\(\int \frac {e^{-5-\frac {x}{e^5}} (e^{e^2} (-2 e^5-x)-24 x-x^4+e^5 (-48+x^3))}{x^3} \, dx\) [6708]

Optimal result

Integrand size = 47, antiderivative size = 25 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=e^{-\frac {x}{e^5}} \left (-\frac {-24-e^{e^2}}{x^2}+x\right ) \]

[Out]

(x-(-24-exp(exp(2)))/x^2)/exp(x/exp(5))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {2230, 2225, 2208, 2209, 2207} \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {\left (24+e^{e^2}\right ) e^{-\frac {x}{e^5}}}{x^2}+e^{-\frac {x}{e^5}} x \]

[In]

Int[(E^(-5 - x/E^5)*(E^E^2*(-2*E^5 - x) - 24*x - x^4 + E^5*(-48 + x^3)))/x^3,x]

[Out]

(24 + E^E^2)/(E^(x/E^5)*x^2) + x/E^(x/E^5)

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2230

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{-\frac {x}{e^5}}-\frac {2 e^{-\frac {x}{e^5}} \left (24+e^{e^2}\right )}{x^3}-\frac {e^{-5-\frac {x}{e^5}} \left (24+e^{e^2}\right )}{x^2}-e^{-5-\frac {x}{e^5}} x\right ) \, dx \\ & = \left (-24-e^{e^2}\right ) \int \frac {e^{-5-\frac {x}{e^5}}}{x^2} \, dx-\left (2 \left (24+e^{e^2}\right )\right ) \int \frac {e^{-\frac {x}{e^5}}}{x^3} \, dx+\int e^{-\frac {x}{e^5}} \, dx-\int e^{-5-\frac {x}{e^5}} x \, dx \\ & = -e^{5-\frac {x}{e^5}}+\frac {e^{-\frac {x}{e^5}} \left (24+e^{e^2}\right )}{x^2}+\frac {e^{-5-\frac {x}{e^5}} \left (24+e^{e^2}\right )}{x}+e^{-\frac {x}{e^5}} x-e^5 \int e^{-5-\frac {x}{e^5}} \, dx+\frac {\left (24+e^{e^2}\right ) \int \frac {e^{-\frac {x}{e^5}}}{x^2} \, dx}{e^5}+\frac {\left (24+e^{e^2}\right ) \int \frac {e^{-5-\frac {x}{e^5}}}{x} \, dx}{e^5} \\ & = \frac {e^{-\frac {x}{e^5}} \left (24+e^{e^2}\right )}{x^2}+e^{-\frac {x}{e^5}} x+\frac {\left (24+e^{e^2}\right ) \text {Ei}\left (-\frac {x}{e^5}\right )}{e^{10}}-\frac {\left (24+e^{e^2}\right ) \int \frac {e^{-\frac {x}{e^5}}}{x} \, dx}{e^{10}} \\ & = \frac {e^{-\frac {x}{e^5}} \left (24+e^{e^2}\right )}{x^2}+e^{-\frac {x}{e^5}} x \\ \end{align*}

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {e^{-\frac {x}{e^5}} \left (24+e^{e^2}+x^3\right )}{x^2} \]

[In]

Integrate[(E^(-5 - x/E^5)*(E^E^2*(-2*E^5 - x) - 24*x - x^4 + E^5*(-48 + x^3)))/x^3,x]

[Out]

