\(\int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} (-4 x^7+e^{2 e^{5/3}} (1-4 x+x^2-4 x^3-x^5)+e^{e^{5/3}} (8 x^4+4 x^6))}{x}} (-24 x^7+e^{2 e^{5/3}} (-1+x^2-8 x^3-4 x^5)+e^{e^{5/3}} (24 x^4+20 x^6))}{x^2} \, dx\) [9083]

Optimal result

Integrand size = 130, antiderivative size = 30 \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx=e^{\frac {1}{x}+x-\left (2+x^2-2 e^{-e^{5/3}} x^3\right )^2} \]

[Out]

exp(1/x-(x^2+2-2*x^3/exp(exp(5/3)))^2+x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(30)=60\).

Time = 0.93 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.30, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {6838} \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx=\exp \left (-\frac {e^{-2 e^{5/3}} \left (4 x^7-4 e^{e^{5/3}} \left (x^6+2 x^4\right )-e^{2 e^{5/3}} \left (-x^5-4 x^3+x^2-4 x+1\right )\right )}{x}\right ) \]

[In]

Int[(E^(-2*E^(5/3) + (-4*x^7 + E^(2*E^(5/3))*(1 - 4*x + x^2 - 4*x^3 - x^5) + E^E^(5/3)*(8*x^4 + 4*x^6))/(E^(2*
E^(5/3))*x))*(-24*x^7 + E^(2*E^(5/3))*(-1 + x^2 - 8*x^3 - 4*x^5) + E^E^(5/3)*(24*x^4 + 20*x^6)))/x^2,x]

[Out]

E^(-((4*x^7 - E^(2*E^(5/3))*(1 - 4*x + x^2 - 4*x^3 - x^5) - 4*E^E^(5/3)*(2*x^4 + x^6))/(E^(2*E^(5/3))*x)))

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}& = \exp \left (-\frac {e^{-2 e^{5/3}} \left (4 x^7-e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )-4 e^{e^{5/3}} \left (2 x^4+x^6\right )\right )}{x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx=e^{-4+\frac {1}{x}+x-4 x^2+8 e^{-e^{5/3}} x^3-x^4+4 e^{-e^{5/3}} x^5-4 e^{-2 e^{5/3}} x^6} \]

[In]

Integrate[(E^(-2*E^(5/3) + (-4*x^7 + E^(2*E^(5/3))*(1 - 4*x + x^2 - 4*x^3 - x^5) + E^E^(5/3)*(8*x^4 + 4*x^6))/
(E^(2*E^(5/3))*x))*(-24*x^7 + E^(2*E^(5/3))*(-1 + x^2 - 8*x^3 - 4*x^5) + E^E^(5/3)*(24*x^4 + 20*x^6)))/x^2,x]

[Out]

E^(-4 + x^(-1) + x - 4*x^2 + (8*x^3)/E^E^(5/3) - x^4 + (4*x^5)/E^E^(5/3) - (4*x^6)/E^(2*E^(5/3)))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(25)=50\).

Time = 1.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87

method	result	size
norman	\({\mathrm e}^{\frac {\left (\left (-x^{5}-4 x^{3}+x^{2}-4 x +1\right ) {\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}}+\left (4 x^{6}+8 x^{4}\right ) {\mathrm e}^{{\mathrm e}^{\frac {5}{3}}}-4 x^{7}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{\frac {5}{3}}}}{x}}\)	\(56\)
parallelrisch	\({\mathrm e}^{\frac {\left (\left (-x^{5}-4 x^{3}+x^{2}-4 x +1\right ) {\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}}+\left (4 x^{6}+8 x^{4}\right ) {\mathrm e}^{{\mathrm e}^{\frac {5}{3}}}-4 x^{7}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{\frac {5}{3}}}}{x}}\)	\(56\)
gosper	\({\mathrm e}^{-\frac {\left (-4 \,{\mathrm e}^{{\mathrm e}^{\frac {5}{3}}} x^{6}+4 x^{7}+{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x^{5}-8 \,{\mathrm e}^{{\mathrm e}^{\frac {5}{3}}} x^{4}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x^{3}-{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x^{2}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x -{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{\frac {5}{3}}}}{x}}\)	\(78\)
risch	\({\mathrm e}^{-\frac {\left (-4 \,{\mathrm e}^{{\mathrm e}^{\frac {5}{3}}} x^{6}+4 x^{7}+{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x^{5}-8 \,{\mathrm e}^{{\mathrm e}^{\frac {5}{3}}} x^{4}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x^{3}-{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x^{2}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}} x -{\mathrm e}^{2 \,{\mathrm e}^{\frac {5}{3}}}\right ) {\mathrm e}^{-2 \,{\mathrm e}^{\frac {5}{3}}}}{x}}\)	\(78\)

