\(\int \frac {-768+448 x-64 x^2-4 x^3+x^4+(-192+112 x-20 x^2+x^3) \log (2)+e^{-\frac {4 x}{-4+x}} (64-32 x+20 x^2+(16-8 x+x^2) \log (2))+(-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} (32 x-16 x^2+2 x^3)) \log (-12+e^{-\frac {4 x}{-4+x}}+x)}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} (16-8 x+x^2)} \, dx\) [344]

Optimal result

Integrand size = 160, antiderivative size = 27 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=x (4+\log (2))+x^2 \log \left (-12+e^{\frac {4 x}{4-x}}+x\right ) \] Output:

x^2*ln(exp(4*x/(4-x))-12+x)+x*(4+ln(2))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=x \left (4+\log (2)+x \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )\right ) \] Input:

Integrate[(-768 + 448*x - 64*x^2 - 4*x^3 + x^4 + (-192 + 112*x - 20*x^2 + 
x^3)*Log[2] + (64 - 32*x + 20*x^2 + (16 - 8*x + x^2)*Log[2])/E^((4*x)/(-4 
+ x)) + (-384*x + 224*x^2 - 40*x^3 + 2*x^4 + (32*x - 16*x^2 + 2*x^3)/E^((4 
*x)/(-4 + x)))*Log[-12 + E^((-4*x)/(-4 + x)) + x])/(-192 + 112*x - 20*x^2 
+ x^3 + (16 - 8*x + x^2)/E^((4*x)/(-4 + x))),x]
 

Output:

x*(4 + Log[2] + x*Log[-12 + E^((-4*x)/(-4 + x)) + x])
 

Rubi [A] (verified)

Time = 5.32 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {7293, 2009}

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.

\(\displaystyle \int \frac {x^4-4 x^3-64 x^2+e^{-\frac {4 x}{x-4}} \left (20 x^2+\left (x^2-8 x+16\right ) \log (2)-32 x+64\right )+\left (x^3-20 x^2+112 x-192\right ) \log (2)+\left (2 x^4-40 x^3+224 x^2+e^{-\frac {4 x}{x-4}} \left (2 x^3-16 x^2+32 x\right )-384 x\right ) \log \left (x+e^{-\frac {4 x}{x-4}}-12\right )+448 x-768}{x^3-20 x^2+e^{-\frac {4 x}{x-4}} \left (x^2-8 x+16\right )+112 x-192} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-x^2-2 x^2 \log \left (x+e^{-\frac {4 x}{x-4}}-12\right )+24 x \log \left (x+e^{-\frac {4 x}{x-4}}-12\right )-4 x \left (1+\frac {\log (2)}{4}\right )+48 \left (1+\frac {\log (2)}{4}\right )}{12-x}-\frac {x^2 \left (x^2-24 x+208\right )}{(x-12) (x-4)^2 \left (e^{\frac {4 x}{x-4}} x-12 e^{\frac {4 x}{x-4}}+1\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle x^2 \log \left (x+e^{\frac {4 x}{4-x}}-12\right )-12 x+x (16+\log (2))\)

Input:

Int[(-768 + 448*x - 64*x^2 - 4*x^3 + x^4 + (-192 + 112*x - 20*x^2 + x^3)*L 
og[2] + (64 - 32*x + 20*x^2 + (16 - 8*x + x^2)*Log[2])/E^((4*x)/(-4 + x)) 
+ (-384*x + 224*x^2 - 40*x^3 + 2*x^4 + (32*x - 16*x^2 + 2*x^3)/E^((4*x)/(- 
4 + x)))*Log[-12 + E^((-4*x)/(-4 + x)) + x])/(-192 + 112*x - 20*x^2 + x^3 
+ (16 - 8*x + x^2)/E^((4*x)/(-4 + x))),x]
 

Output:

-12*x + x*(16 + Log[2]) + x^2*Log[-12 + E^((4*x)/(4 - x)) + x]
 

Defintions of rubi rules used

rule 2009

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

rule 7293

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

Maple [A] (verified)

Time = 1.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96

method	result	size
risch	\(\ln \left ({\mathrm e}^{-\frac {4 x}{x -4}}+x -12\right ) x^{2}+x \ln \left (2\right )+4 x\)	\(26\)
parallelrisch	\(\ln \left ({\mathrm e}^{-\frac {4 x}{x -4}}+x -12\right ) x^{2}+x \ln \left (2\right )+16 \ln \left (2\right )+4 x +64\)	\(31\)
norman	\(\frac {\left (4+\ln \left (2\right )\right ) x^{2}+\ln \left ({\mathrm e}^{-\frac {4 x}{x -4}}+x -12\right ) x^{3}-64-4 \ln \left ({\mathrm e}^{-\frac {4 x}{x -4}}+x -12\right ) x^{2}-16 \ln \left (2\right )}{x -4}\)	\(56\)

