3.71.21 \(\int \frac {x^2+e^{e^{\frac {-1-e-x+\log (5)}{x}}+\frac {-1-e-x+\log (5)}{x}} (1+e-\log (5))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{e^{-x+\frac {-1-e-x+x^2+\log (5)}{x}}}+x \]

________________________________________________________________________________________

Rubi [F]  time = 0.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^2+e^{e^{\frac {-1-e-x+\log (5)}{x}}+\frac {-1-e-x+\log (5)}{x}} (1+e-\log (5))}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2 + E^(E^((-1 - E - x + Log[5])/x) + (-1 - E - x + Log[5])/x)*(1 + E - Log[5]))/x^2,x]

[Out]

x - (1 + E - Log[5])*Defer[Subst][Defer[Int][5^x*E^(-1 + 5^x*E^(-1 - x - E*x) + (-1 - E)*x), x], x, x^(-1)]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {5^{\frac {1}{x}} e^{-1+5^{\frac {1}{x}} e^{-1-\frac {1}{x}-\frac {e}{x}}-\frac {1+e}{x}} (1+e-\log (5))}{x^2}\right ) \, dx\\ &=x+(1+e-\log (5)) \int \frac {5^{\frac {1}{x}} e^{-1+5^{\frac {1}{x}} e^{-1-\frac {1}{x}-\frac {e}{x}}-\frac {1+e}{x}}}{x^2} \, dx\\ &=x+(-1-e+\log (5)) \operatorname {Subst}\left (\int 5^x e^{-1+5^x e^{-1-x-e x}+(-1-e) x} \, dx,x,\frac {1}{x}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.25, size = 21, normalized size = 0.78 \begin {gather*} e^{5^{\frac {1}{x}} e^{-\frac {1+e+x}{x}}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + E^(E^((-1 - E - x + Log[5])/x) + (-1 - E - x + Log[5])/x)*(1 + E - Log[5]))/x^2,x]

[Out]

E^(5^x^(-1)/E^((1 + E + x)/x)) + x

________________________________________________________________________________________

fricas [B]  time = 1.09, size = 66, normalized size = 2.44 \begin {gather*} {\left (x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} + e^{\left (\frac {x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} - x - e + \log \relax (5) - 1}{x}\right )}\right )} e^{\left (\frac {x + e - \log \relax (5) + 1}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)+1+exp(1))*exp((log(5)-exp(1)-x-1)/x)*exp(exp((log(5)-exp(1)-x-1)/x))+x^2)/x^2,x, algorithm
="fricas")

[Out]

(x*e^(-(x + e - log(5) + 1)/x) + e^((x*e^(-(x + e - log(5) + 1)/x) - x - e + log(5) - 1)/x))*e^((x + e - log(5
) + 1)/x)

________________________________________________________________________________________

giac [B]  time = 0.29, size = 219, normalized size = 8.11 \begin {gather*} \frac {x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} \log \relax (5) - x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x} + 1\right )} - x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} + e^{\left (\frac {x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} - x - e + \log \relax (5) - 1}{x}\right )} \log \relax (5) - e^{\left (\frac {x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} - x - e + \log \relax (5) - 1}{x}\right )} - e^{\left (\frac {x e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} - e + \log \relax (5) - 1}{x}\right )}}{e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )} \log \relax (5) - e^{\left (-\frac {x + e - \log \relax (5) + 1}{x} + 1\right )} - e^{\left (-\frac {x + e - \log \relax (5) + 1}{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)+1+exp(1))*exp((log(5)-exp(1)-x-1)/x)*exp(exp((log(5)-exp(1)-x-1)/x))+x^2)/x^2,x, algorithm
="giac")

[Out]

(x*e^(-(x + e - log(5) + 1)/x)*log(5) - x*e^(-(x + e - log(5) + 1)/x + 1) - x*e^(-(x + e - log(5) + 1)/x) + e^
((x*e^(-(x + e - log(5) + 1)/x) - x - e + log(5) - 1)/x)*log(5) - e^((x*e^(-(x + e - log(5) + 1)/x) - x - e +
log(5) - 1)/x) - e^((x*e^(-(x + e - log(5) + 1)/x) - e + log(5) - 1)/x))/(e^(-(x + e - log(5) + 1)/x)*log(5) -
e^(-(x + e - log(5) + 1)/x + 1) - e^(-(x + e - log(5) + 1)/x))

