3.12.11 \(\int \frac {32 e^{8 x} x+2 x^4+(-120 x^3-6 x^4+e^{8 x} (3840-15168 x-768 x^2)+(40 x^3+2 x^4+e^{8 x} (-1280+5056 x+256 x^2)) \log (\frac {3}{20+x})) \log (3-\log (\frac {3}{20+x}))}{(-60 x^4-3 x^5+e^{8 x} (-960 x-48 x^2)+(20 x^4+x^5+e^{8 x} (320 x+16 x^2)) \log (\frac {3}{20+x})) \log (3-\log (\frac {3}{20+x}))} \, dx\) [1111]

3.12.11.1 Optimal result

Integrand size = 168, antiderivative size = 31 \begin {dmath*} \int \frac {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx=2 \log \left (\frac {\frac {16 e^{8 x}}{x^2}+x}{\log \left (3-\log \left (\frac {3}{20+x}\right )\right )}\right ) \end {dmath*}

output

2*ln((16*exp(4*x)^2/x^2+x)/ln(-ln(3/(20+x))+3))
 

3.12.11.2 Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \begin {dmath*} \int \frac {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx=2 \left (-2 \log (x)+\log \left (16 e^{8 x}+x^3\right )-\log \left (\log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )\right ) \end {dmath*}

input

Integrate[(32*E^(8*x)*x + 2*x^4 + (-120*x^3 - 6*x^4 + E^(8*x)*(3840 - 1516 
8*x - 768*x^2) + (40*x^3 + 2*x^4 + E^(8*x)*(-1280 + 5056*x + 256*x^2))*Log 
[3/(20 + x)])*Log[3 - Log[3/(20 + x)]])/((-60*x^4 - 3*x^5 + E^(8*x)*(-960* 
x - 48*x^2) + (20*x^4 + x^5 + E^(8*x)*(320*x + 16*x^2))*Log[3/(20 + x)])*L 
og[3 - Log[3/(20 + x)]]),x]
 

output

2*(-2*Log[x] + Log[16*E^(8*x) + x^3] - Log[Log[3 - Log[3/(20 + x)]]])
 

3.12.11.3 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.

\(\displaystyle \int \frac {2 x^4+\left (-6 x^4-120 x^3+e^{8 x} \left (-768 x^2-15168 x+3840\right )+\left (2 x^4+40 x^3+e^{8 x} \left (256 x^2+5056 x-1280\right )\right ) \log \left (\frac {3}{x+20}\right )\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )+32 e^{8 x} x}{\left (-3 x^5-60 x^4+e^{8 x} \left (-48 x^2-960 x\right )+\left (x^5+20 x^4+e^{8 x} \left (16 x^2+320 x\right )\right ) \log \left (\frac {3}{x+20}\right )\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-2 x^4-\left (-6 x^4-120 x^3+e^{8 x} \left (-768 x^2-15168 x+3840\right )+\left (2 x^4+40 x^3+e^{8 x} \left (256 x^2+5056 x-1280\right )\right ) \log \left (\frac {3}{x+20}\right )\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )-32 e^{8 x} x}{x (x+20) \left (x^3+16 e^{8 x}\right ) \left (3-\log \left (\frac {3}{x+20}\right )\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (8 x^2 \log \left (\frac {3}{x+20}\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )-24 x^2 \log \left (3-\log \left (\frac {3}{x+20}\right )\right )+x+158 x \log \left (\frac {3}{x+20}\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )-474 x \log \left (3-\log \left (\frac {3}{x+20}\right )\right )-40 \log \left (\frac {3}{x+20}\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )+120 \log \left (3-\log \left (\frac {3}{x+20}\right )\right )\right )}{x (x+20) \left (\log \left (\frac {3}{x+20}\right )-3\right ) \log \left (3-\log \left (\frac {3}{x+20}\right )\right )}-\frac {2 x^2 (8 x-3)}{x^3+16 e^{8 x}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -16 \int \frac {x^3}{x^3+16 e^{8 x}}dx+6 \int \frac {x^2}{x^3+16 e^{8 x}}dx+16 x-4 \log (x)-2 \log \left (\log \left (3-\log \left (\frac {3}{x+20}\right )\right )\right )\)

input

Int[(32*E^(8*x)*x + 2*x^4 + (-120*x^3 - 6*x^4 + E^(8*x)*(3840 - 15168*x - 
768*x^2) + (40*x^3 + 2*x^4 + E^(8*x)*(-1280 + 5056*x + 256*x^2))*Log[3/(20 
 + x)])*Log[3 - Log[3/(20 + x)]])/((-60*x^4 - 3*x^5 + E^(8*x)*(-960*x - 48 
*x^2) + (20*x^4 + x^5 + E^(8*x)*(320*x + 16*x^2))*Log[3/(20 + x)])*Log[3 - 
 Log[3/(20 + x)]]),x]
 

output

$Aborted
 

3.12.11.3.1 Defintions of rubi rules used

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] 
]
 

3.12.11.4 Maple [A] (verified)

Time = 12.36 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06

method	result	size
risch	\(-4 \ln \left (x \right )+2 \ln \left (\frac {x^{3}}{16}+{\mathrm e}^{8 x}\right )-2 \ln \left (\ln \left (-\ln \left (3\right )+\ln \left (20+x \right )+3\right )\right )\)	\(33\)
parallelrisch	\(-4 \ln \left (x \right )-2 \ln \left (\ln \left (-\ln \left (\frac {3}{20+x}\right )+3\right )\right )+2 \ln \left (x^{3}+16 \,{\mathrm e}^{8 x}\right )\)	\(37\)

input

int(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*ln(3/(20+x))+(-768*x 
^2-15168*x+3840)*exp(4*x)^2-6*x^4-120*x^3)*ln(-ln(3/(20+x))+3)+32*x*exp(4* 
x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*ln(3/(20+x))+(-48*x^2- 
960*x)*exp(4*x)^2-3*x^5-60*x^4)/ln(-ln(3/(20+x))+3),x,method=_RETURNVERBOS 
E)
 

output

-4*ln(x)+2*ln(1/16*x^3+exp(8*x))-2*ln(ln(-ln(3)+ln(20+x)+3))
 

