3.13.70 \(\int \frac {(-32 x-32 e^{-x} x-192 x^6) \log (1-e^{-x}+x+x^6)+(16-16 e^{-x}+16 x+16 x^6) \log ^2(1-e^{-x}+x+x^6)}{-x^2+e^{-x} x^2-x^3-x^8} \, dx\) [1270]

3.13.70.1 Optimal result

Integrand size = 95, antiderivative size = 21 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \log ^2\left (1-e^{-x}+x+x^6\right )}{x} \]

output

16*ln(-exp(-x)+x^6+x+1)^2/x
 

3.13.70.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(21)=42\).

Time = 0.73 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.76 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=16 \left (2 x+2 \log \left (1-e^{-x}+x+x^6\right )+\frac {\log ^2\left (1-e^{-x}+x+x^6\right )}{x}-2 \log \left (1-e^x \left (1+x+x^6\right )\right )\right ) \]

input

Integrate[((-32*x - (32*x)/E^x - 192*x^6)*Log[1 - E^(-x) + x + x^6] + (16 
- 16/E^x + 16*x + 16*x^6)*Log[1 - E^(-x) + x + x^6]^2)/(-x^2 + x^2/E^x - x 
^3 - x^8),x]
 

output

16*(2*x + 2*Log[1 - E^(-x) + x + x^6] + Log[1 - E^(-x) + x + x^6]^2/x - 2* 
Log[1 - E^x*(1 + x + x^6)])
 

3.13.70.3 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.

\(\displaystyle \int \frac {\left (16 x^6+16 x-16 e^{-x}+16\right ) \log ^2\left (x^6+x-e^{-x}+1\right )+\left (-192 x^6-32 e^{-x} x-32 x\right ) \log \left (x^6+x-e^{-x}+1\right )}{-x^8-x^3+e^{-x} x^2-x^2} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^x \left (\left (16 x^6+16 x-16 e^{-x}+16\right ) \log ^2\left (x^6+x-e^{-x}+1\right )+\left (-192 x^6-32 e^{-x} x-32 x\right ) \log \left (x^6+x-e^{-x}+1\right )\right )}{x^2 \left (-e^x x^6-e^x x-e^x+1\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {32 e^x \left (x^6+6 x^5+x+2\right ) \log \left (x^6+x-e^{-x}+1\right )}{x \left (e^x x^6+e^x x+e^x-1\right )}-\frac {16 \log \left (x^6+x-e^{-x}+1\right ) \left (\log \left (x^6+x-e^{-x}+1\right )+2 x\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 32 \log \left (x^6+x-e^{-x}+1\right ) \int \frac {e^x}{e^x x^6+e^x x+e^x-1}dx+64 \log \left (x^6+x-e^{-x}+1\right ) \int \frac {e^x}{x \left (e^x x^6+e^x x+e^x-1\right )}dx+192 \log \left (x^6+x-e^{-x}+1\right ) \int \frac {e^x x^4}{e^x x^6+e^x x+e^x-1}dx+32 \log \left (x^6+x-e^{-x}+1\right ) \int \frac {e^x x^5}{e^x x^6+e^x x+e^x-1}dx-32 \int \frac {\log \left (x^6+x-e^{-x}+1\right )}{x}dx-16 \int \frac {\log ^2\left (x^6+x-e^{-x}+1\right )}{x^2}dx-32 \int \frac {\int \frac {e^x}{e^x \left (x^6+x+1\right )-1}dx}{x^6+x+1}dx-192 \int \frac {x^5 \int \frac {e^x}{e^x \left (x^6+x+1\right )-1}dx}{x^6+x+1}dx-32 \int \frac {\int \frac {e^x}{e^x \left (x^6+x+1\right )-1}dx}{e^x x^6+e^x x+e^x-1}dx-32 \int \frac {\int \frac {e^x}{e^x \left (x^6+x+1\right )-1}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx-192 \int \frac {x^5 \int \frac {e^x}{e^x \left (x^6+x+1\right )-1}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx-64 \int \frac {\int \frac {e^x}{x \left (e^x \left (x^6+x+1\right )-1\right )}dx}{x^6+x+1}dx-384 \int \frac {x^5 \int \frac {e^x}{x \left (e^x \left (x^6+x+1\right )-1\right )}dx}{x^6+x+1}dx-64 \int \frac {\int \frac {e^x}{x \left (e^x \left (x^6+x+1\right )-1\right )}dx}{e^x x^6+e^x x+e^x-1}dx-64 \int \frac {\int \frac {e^x}{x \left (e^x \left (x^6+x+1\right )-1\right )}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx-384 \int \frac {x^5 \int \frac {e^x}{x \left (e^x \left (x^6+x+1\right )-1\right )}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx-192 \int \frac {\int \frac {e^x x^4}{e^x \left (x^6+x+1\right )-1}dx}{x^6+x+1}dx-1152 \int \frac {x^5 \int \frac {e^x x^4}{e^x \left (x^6+x+1\right )-1}dx}{x^6+x+1}dx-192 \int \frac {\int \frac {e^x x^4}{e^x \left (x^6+x+1\right )-1}dx}{e^x x^6+e^x x+e^x-1}dx-192 \int \frac {\int \frac {e^x x^4}{e^x \left (x^6+x+1\right )-1}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx-1152 \int \frac {x^5 \int \frac {e^x x^4}{e^x \left (x^6+x+1\right )-1}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx-32 \int \frac {\int \frac {e^x x^5}{e^x \left (x^6+x+1\right )-1}dx}{x^6+x+1}dx-192 \int \frac {x^5 \int \frac {e^x x^5}{e^x \left (x^6+x+1\right )-1}dx}{x^6+x+1}dx-32 \int \frac {\int \frac {e^x x^5}{e^x \left (x^6+x+1\right )-1}dx}{e^x x^6+e^x x+e^x-1}dx-32 \int \frac {\int \frac {e^x x^5}{e^x \left (x^6+x+1\right )-1}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx-192 \int \frac {x^5 \int \frac {e^x x^5}{e^x \left (x^6+x+1\right )-1}dx}{\left (x^6+x+1\right ) \left (e^x x^6+e^x x+e^x-1\right )}dx\)