________________________________________________________________________________________

maple [A]  time = 0.27, size = 19, normalized size = 0.70

method	result	size
risch	\(x +{\mathrm e}^{{\mathrm e}^{-\frac {-\ln \relax (5)+{\mathrm e}+x +1}{x}}}\)	\(19\)
norman	\(\frac {x^{2}+x \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \relax (5)-{\mathrm e}-x -1}{x}}}}{x}\)	\(28\)
default	\(x -\frac {{\mathrm e}^{-1} {\mathrm e} \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \relax (5)}{x}} {\mathrm e}^{-\frac {{\mathrm e}}{x}} {\mathrm e}^{-1} {\mathrm e}^{-\frac {1}{x}}}}{\ln \relax (5)-{\mathrm e}-1}-\frac {{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \relax (5)}{x}} {\mathrm e}^{-\frac {{\mathrm e}}{x}} {\mathrm e}^{-1} {\mathrm e}^{-\frac {1}{x}}}}{\ln \relax (5)-{\mathrm e}-1}+\frac {{\mathrm e}^{-1} {\mathrm e} \ln \relax (5) {\mathrm e}^{{\mathrm e}^{\frac {\ln \relax (5)}{x}} {\mathrm e}^{-\frac {{\mathrm e}}{x}} {\mathrm e}^{-1} {\mathrm e}^{-\frac {1}{x}}}}{\ln \relax (5)-{\mathrm e}-1}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(5)+1+exp(1))*exp((ln(5)-exp(1)-x-1)/x)*exp(exp((ln(5)-exp(1)-x-1)/x))+x^2)/x^2,x,method=_RETURNVERBO
SE)

[Out]

x+exp(exp(-(-ln(5)+exp(1)+x+1)/x))

________________________________________________________________________________________

maxima [B]  time = 0.49, size = 106, normalized size = 3.93 \begin {gather*} x - \frac {e^{\left (e^{\left (-\frac {e}{x} + \frac {\log \relax (5)}{x} - \frac {1}{x} - 1\right )}\right )} \log \relax (5)}{e - \log \relax (5) + 1} + \frac {e^{\left (e^{\left (-\frac {e}{x} + \frac {\log \relax (5)}{x} - \frac {1}{x} - 1\right )} + 1\right )}}{e - \log \relax (5) + 1} + \frac {e^{\left (e^{\left (-\frac {e}{x} + \frac {\log \relax (5)}{x} - \frac {1}{x} - 1\right )}\right )}}{e - \log \relax (5) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(5)+1+exp(1))*exp((log(5)-exp(1)-x-1)/x)*exp(exp((log(5)-exp(1)-x-1)/x))+x^2)/x^2,x, algorithm
="maxima")

[Out]

x - e^(e^(-e/x + log(5)/x - 1/x - 1))*log(5)/(e - log(5) + 1) + e^(e^(-e/x + log(5)/x - 1/x - 1) + 1)/(e - log
(5) + 1) + e^(e^(-e/x + log(5)/x - 1/x - 1))/(e - log(5) + 1)

________________________________________________________________________________________

mupad [B]  time = 4.31, size = 25, normalized size = 0.93 \begin {gather*} x+{\mathrm {e}}^{5^{1/x}\,{\mathrm {e}}^{-\frac {\mathrm {e}}{x}}\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-\frac {1}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + exp(exp(-(x + exp(1) - log(5) + 1)/x))*exp(-(x + exp(1) - log(5) + 1)/x)*(exp(1) - log(5) + 1))/x^2
,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.35, size = 15, normalized size = 0.56 \begin {gather*} x + e^{e^{\frac {- x - e - 1 + \log {\relax (5 )}}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(5)+1+exp(1))*exp((ln(5)-exp(1)-x-1)/x)*exp(exp((ln(5)-exp(1)-x-1)/x))+x**2)/x**2,x)

[Out]

x + exp(exp((-x - E - 1 + log(5))/x))

________________________________________________________________________________________