3.12.11.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \begin {dmath*} \int \frac {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx=2 \, \log \left (x^{3} + 16 \, e^{\left (8 \, x\right )}\right ) - 4 \, \log \left (x\right ) - 2 \, \log \left (\log \left (-\log \left (\frac {3}{x + 20}\right ) + 3\right )\right ) \end {dmath*}

input

integrate(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*log(3/(20+x))+ 
(-768*x^2-15168*x+3840)*exp(4*x)^2-6*x^4-120*x^3)*log(-log(3/(20+x))+3)+32 
*x*exp(4*x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*log(3/(20+x)) 
+(-48*x^2-960*x)*exp(4*x)^2-3*x^5-60*x^4)/log(-log(3/(20+x))+3),x, algorit 
hm=\
 

output

2*log(x^3 + 16*e^(8*x)) - 4*log(x) - 2*log(log(-log(3/(x + 20)) + 3))
 

3.12.11.6 Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx=- 4 \log {\left (x \right )} + 2 \log {\left (\frac {x^{3}}{16} + e^{8 x} \right )} - 2 \log {\left (\log {\left (3 - \log {\left (\frac {3}{x + 20} \right )} \right )} \right )} \end {dmath*}

input

integrate(((((256*x**2+5056*x-1280)*exp(4*x)**2+2*x**4+40*x**3)*ln(3/(20+x 
))+(-768*x**2-15168*x+3840)*exp(4*x)**2-6*x**4-120*x**3)*ln(-ln(3/(20+x))+ 
3)+32*x*exp(4*x)**2+2*x**4)/(((16*x**2+320*x)*exp(4*x)**2+x**5+20*x**4)*ln 
(3/(20+x))+(-48*x**2-960*x)*exp(4*x)**2-3*x**5-60*x**4)/ln(-ln(3/(20+x))+3 
),x)
 

output

-4*log(x) + 2*log(x**3/16 + exp(8*x)) - 2*log(log(3 - log(3/(x + 20))))
 

3.12.11.7 Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \begin {dmath*} \int \frac {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx=2 \, \log \left (\frac {1}{16} \, x^{3} + e^{\left (8 \, x\right )}\right ) - 4 \, \log \left (x\right ) - 2 \, \log \left (\log \left (-\log \left (3\right ) + \log \left (x + 20\right ) + 3\right )\right ) \end {dmath*}

input

integrate(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*log(3/(20+x))+ 
(-768*x^2-15168*x+3840)*exp(4*x)^2-6*x^4-120*x^3)*log(-log(3/(20+x))+3)+32 
*x*exp(4*x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*log(3/(20+x)) 
+(-48*x^2-960*x)*exp(4*x)^2-3*x^5-60*x^4)/log(-log(3/(20+x))+3),x, algorit 
hm=\
 

output

2*log(1/16*x^3 + e^(8*x)) - 4*log(x) - 2*log(log(-log(3) + log(x + 20) + 3 
))
 

3.12.11.8 Giac [A] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \begin {dmath*} \int \frac {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx=2 \, \log \left (-x^{3} - 16 \, e^{\left (8 \, x\right )}\right ) - 4 \, \log \left (x\right ) - 2 \, \log \left (\log \left (-\log \left (3\right ) + \log \left (x + 20\right ) + 3\right )\right ) \end {dmath*}

input

integrate(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*log(3/(20+x))+ 
(-768*x^2-15168*x+3840)*exp(4*x)^2-6*x^4-120*x^3)*log(-log(3/(20+x))+3)+32 
*x*exp(4*x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*log(3/(20+x)) 
+(-48*x^2-960*x)*exp(4*x)^2-3*x^5-60*x^4)/log(-log(3/(20+x))+3),x, algorit 
hm=\
 

output

2*log(-x^3 - 16*e^(8*x)) - 4*log(x) - 2*log(log(-log(3) + log(x + 20) + 3) 
)
 

3.12.11.9 Mupad [B] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \begin {dmath*} \int \frac {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx=2\,\ln \left ({\mathrm {e}}^{8\,x}+\frac {x^3}{16}\right )-2\,\ln \left (\ln \left (3-\ln \left (\frac {3}{x+20}\right )\right )\right )-4\,\ln \left (x\right ) \end {dmath*}

input

int(-(32*x*exp(8*x) - log(3 - log(3/(x + 20)))*(exp(8*x)*(15168*x + 768*x^ 
2 - 3840) - log(3/(x + 20))*(exp(8*x)*(5056*x + 256*x^2 - 1280) + 40*x^3 + 
 2*x^4) + 120*x^3 + 6*x^4) + 2*x^4)/(log(3 - log(3/(x + 20)))*(exp(8*x)*(9 
60*x + 48*x^2) + 60*x^4 + 3*x^5 - log(3/(x + 20))*(exp(8*x)*(320*x + 16*x^ 
2) + 20*x^4 + x^5))),x)
 

output

2*log(exp(8*x) + x^3/16) - 2*log(log(3 - log(3/(x + 20)))) - 4*log(x)