Integrand size = 47, antiderivative size = 25 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=e^{-\frac {x}{e^5}} \left (-\frac {-24-e^{e^2}}{x^2}+x\right ) \] Output:
(x-(-24-exp(exp(2)))/x^2)/exp(x/exp(5))
Time = 1.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {e^{-\frac {x}{e^5}} \left (24+e^{e^2}+x^3\right )}{x^2} \] Input:
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]
Output:
(24 + E^E^2 + x^3)/(E^(x/E^5)*x^2)
Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2629, 2009}
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.
\(\displaystyle \int \frac {e^{-\frac {x}{e^5}-5} \left (-x^4+e^5 \left (x^3-48\right )-24 x+e^{e^2} \left (-x-2 e^5\right )\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \int \left (-\frac {2 \left (24+e^{e^2}\right ) e^{-\frac {x}{e^5}}}{x^3}-\frac {\left (24+e^{e^2}\right ) e^{-\frac {x}{e^5}-5}}{x^2}-e^{-\frac {x}{e^5}-5} x+e^{-\frac {x}{e^5}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (24+e^{e^2}\right ) e^{-\frac {x}{e^5}}}{x^2}+e^{-\frac {x}{e^5}} x\) |
Input:
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]
Output:
(24 + E^E^2)/(E^(x/E^5)*x^2) + x/E^(x/E^5)
Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.58 (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 | \(-\frac {{\mathrm e}^{-5} \left (-x^{3} {\mathrm e}^{5}-{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5}-24 \,{\mathrm e}^{5}\right ) {\mathrm e}^{-{\mathrm e}^{-5} x}}{x^{2}}\) | \(38\) |
derivativedivides | \({\mathrm e}^{-15} \left (-{\mathrm e}^{-{\mathrm e}^{-5} x} {\mathrm e}^{20}-48 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-24 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )\right )-{\mathrm e}^{20} \left (-x \,{\mathrm e}^{-5} {\mathrm e}^{-{\mathrm e}^{-5} x}-{\mathrm e}^{-{\mathrm e}^{-5} x}\right )-2 \,{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )\right )\right )\) | \(218\) |
default | \({\mathrm e}^{-15} \left (-{\mathrm e}^{-{\mathrm e}^{-5} x} {\mathrm e}^{20}-48 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-24 \,{\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )\right )-{\mathrm e}^{20} \left (-x \,{\mathrm e}^{-5} {\mathrm e}^{-{\mathrm e}^{-5} x}-{\mathrm e}^{-{\mathrm e}^{-5} x}\right )-2 \,{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{10} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x^{2}}+\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{2 x}-\frac {\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-{\mathrm e}^{{\mathrm e}^{2}} {\mathrm e}^{5} \left (-\frac {{\mathrm e}^{5} {\mathrm e}^{-{\mathrm e}^{-5} x}}{x}+\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )\right )\right )\) | \(218\) |
meijerg | \({\mathrm e}^{-10} \left (-24-{\mathrm e}^{{\mathrm e}^{2}}\right ) \left (-\frac {{\mathrm e}^{5}}{x}+6-\ln \left (x \right )+\frac {{\mathrm e}^{5} \left (2-2 \,{\mathrm e}^{-5} x \right )}{2 x}-\frac {{\mathrm e}^{5-{\mathrm e}^{-5} x}}{x}+\ln \left ({\mathrm e}^{-5} x \right )+\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )\right )+{\mathrm e}^{5} \left (1-{\mathrm e}^{-{\mathrm e}^{-5} x}\right )-{\mathrm e}^{5} \left (1-\frac {\left (2+2 \,{\mathrm e}^{-5} x \right ) {\mathrm e}^{-{\mathrm e}^{-5} x}}{2}\right )-2 \,{\mathrm e}^{{\mathrm e}^{2}-10} \left (-\frac {{\mathrm e}^{10}}{2 x^{2}}+\frac {{\mathrm e}^{5}}{x}-\frac {13}{4}+\frac {\ln \left (x \right )}{2}+\frac {{\mathrm e}^{10} \left (9 \,{\mathrm e}^{-10} x^{2}-12 \,{\mathrm