3.14.59 \(\int \frac {4^{-x} (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2})^x (1-2 \log (x)+(4 x^2+x^2 \log (7)+\log (x)) \log (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}))}{4 x^2+x^2 \log (7)+\log (x)} \, dx\) [1359]

3.14.59.1 Optimal result

Integrand size = 85, antiderivative size = 18 \[ \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx=4^{-x} \left (4+\log (7)+\frac {\log (x)}{x^2}\right )^x \]

output

exp(ln(1+1/4*ln(7)+1/4*ln(x)/x^2)*x)
 

3.14.59.2 Mathematica [F]

\[ \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx=\int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx \]

input

Integrate[(((4*x^2 + x^2*Log[7] + Log[x])/x^2)^x*(1 - 2*Log[x] + (4*x^2 + 
x^2*Log[7] + Log[x])*Log[(4*x^2 + x^2*Log[7] + Log[x])/(4*x^2)]))/(4^x*(4* 
x^2 + x^2*Log[7] + Log[x])),x]
 

output

Integrate[(((4*x^2 + x^2*Log[7] + Log[x])/x^2)^x*(1 - 2*Log[x] + (4*x^2 + 
x^2*Log[7] + Log[x])*Log[(4*x^2 + x^2*Log[7] + Log[x])/(4*x^2)]))/(4^x*(4* 
x^2 + x^2*Log[7] + Log[x])), x]
 

3.14.59.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )-2 \log (x)+1\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )-2 \log (x)+1\right )}{x^2 (4+\log (7))+\log (x)}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {4^{-x} \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )^x \left (\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )-2 \log (x)+1\right )}{x^2 (4+\log (7))+\log (x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (4^{-x} \log \left (\frac {1}{4} \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )\right ) \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )^x+\frac {4^{-x} (1-2 \log (x)) \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )^x}{4 x^2 \left (1+\frac {\log (7)}{4}\right )+\log (x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int 2^{1-2 x} \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )^xdx+\int \frac {4^{-x} \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )^x}{4 \left (1+\frac {\log (7)}{4}\right ) x^2+\log (x)}dx+(4+\log (7)) \int \frac {2^{1-2 x} x^2 \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )^x}{4 \left (1+\frac {\log (7)}{4}\right ) x^2+\log (x)}dx+\int 4^{-x} \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )^x \log \left (\frac {1}{4} \left (\frac {\log (x)}{x^2}+4 \left (1+\frac {\log (7)}{4}\right )\right )\right )dx\)

input

Int[(((4*x^2 + x^2*Log[7] + Log[x])/x^2)^x*(1 - 2*Log[x] + (4*x^2 + x^2*Lo 
g[7] + Log[x])*Log[(4*x^2 + x^2*Log[7] + Log[x])/(4*x^2)]))/(4^x*(4*x^2 + 
x^2*Log[7] + Log[x])),x]
 

output

$Aborted
 

3.14.59.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

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

rule 7293

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

3.14.59.4 Maple [A] (verified)

Time = 3.98 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33

method	result	size
parallelrisch	\({\mathrm e}^{x \ln \left (\frac {\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}}{4 x^{2}}\right )}\)	\(24\)
risch	\(x^{-2 x} \left (\frac {1}{4}\right )^{x} \left (\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}\right )^{x} {\mathrm e}^{\frac {i \pi x \left (\operatorname {csgn}\left (i x^{2}\right )^{3}-2 \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-{\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}\right )}{x^{2}}\right )}^{3}+{\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}\right )}{x^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x^{2}}\right )+{\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}\right )}{x^{2}}\right )}^{2} \operatorname {csgn}\left (i \left (\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}\right )\right )-\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}\right )}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i \left (\ln \left (x \right )+x^{2} \ln \left (7\right )+4 x^{2}\right )\right )\right )}{2}}\)	\(219\)

input

int(((ln(x)+x^2*ln(7)+4*x^2)*ln(1/4*(ln(x)+x^2*ln(7)+4*x^2)/x^2)-2*ln(x)+1 
)*exp(x*ln(1/4*(ln(x)+x^2*ln(7)+4*x^2)/x^2))/(ln(x)+x^2*ln(7)+4*x^2),x,met 
hod=_RETURNVERBOSE)
 

output

exp(x*ln(1/4*(ln(x)+x^2*ln(7)+4*x^2)/x^2))
 

