\(\int \frac {(\frac {1}{3} (-20+4 e^2))^{\frac {1}{x^2}} (\frac {\log (x)}{x})^{\frac {1}{x^2}} (1-\log (x)-2 \log (x) \log (\frac {(-20+4 e^2) \log (x)}{3 x}))}{x^3 \log (x)} \, dx\) [7244]

Optimal result

Integrand size = 60, antiderivative size = 24 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\left (\frac {4}{3} \left (-5+e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \]

[Out]

exp(ln(4/3*ln(x)*(exp(2)-5)/x)/x^2)

Rubi [F]

\[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx \]

[In]

Int[(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2)*(1 - Log[x] - 2*Log[x]*Log[((-20 + 4*E^2)*Log[x])/(3*x)]))/(x^
3*Log[x]),x]

[Out]

-Defer[Int][(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2))/x^3, x] + Defer[Int][(((-20 + 4*E^2)/3)^x^(-2)*(Log[x
]/x)^x^(-2))/(x^3*Log[x]), x] - 2*Defer[Int][(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2)*Log[(4*(-5 + E^2)*Log
[x])/(3*x)])/x^3, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} (1-\log (x)) \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)}-\frac {2 \left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3}\right ) \, dx \\ & = -\left (2 \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3} \, dx\right )+\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} (1-\log (x)) \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)} \, dx \\ & = -\left (2 \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3} \, dx\right )+\int \left (-\frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3}+\frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)}\right ) \, dx \\ & = -\left (2 \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3} \, dx\right )-\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3} \, dx+\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx \]

[In]

Integrate[(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2)*(1 - Log[x] - 2*Log[x]*Log[((-20 + 4*E^2)*Log[x])/(3*x)]
))/(x^3*Log[x]),x]

[Out]

Integrate[(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2)*(1 - Log[x] - 2*Log[x]*Log[((-20 + 4*E^2)*Log[x])/(3*x)]
))/(x^3*Log[x]), x]

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42

method	result	size
risch	\(\left (\frac {4}{3}\right )^{\frac {1}{x^{2}}} x^{-\frac {1}{x^{2}}} \ln \left (x \right )^{\frac {1}{x^{2}}} \left ({\mathrm e}^{2}-5\right )^{\frac {1}{x^{2}}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right ) \left (\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )-\operatorname {csgn}\left (\frac {i}{x}\right )\right )}{2 x^{2}}}\)	\(82\)

[In]

int((-2*ln(x)*ln(1/3*(4*exp(2)-20)*ln(x)/x)+1-ln(x))*exp(ln(1/3*(4*exp(2)-20)*ln(x)/x)/x^2)/x^3/ln(x),x,method
=_RETURNVERBOSE)

[Out]

(4/3)^(1/x^2)*x^(-1/x^2)*ln(x)^(1/x^2)*(exp(2)-5)^(1/x^2)*exp(1/2*I*Pi*csgn(I*ln(x)/x)*(-csgn(I*ln(x)/x)+csgn(
I*ln(x)))*(csgn(I*ln(x)/x)-csgn(I/x))/x^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\left (\frac {4 \, {\left (e^{2} - 5\right )} \log \left (x\right )}{3 \, x}\right )^{\left (\frac {1}{x^{2}}\right )} \]

[In]

integrate((-2*log(x)*log(1/3*(4*exp(2)-20)*log(x)/x)+1-log(x))*exp(log(1/3*(4*exp(2)-20)*log(x)/x)/x^2)/x^3/lo
g(x),x, algorithm="fricas")

[Out]

(4/3*(e^2 - 5)*log(x)/x)^(x^(-2))

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=e^{\frac {\log {\left (\frac {\left (- \frac {20}{3} + \frac {4 e^{2}}{3}\right ) \log {\left (x \right )}}{x} \right )}}{x^{2}}} \]

[In]

integrate((-2*ln(x)*ln(1/3*(4*exp(2)-20)*ln(x)/x)+1-ln(x))*exp(ln(1/3*(4*exp(2)-20)*ln(x)/x)/x**2)/x**3/ln(x),
x)

[Out]

exp(log((-20/3 + 4*exp(2)/3)*log(x)/x)/x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).

Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=e^{\left (-\frac {\log \left (3\right )}{x^{2}} + \frac {2 \, \log \left (2\right )}{x^{2}} - \frac {\log \left (x\right )}{x^{2}} + \frac {\log \left (e^{2} - 5\right )}{x^{2}} + \frac {\log \left (\log \left (x\right )\right )}{x^{2}}\right )} \]

[In]

integrate((-2*log(x)*log(1/3*(4*exp(2)-20)*log(x)/x)+1-log(x))*exp(log(1/3*(4*exp(2)-20)*log(x)/x)/x^2)/x^3/lo
g(x),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\int { -\frac {{\left (2 \, \log \left (x\right ) \log \left (\frac {4 \, {\left (e^{2} - 5\right )} \log \left (x\right )}{3 \, x}\right ) + \log \left (x\right ) - 1\right )} \left (\frac {4 \, {\left (e^{2} - 5\right )} \log \left (x\right )}{3 \, x}\right )^{\left (\frac {1}{x^{2}}\right )}}{x^{3} \log \left (x\right )} \,d x } \]

[In]

integrate((-2*log(x)*log(1/3*(4*exp(2)-20)*log(x)/x)+1-log(x))*exp(log(1/3*(4*exp(2)-20)*log(x)/x)/x^2)/x^3/lo
g(x),x, algorithm="giac")

[Out]

integrate(-(2*log(x)*log(4/3*(e^2 - 5)*log(x)/x) + log(x) - 1)*(4/3*(e^2 - 5)*log(x)/x)^(x^(-2))/(x^3*log(x)),
x)

Mupad [B] (verification not implemented)

Time = 12.70 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx={\left (-\frac {20\,\ln \left (x\right )-4\,{\mathrm {e}}^2\,\ln \left (x\right )}{3\,x}\right )}^{\frac {1}{x^2}} \]

[In]

int(-(exp(log((log(x)*(4*exp(2) - 20))/(3*x))/x^2)*(log(x) + 2*log((log(x)*(4*exp(2) - 20))/(3*x))*log(x) - 1)
)/(x^3*log(x)),x)

[Out]

(-(20*log(x) - 4*exp(2)*log(x))/(3*x))^(1/x^2)