input

Int[((-32*x - (32*x)/E^x - 192*x^6)*Log[1 - E^(-x) + x + x^6] + (16 - 16/E 
^x + 16*x + 16*x^6)*Log[1 - E^(-x) + x + x^6]^2)/(-x^2 + x^2/E^x - x^3 - x 
^8),x]
 

output

$Aborted
 

3.13.70.3.1 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] 
]
 

3.13.70.4 Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

method	result	size
risch	\(\frac {16 \ln \left (-{\mathrm e}^{-x}+x^{6}+x +1\right )^{2}}{x}\)	\(21\)
parallelrisch	\(\frac {16 \ln \left (-{\mathrm e}^{-x}+x^{6}+x +1\right )^{2}}{x}\)	\(21\)

input

int(((-16*exp(-x)+16*x^6+16*x+16)*ln(-exp(-x)+x^6+x+1)^2+(-32*x*exp(-x)-19 
2*x^6-32*x)*ln(-exp(-x)+x^6+x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x,method=_RETU 
RNVERBOSE)
 

output

16*ln(-exp(-x)+x^6+x+1)^2/x
 

3.13.70.5 Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \, \log \left (x^{6} + x - e^{\left (-x\right )} + 1\right )^{2}}{x} \]

input

integrate(((-16*exp(-x)+16*x^6+16*x+16)*log(-exp(-x)+x^6+x+1)^2+(-32*x*exp 
(-x)-192*x^6-32*x)*log(-exp(-x)+x^6+x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x, alg 
orithm=\
 

output

16*log(x^6 + x - e^(-x) + 1)^2/x
 

3.13.70.6 Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \log {\left (x^{6} + x + 1 - e^{- x} \right )}^{2}}{x} \]

input

integrate(((-16*exp(-x)+16*x**6+16*x+16)*ln(-exp(-x)+x**6+x+1)**2+(-32*x*e 
xp(-x)-192*x**6-32*x)*ln(-exp(-x)+x**6+x+1))/(x**2*exp(-x)-x**8-x**3-x**2) 
,x)
 

output

16*log(x**6 + x + 1 - exp(-x))**2/x
 

3.13.70.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.67 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \, {\left (x^{2} + \log \left ({\left (x^{6} + x + 1\right )} e^{x} - 1\right )^{2}\right )}}{x} - 32 \, \log \left (x^{6} + x + 1\right ) - 32 \, \log \left (\frac {{\left (x^{6} + x + 1\right )} e^{x} - 1}{x^{6} + x + 1}\right ) \]

input

integrate(((-16*exp(-x)+16*x^6+16*x+16)*log(-exp(-x)+x^6+x+1)^2+(-32*x*exp 
(-x)-192*x^6-32*x)*log(-exp(-x)+x^6+x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x, alg 
orithm=\
 

output

16*(x^2 + log((x^6 + x + 1)*e^x - 1)^2)/x - 32*log(x^6 + x + 1) - 32*log(( 
(x^6 + x + 1)*e^x - 1)/(x^6 + x + 1))
 

3.13.70.8 Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \, \log \left (x^{6} + x - e^{\left (-x\right )} + 1\right )^{2}}{x} \]

input

integrate(((-16*exp(-x)+16*x^6+16*x+16)*log(-exp(-x)+x^6+x+1)^2+(-32*x*exp 
(-x)-192*x^6-32*x)*log(-exp(-x)+x^6+x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x, alg 
orithm=\
 

output

16*log(x^6 + x - e^(-x) + 1)^2/x
 

3.13.70.9 Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16\,{\ln \left (x-{\mathrm {e}}^{-x}+x^6+1\right )}^2}{x} \]

input

int(-(log(x - exp(-x) + x^6 + 1)^2*(16*x - 16*exp(-x) + 16*x^6 + 16) - log 
(x - exp(-x) + x^6 + 1)*(32*x + 32*x*exp(-x) + 192*x^6))/(x^2 - x^2*exp(-x 
) + x^3 + x^8),x)
 

output

(16*log(x - exp(-x) + x^6 + 1)^2)/x