3.25.22 \(\int \frac {e^{2 e^4} (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 (-4+x^2))}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 (-128 x^3+128 x^4-32 x^5)} \, dx\) [2422]

3.25.22.1 Optimal result

Integrand size = 96, antiderivative size = 27 \[ \int \frac {e^{2 e^4} \left (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx=\frac {e^{2 e^4}}{\left (-16+\frac {e^8}{(-2+x)^2 x}\right ) x^2} \]

output

exp(2*exp(4))/(exp(4)^2/(-2+x)^2/x-16)/x^2
 

3.25.22.2 Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {e^{2 e^4} \left (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx=-\frac {e^{2 e^4} (-2+x)^2}{x \left (-e^8+16 (-2+x)^2 x\right )} \]

input

Integrate[(E^(2*E^4)*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + E^8* 
(-4 + x^2)))/(E^16*x^2 + 4096*x^4 - 8192*x^5 + 6144*x^6 - 2048*x^7 + 256*x 
^8 + E^8*(-128*x^3 + 128*x^4 - 32*x^5)),x]
 

output

-((E^(2*E^4)*(-2 + x)^2)/(x*(-E^8 + 16*(-2 + x)^2*x)))
 

3.25.22.3 Rubi [F]

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.

\(\displaystyle \int \frac {e^{2 e^4} \left (32 x^5-256 x^4+768 x^3-1024 x^2+e^8 \left (x^2-4\right )+512 x\right )}{256 x^8-2048 x^7+6144 x^6-8192 x^5+4096 x^4+e^{16} x^2+e^8 \left (-32 x^5+128 x^4-128 x^3\right )} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle e^{2 e^4} \int \frac {32 x^5-256 x^4+768 x^3-1024 x^2+512 x-e^8 \left (4-x^2\right )}{256 x^8-2048 x^7+6144 x^6-8192 x^5+4096 x^4+e^{16} x^2-32 e^8 \left (x^5-4 x^4+4 x^3\right )}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle e^{2 e^4} \int \frac {32 x^5-256 x^4+768 x^3-1024 x^2+512 x-e^8 \left (4-x^2\right )}{x^2 \left (256 x^6-2048 x^5+6144 x^4-32 \left (256+e^8\right ) x^3+128 \left (32+e^8\right ) x^2-128 e^8 x+e^{16}\right )}dx\)

\(\Big \downarrow \) 2462

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle e^{2 e^4} \int \left (-\frac {2 \left (32 x+e^8-128\right )}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )}-\frac {4}{e^8 x^2}+\frac {128 \left (32-e^8\right ) x^2-64 \left (256-9 e^8\right ) x+3 e^{16}-768 e^8+16384}{e^8 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle e^{2 e^4} \int \frac {(2-x) \left (-32 x^4+192 x^3-384 x^2+\left (256-e^8\right ) x-2 e^8\right )}{x^2 \left (-16 x^3+64 x^2-64 x+e^8\right )^2}dx\)

input

Int[(E^(2*E^4)*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + E^8*(-4 + 
x^2)))/(E^16*x^2 + 4096*x^4 - 8192*x^5 + 6144*x^6 - 2048*x^7 + 256*x^8 + E 
^8*(-128*x^3 + 128*x^4 - 32*x^5)),x]
 

output

$Aborted
 

3.25.22.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 2462

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u*Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ 
[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 0 
] && RationalFunctionQ[u, x]
 

rule 7239

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

rule 7293

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

3.25.22.4 Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30

method	result	size
gosper	\(\frac {\left (-2+x \right )^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}}{x \left (-16 x^{3}+{\mathrm e}^{8}+64 x^{2}-64 x \right )}\)	\(35\)
risch	\(\frac {{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (x^{2}-4 x +4\right )}{x \left (-16 x^{3}+{\mathrm e}^{8}+64 x^{2}-64 x \right )}\)	\(36\)
parallelrisch	\(-\frac {{\mathrm e}^{2 \,{\mathrm e}^{4}} \left (-16 x^{2}+64 x -64\right )}{16 x \left (-16 x^{3}+{\mathrm e}^{8}+64 x^{2}-64 x \right )}\)	\(41\)
norman	\(\frac {x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}}-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x +4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}}{x \left (-16 x^{3}+{\mathrm e}^{8}+64 x^{2}-64 x \right )}\)	\(50\)