3.14.59.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx=\left (\frac {x^{2} \log \left (7\right ) + 4 \, x^{2} + \log \left (x\right )}{4 \, x^{2}}\right )^{x} \]

input

integrate(((log(x)+x^2*log(7)+4*x^2)*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2 
)-2*log(x)+1)*exp(x*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2))/(log(x)+x^2*lo 
g(7)+4*x^2),x, algorithm=\
 

output

(1/4*(x^2*log(7) + 4*x^2 + log(x))/x^2)^x
 

3.14.59.6 Sympy [A] (verification not implemented)

Time = 0.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx=e^{x \log {\left (\frac {\frac {x^{2} \log {\left (7 \right )}}{4} + x^{2} + \frac {\log {\left (x \right )}}{4}}{x^{2}} \right )}} \]

input

integrate(((ln(x)+x**2*ln(7)+4*x**2)*ln(1/4*(ln(x)+x**2*ln(7)+4*x**2)/x**2 
)-2*ln(x)+1)*exp(x*ln(1/4*(ln(x)+x**2*ln(7)+4*x**2)/x**2))/(ln(x)+x**2*ln( 
7)+4*x**2),x)
 

output

exp(x*log((x**2*log(7)/4 + x**2 + log(x)/4)/x**2))
 

3.14.59.7 Maxima [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx=e^{\left (-2 \, x \log \left (2\right ) + x \log \left (x^{2} {\left (\log \left (7\right ) + 4\right )} + \log \left (x\right )\right ) - 2 \, x \log \left (x\right )\right )} \]

input

integrate(((log(x)+x^2*log(7)+4*x^2)*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2 
)-2*log(x)+1)*exp(x*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2))/(log(x)+x^2*lo 
g(7)+4*x^2),x, algorithm=\
 

output

e^(-2*x*log(2) + x*log(x^2*(log(7) + 4) + log(x)) - 2*x*log(x))
 

3.14.59.8 Giac [F]

\[ \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx=\int { \frac {{\left ({\left (x^{2} \log \left (7\right ) + 4 \, x^{2} + \log \left (x\right )\right )} \log \left (\frac {x^{2} \log \left (7\right ) + 4 \, x^{2} + \log \left (x\right )}{4 \, x^{2}}\right ) - 2 \, \log \left (x\right ) + 1\right )} \left (\frac {x^{2} \log \left (7\right ) + 4 \, x^{2} + \log \left (x\right )}{4 \, x^{2}}\right )^{x}}{x^{2} \log \left (7\right ) + 4 \, x^{2} + \log \left (x\right )} \,d x } \]

input

integrate(((log(x)+x^2*log(7)+4*x^2)*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2 
)-2*log(x)+1)*exp(x*log(1/4*(log(x)+x^2*log(7)+4*x^2)/x^2))/(log(x)+x^2*lo 
g(7)+4*x^2),x, algorithm=\
 

output

integrate(((x^2*log(7) + 4*x^2 + log(x))*log(1/4*(x^2*log(7) + 4*x^2 + log 
(x))/x^2) - 2*log(x) + 1)*(1/4*(x^2*log(7) + 4*x^2 + log(x))/x^2)^x/(x^2*l 
og(7) + 4*x^2 + log(x)), x)
 

3.14.59.9 Mupad [B] (verification not implemented)

Time = 10.13 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.61 \[ \int \frac {4^{-x} \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{x^2}\right )^x \left (1-2 \log (x)+\left (4 x^2+x^2 \log (7)+\log (x)\right ) \log \left (\frac {4 x^2+x^2 \log (7)+\log (x)}{4 x^2}\right )\right )}{4 x^2+x^2 \log (7)+\log (x)} \, dx=\frac {{\left (\frac {1}{x^2}\right )}^x\,{\left (\ln \left (x\right )+x^2\,\ln \left (7\right )+4\,x^2\right )}^x}{2^{2\,x}} \]

input

int((exp(x*log((log(x)/4 + (x^2*log(7))/4 + x^2)/x^2))*(log((log(x)/4 + (x 
^2*log(7))/4 + x^2)/x^2)*(log(x) + x^2*log(7) + 4*x^2) - 2*log(x) + 1))/(l 
og(x) + x^2*log(7) + 4*x^2),x)
 

output

((1/x^2)^x*(log(x) + x^2*log(7) + 4*x^2)^x)/2^(2*x)