\(\int \frac {2 x^2 \log (\frac {2}{x})+e^{\frac {1}{x+\log (\frac {3}{\log (\frac {2}{x})})}} (-1-x \log (\frac {2}{x})) \log (x)+4 x \log (\frac {2}{x}) \log (\frac {3}{\log (\frac {2}{x})})+2 \log (\frac {2}{x}) \log ^2(\frac {3}{\log (\frac {2}{x})})}{x^3 \log (\frac {2}{x}) \log (x)+2 x^2 \log (\frac {2}{x}) \log (x) \log (\frac {3}{\log (\frac {2}{x})})+x \log (\frac {2}{x}) \log (x) \log ^2(\frac {3}{\log (\frac {2}{x})})} \, dx\) [1899]

Optimal result

Integrand size = 147, antiderivative size = 23 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+\log \left (\log ^2(x)\right ) \]

[Out]

ln(ln(x)^2)+exp(1/(ln(3/ln(2/x))+x))

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {6820, 6874, 2339, 29, 6838} \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=2 \log (\log (x))+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \]

[In]

Int[(2*x^2*Log[2/x] + E^(x + Log[3/Log[2/x]])^(-1)*(-1 - x*Log[2/x])*Log[x] + 4*x*Log[2/x]*Log[3/Log[2/x]] + 2
*Log[2/x]*Log[3/Log[2/x]]^2)/(x^3*Log[2/x]*Log[x] + 2*x^2*Log[2/x]*Log[x]*Log[3/Log[2/x]] + x*Log[2/x]*Log[x]*
Log[3/Log[2/x]]^2),x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}}+2 \log (\log (x)) \]

[In]

Integrate[(2*x^2*Log[2/x] + E^(x + Log[3/Log[2/x]])^(-1)*(-1 - x*Log[2/x])*Log[x] + 4*x*Log[2/x]*Log[3/Log[2/x
]] + 2*Log[2/x]*Log[3/Log[2/x]]^2)/(x^3*Log[2/x]*Log[x] + 2*x^2*Log[2/x]*Log[x]*Log[3/Log[2/x]] + x*Log[2/x]*L
og[x]*Log[3/Log[2/x]]^2),x]

[Out]

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

Maple [A] (verified)

Time = 103.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(2 \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{\frac {1}{\ln \left (\frac {3}{\ln \left (\frac {2}{x}\right )}\right )+x}}\)	\(23\)
risch	\(2 \ln \left (\ln \left (x \right )\right )+{\mathrm e}^{\frac {2}{i \pi \operatorname {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )^{3}+i \pi \operatorname {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )-i \pi \operatorname {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )^{2}-i \pi \,\operatorname {csgn}\left (\frac {1}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right ) \operatorname {csgn}\left (\frac {i}{2 i \ln \left (2\right )-2 i \ln \left (x \right )}\right )+i \pi +2 \ln \left (6\right )-2 \ln \left (2 i \ln \left (2\right )-2 i \ln \left (x \right )\right )+2 x}}\)	\(150\)

[In]

int(((-x*ln(2/x)-1)*ln(x)*exp(1/(ln(3/ln(2/x))+x))+2*ln(2/x)*ln(3/ln(2/x))^2+4*x*ln(2/x)*ln(3/ln(2/x))+2*x^2*l
n(2/x))/(x*ln(2/x)*ln(x)*ln(3/ln(2/x))^2+2*x^2*ln(2/x)*ln(x)*ln(3/ln(2/x))+x^3*ln(2/x)*ln(x)),x,method=_RETURN
VERBOSE)

[Out]

2*ln(ln(x))+exp(1/(ln(3/ln(2/x))+x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\left (\frac {1}{x + \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}\right )} + 2 \, \log \left (-\log \left (2\right ) + \log \left (\frac {2}{x}\right )\right ) \]

[In]

integrate(((-x*log(2/x)-1)*log(x)*exp(1/(log(3/log(2/x))+x))+2*log(2/x)*log(3/log(2/x))^2+4*x*log(2/x)*log(3/l
og(2/x))+2*x^2*log(2/x))/(x*log(2/x)*log(x)*log(3/log(2/x))^2+2*x^2*log(2/x)*log(x)*log(3/log(2/x))+x^3*log(2/
x)*log(x)),x, algorithm="fricas")

[Out]

e^(1/(x + log(3/log(2/x)))) + 2*log(-log(2) + log(2/x))

Sympy [A] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\frac {1}{x + \log {\left (\frac {3}{- \log {\left (x \right )} + \log {\left (2 \right )}} \right )}}} + 2 \log {\left (\log {\left (x \right )} \right )} \]

