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
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(21)=42\).
Time = 0.48 (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)])
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
Time = 1.72 (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
Time = 0.07 (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="fricas")
Output:
16*log(x^6 + x - e^(-x) + 1)^2/x
Time = 0.15 (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
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
Time = 0.09 (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="maxima")
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))
Time = 0.28 (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="giac")
Output:
16*log(x^6 + x - e^(-x) + 1)^2/x
Time = 1.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16\,{\ln \left (x-{\mathrm {e}}^{-x}+x^6+1\right )}^2}{x} \] 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
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.48 \[ \int \frac {\left (-32 x-32 e^{-x} x-192 x^6\right ) \log \left (1-e^{-x}+x+x^6\right )+\left (16-16 e^{-x}+16 x+16 x^6\right ) \log ^2\left (1-e^{-x}+x+x^6\right )}{-x^2+e^{-x} x^2-x^3-x^8} \, dx=\frac {16 \mathrm {log}\left (\frac {e^{x} x^{6}+e^{x} x +e^{x}-1}{e^{x}}\right )^{2}}{x} \] Input:
int(((-16*exp(-x)+16*x^6+16*x+16)*log(-exp(-x)+x^6+x+1)^2+(-32*x*exp(-x)-1 92*x^6-32*x)*log(-exp(-x)+x^6+x+1))/(x^2*exp(-x)-x^8-x^3-x^2),x)
Output:
(16*log((e**x*x**6 + e**x*x + e**x - 1)/e**x)**2)/x