input

int(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*exp(4)) 
/(x^2*exp(4)^4+(-32*x^5+128*x^4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6144*x^ 
6-8192*x^5+4096*x^4),x,method=_RETURNVERBOSE)
 

output

1/x*(-2+x)^2*exp(2*exp(4))/(-16*x^3+exp(4)^2+64*x^2-64*x)
 

3.25.22.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{2 e^4} \left (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx=-\frac {{\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e^{4}\right )}}{16 \, x^{4} - 64 \, x^{3} + 64 \, x^{2} - x e^{8}} \]

input

integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*e 
xp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6 
144*x^6-8192*x^5+4096*x^4),x, algorithm=\
 

output

-(x^2 - 4*x + 4)*e^(2*e^4)/(16*x^4 - 64*x^3 + 64*x^2 - x*e^8)
 

3.25.22.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).

Time = 0.90 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {e^{2 e^4} \left (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx=\frac {- x^{2} e^{2 e^{4}} + 4 x e^{2 e^{4}} - 4 e^{2 e^{4}}}{16 x^{4} - 64 x^{3} + 64 x^{2} - x e^{8}} \]

input

integrate(((x**2-4)*exp(4)**2+32*x**5-256*x**4+768*x**3-1024*x**2+512*x)*e 
xp(2*exp(4))/(x**2*exp(4)**4+(-32*x**5+128*x**4-128*x**3)*exp(4)**2+256*x* 
*8-2048*x**7+6144*x**6-8192*x**5+4096*x**4),x)
 

output

(-x**2*exp(2*exp(4)) + 4*x*exp(2*exp(4)) - 4*exp(2*exp(4)))/(16*x**4 - 64* 
x**3 + 64*x**2 - x*exp(8))
 

3.25.22.7 Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{2 e^4} \left (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx=-\frac {{\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e^{4}\right )}}{16 \, x^{4} - 64 \, x^{3} + 64 \, x^{2} - x e^{8}} \]

input

integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*e 
xp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6 
144*x^6-8192*x^5+4096*x^4),x, algorithm=\
 

output

-(x^2 - 4*x + 4)*e^(2*e^4)/(16*x^4 - 64*x^3 + 64*x^2 - x*e^8)
 

3.25.22.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{2 e^4} \left (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx=-\frac {{\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, e^{4}\right )}}{16 \, x^{4} - 64 \, x^{3} + 64 \, x^{2} - x e^{8}} \]

input

integrate(((x^2-4)*exp(4)^2+32*x^5-256*x^4+768*x^3-1024*x^2+512*x)*exp(2*e 
xp(4))/(x^2*exp(4)^4+(-32*x^5+128*x^4-128*x^3)*exp(4)^2+256*x^8-2048*x^7+6 
144*x^6-8192*x^5+4096*x^4),x, algorithm=\
 

output

-(x^2 - 4*x + 4)*e^(2*e^4)/(16*x^4 - 64*x^3 + 64*x^2 - x*e^8)
 

3.25.22.9 Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 e^4} \left (512 x-1024 x^2+768 x^3-256 x^4+32 x^5+e^8 \left (-4+x^2\right )\right )}{e^{16} x^2+4096 x^4-8192 x^5+6144 x^6-2048 x^7+256 x^8+e^8 \left (-128 x^3+128 x^4-32 x^5\right )} \, dx=\text {Hanged} \]

input

int((exp(2*exp(4))*(512*x - 1024*x^2 + 768*x^3 - 256*x^4 + 32*x^5 + exp(8) 
*(x^2 - 4)))/(x^2*exp(16) - exp(8)*(128*x^3 - 128*x^4 + 32*x^5) + 4096*x^4 
 - 8192*x^5 + 6144*x^6 - 2048*x^7 + 256*x^8),x)
 

output

\text{Hanged}