3.13.59 \(\int \frac {e^{-\frac {x}{\log (\frac {48 x}{4+3 x})}} (1036+(-1036-777 x) \log (\frac {48 x}{4+3 x}))}{(4+3 x) \log ^2(\frac {48 x}{4+3 x})} \, dx\) [1259]

3.13.59.1 Optimal result

Integrand size = 58, antiderivative size = 20 \[ \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \left (1036+(-1036-777 x) \log \left (\frac {48 x}{4+3 x}\right )\right )}{(4+3 x) \log ^2\left (\frac {48 x}{4+3 x}\right )} \, dx=259 e^{-\frac {x}{\log \left (\frac {16 x}{\frac {4}{3}+x}\right )}} \]

output

259/exp(x/ln(x/(1/16*x+1/12)))
 

3.13.59.2 Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \left (1036+(-1036-777 x) \log \left (\frac {48 x}{4+3 x}\right )\right )}{(4+3 x) \log ^2\left (\frac {48 x}{4+3 x}\right )} \, dx=259 e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \]

input

Integrate[(1036 + (-1036 - 777*x)*Log[(48*x)/(4 + 3*x)])/(E^(x/Log[(48*x)/ 
(4 + 3*x)])*(4 + 3*x)*Log[(48*x)/(4 + 3*x)]^2),x]
 

output

259/E^(x/Log[(48*x)/(4 + 3*x)])
 

3.13.59.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {7257}

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.

\(\displaystyle \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{3 x+4}\right )}} \left ((-777 x-1036) \log \left (\frac {48 x}{3 x+4}\right )+1036\right )}{(3 x+4) \log ^2\left (\frac {48 x}{3 x+4}\right )} \, dx\)

\(\Big \downarrow \) 7257

\(\displaystyle 259 e^{-\frac {x}{\log \left (\frac {48 x}{3 x+4}\right )}}\)

input

Int[(1036 + (-1036 - 777*x)*Log[(48*x)/(4 + 3*x)])/(E^(x/Log[(48*x)/(4 + 3 
*x)])*(4 + 3*x)*Log[(48*x)/(4 + 3*x)]^2),x]
 

output

259/E^(x/Log[(48*x)/(4 + 3*x)])
 

3.13.59.3.1 Defintions of rubi rules used

rule 7257

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim 
p[q*(F^v/Log[F]), x] /;  !FalseQ[q]] /; FreeQ[F, x]
 

3.13.59.4 Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

method	result	size
risch	\(259 \,{\mathrm e}^{-\frac {x}{\ln \left (\frac {48 x}{4+3 x}\right )}}\)	\(20\)
norman	\(259 \,{\mathrm e}^{-\frac {x}{\ln \left (\frac {48 x}{4+3 x}\right )}}\)	\(21\)
parallelrisch	\(259 \,{\mathrm e}^{-\frac {x}{\ln \left (\frac {48 x}{4+3 x}\right )}}\)	\(21\)

input

int(((-777*x-1036)*ln(48*x/(4+3*x))+1036)/(4+3*x)/ln(48*x/(4+3*x))^2/exp(x 
/ln(48*x/(4+3*x))),x,method=_RETURNVERBOSE)
 

output

259*exp(-x/ln(48*x/(4+3*x)))
 

3.13.59.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \left (1036+(-1036-777 x) \log \left (\frac {48 x}{4+3 x}\right )\right )}{(4+3 x) \log ^2\left (\frac {48 x}{4+3 x}\right )} \, dx=259 \, e^{\left (-\frac {x}{\log \left (\frac {48 \, x}{3 \, x + 4}\right )}\right )} \]

input

integrate(((-777*x-1036)*log(48*x/(4+3*x))+1036)/(4+3*x)/log(48*x/(4+3*x)) 
^2/exp(x/log(48*x/(4+3*x))),x, algorithm=\
 

output

259*e^(-x/log(48*x/(3*x + 4)))
 

3.13.59.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \left (1036+(-1036-777 x) \log \left (\frac {48 x}{4+3 x}\right )\right )}{(4+3 x) \log ^2\left (\frac {48 x}{4+3 x}\right )} \, dx=\text {Exception raised: TypeError} \]

input

integrate(((-777*x-1036)*ln(48*x/(4+3*x))+1036)/(4+3*x)/ln(48*x/(4+3*x))** 
2/exp(x/ln(48*x/(4+3*x))),x)
 

output

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

3.13.59.7 Maxima [F]

\[ \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \left (1036+(-1036-777 x) \log \left (\frac {48 x}{4+3 x}\right )\right )}{(4+3 x) \log ^2\left (\frac {48 x}{4+3 x}\right )} \, dx=\int { -\frac {259 \, {\left ({\left (3 \, x + 4\right )} \log \left (\frac {48 \, x}{3 \, x + 4}\right ) - 4\right )} e^{\left (-\frac {x}{\log \left (\frac {48 \, x}{3 \, x + 4}\right )}\right )}}{{\left (3 \, x + 4\right )} \log \left (\frac {48 \, x}{3 \, x + 4}\right )^{2}} \,d x } \]

input

integrate(((-777*x-1036)*log(48*x/(4+3*x))+1036)/(4+3*x)/log(48*x/(4+3*x)) 
^2/exp(x/log(48*x/(4+3*x))),x, algorithm=\
 

output

-259*integrate(((3*x + 4)*log(48*x/(3*x + 4)) - 4)*e^(-x/log(48*x/(3*x + 4 
)))/((3*x + 4)*log(48*x/(3*x + 4))^2), x)
 

3.13.59.8 Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \left (1036+(-1036-777 x) \log \left (\frac {48 x}{4+3 x}\right )\right )}{(4+3 x) \log ^2\left (\frac {48 x}{4+3 x}\right )} \, dx=259 \, e^{\left (-\frac {x}{\log \left (\frac {48 \, x}{3 \, x + 4}\right )}\right )} \]

input

integrate(((-777*x-1036)*log(48*x/(4+3*x))+1036)/(4+3*x)/log(48*x/(4+3*x)) 
^2/exp(x/log(48*x/(4+3*x))),x, algorithm=\
 

output

259*e^(-x/log(48*x/(3*x + 4)))
 

3.13.59.9 Mupad [B] (verification not implemented)

Time = 9.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {e^{-\frac {x}{\log \left (\frac {48 x}{4+3 x}\right )}} \left (1036+(-1036-777 x) \log \left (\frac {48 x}{4+3 x}\right )\right )}{(4+3 x) \log ^2\left (\frac {48 x}{4+3 x}\right )} \, dx=259\,{\mathrm {e}}^{-\frac {x}{\ln \left (\frac {48\,x}{3\,x+4}\right )}} \]

input

int(-(exp(-x/log((48*x)/(3*x + 4)))*(log((48*x)/(3*x + 4))*(777*x + 1036) 
- 1036))/(log((48*x)/(3*x + 4))^2*(3*x + 4)),x)
 

output

259*exp(-x/log((48*x)/(3*x + 4)))