\(\int \frac {e^{-\frac {2 e^8}{x^4}} (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} (12 e^8 x+3 x^5))}{3 x^5} \, dx\) [6540]

Optimal result

Integrand size = 59, antiderivative size = 20 \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=\frac {1}{6} \left (3 e^{-\frac {e^8}{x^4}}+x\right )^2 \]

[Out]

1/6*(3/exp(exp(1)^8/x^4)+x)^2

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {12, 6873, 6874, 2240, 2326} \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=e^{-\frac {e^8}{x^4}} x+\frac {3}{2} e^{-\frac {2 e^8}{x^4}}+\frac {x^2}{6} \]

[In]

Int[(36*E^8 + E^((2*E^8)/x^4)*x^6 + E^(E^8/x^4)*(12*E^8*x + 3*x^5))/(3*E^((2*E^8)/x^4)*x^5),x]

[Out]

3/(2*E^((2*E^8)/x^4)) + x/E^(E^8/x^4) + x^2/6

Rule 12

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

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{x^5} \, dx \\ & = \frac {1}{3} \int \frac {e^{-\frac {2 e^8}{x^4}} \left (3+e^{\frac {e^8}{x^4}} x\right ) \left (12 e^8+e^{\frac {e^8}{x^4}} x^5\right )}{x^5} \, dx \\ & = \frac {1}{3} \int \left (\frac {36 e^{8-\frac {2 e^8}{x^4}}}{x^5}+x+\frac {3 e^{-\frac {e^8}{x^4}} \left (4 e^8+x^4\right )}{x^4}\right ) \, dx \\ & = \frac {x^2}{6}+12 \int \frac {e^{8-\frac {2 e^8}{x^4}}}{x^5} \, dx+\int \frac {e^{-\frac {e^8}{x^4}} \left (4 e^8+x^4\right )}{x^4} \, dx \\ & = \frac {3}{2} e^{-\frac {2 e^8}{x^4}}+e^{-\frac {e^8}{x^4}} x+\frac {x^2}{6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=\frac {1}{6} e^{-\frac {2 e^8}{x^4}} \left (3+e^{\frac {e^8}{x^4}} x\right )^2 \]

[In]

Integrate[(36*E^8 + E^((2*E^8)/x^4)*x^6 + E^(E^8/x^4)*(12*E^8*x + 3*x^5))/(3*E^((2*E^8)/x^4)*x^5),x]

[Out]

(3 + E^(E^8/x^4)*x)^2/(6*E^((2*E^8)/x^4))

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35

method	result	size
risch	\(\frac {x^{2}}{6}+x \,{\mathrm e}^{-\frac {{\mathrm e}^{8}}{x^{4}}}+\frac {3 \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{8}}{x^{4}}}}{2}\)	\(27\)
parts	\(\frac {x^{2}}{6}+x \,{\mathrm e}^{-\frac {{\mathrm e}^{8}}{x^{4}}}+\frac {3 \,{\mathrm e}^{-\frac {2 \,{\mathrm e}^{8}}{x^{4}}}}{2}\)	\(33\)
norman	\(\frac {\left ({\mathrm e}^{\frac {{\mathrm e}^{8}}{x^{4}}} x^{5}+\frac {3 x^{4}}{2}+\frac {x^{6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{8}}{x^{4}}}}{6}\right ) {\mathrm e}^{-\frac {2 \,{\mathrm e}^{8}}{x^{4}}}}{x^{4}}\)	\(51\)
parallelrisch	\(-\frac {\left (-x^{6} {\mathrm e}^{\frac {2 \,{\mathrm e}^{8}}{x^{4}}}-6 \,{\mathrm e}^{\frac {{\mathrm e}^{8}}{x^{4}}} x^{5}-9 x^{4}\right ) {\mathrm e}^{-\frac {2 \,{\mathrm e}^{8}}{x^{4}}}}{6 x^{4}}\)	\(53\)

[In]

int(1/3*(x^6*exp(exp(1)^8/x^4)^2+(12*x*exp(1)^8+3*x^5)*exp(exp(1)^8/x^4)+36*exp(1)^8)/x^5/exp(exp(1)^8/x^4)^2,
x,method=_RETURNVERBOSE)

[Out]

1/6*x^2+x*exp(-exp(8)/x^4)+3/2*exp(-2*exp(8)/x^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (16) = 32\).

