Integrand size = 137, antiderivative size = 24 \[ \int \frac {32 e^4 x^3+2 x^4+e^{32-16 x+2 x^2} \left (-16 x^3+4 x^4+e^8 \left (-512-4096 x+1024 x^2\right )+e^4 \left (-32 x-512 x^2+128 x^3\right )\right )+e^{16-8 x+x^2} \left (2 x^3-16 x^4+4 x^5+e^8 \left (-512 x-4096 x^2+1024 x^3\right )+e^4 \left (-512 x^3+128 x^4\right )\right )}{e^8 x^3} \, dx=\frac {\left (e^{(-4+x)^2}+x\right )^2 \left (16+\frac {x}{e^4}\right )^2}{x^2} \]
Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(24)=48\).
Time = 6.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \frac {32 e^4 x^3+2 x^4+e^{32-16 x+2 x^2} \left (-16 x^3+4 x^4+e^8 \left (-512-4096 x+1024 x^2\right )+e^4 \left (-32 x-512 x^2+128 x^3\right )\right )+e^{16-8 x+x^2} \left (2 x^3-16 x^4+4 x^5+e^8 \left (-512 x-4096 x^2+1024 x^3\right )+e^4 \left (-512 x^3+128 x^4\right )\right )}{e^8 x^3} \, dx=\frac {32 e^4 x^3+x^4+e^{2 (-4+x)^2} \left (16 e^4+x\right )^2+2 e^{(-4+x)^2} x \left (16 e^4+x\right )^2}{e^8 x^2} \]
Integrate[(32*E^4*x^3 + 2*x^4 + E^(32 - 16*x + 2*x^2)*(-16*x^3 + 4*x^4 + E ^8*(-512 - 4096*x + 1024*x^2) + E^4*(-32*x - 512*x^2 + 128*x^3)) + E^(16 - 8*x + x^2)*(2*x^3 - 16*x^4 + 4*x^5 + E^8*(-512*x - 4096*x^2 + 1024*x^3) + E^4*(-512*x^3 + 128*x^4)))/(E^8*x^3),x]
(32*E^4*x^3 + x^4 + E^(2*(-4 + x)^2)*(16*E^4 + x)^2 + 2*E^(-4 + x)^2*x*(16 *E^4 + x)^2)/(E^8*x^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.85 (sec) , antiderivative size = 211, normalized size of antiderivative = 8.79, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {27, 27, 2010, 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.
\(\displaystyle \int \frac {2 x^4+32 e^4 x^3+e^{2 x^2-16 x+32} \left (4 x^4-16 x^3+e^8 \left (1024 x^2-4096 x-512\right )+e^4 \left (128 x^3-512 x^2-32 x\right )\right )+e^{x^2-8 x+16} \left (4 x^5-16 x^4+2 x^3+e^4 \left (128 x^4-512 x^3\right )+e^8 \left (1024 x^3-4096 x^2-512 x\right )\right )}{e^8 x^3} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {2 \left (x^4+16 e^4 x^3-2 e^{2 x^2-16 x+32} \left (-x^4+4 x^3+128 e^8 \left (-2 x^2+8 x+1\right )+8 e^4 \left (-4 x^3+16 x^2+x\right )\right )+e^{x^2-8 x+16} \left (2 x^5-8 x^4+x^3-256 e^8 \left (-2 x^3+8 x^2+x\right )-64 e^4 \left (4 x^3-x^4\right )\right )\right )}{x^3}dx}{e^8}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \int \frac {x^4+16 e^4 x^3-2 e^{2 x^2-16 x+32} \left (-x^4+4 x^3+128 e^8 \left (-2 x^2+8 x+1\right )+8 e^4 \left (-4 x^3+16 x^2+x\right )\right )+e^{x^2-8 x+16} \left (2 x^5-8 x^4+x^3-256 e^8 \left (-2 x^3+8 x^2+x\right )-64 e^4 \left (4 x^3-x^4\right )\right )}{x^3}dx}{e^8}\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \frac {2 \int \left (x+16 e^4+\frac {e^{x^2-8 x+16} \left (x+16 e^4\right ) \left (2 x^3-8 \left (1-4 e^4\right ) x^2+\left (1-128 e^4\right ) x-16 e^4\right )}{x^2}+\frac {2 e^{2 (x-4)^2} \left (x+16 e^4\right ) \left (x^3-4 \left (1-4 e^4\right ) x^2-64 e^4 x-8 e^4\right )}{x^3}\right )dx}{e^8}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (-\frac {1}{2} \left (1-256 e^4+512 e^8\right ) \sqrt {\pi } \text {erfi}(4-x)+16 \left (1-8 e^4\right ) \sqrt {\pi } \text {erfi}(4-x)+256 e^8 \sqrt {\pi } \text {erfi}(4-x)-\frac {31}{2} \sqrt {\pi } \text {erfi}(4-x)+\frac {x^2}{2}+e^{x^2-8 x+16} x+4 e^{x^2-8 x+16}-4 \left (1-8 e^4\right ) e^{x^2-8 x+16}+\frac {256 e^{x^2-8 x+24}}{x}+\frac {e^{2 (4-x)^2} \left (x+16 e^4\right ) \left (-x^3+4 \left (1-4 e^4\right ) x^2+64 e^4 x\right )}{2 (4-x) x^3}+16 e^4 x\right )}{e^8}\) |
Int[(32*E^4*x^3 + 2*x^4 + E^(32 - 16*x + 2*x^2)*(-16*x^3 + 4*x^4 + E^8*(-5 12 - 4096*x + 1024*x^2) + E^4*(-32*x - 512*x^2 + 128*x^3)) + E^(16 - 8*x + x^2)*(2*x^3 - 16*x^4 + 4*x^5 + E^8*(-512*x - 4096*x^2 + 1024*x^3) + E^4*( -512*x^3 + 128*x^4)))/(E^8*x^3),x]
(2*(4*E^(16 - 8*x + x^2) - 4*E^(16 - 8*x + x^2)*(1 - 8*E^4) + (256*E^(24 - 8*x + x^2))/x + 16*E^4*x + E^(16 - 8*x + x^2)*x + x^2/2 + (E^(2*(4 - x)^2 )*(16*E^4 + x)*(64*E^4*x + 4*(1 - 4*E^4)*x^2 - x^3))/(2*(4 - x)*x^3) - (31 *Sqrt[Pi]*Erfi[4 - x])/2 + 256*E^8*Sqrt[Pi]*Erfi[4 - x] + 16*(1 - 8*E^4)*S qrt[Pi]*Erfi[4 - x] - ((1 - 256*E^4 + 512*E^8)*Sqrt[Pi]*Erfi[4 - x])/2))/E ^8
3.4.64.3.1 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]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(24)=48\).
Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.75
method | result | size |
risch | \(32 x \,{\mathrm e}^{-4}+{\mathrm e}^{-8} x^{2}+\frac {\left (256 \,{\mathrm e}^{8}+32 x \,{\mathrm e}^{4}+x^{2}\right ) {\mathrm e}^{2 \left (-2+x \right ) \left (-6+x \right )}}{x^{2}}+\frac {2 \left (256 \,{\mathrm e}^{8}+32 x \,{\mathrm e}^{4}+x^{2}\right ) {\mathrm e}^{x^{2}-8 x +8}}{x}\) | \(66\) |
parallelrisch | \(\frac {{\mathrm e}^{-8} \left (512 \,{\mathrm e}^{x^{2}-8 x +16} {\mathrm e}^{8} x +64 \,{\mathrm e}^{4} {\mathrm e}^{x^{2}-8 x +16} x^{2}+2 \,{\mathrm e}^{x^{2}-8 x +16} x^{3}+32 x^{3} {\mathrm e}^{4}+x^{4}+256 \,{\mathrm e}^{8} {\mathrm e}^{2 x^{2}-16 x +32}+32 \,{\mathrm e}^{4} {\mathrm e}^{2 x^{2}-16 x +32} x +x^{2} {\mathrm e}^{2 x^{2}-16 x +32}\right )}{x^{2}}\) | \(120\) |
