\(\int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} (-8+64 x^4))}{x^5} \, dx\) [9579]

Optimal result

Integrand size = 70, antiderivative size = 22 \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx=e^{2 \left (-\frac {e^{4 x^4}}{x^2}+x^2\right )^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {6838} \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx=e^{\frac {2 \left (x^8-2 e^{4 x^4} x^4+e^{8 x^4}\right )}{x^4}} \]

[In]

Int[(E^((2*E^(8*x^4) - 4*E^(4*x^4)*x^4 + 2*x^8)/x^4)*(8*x^8 - 64*E^(4*x^4)*x^8 + E^(8*x^4)*(-8 + 64*x^4)))/x^5
,x]

[Out]

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

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = e^{\frac {2 \left (e^{8 x^4}-2 e^{4 x^4} x^4+x^8\right )}{x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx=e^{\frac {2 \left (e^{4 x^4}-x^4\right )^2}{x^4}} \]

[In]

Integrate[(E^((2*E^(8*x^4) - 4*E^(4*x^4)*x^4 + 2*x^8)/x^4)*(8*x^8 - 64*E^(4*x^4)*x^8 + E^(8*x^4)*(-8 + 64*x^4)
))/x^5,x]

[Out]

E^((2*(E^(4*x^4) - x^4)^2)/x^4)

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27

method	result	size
risch	\({\mathrm e}^{\frac {2 \,{\mathrm e}^{8 x^{4}}-4 x^{4} {\mathrm e}^{4 x^{4}}+2 x^{8}}{x^{4}}}\)	\(28\)
parallelrisch	\({\mathrm e}^{\frac {2 \,{\mathrm e}^{8 x^{4}}-4 x^{4} {\mathrm e}^{4 x^{4}}+2 x^{8}}{x^{4}}}\)	\(30\)

[In]

int(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x^5,x,m
ethod=_RETURNVERBOSE)

[Out]

exp(2*(x^8-2*x^4*exp(4*x^4)+exp(8*x^4))/x^4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx=e^{\left (\frac {2 \, {\left (x^{8} - 2 \, x^{4} e^{\left (4 \, x^{4}\right )} + e^{\left (8 \, x^{4}\right )}\right )}}{x^{4}}\right )} \]

[In]

integrate(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x
^5,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx=e^{\frac {2 x^{8} - 4 x^{4} e^{4 x^{4}} + 2 e^{8 x^{4}}}{x^{4}}} \]

[In]

integrate(((64*x**4-8)*exp(4*x**4)**2-64*x**8*exp(4*x**4)+8*x**8)*exp((2*exp(4*x**4)**2-4*x**4*exp(4*x**4)+2*x
**8)/x**4)/x**5,x)

[Out]

exp((2*x**8 - 4*x**4*exp(4*x**4) + 2*exp(8*x**4))/x**4)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx=e^{\left (2 \, x^{4} + \frac {2 \, e^{\left (8 \, x^{4}\right )}}{x^{4}} - 4 \, e^{\left (4 \, x^{4}\right )}\right )} \]

[In]

integrate(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x
^5,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx=e^{\left (2 \, x^{4} + \frac {2 \, e^{\left (8 \, x^{4}\right )}}{x^{4}} - 4 \, e^{\left (4 \, x^{4}\right )}\right )} \]

[In]

integrate(((64*x^4-8)*exp(4*x^4)^2-64*x^8*exp(4*x^4)+8*x^8)*exp((2*exp(4*x^4)^2-4*x^4*exp(4*x^4)+2*x^8)/x^4)/x
^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.54 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\frac {2 e^{8 x^4}-4 e^{4 x^4} x^4+2 x^8}{x^4}} \left (8 x^8-64 e^{4 x^4} x^8+e^{8 x^4} \left (-8+64 x^4\right )\right )}{x^5} \, dx={\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{8\,x^4}}{x^4}}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^{4\,x^4}}\,{\mathrm {e}}^{2\,x^4} \]

[In]

int((exp((2*exp(8*x^4) - 4*x^4*exp(4*x^4) + 2*x^8)/x^4)*(exp(8*x^4)*(64*x^4 - 8) - 64*x^8*exp(4*x^4) + 8*x^8))
/x^5,x)

[Out]

exp((2*exp(8*x^4))/x^4)*exp(-4*exp(4*x^4))*exp(2*x^4)