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=\frac {1}{6} \, {\left (x^{2} e^{\left (\frac {2 \, e^{8}}{x^{4}}\right )} + 6 \, x e^{\left (\frac {e^{8}}{x^{4}}\right )} + 9\right )} e^{\left (-\frac {2 \, e^{8}}{x^{4}}\right )} \]

[In]

integrate(1/3*(x^6*exp(exp(1)^8/x^4)^2+(12*x*exp(1)^8+3*x^5)*exp(exp(1)^8/x^4)+36*exp(1)^8)/x^5/exp(exp(1)^8/x
^4)^2,x, algorithm="fricas")

[Out]

1/6*(x^2*e^(2*e^8/x^4) + 6*x*e^(e^8/x^4) + 9)*e^(-2*e^8/x^4)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=\frac {x^{2}}{6} + x e^{- \frac {e^{8}}{x^{4}}} + \frac {3 e^{- \frac {2 e^{8}}{x^{4}}}}{2} \]

[In]

integrate(1/3*(x**6*exp(exp(1)**8/x**4)**2+(12*x*exp(1)**8+3*x**5)*exp(exp(1)**8/x**4)+36*exp(1)**8)/x**5/exp(
exp(1)**8/x**4)**2,x)

[Out]

x**2/6 + x*exp(-exp(8)/x**4) + 3*exp(-2*exp(8)/x**4)/2

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.65 \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=\frac {1}{4} \, x \frac {1}{x^{4}}^{\frac {1}{4}} e^{2} \Gamma \left (-\frac {1}{4}, \frac {e^{8}}{x^{4}}\right ) + \frac {1}{6} \, x^{2} + \frac {{\left (x^{4}\right )}^{\frac {3}{4}} e^{2} \Gamma \left (\frac {3}{4}, \frac {e^{8}}{x^{4}}\right )}{x^{3}} + \frac {3}{2} \, e^{\left (-\frac {2 \, e^{8}}{x^{4}}\right )} \]

[In]

integrate(1/3*(x^6*exp(exp(1)^8/x^4)^2+(12*x*exp(1)^8+3*x^5)*exp(exp(1)^8/x^4)+36*exp(1)^8)/x^5/exp(exp(1)^8/x
^4)^2,x, algorithm="maxima")

[Out]

1/4*x*(x^(-4))^(1/4)*e^2*gamma(-1/4, e^8/x^4) + 1/6*x^2 + (x^4)^(3/4)*e^2*gamma(3/4, e^8/x^4)/x^3 + 3/2*e^(-2*
e^8/x^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75 \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=\frac {1}{6} \, x^{2} + x e^{\left (\frac {8 \, x^{4} - e^{8}}{x^{4}} - 8\right )} + \frac {3}{2} \, e^{\left (-\frac {2 \, e^{8}}{x^{4}}\right )} \]

[In]

integrate(1/3*(x^6*exp(exp(1)^8/x^4)^2+(12*x*exp(1)^8+3*x^5)*exp(exp(1)^8/x^4)+36*exp(1)^8)/x^5/exp(exp(1)^8/x
^4)^2,x, algorithm="giac")

[Out]

1/6*x^2 + x*e^((8*x^4 - e^8)/x^4 - 8) + 3/2*e^(-2*e^8/x^4)

Mupad [B] (verification not implemented)

Time = 12.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-\frac {2 e^8}{x^4}} \left (36 e^8+e^{\frac {2 e^8}{x^4}} x^6+e^{\frac {e^8}{x^4}} \left (12 e^8 x+3 x^5\right )\right )}{3 x^5} \, dx=\frac {{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^8}{x^4}}\,{\left (x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^8}{x^4}}+3\right )}^2}{6} \]

[In]

int((exp(-(2*exp(8))/x^4)*(12*exp(8) + (exp(exp(8)/x^4)*(12*x*exp(8) + 3*x^5))/3 + (x^6*exp((2*exp(8))/x^4))/3
))/x^5,x)

[Out]

(exp(-(2*exp(8))/x^4)*(x*exp(exp(8)/x^4) + 3)^2)/6