norman | \(\frac {\left ({\mathrm e}^{-2} x^{4}+x^{2} {\mathrm e}^{-2} {\mathrm e}^{2 x^{2}-16 x +32}+32 x^{3} {\mathrm e}^{2}+256 \,{\mathrm e}^{6} {\mathrm e}^{2 x^{2}-16 x +32}+512 x \,{\mathrm e}^{6} {\mathrm e}^{x^{2}-8 x +16}+64 x^{2} {\mathrm e}^{2} {\mathrm e}^{x^{2}-8 x +16}+2 \,{\mathrm e}^{-2} x^{3} {\mathrm e}^{x^{2}-8 x +16}+32 \,{\mathrm e}^{2} x \,{\mathrm e}^{2 x^{2}-16 x +32}\right ) {\mathrm e}^{-6}}{x^{2}}\) | \(127\) |
parts | \(2 \,{\mathrm e}^{-8} \left (16 x \,{\mathrm e}^{4}+\frac {x^{2}}{2}\right )+\frac {\left (512 \,{\mathrm e}^{6} {\mathrm e}^{x^{2}-8 x +16}+2 \,{\mathrm e}^{-2} x^{2} {\mathrm e}^{x^{2}-8 x +16}+64 \,{\mathrm e}^{2} x \,{\mathrm e}^{x^{2}-8 x +16}\right ) {\mathrm e}^{-6}}{x}+\frac {\left (x^{2} {\mathrm e}^{-2} {\mathrm e}^{2 x^{2}-16 x +32}+256 \,{\mathrm e}^{6} {\mathrm e}^{2 x^{2}-16 x +32}+32 \,{\mathrm e}^{2} x \,{\mathrm e}^{2 x^{2}-16 x +32}\right ) {\mathrm e}^{-6}}{x^{2}}\) | \(138\) |
int((((1024*x^2-4096*x-512)*exp(2)^4+(128*x^3-512*x^2-32*x)*exp(2)^2+4*x^4 -16*x^3)*exp(x^2-8*x+16)^2+((1024*x^3-4096*x^2-512*x)*exp(2)^4+(128*x^4-51 2*x^3)*exp(2)^2+4*x^5-16*x^4+2*x^3)*exp(x^2-8*x+16)+32*x^3*exp(2)^2+2*x^4) /x^3/exp(2)^4,x,method=_RETURNVERBOSE)
32*x*exp(-4)+exp(-8)*x^2+1/x^2*(256*exp(8)+32*x*exp(4)+x^2)*exp(2*(-2+x)*( -6+x))+2/x*(256*exp(8)+32*x*exp(4)+x^2)*exp(x^2-8*x+8)
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.88 \[ \int \frac {32 e^4 x^3+2 x^4+e^{32-16 x+2 x^2} \left (-16 x^3+4 x^4+e^8 \left (-512-4096 x+1024 x^2\right )+e^4 \left (-32 x-512 x^2+128 x^3\right )\right )+e^{16-8 x+x^2} \left (2 x^3-16 x^4+4 x^5+e^8 \left (-512 x-4096 x^2+1024 x^3\right )+e^4 \left (-512 x^3+128 x^4\right )\right )}{e^8 x^3} \, dx=\frac {{\left (x^{4} + 32 \, x^{3} e^{4} + {\left (x^{2} + 32 \, x e^{4} + 256 \, e^{8}\right )} e^{\left (2 \, x^{2} - 16 \, x + 32\right )} + 2 \, {\left (x^{3} + 32 \, x^{2} e^{4} + 256 \, x e^{8}\right )} e^{\left (x^{2} - 8 \, x + 16\right )}\right )} e^{\left (-8\right )}}{x^{2}} \]
integrate((((1024*x^2-4096*x-512)*exp(2)^4+(128*x^3-512*x^2-32*x)*exp(2)^2 +4*x^4-16*x^3)*exp(x^2-8*x+16)^2+((1024*x^3-4096*x^2-512*x)*exp(2)^4+(128* x^4-512*x^3)*exp(2)^2+4*x^5-16*x^4+2*x^3)*exp(x^2-8*x+16)+32*x^3*exp(2)^2+ 2*x^4)/x^3/exp(2)^4,x, algorithm=\