(24 + E^E^2 + x^3)/(E^(x/E^5)*x^2)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76

method	result	size
risch	\(\frac {\left (x^{3}+{\mathrm e}^{{\mathrm e}^{2}}+24\right ) {\mathrm e}^{-{\mathrm e}^{-5} x}}{x^{2}}\)	\(19\)
gosper	\(\frac {\left (x^{3}+{\mathrm e}^{{\mathrm e}^{2}}+24\right ) {\mathrm e}^{-{\mathrm e}^{-5} x}}{x^{2}}\)	\(22\)
norman	\(\frac {\left (x^{3}+{\mathrm e}^{{\mathrm e}^{2}}+24\right ) {\mathrm e}^{-{\mathrm e}^{-5} x}}{x^{2}}\)	\(22\)
parallelrisch	\(\frac {{\mathrm e}^{-5} \left (x^{3} {\mathrm e}^{5}+{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5}+24 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-{\mathrm e}^{-5} x}}{x^{2}}\)	\(35\)
derivativedivides	\({\mathrm e}^{-15} \left (-{\mathrm e}^{-{\mathrm e}^{-5} x} {\mathrm e}^{20}-48 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-24 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )\right )-{\mathrm e}^{20} \left (-x \,{\mathrm e}^{-5} {\mathrm e}^{-{\mathrm e}^{-5} x}-{\mathrm e}^{-{\mathrm e}^{-5} x}\right )-2 \,{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )\right )\right )\)	\(218\)
default	\({\mathrm e}^{-15} \left (-{\mathrm e}^{-{\mathrm e}^{-5} x} {\mathrm e}^{20}-48 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-24 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )\right )-{\mathrm e}^{20} \left (-x \,{\mathrm e}^{-5} {\mathrm e}^{-{\mathrm e}^{-5} x}-{\mathrm e}^{-{\mathrm e}^{-5} x}\right )-2 \,{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )\right )\right )\)	\(218\)
meijerg	\({\mathrm e}^{5} \left (1-{\mathrm e}^{-{\mathrm e}^{-5} x}\right )-{\mathrm e}^{5} \left (1-\frac {\left (2+2 \,{\mathrm e}^{-5} x \right ) {\mathrm e}^{-{\mathrm e}^{-5} x}}{2}\right )+{\mathrm e}^{-10} \left (-24-{\mathrm e}^{{\mathrm e}^{2}}\right ) \left (-\frac {{\mathrm e}^{5}}{x}+6-\ln \left (x \right )+\frac {{\mathrm e}^{5} \left (2-2 \,{\mathrm e}^{-5} x \right )}{2 x}-\frac {{\mathrm e}^{5-{\mathrm e}^{-5} x}}{x}+\ln \left ({\mathrm e}^{-5} x \right )+\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )\right )-2 \,{\mathrm e}^{{\mathrm e}^{2}-10} \left (-\frac {{\mathrm e}^{10}}{2 x^{2}}+\frac {{\mathrm e}^{5}}{x}-\frac {13}{4}+\frac {\ln \left (x \right )}{2}+\frac {{\mathrm e}^{10} \left (9 x^{2} {\mathrm e}^{-10}-12 \,{\mathrm e}^{-5} x +6\right )}{12 x^{2}}-\frac {{\mathrm e}^{10-{\mathrm e}^{-5} x} \left (3-3 \,{\mathrm e}^{-5} x \right )}{6 x^{2}}-\frac {\ln \left ({\mathrm e}^{-5} x \right )}{2}-\frac {\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-48 \,{\mathrm e}^{-10} \left (-\frac {{\mathrm e}^{10}}{2 x^{2}}+\frac {{\mathrm e}^{5}}{x}-\frac {13}{4}+\frac {\ln \left (x \right )}{2}+\frac {{\mathrm e}^{10} \left (9 x^{2} {\mathrm e}^{-10}-12 \,{\mathrm e}^{-5} x +6\right )}{12 x^{2}}-\frac {{\mathrm e}^{10-{\mathrm e}^{-5} x} \left (3-3 \,{\mathrm e}^{-5} x \right )}{6 x^{2}}-\frac {\ln \left ({\mathrm e}^{-5} x \right )}{2}-\frac {\operatorname {Ei}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )\)	\(258\)

[In]

int(((-2*exp(5)-x)*exp(exp(2))+(x^3-48)*exp(5)-x^4-24*x)/x^3/exp(5)/exp(x/exp(5)),x,method=_RETURNVERBOSE)