e}^{-5} x +6\right )}{12 x^{2}}-\frac {{\mathrm e}^{10-{\mathrm e}^{-5} x} \left (3-3 \,{\mathrm e}^{-5} x \right )}{6 x^{2}}-\frac {\ln \left ({\mathrm e}^{-5} x \right )}{2}-\frac {\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )-48 \,{\mathrm e}^{-10} \left (-\frac {{\mathrm e}^{10}}{2 x^{2}}+\frac {{\mathrm e}^{5}}{x}-\frac {13}{4}+\frac {\ln \left (x \right )}{2}+\frac {{\mathrm e}^{10} \left (9 \,{\mathrm e}^{-10} x^{2}-12 \,{\mathrm e}^{-5} x +6\right )}{12 x^{2}}-\frac {{\mathrm e}^{10-{\mathrm e}^{-5} x} \left (3-3 \,{\mathrm e}^{-5} x \right )}{6 x^{2}}-\frac {\ln \left ({\mathrm e}^{-5} x \right )}{2}-\frac {\operatorname {expIntegral}_{1}\left ({\mathrm e}^{-5} x \right )}{2}\right )\) | \(258\) |
Input:
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)
Output:
(x^3+exp(exp(2))+24)/x^2*exp(-exp(-5)*x)
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {{\left (x^{3} + 24\right )} e^{\left (-{\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )} + 5\right )} + e^{\left (-{\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )} + e^{2} + 5\right )}}{x^{2}} \] Input:
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")
Output:
((x^3 + 24)*e^(-(x + 5*e^5)*e^(-5) + 5) + e^(-(x + 5*e^5)*e^(-5) + e^2 + 5 ))/x^2
Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {\left (x^{3} + 24 + e^{e^{2}}\right ) e^{- \frac {x}{e^{5}}}}{x^{2}} \] Input:
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)
Output:
(x**3 + 24 + exp(exp(2)))*exp(-x*exp(-5))/x**2
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx={\left (x + e^{5}\right )} e^{\left (-x e^{\left (-5\right )}\right )} + 24 \, e^{\left (-10\right )} \Gamma \left (-1, x e^{\left (-5\right )}\right ) + e^{\left (e^{2} - 10\right )} \Gamma \left (-1, x e^{\left (-5\right )}\right ) + 48 \, e^{\left (-10\right )} \Gamma \left (-2, x e^{\left (-5\right )}\right ) + 2 \, e^{\left (e^{2} - 10\right )} \Gamma \left (-2, x e^{\left (-5\right )}\right ) - e^{\left (-x e^{\left (-5\right )} + 5\right )} \] Input:
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")
Output:
(x + e^5)*e^(-x*e^(-5)) + 24*e^(-10)*gamma(-1, x*e^(-5)) + e^(e^2 - 10)*ga mma(-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)
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (18) = 36\).
Time = 0.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {{\left (x^{3} e^{\left (-x e^{\left (-5\right )} + 10\right )} + e^{\left (-x e^{\left (-5\right )} + e^{2} + 10\right )} + 24 \, e^{\left (-x e^{\left (-5\right )} + 10\right )}\right )} e^{\left (-10\right )}}{x^{2}} \] Input:
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")
Output:
(x^3*e^(-x*e^(-5) + 10) + e^(-x*e^(-5) + e^2 + 10) + 24*e^(-x*e^(-5) + 10) )*e^(-10)/x^2
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {{\mathrm {e}}^{-x\,{\mathrm {e}}^{-5}-5}\,\left ({\mathrm {e}}^5\,x^3+{\mathrm {e}}^{{\mathrm {e}}^2+5}+24\,{\mathrm {e}}^5\right )}{x^2} \] Input:
int(-(exp(-5)*exp(-x*exp(-5))*(24*x + exp(exp(2))*(x + 2*exp(5)) + x^4 - e xp(5)*(x^3 - 48)))/x^3,x)
Output:
(exp(- x*exp(-5) - 5)*(exp(exp(2) + 5) + 24*exp(5) + x^3*exp(5)))/x^2
Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-5-\frac {x}{e^5}} \left (e^{e^2} \left (-2 e^5-x\right )-24 x-x^4+e^5 \left (-48+x^3\right )\right )}{x^3} \, dx=\frac {e^{e^{2}}+x^{3}+24}{e^{\frac {x}{e^{5}}} x^{2}} \] Input:
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)
Output:
(e**(e**2) + x**3 + 24)/(e**(x/e**5)*x**2)