\(\int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} (\frac {1}{x})^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} (-10+10 \log (\frac {1}{4 x}))}{(i \pi +\log (\log (625)))^2} \, dx\) [7729]

Optimal result

Integrand size = 58, antiderivative size = 35 \[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx=4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \]

[Out]

exp(5*x*ln(1/4/x)/ln(-4*ln(5))^2)^2

Rubi [F]

\[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx=\int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx \]

[In]

Int[((x^(-1))^((10*x)/(I*Pi + Log[Log[625]])^2)*(-10 + 10*Log[1/(4*x)]))/(4^((10*x)/(I*Pi + Log[Log[625]])^2)*
(I*Pi + Log[Log[625]])^2),x]

[Out]

(-5*Defer[Int][2^(1 - (20*x)/(I*Pi + Log[Log[625]])^2)*(x^(-1))^((10*x)/(I*Pi + Log[Log[625]])^2), x])/(I*Pi +
Log[Log[625]])^2 + (5*Log[1/(4*x)]*Defer[Int][2^(1 - (20*x)/(I*Pi + Log[Log[625]])^2)*(x^(-1))^((10*x)/(I*Pi
+ Log[Log[625]])^2), x])/(I*Pi + Log[Log[625]])^2 + (5*Defer[Int][Defer[Int][2^(1 + (20*x)/(Pi - I*Log[Log[625
]])^2)*(x^(-1))^((10*x)/(I*Pi + Log[Log[625]])^2), x]/x, x])/(I*Pi + Log[Log[625]])^2

Rubi steps \begin{align*} \text {integral}& = \frac {\int 4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right ) \, dx}{(i \pi +\log (\log (625)))^2} \\ & = \frac {\int 5\ 2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-1+\log \left (\frac {1}{4 x}\right )\right ) \, dx}{(i \pi +\log (\log (625)))^2} \\ & = \frac {5 \int 2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-1+\log \left (\frac {1}{4 x}\right )\right ) \, dx}{(i \pi +\log (\log (625)))^2} \\ & = \frac {5 \int \left (-2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}}+2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \log \left (\frac {1}{4 x}\right )\right ) \, dx}{(i \pi +\log (\log (625)))^2} \\ & = -\frac {5 \int 2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \, dx}{(i \pi +\log (\log (625)))^2}+\frac {5 \int 2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \log \left (\frac {1}{4 x}\right ) \, dx}{(i \pi +\log (\log (625)))^2} \\ & = -\frac {5 \int 2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \, dx}{(i \pi +\log (\log (625)))^2}+\frac {5 \int \frac {\int 2^{1+\frac {20 x}{(\pi -i \log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \, dx}{x} \, dx}{(i \pi +\log (\log (625)))^2}+\frac {\left (5 \log \left (\frac {1}{4 x}\right )\right ) \int 2^{1-\frac {20 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \, dx}{(i \pi +\log (\log (625)))^2} \\ \end{align*}

Mathematica [F]

\[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx=\int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx \]

[In]

Integrate[((x^(-1))^((10*x)/(I*Pi + Log[Log[625]])^2)*(-10 + 10*Log[1/(4*x)]))/(4^((10*x)/(I*Pi + Log[Log[625]
])^2)*(I*Pi + Log[Log[625]])^2),x]

[Out]

Integrate[((x^(-1))^((10*x)/(I*Pi + Log[Log[625]])^2)*(-10 + 10*Log[1/(4*x)]))/4^((10*x)/(I*Pi + Log[Log[625]]
)^2), x]/(I*Pi + Log[Log[625]])^2