[In]

integrate(((-x*ln(2/x)-1)*ln(x)*exp(1/(ln(3/ln(2/x))+x))+2*ln(2/x)*ln(3/ln(2/x))**2+4*x*ln(2/x)*ln(3/ln(2/x))+
2*x**2*ln(2/x))/(x*ln(2/x)*ln(x)*ln(3/ln(2/x))**2+2*x**2*ln(2/x)*ln(x)*ln(3/ln(2/x))+x**3*ln(2/x)*ln(x)),x)

[Out]

exp(1/(x + log(3/(-log(x) + log(2))))) + 2*log(log(x))

Maxima [F]

\[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=\int { -\frac {{\left (x \log \left (\frac {2}{x}\right ) + 1\right )} e^{\left (\frac {1}{x + \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}\right )} \log \left (x\right ) - 2 \, x^{2} \log \left (\frac {2}{x}\right ) - 4 \, x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right ) - 2 \, \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )^{2}}{x^{3} \log \left (x\right ) \log \left (\frac {2}{x}\right ) + 2 \, x^{2} \log \left (x\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right ) + x \log \left (x\right ) \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )^{2}} \,d x } \]

[In]

integrate(((-x*log(2/x)-1)*log(x)*exp(1/(log(3/log(2/x))+x))+2*log(2/x)*log(3/log(2/x))^2+4*x*log(2/x)*log(3/l
og(2/x))+2*x^2*log(2/x))/(x*log(2/x)*log(x)*log(3/log(2/x))^2+2*x^2*log(2/x)*log(x)*log(3/log(2/x))+x^3*log(2/
x)*log(x)),x, algorithm="maxima")

[Out]

-integrate((x*log(2) - x*log(x) + 1)*e^(1/(x + log(3) - log(log(2) - log(x))))/(x^3*log(2) + 2*x^2*log(3)*log(
2) + x*log(3)^2*log(2) + (x*log(2) - x*log(x))*log(log(2) - log(x))^2 - (x^3 + 2*x^2*log(3) + x*log(3)^2)*log(
x) - 2*(x^2*log(2) + x*log(3)*log(2) - (x^2 + x*log(3))*log(x))*log(log(2) - log(x))), x) + 2*log(log(x))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=e^{\left (\frac {1}{x + \log \left (3\right ) - \log \left (\log \left (2\right ) - \log \left (x\right )\right )}\right )} + 2 \, \log \left (\log \left (x\right )\right ) \]

[In]

integrate(((-x*log(2/x)-1)*log(x)*exp(1/(log(3/log(2/x))+x))+2*log(2/x)*log(3/log(2/x))^2+4*x*log(2/x)*log(3/l
og(2/x))+2*x^2*log(2/x))/(x*log(2/x)*log(x)*log(3/log(2/x))^2+2*x^2*log(2/x)*log(x)*log(3/log(2/x))+x^3*log(2/
x)*log(x)),x, algorithm="giac")

[Out]

e^(1/(x + log(3) - log(log(2) - log(x)))) + 2*log(log(x))

Mupad [B] (verification not implemented)

Time = 9.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {2 x^2 \log \left (\frac {2}{x}\right )+e^{\frac {1}{x+\log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}} \left (-1-x \log \left (\frac {2}{x}\right )\right ) \log (x)+4 x \log \left (\frac {2}{x}\right ) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+2 \log \left (\frac {2}{x}\right ) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )}{x^3 \log \left (\frac {2}{x}\right ) \log (x)+2 x^2 \log \left (\frac {2}{x}\right ) \log (x) \log \left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )+x \log \left (\frac {2}{x}\right ) \log (x) \log ^2\left (\frac {3}{\log \left (\frac {2}{x}\right )}\right )} \, dx=2\,\ln \left (\ln \left (x\right )\right )+{\mathrm {e}}^{\frac {1}{x+\ln \left (\frac {3}{\ln \left (\frac {2}{x}\right )}\right )}} \]

[In]

int((2*log(2/x)*log(3/log(2/x))^2 + 2*x^2*log(2/x) + 4*x*log(2/x)*log(3/log(2/x)) - exp(1/(x + log(3/log(2/x))
))*log(x)*(x*log(2/x) + 1))/(x^3*log(2/x)*log(x) + x*log(2/x)*log(3/log(2/x))^2*log(x) + 2*x^2*log(2/x)*log(3/
log(2/x))*log(x)),x)

[Out]

2*log(log(x)) + exp(1/(x + log(3/log(2/x))))