Input:

int((((2*x^3-16*x^2+32*x)*exp(-4*x/(x-4))+2*x^4-40*x^3+224*x^2-384*x)*ln(e 
xp(-4*x/(x-4))+x-12)+((x^2-8*x+16)*ln(2)+20*x^2-32*x+64)*exp(-4*x/(x-4))+( 
x^3-20*x^2+112*x-192)*ln(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2-8*x+16)*exp( 
-4*x/(x-4))+x^3-20*x^2+112*x-192),x,method=_RETURNVERBOSE)
 

Output:

ln(exp(-4*x/(x-4))+x-12)*x^2+x*ln(2)+4*x
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=x^{2} \log \left (x + e^{\left (-\frac {4 \, x}{x - 4}\right )} - 12\right ) + x \log \left (2\right ) + 4 \, x \] Input:

integrate((((2*x^3-16*x^2+32*x)*exp(-4*x/(-4+x))+2*x^4-40*x^3+224*x^2-384* 
x)*log(exp(-4*x/(-4+x))+x-12)+((x^2-8*x+16)*log(2)+20*x^2-32*x+64)*exp(-4* 
x/(-4+x))+(x^3-20*x^2+112*x-192)*log(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2- 
8*x+16)*exp(-4*x/(-4+x))+x^3-20*x^2+112*x-192),x, algorithm="fricas")
 

Output:

x^2*log(x + e^(-4*x/(x - 4)) - 12) + x*log(2) + 4*x
 

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=x^{2} \log {\left (x - 12 + e^{- \frac {4 x}{x - 4}} \right )} + x \left (\log {\left (2 \right )} + 4\right ) \] Input:

integrate((((2*x**3-16*x**2+32*x)*exp(-4*x/(-4+x))+2*x**4-40*x**3+224*x**2 
-384*x)*ln(exp(-4*x/(-4+x))+x-12)+((x**2-8*x+16)*ln(2)+20*x**2-32*x+64)*ex 
p(-4*x/(-4+x))+(x**3-20*x**2+112*x-192)*ln(2)+x**4-4*x**3-64*x**2+448*x-76 
8)/((x**2-8*x+16)*exp(-4*x/(-4+x))+x**3-20*x**2+112*x-192),x)
                                                                                    
                                                                                    
 

Output:

x**2*log(x - 12 + exp(-4*x/(x - 4))) + x*(log(2) + 4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (24) = 48\).

Time = 0.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=-\frac {4 \, x^{3} - x^{2} {\left (\log \left (2\right ) + 4\right )} + 4 \, x {\left (\log \left (2\right ) - 12\right )} - {\left (x^{3} - 4 \, x^{2}\right )} \log \left ({\left (x e^{4} - 12 \, e^{4}\right )} e^{\left (\frac {16}{x - 4}\right )} + 1\right ) + 256}{x - 4} \] Input:

integrate((((2*x^3-16*x^2+32*x)*exp(-4*x/(-4+x))+2*x^4-40*x^3+224*x^2-384* 
x)*log(exp(-4*x/(-4+x))+x-12)+((x^2-8*x+16)*log(2)+20*x^2-32*x+64)*exp(-4* 
x/(-4+x))+(x^3-20*x^2+112*x-192)*log(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2- 
8*x+16)*exp(-4*x/(-4+x))+x^3-20*x^2+112*x-192),x, algorithm="maxima")
 

Output:

-(4*x^3 - x^2*(log(2) + 4) + 4*x*(log(2) - 12) - (x^3 - 4*x^2)*log((x*e^4 
- 12*e^4)*e^(16/(x - 4)) + 1) + 256)/(x - 4)
 

Giac [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=x^{2} \log \left (x + e^{\left (-\frac {4 \, x}{x - 4}\right )} - 12\right ) + x \log \left (2\right ) + 4 \, x \] Input:

integrate((((2*x^3-16*x^2+32*x)*exp(-4*x/(-4+x))+2*x^4-40*x^3+224*x^2-384* 
x)*log(exp(-4*x/(-4+x))+x-12)+((x^2-8*x+16)*log(2)+20*x^2-32*x+64)*exp(-4* 
x/(-4+x))+(x^3-20*x^2+112*x-192)*log(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2- 
8*x+16)*exp(-4*x/(-4+x))+x^3-20*x^2+112*x-192),x, algorithm="giac")
 

Output:

x^2*log(x + e^(-4*x/(x - 4)) - 12) + x*log(2) + 4*x
 

Mupad [B] (verification not implemented)

Time = 4.17 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=x\,\left (\ln \left (2\right )+4\right )-\frac {\ln \left (x+{\mathrm {e}}^{-\frac {4\,x}{x-4}}-12\right )\,\left (4\,x^2-x^3\right )}{x-4} \] Input:

int((448*x + log(2)*(112*x - 20*x^2 + x^3 - 192) + exp(-(4*x)/(x - 4))*(20 
*x^2 - 32*x + log(2)*(x^2 - 8*x + 16) + 64) + log(x + exp(-(4*x)/(x - 4)) 
- 12)*(exp(-(4*x)/(x - 4))*(32*x - 16*x^2 + 2*x^3) - 384*x + 224*x^2 - 40* 
x^3 + 2*x^4) - 64*x^2 - 4*x^3 + x^4 - 768)/(112*x - 20*x^2 + x^3 + exp(-(4 
*x)/(x - 4))*(x^2 - 8*x + 16) - 192),x)
 