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57

method	result	size
default	\({\mathrm e}^{\frac {10 x \ln \left (\frac {1}{4 x}\right )}{\ln \left (-4 \ln \left (5\right )\right )^{2}}}\)	\(20\)
parallelrisch	\({\mathrm e}^{\frac {10 x \ln \left (\frac {1}{4 x}\right )}{\ln \left (-4 \ln \left (5\right )\right )^{2}}}\)	\(20\)
norman	\(\frac {\left (4 \ln \left (2\right )^{2}+4 \ln \left (2\right ) \ln \left (\ln \left (5\right )\right )+4 i \ln \left (2\right ) \pi +\ln \left (\ln \left (5\right )\right )^{2}+2 i \ln \left (\ln \left (5\right )\right ) \pi -\pi ^{2}\right ) {\mathrm e}^{\frac {10 x \ln \left (\frac {1}{4 x}\right )}{\ln \left (-4 \ln \left (5\right )\right )^{2}}}}{\left (2 \ln \left (2\right )+\ln \left (\ln \left (5\right )\right )+i \pi \right ) \ln \left (-4 \ln \left (5\right )\right )}\)	\(79\)
risch	\(\frac {4 \left (\frac {1}{4 x}\right )^{\frac {10 x}{\left (i \pi +2 \ln \left (2 \sqrt {\ln \left (5\right )}\right )\right )^{2}}} \ln \left (2\right )^{2}}{\left (2 \ln \left (2\right )+\ln \left (\ln \left (5\right )\right )+i \pi \right )^{2}}+\frac {4 \left (\frac {1}{4 x}\right )^{\frac {10 x}{\left (i \pi +2 \ln \left (2 \sqrt {\ln \left (5\right )}\right )\right )^{2}}} \ln \left (2\right ) \ln \left (\ln \left (5\right )\right )}{\left (2 \ln \left (2\right )+\ln \left (\ln \left (5\right )\right )+i \pi \right )^{2}}+\frac {\left (\frac {1}{4 x}\right )^{\frac {10 x}{\left (i \pi +2 \ln \left (2 \sqrt {\ln \left (5\right )}\right )\right )^{2}}} \ln \left (\ln \left (5\right )\right )^{2}}{\left (2 \ln \left (2\right )+\ln \left (\ln \left (5\right )\right )+i \pi \right )^{2}}+\frac {4 i \left (\frac {1}{4 x}\right )^{\frac {10 x}{\left (i \pi +2 \ln \left (2 \sqrt {\ln \left (5\right )}\right )\right )^{2}}} \ln \left (2\right ) \pi }{\left (2 \ln \left (2\right )+\ln \left (\ln \left (5\right )\right )+i \pi \right )^{2}}+\frac {2 i \left (\frac {1}{4 x}\right )^{\frac {10 x}{\left (i \pi +2 \ln \left (2 \sqrt {\ln \left (5\right )}\right )\right )^{2}}} \ln \left (\ln \left (5\right )\right ) \pi }{\left (2 \ln \left (2\right )+\ln \left (\ln \left (5\right )\right )+i \pi \right )^{2}}-\frac {\left (\frac {1}{4 x}\right )^{\frac {10 x}{\left (i \pi +2 \ln \left (2 \sqrt {\ln \left (5\right )}\right )\right )^{2}}} \pi ^{2}}{\left (2 \ln \left (2\right )+\ln \left (\ln \left (5\right )\right )+i \pi \right )^{2}}\)	\(273\)

[In]

int((10*ln(1/4/x)-10)*exp(5*x*ln(1/4/x)/ln(-4*ln(5))^2)^2/ln(-4*ln(5))^2,x,method=_RETURNVERBOSE)

[Out]

exp(5*x*ln(1/4/x)/ln(-4*ln(5))^2)^2

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.46 \[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx=\left (\frac {1}{4 \, x}\right )^{\frac {10 \, x}{\log \left (-4 \, \log \left (5\right )\right )^{2}}} \]

[In]

integrate((10*log(1/4/x)-10)*exp(5*x*log(1/4/x)/log(-4*log(5))^2)^2/log(-4*log(5))^2,x, algorithm="fricas")

[Out]

(1/4/x)^(10*x/log(-4*log(5))^2)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (22) = 44\).

Time = 9.71 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.11 \[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx=e^{- \frac {20 x \log {\left (2 \right )}}{- \pi ^{2} + \log {\left (\log {\left (5 \right )} \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 4 \log {\left (2 \right )}^{2} + 2 i \pi \log {\left (\log {\left (5 \right )} \right )} + 4 i \pi \log {\left (2 \right )}}} e^{\frac {10 x \log {\left (\frac {1}{x} \right )}}{- \pi ^{2} + \log {\left (\log {\left (5 \right )} \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 4 \log {\left (2 \right )}^{2} + 2 i \pi \log {\left (\log {\left (5 \right )} \right )} + 4 i \pi \log {\left (2 \right )}}} \]

[In]

integrate((10*ln(1/4/x)-10)*exp(5*x*ln(1/4/x)/ln(-4*ln(5))**2)**2/ln(-4*ln(5))**2,x)

[Out]

exp(-20*x*log(2)/(-pi**2 + log(log(5))**2 + 4*log(2)*log(log(5)) + 4*log(2)**2 + 2*I*pi*log(log(5)) + 4*I*pi*l
og(2)))*exp(10*x*log(1/x)/(-pi**2 + log(log(5))**2 + 4*log(2)*log(log(5)) + 4*log(2)**2 + 2*I*pi*log(log(5)) +
4*I*pi*log(2)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (16) = 32\).