[In]

int(((-4*x^5-8*x^3+x^2-1)*exp(exp(5/3))^2+(20*x^6+24*x^4)*exp(exp(5/3))-24*x^7)*exp(((-x^5-4*x^3+x^2-4*x+1)*ex
p(exp(5/3))^2+(4*x^6+8*x^4)*exp(exp(5/3))-4*x^7)/x/exp(exp(5/3))^2)/x^2/exp(exp(5/3))^2,x,method=_RETURNVERBOS
E)

[Out]

exp(((-x^5-4*x^3+x^2-4*x+1)*exp(exp(5/3))^2+(4*x^6+8*x^4)*exp(exp(5/3))-4*x^7)/x/exp(exp(5/3))^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).

Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx=e^{\left (-\frac {{\left (4 \, x^{7} + {\left (x^{5} + 4 \, x^{3} - x^{2} + 2 \, x e^{\frac {5}{3}} + 4 \, x - 1\right )} e^{\left (2 \, e^{\frac {5}{3}}\right )} - 4 \, {\left (x^{6} + 2 \, x^{4}\right )} e^{\left (e^{\frac {5}{3}}\right )}\right )} e^{\left (-2 \, e^{\frac {5}{3}}\right )}}{x} + 2 \, e^{\frac {5}{3}}\right )} \]

[In]

integrate(((-4*x^5-8*x^3+x^2-1)*exp(exp(5/3))^2+(20*x^6+24*x^4)*exp(exp(5/3))-24*x^7)*exp(((-x^5-4*x^3+x^2-4*x
+1)*exp(exp(5/3))^2+(4*x^6+8*x^4)*exp(exp(5/3))-4*x^7)/x/exp(exp(5/3))^2)/x^2/exp(exp(5/3))^2,x, algorithm="fr
icas")

[Out]

e^(-(4*x^7 + (x^5 + 4*x^3 - x^2 + 2*x*e^(5/3) + 4*x - 1)*e^(2*e^(5/3)) - 4*(x^6 + 2*x^4)*e^(e^(5/3)))*e^(-2*e^
(5/3))/x + 2*e^(5/3))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).

Time = 0.14 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.93 \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx=e^{\frac {- 4 x^{7} + \left (4 x^{6} + 8 x^{4}\right ) e^{e^{\frac {5}{3}}} + \left (- x^{5} - 4 x^{3} + x^{2} - 4 x + 1\right ) e^{2 e^{\frac {5}{3}}}}{x e^{2 e^{\frac {5}{3}}}}} \]

[In]

integrate(((-4*x**5-8*x**3+x**2-1)*exp(exp(5/3))**2+(20*x**6+24*x**4)*exp(exp(5/3))-24*x**7)*exp(((-x**5-4*x**
3+x**2-4*x+1)*exp(exp(5/3))**2+(4*x**6+8*x**4)*exp(exp(5/3))-4*x**7)/x/exp(exp(5/3))**2)/x**2/exp(exp(5/3))**2
,x)

[Out]

exp((-4*x**7 + (4*x**6 + 8*x**4)*exp(exp(5/3)) + (-x**5 - 4*x**3 + x**2 - 4*x + 1)*exp(2*exp(5/3)))*exp(-2*exp
(5/3))/x)