Output:

x*(log(2) + 4) - (log(x + exp(-(4*x)/(x - 4)) - 12)*(4*x^2 - x^3))/(x - 4)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 454, normalized size of antiderivative = 16.81 \[ \int \frac {-768+448 x-64 x^2-4 x^3+x^4+\left (-192+112 x-20 x^2+x^3\right ) \log (2)+e^{-\frac {4 x}{-4+x}} \left (64-32 x+20 x^2+\left (16-8 x+x^2\right ) \log (2)\right )+\left (-384 x+224 x^2-40 x^3+2 x^4+e^{-\frac {4 x}{-4+x}} \left (32 x-16 x^2+2 x^3\right )\right ) \log \left (-12+e^{-\frac {4 x}{-4+x}}+x\right )}{-192+112 x-20 x^2+x^3+e^{-\frac {4 x}{-4+x}} \left (16-8 x+x^2\right )} \, dx=\frac {-\mathrm {log}\left (e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1\right ) \mathrm {log}\left (2\right ) x +4 \,\mathrm {log}\left (e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1\right ) \mathrm {log}\left (2\right )-4 \,\mathrm {log}\left (e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1\right ) x +16 \,\mathrm {log}\left (e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1\right )+\mathrm {log}\left (\frac {e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1}{e^{\frac {16}{x -4}} e^{4}}\right ) \mathrm {log}\left (2\right ) x -4 \,\mathrm {log}\left (\frac {e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1}{e^{\frac {16}{x -4}} e^{4}}\right ) \mathrm {log}\left (2\right )+\mathrm {log}\left (\frac {e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1}{e^{\frac {16}{x -4}} e^{4}}\right ) x^{3}-4 \,\mathrm {log}\left (\frac {e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1}{e^{\frac {16}{x -4}} e^{4}}\right ) x^{2}+4 \,\mathrm {log}\left (\frac {e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1}{e^{\frac {16}{x -4}} e^{4}}\right ) x -16 \,\mathrm {log}\left (\frac {e^{\frac {16}{x -4}} e^{4} x -12 e^{\frac {16}{x -4}} e^{4}+1}{e^{\frac {16}{x -4}} e^{4}}\right )+\mathrm {log}\left (2\right ) x^{2}+4 x^{2}}{x -4} \] Input:

int((((2*x^3-16*x^2+32*x)*exp(-4*x/(-4+x))+2*x^4-40*x^3+224*x^2-384*x)*log 
(exp(-4*x/(-4+x))+x-12)+((x^2-8*x+16)*log(2)+20*x^2-32*x+64)*exp(-4*x/(-4+ 
x))+(x^3-20*x^2+112*x-192)*log(2)+x^4-4*x^3-64*x^2+448*x-768)/((x^2-8*x+16 
)*exp(-4*x/(-4+x))+x^3-20*x^2+112*x-192),x)
 

Output:

( - log(e**(16/(x - 4))*e**4*x - 12*e**(16/(x - 4))*e**4 + 1)*log(2)*x + 4 
*log(e**(16/(x - 4))*e**4*x - 12*e**(16/(x - 4))*e**4 + 1)*log(2) - 4*log( 
e**(16/(x - 4))*e**4*x - 12*e**(16/(x - 4))*e**4 + 1)*x + 16*log(e**(16/(x 
 - 4))*e**4*x - 12*e**(16/(x - 4))*e**4 + 1) + log((e**(16/(x - 4))*e**4*x 
 - 12*e**(16/(x - 4))*e**4 + 1)/(e**(16/(x - 4))*e**4))*log(2)*x - 4*log(( 
e**(16/(x - 4))*e**4*x - 12*e**(16/(x - 4))*e**4 + 1)/(e**(16/(x - 4))*e** 
4))*log(2) + log((e**(16/(x - 4))*e**4*x - 12*e**(16/(x - 4))*e**4 + 1)/(e 
**(16/(x - 4))*e**4))*x**3 - 4*log((e**(16/(x - 4))*e**4*x - 12*e**(16/(x 
- 4))*e**4 + 1)/(e**(16/(x - 4))*e**4))*x**2 + 4*log((e**(16/(x - 4))*e**4 
*x - 12*e**(16/(x - 4))*e**4 + 1)/(e**(16/(x - 4))*e**4))*x - 16*log((e**( 
16/(x - 4))*e**4*x - 12*e**(16/(x - 4))*e**4 + 1)/(e**(16/(x - 4))*e**4)) 
+ log(2)*x**2 + 4*x**2)/(x - 4)