\(\int \frac {e^{\frac {5}{3}+\frac {x-\log (x^2)}{x}} x+x \log (x)+(x \log (\frac {5}{x})+e^{\frac {x-\log (x^2)}{x}} (-2 e^{5/3} \log (\frac {5}{x})+e^{5/3} \log (\frac {5}{x}) \log (x^2))) \log (\log (\frac {5}{x}))}{x^2 \log (\frac {5}{x}) \log ^2(\log (\frac {5}{x}))} \, dx\) [604]

Optimal result

Integrand size = 108, antiderivative size = 31 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}}+\log (x)}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx \]

[In]

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

[Out]

-LogIntegral[Log[5/x]] + E^(8/3)*Defer[Int][(x*(x^2)^(-1 - x^(-1)))/(Log[5/x]*Log[Log[5/x]]^2), x] + Defer[Int
][Log[x]/(x*Log[5/x]*Log[Log[5/x]]^2), x] - 2*E^(8/3)*Defer[Int][(x^2)^(-1 - x^(-1))/Log[Log[5/x]], x] + E^(8/
3)*Defer[Int][((x^2)^(-1 - x^(-1))*Log[x^2])/Log[Log[5/x]], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log (x)+\log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}+\frac {e^{8/3} \left (x^2\right )^{-1-\frac {1}{x}} \left (x-2 \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )+\log \left (\frac {5}{x}\right ) \log \left (x^2\right ) \log \left (\log \left (\frac {5}{x}\right )\right )\right )}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx \\ & = e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (x-2 \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )+\log \left (\frac {5}{x}\right ) \log \left (x^2\right ) \log \left (\log \left (\frac {5}{x}\right )\right )\right )}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)+\log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx \\ & = e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (x+\log \left (\frac {5}{x}\right ) \left (-2+\log \left (x^2\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )\right )}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \left (\frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}+\frac {1}{x \log \left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx \\ & = e^{8/3} \int \left (\frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}+\frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (-2+\log \left (x^2\right )\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {1}{x \log \left (\log \left (\frac {5}{x}\right )\right )} \, dx \\ & = e^{8/3} \int \frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \left (-2+\log \left (x^2\right )\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx-\text {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,\log \left (\frac {5}{x}\right )\right ) \\ & = -\operatorname {LogIntegral}\left (\log \left (\frac {5}{x}\right )\right )+e^{8/3} \int \left (-\frac {2 \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\log \left (\frac {5}{x}\right )\right )}+\frac {\left (x^2\right )^{-1-\frac {1}{x}} \log \left (x^2\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+e^{8/3} \int \frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx \\ & = -\operatorname {LogIntegral}\left (\log \left (\frac {5}{x}\right )\right )+e^{8/3} \int \frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx+e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \log \left (x^2\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )} \, dx-\left (2 e^{8/3}\right ) \int \frac {\left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{8/3} \left (x^2\right )^{-1/x}+\log (x)}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]

[In]

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

[Out]

(E^(8/3)/(x^2)^x^(-1) + Log[x])/Log[Log[5/x]]

Maple [A] (verified)

Time = 274.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{\frac {5}{3}} {\mathrm e}^{-\frac {\ln \left (x^{2}\right )-x}{x}}+\ln \left (x \right )}{\ln \left (\ln \left (\frac {5}{x}\right )\right )}\)	\(31\)
risch	\(\frac {x^{-\frac {2}{x}} {\mathrm e}^{\frac {3 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+3 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+16 x}{6 x}}+\ln \left (x \right )}{\ln \left (\ln \left (5\right )-\ln \left (x \right )\right )}\)	\(82\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{\left (\frac {2 \, {\left (4 \, x - 3 \, \log \left (5\right ) + 3 \, \log \left (\frac {5}{x}\right )\right )}}{3 \, x}\right )} + \log \left (5\right ) - \log \left (\frac {5}{x}\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]

[In]

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

[Out]

(e^(2/3*(4*x - 3*log(5) + 3*log(5/x))/x) + log(5) - log(5/x))/log(log(5/x))

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{\frac {5}{3}} e^{\frac {x - 2 \log {\left (x \right )}}{x}}}{\log {\left (- \log {\left (x \right )} + \log {\left (5 \right )} \right )}} + \frac {\log {\left (x \right )}}{\log {\left (- \log {\left (x \right )} + \log {\left (5 \right )} \right )}} \]

[In]

integrate((((exp(5/3)*ln(5/x)*ln(x**2)-2*exp(5/3)*ln(5/x))*exp((-ln(x**2)+x)/x)+x*ln(5/x))*ln(ln(5/x))+x*exp(5
/3)*exp((-ln(x**2)+x)/x)+x*ln(x))/x**2/ln(5/x)/ln(ln(5/x))**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {x^{\frac {2}{x}} \log \left (x\right ) + e^{\frac {8}{3}}}{x^{\frac {2}{x}} \log \left (\log \left (5\right ) - \log \left (x\right )\right )} \]

[In]

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

[Out]

(x^(2/x)*log(x) + e^(8/3))/(x^(2/x)*log(log(5) - log(x)))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {\log \left (x\right )}{\log \left (\log \left (5\right ) - \log \left (x\right )\right )} + \frac {e^{\frac {8}{3}}}{x^{\frac {2}{x}} \log \left (\log \left (5\right ) - \log \left (x\right )\right )} \]

[In]

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

[Out]

log(x)/log(log(5) - log(x)) + e^(8/3)/(x^(2/x)*log(log(5) - log(x)))

Mupad [B] (verification not implemented)

Time = 8.59 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\ln \left (\frac {1}{x}\right )+\ln \left (x\right )+\frac {\ln \left (x\right )}{\ln \left (\ln \left (\frac {1}{x}\right )+\ln \left (5\right )\right )}+\frac {{\mathrm {e}}^{8/3}}{\ln \left (\ln \left (\frac {1}{x}\right )+\ln \left (5\right )\right )\,{\left (x^2\right )}^{1/x}} \]

[In]

int((log(log(5/x))*(x*log(5/x) - exp((x - log(x^2))/x)*(2*exp(5/3)*log(5/x) - log(x^2)*exp(5/3)*log(5/x))) + x
*log(x) + x*exp((x - log(x^2))/x)*exp(5/3))/(x^2*log(log(5/x))^2*log(5/x)),x)

[Out]

log(1/x) + log(x) + log(x)/log(log(1/x) + log(5)) + exp(8/3)/(log(log(1/x) + log(5))*(x^2)^(1/x))