Maxima [A] (verification not implemented)

none

Time = 0.58 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx=e^{\left (-4 \, x^{6} e^{\left (-2 \, e^{\frac {5}{3}}\right )} + 4 \, x^{5} e^{\left (-e^{\frac {5}{3}}\right )} - x^{4} + 8 \, x^{3} e^{\left (-e^{\frac {5}{3}}\right )} - 4 \, x^{2} + x + \frac {1}{x} - 4\right )} \]

[In]

integrate(((-4*x^5-8*x^3+x^2-1)*exp(exp(5/3))^2+(20*x^6+24*x^4)*exp(exp(5/3))-24*x^7)*exp(((-x^5-4*x^3+x^2-4*x
+1)*exp(exp(5/3))^2+(4*x^6+8*x^4)*exp(exp(5/3))-4*x^7)/x/exp(exp(5/3))^2)/x^2/exp(exp(5/3))^2,x, algorithm="ma
xima")

[Out]

e^(-4*x^6*e^(-2*e^(5/3)) + 4*x^5*e^(-e^(5/3)) - x^4 + 8*x^3*e^(-e^(5/3)) - 4*x^2 + x + 1/x - 4)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx=e^{\left (-4 \, x^{6} e^{\left (-2 \, e^{\frac {5}{3}}\right )} + 4 \, x^{5} e^{\left (-e^{\frac {5}{3}}\right )} - x^{4} + 8 \, x^{3} e^{\left (-e^{\frac {5}{3}}\right )} - 4 \, x^{2} + x + \frac {1}{x} - 4\right )} \]

[In]

integrate(((-4*x^5-8*x^3+x^2-1)*exp(exp(5/3))^2+(20*x^6+24*x^4)*exp(exp(5/3))-24*x^7)*exp(((-x^5-4*x^3+x^2-4*x
+1)*exp(exp(5/3))^2+(4*x^6+8*x^4)*exp(exp(5/3))-4*x^7)/x/exp(exp(5/3))^2)/x^2/exp(exp(5/3))^2,x, algorithm="gi
ac")

[Out]

e^(-4*x^6*e^(-2*e^(5/3)) + 4*x^5*e^(-e^(5/3)) - x^4 + 8*x^3*e^(-e^(5/3)) - 4*x^2 + x + 1/x - 4)

Mupad [B] (verification not implemented)

Time = 13.74 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {e^{-2 e^{5/3}+\frac {e^{-2 e^{5/3}} \left (-4 x^7+e^{2 e^{5/3}} \left (1-4 x+x^2-4 x^3-x^5\right )+e^{e^{5/3}} \left (8 x^4+4 x^6\right )\right )}{x}} \left (-24 x^7+e^{2 e^{5/3}} \left (-1+x^2-8 x^3-4 x^5\right )+e^{e^{5/3}} \left (24 x^4+20 x^6\right )\right )}{x^2} \, dx={\mathrm {e}}^{1/x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{-x^4}\,{\mathrm {e}}^{-4\,x^2}\,{\mathrm {e}}^{4\,x^5\,{\mathrm {e}}^{-{\mathrm {e}}^{5/3}}}\,{\mathrm {e}}^{8\,x^3\,{\mathrm {e}}^{-{\mathrm {e}}^{5/3}}}\,{\mathrm {e}}^{-4\,x^6\,{\mathrm {e}}^{-2\,{\mathrm {e}}^{5/3}}}\,{\mathrm {e}}^x \]

[In]

int(-(exp(-2*exp(5/3))*exp(-(exp(-2*exp(5/3))*(exp(2*exp(5/3))*(4*x - x^2 + 4*x^3 + x^5 - 1) + 4*x^7 - exp(exp
(5/3))*(8*x^4 + 4*x^6)))/x)*(exp(2*exp(5/3))*(8*x^3 - x^2 + 4*x^5 + 1) + 24*x^7 - exp(exp(5/3))*(24*x^4 + 20*x
^6)))/x^2,x)

[Out]

exp(1/x)*exp(-4)*exp(-x^4)*exp(-4*x^2)*exp(4*x^5*exp(-exp(5/3)))*exp(8*x^3*exp(-exp(5/3)))*exp(-4*x^6*exp(-2*e
xp(5/3)))*exp(x)