp(x),x)
Output:
16*x*(e**((4*e + 16)/e**x) + 4*e**((3*e + 12)/e**x)*x + 16*e**((3*e + 12)/ e**x) + 6*e**((2*e + 8)/e**x)*x**2 + 48*e**((2*e + 8)/e**x)*x + 96*e**((2* e + 8)/e**x) + 4*e**((e + 4)/e**x)*x**3 + 48*e**((e + 4)/e**x)*x**2 + 192* e**((e + 4)/e**x)*x + 256*e**((e + 4)/e**x) + x**4 + 16*x**3 + 96*x**2 + 2 56*x + 256)