(x^4 + 32*x^3*e^4 + (x^2 + 32*x*e^4 + 256*e^8)*e^(2*x^2 - 16*x + 32) + 2*( x^3 + 32*x^2*e^4 + 256*x*e^8)*e^(x^2 - 8*x + 16))*e^(-8)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.67 \[ \int \frac {32 e^4 x^3+2 x^4+e^{32-16 x+2 x^2} \left (-16 x^3+4 x^4+e^8 \left (-512-4096 x+1024 x^2\right )+e^4 \left (-32 x-512 x^2+128 x^3\right )\right )+e^{16-8 x+x^2} \left (2 x^3-16 x^4+4 x^5+e^8 \left (-512 x-4096 x^2+1024 x^3\right )+e^4 \left (-512 x^3+128 x^4\right )\right )}{e^8 x^3} \, dx=\frac {x^{2}}{e^{8}} + \frac {32 x}{e^{4}} + \frac {\left (x^{3} e^{8} + 32 x^{2} e^{12} + 256 x e^{16}\right ) e^{2 x^{2} - 16 x + 32} + \left (2 x^{4} e^{8} + 64 x^{3} e^{12} + 512 x^{2} e^{16}\right ) e^{x^{2} - 8 x + 16}}{x^{3} e^{16}} \]
integrate((((1024*x**2-4096*x-512)*exp(2)**4+(128*x**3-512*x**2-32*x)*exp( 2)**2+4*x**4-16*x**3)*exp(x**2-8*x+16)**2+((1024*x**3-4096*x**2-512*x)*exp (2)**4+(128*x**4-512*x**3)*exp(2)**2+4*x**5-16*x**4+2*x**3)*exp(x**2-8*x+1 6)+32*x**3*exp(2)**2+2*x**4)/x**3/exp(2)**4,x)
x**2*exp(-8) + 32*x*exp(-4) + ((x**3*exp(8) + 32*x**2*exp(12) + 256*x*exp( 16))*exp(2*x**2 - 16*x + 32) + (2*x**4*exp(8) + 64*x**3*exp(12) + 512*x**2 *exp(16))*exp(x**2 - 8*x + 16))*exp(-16)/x**3
\[ \int \frac {32 e^4 x^3+2 x^4+e^{32-16 x+2 x^2} \left (-16 x^3+4 x^4+e^8 \left (-512-4096 x+1024 x^2\right )+e^4 \left (-32 x-512 x^2+128 x^3\right )\right )+e^{16-8 x+x^2} \left (2 x^3-16 x^4+4 x^5+e^8 \left (-512 x-4096 x^2+1024 x^3\right )+e^4 \left (-512 x^3+128 x^4\right )\right )}{e^8 x^3} \, dx=\int { \frac {2 \, {\left (x^{4} + 16 \, x^{3} e^{4} + 2 \, {\left (x^{4} - 4 \, x^{3} + 128 \, {\left (2 \, x^{2} - 8 \, x - 1\right )} e^{8} + 8 \, {\left (4 \, x^{3} - 16 \, x^{2} - x\right )} e^{4}\right )} e^{\left (2 \, x^{2} - 16 \, x + 32\right )} + {\left (2 \, x^{5} - 8 \, x^{4} + x^{3} + 256 \, {\left (2 \, x^{3} - 8 \, x^{2} - x\right )} e^{8} + 64 \, {\left (x^{4} - 4 \, x^{3}\right )} e^{4}\right )} e^{\left (x^{2} - 8 \, x + 16\right )}\right )} e^{\left (-8\right )}}{x^{3}} \,d x } \]
integrate((((1024*x^2-4096*x-512)*exp(2)^4+(128*x^3-512*x^2-32*x)*exp(2)^2 +4*x^4-16*x^3)*exp(x^2-8*x+16)^2+((1024*x^3-4096*x^2-512*x)*exp(2)^4+(128* x^4-512*x^3)*exp(2)^2+4*x^5-16*x^4+2*x^3)*exp(x^2-8*x+16)+32*x^3*exp(2)^2+ 2*x^4)/x^3/exp(2)^4,x, algorithm=\
-(512*I*sqrt(pi)*erf(I*x - 4*I)*e^8 - 256*I*sqrt(pi)*erf(I*x - 4*I)*e^4 + 2*(x - 4)^3*gamma(3/2, -(x - 4)^2)/(-(x - 4)^2)^(3/2) - x^2 - 64*(4*sqrt(p i)*(x - 4)*(erf(sqrt(-(x - 4)^2)) - 1)/sqrt(-(x - 4)^2) + e^((x - 4)^2))*e ^4 - 32*x*e^4 + I*sqrt(pi)*erf(I*x - 4*I) - (x^2*e^32 + 32*x*e^36 + 256*e^ 40)*e^(2*x^2 - 16*x)/x^2 - 8*e^((x - 4)^2) + 2*integrate(256*(8*x*e^24 + e ^24)*e^(x^2 - 8*x)/x^2, x))*e^(-8)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (22) = 44\).