[Out]

(x^3+exp(exp(2))+24)/x^2*exp(-exp(-5)*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {{\left (x^{3} + 24\right )} e^{\left (-{\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )} + 5\right )} + e^{\left (-{\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )} + e^{2} + 5\right )}}{x^{2}} \]

[In]

integrate(((-2*exp(5)-x)*exp(exp(2))+(x^3-48)*exp(5)-x^4-24*x)/x^3/exp(5)/exp(x/exp(5)),x, algorithm="fricas")

[Out]

((x^3 + 24)*e^(-(x + 5*e^5)*e^(-5) + 5) + e^(-(x + 5*e^5)*e^(-5) + e^2 + 5))/x^2

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {\left (x^{3} + 24 + e^{e^{2}}\right ) e^{- \frac {x}{e^{5}}}}{x^{2}} \]

[In]

integrate(((-2*exp(5)-x)*exp(exp(2))+(x**3-48)*exp(5)-x**4-24*x)/x**3/exp(5)/exp(x/exp(5)),x)

[Out]

(x**3 + 24 + exp(exp(2)))*exp(-x*exp(-5))/x**2

Maxima [C] (verification not implemented)

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

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx={\left (x + e^{5}\right )} e^{\left (-x e^{\left (-5\right )}\right )} + 24 \, e^{\left (-10\right )} \Gamma \left (-1, x e^{\left (-5\right )}\right ) + e^{\left (e^{2} - 10\right )} \Gamma \left (-1, x e^{\left (-5\right )}\right ) + 48 \, e^{\left (-10\right )} \Gamma \left (-2, x e^{\left (-5\right )}\right ) + 2 \, e^{\left (e^{2} - 10\right )} \Gamma \left (-2, x e^{\left (-5\right )}\right ) - e^{\left (-x e^{\left (-5\right )} + 5\right )} \]

[In]

integrate(((-2*exp(5)-x)*exp(exp(2))+(x^3-48)*exp(5)-x^4-24*x)/x^3/exp(5)/exp(x/exp(5)),x, algorithm="maxima")

[Out]

(x + e^5)*e^(-x*e^(-5)) + 24*e^(-10)*gamma(-1, x*e^(-5)) + e^(e^2 - 10)*gamma(-1, x*e^(-5)) + 48*e^(-10)*gamma
(-2, x*e^(-5)) + 2*e^(e^2 - 10)*gamma(-2, x*e^(-5)) - e^(-x*e^(-5) + 5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {{\left (x^{3} e^{\left (-x e^{\left (-5\right )} + 10\right )} + e^{\left (-x e^{\left (-5\right )} + e^{2} + 10\right )} + 24 \, e^{\left (-x e^{\left (-5\right )} + 10\right )}\right )} e^{\left (-10\right )}}{x^{2}} \]

[In]

integrate(((-2*exp(5)-x)*exp(exp(2))+(x^3-48)*exp(5)-x^4-24*x)/x^3/exp(5)/exp(x/exp(5)),x, algorithm="giac")

[Out]

(x^3*e^(-x*e^(-5) + 10) + e^(-x*e^(-5) + e^2 + 10) + 24*e^(-x*e^(-5) + 10))*e^(-10)/x^2

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {{\mathrm {e}}^{-x\,{\mathrm {e}}^{-5}-5}\,\left ({\mathrm {e}}^5\,x^3+{\mathrm {e}}^{{\mathrm {e}}^2+5}+24\,{\mathrm {e}}^5\right )}{x^2} \]

[In]

int(-(exp(-5)*exp(-x*exp(-5))*(24*x + exp(exp(2))*(x + 2*exp(5)) + x^4 - exp(5)*(x^3 - 48)))/x^3,x)

[Out]

(exp(- x*exp(-5) - 5)*(exp(exp(2) + 5) + 24*exp(5) + x^3*exp(5)))/x^2