Time = 0.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.66 \[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx=\frac {{\left (4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (-\log \left (5\right )\right ) + \log \left (-\log \left (5\right )\right )^{2}\right )} e^{\left (-\frac {20 \, x \log \left (2\right )}{4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (-\log \left (5\right )\right ) + \log \left (-\log \left (5\right )\right )^{2}} - \frac {10 \, x \log \left (x\right )}{4 \, \log \left (2\right )^{2} + 4 \, \log \left (2\right ) \log \left (-\log \left (5\right )\right ) + \log \left (-\log \left (5\right )\right )^{2}}\right )}}{\log \left (-4 \, \log \left (5\right )\right )^{2}} \]

[In]

integrate((10*log(1/4/x)-10)*exp(5*x*log(1/4/x)/log(-4*log(5))^2)^2/log(-4*log(5))^2,x, algorithm="maxima")

[Out]

(4*log(2)^2 + 4*log(2)*log(-log(5)) + log(-log(5))^2)*e^(-20*x*log(2)/(4*log(2)^2 + 4*log(2)*log(-log(5)) + lo
g(-log(5))^2) - 10*x*log(x)/(4*log(2)^2 + 4*log(2)*log(-log(5)) + log(-log(5))^2))/log(-4*log(5))^2

Giac [F]

\[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx=\int { \frac {10 \, \left (\frac {1}{4 \, x}\right )^{\frac {10 \, x}{\log \left (-4 \, \log \left (5\right )\right )^{2}}} {\left (\log \left (\frac {1}{4 \, x}\right ) - 1\right )}}{\log \left (-4 \, \log \left (5\right )\right )^{2}} \,d x } \]

[In]

integrate((10*log(1/4/x)-10)*exp(5*x*log(1/4/x)/log(-4*log(5))^2)^2/log(-4*log(5))^2,x, algorithm="giac")

[Out]

integrate(10*(1/4/x)^(10*x/log(-4*log(5))^2)*(log(1/4/x) - 1)/log(-4*log(5))^2, x)

Mupad [B] (verification not implemented)

Time = 14.48 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.66 \[ \int \frac {4^{-\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (\frac {1}{x}\right )^{\frac {10 x}{(i \pi +\log (\log (625)))^2}} \left (-10+10 \log \left (\frac {1}{4 x}\right )\right )}{(i \pi +\log (\log (625)))^2} \, dx={\mathrm {e}}^{-\frac {20\,x\,\ln \left (2\right )}{{\ln \left (\ln \left (5\right )\right )}^2+4\,\ln \left (2\right )\,\ln \left (\ln \left (5\right )\right )-\pi ^2+4\,{\ln \left (2\right )}^2+\pi \,\ln \left (2\right )\,4{}\mathrm {i}+\pi \,\ln \left (\ln \left (5\right )\right )\,2{}\mathrm {i}}}\,{\mathrm {e}}^{\frac {10\,x\,\ln \left (\frac {1}{x}\right )}{{\ln \left (\ln \left (5\right )\right )}^2+4\,\ln \left (2\right )\,\ln \left (\ln \left (5\right )\right )-\pi ^2+4\,{\ln \left (2\right )}^2+\pi \,\ln \left (2\right )\,4{}\mathrm {i}+\pi \,\ln \left (\ln \left (5\right )\right )\,2{}\mathrm {i}}} \]

[In]

int((exp((10*x*log(1/(4*x)))/log(-4*log(5))^2)*(10*log(1/(4*x)) - 10))/log(-4*log(5))^2,x)

[Out]

exp(-(20*x*log(2))/(pi*log(2)*4i + log(log(5))^2 + pi*log(log(5))*2i + 4*log(2)*log(log(5)) - pi^2 + 4*log(2)^
2))*exp((10*x*log(1/x))/(pi*log(2)*4i + log(log(5))^2 + pi*log(log(5))*2i + 4*log(2)*log(log(5)) - pi^2 + 4*lo
g(2)^2))