Time = 0.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.12 \[ \int \frac {32 e^4 x^3+2 x^4+e^{32-16 x+2 x^2} \left (-16 x^3+4 x^4+e^8 \left (-512-4096 x+1024 x^2\right )+e^4 \left (-32 x-512 x^2+128 x^3\right )\right )+e^{16-8 x+x^2} \left (2 x^3-16 x^4+4 x^5+e^8 \left (-512 x-4096 x^2+1024 x^3\right )+e^4 \left (-512 x^3+128 x^4\right )\right )}{e^8 x^3} \, dx=\frac {{\left (x^{4} + 32 \, x^{3} e^{4} + 2 \, x^{3} e^{\left (x^{2} - 8 \, x + 16\right )} + x^{2} e^{\left (2 \, x^{2} - 16 \, x + 32\right )} + 64 \, x^{2} e^{\left (x^{2} - 8 \, x + 20\right )} + 32 \, x e^{\left (2 \, x^{2} - 16 \, x + 36\right )} + 512 \, x e^{\left (x^{2} - 8 \, x + 24\right )} + 256 \, e^{\left (2 \, x^{2} - 16 \, x + 40\right )}\right )} e^{\left (-8\right )}}{x^{2}} \]
integrate((((1024*x^2-4096*x-512)*exp(2)^4+(128*x^3-512*x^2-32*x)*exp(2)^2 +4*x^4-16*x^3)*exp(x^2-8*x+16)^2+((1024*x^3-4096*x^2-512*x)*exp(2)^4+(128* x^4-512*x^3)*exp(2)^2+4*x^5-16*x^4+2*x^3)*exp(x^2-8*x+16)+32*x^3*exp(2)^2+ 2*x^4)/x^3/exp(2)^4,x, algorithm=\
(x^4 + 32*x^3*e^4 + 2*x^3*e^(x^2 - 8*x + 16) + x^2*e^(2*x^2 - 16*x + 32) + 64*x^2*e^(x^2 - 8*x + 20) + 32*x*e^(2*x^2 - 16*x + 36) + 512*x*e^(x^2 - 8 *x + 24) + 256*e^(2*x^2 - 16*x + 40))*e^(-8)/x^2
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 4.58 \[ \int \frac {32 e^4 x^3+2 x^4+e^{32-16 x+2 x^2} \left (-16 x^3+4 x^4+e^8 \left (-512-4096 x+1024 x^2\right )+e^4 \left (-32 x-512 x^2+128 x^3\right )\right )+e^{16-8 x+x^2} \left (2 x^3-16 x^4+4 x^5+e^8 \left (-512 x-4096 x^2+1024 x^3\right )+e^4 \left (-512 x^3+128 x^4\right )\right )}{e^8 x^3} \, dx={\mathrm {e}}^{-16\,x-8}\,\left (64\,{\mathrm {e}}^{x^2+8\,x+20}+{\mathrm {e}}^{2\,x^2+32}\right )+x^2\,{\mathrm {e}}^{-8}+\frac {256\,{\mathrm {e}}^{2\,x^2-16\,x+32}+x\,{\mathrm {e}}^{-16\,x-8}\,\left (512\,{\mathrm {e}}^{x^2+8\,x+24}+32\,{\mathrm {e}}^{2\,x^2+36}\right )}{x^2}+x\,{\mathrm {e}}^{-16\,x-8}\,\left (2\,{\mathrm {e}}^{x^2+8\,x+16}+32\,{\mathrm {e}}^{16\,x+4}\right ) \]
int(-(exp(-8)*(exp(2*x^2 - 16*x + 32)*(exp(8)*(4096*x - 1024*x^2 + 512) + exp(4)*(32*x + 512*x^2 - 128*x^3) + 16*x^3 - 4*x^4) - 32*x^3*exp(4) + exp( x^2 - 8*x + 16)*(exp(8)*(512*x + 4096*x^2 - 1024*x^3) + exp(4)*(512*x^3 - 128*x^4) - 2*x^3 + 16*x^4 - 4*x^5) - 2*x^4))/x^3,x)