Integrand size = 101, antiderivative size = 23 \[ \int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx=\left (-1+(8-x)^4+x\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right ) \]
Time = 2.85 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx=4095 \log \left (1-e^x-x\right )+x \left (-2047+384 x-32 x^2+x^3\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right ) \]
Integrate[(4095 - 2047*x + 384*x^2 - 32*x^3 + x^4 + E^x*(4095 - 2047*x + 3 84*x^2 - 32*x^3 + x^4) + (2047 - 2815*x + 864*x^2 - 100*x^3 + 4*x^4 + E^x* (-2047 + 768*x - 96*x^2 + 4*x^3))*Log[(-1 + E^x + x)/Log[3]])/(-1 + E^x + x),x]
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.
\(\displaystyle \int \frac {x^4-32 x^3+384 x^2+e^x \left (x^4-32 x^3+384 x^2-2047 x+4095\right )+\left (4 x^4-100 x^3+864 x^2+e^x \left (4 x^3-96 x^2+768 x-2047\right )-2815 x+2047\right ) \log \left (\frac {x+e^x-1}{\log (3)}\right )-2047 x+4095}{x+e^x-1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (x^4-32 x^3+4 x^3 \log \left (\frac {x+e^x-1}{\log (3)}\right )+384 x^2-96 x^2 \log \left (\frac {x+e^x-1}{\log (3)}\right )-\frac {x^5-34 x^4+448 x^3-2815 x^2+8189 x-8190}{x+e^x-1}-2047 x+768 x \log \left (\frac {x+e^x-1}{\log (3)}\right )-2047 \log \left (\frac {x+e^x-1}{\log (3)}\right )+4095\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 8190 \int \frac {1}{x+e^x-1}dx-4095 \int \frac {x}{x+e^x-1}dx+x^4 \log \left (-\frac {-x-e^x+1}{\log (3)}\right )-32 x^3 \log \left (-\frac {-x-e^x+1}{\log (3)}\right )+384 x^2 \log \left (-\frac {-x-e^x+1}{\log (3)}\right )+4095 x-2047 x \log \left (-\frac {-x-e^x+1}{\log (3)}\right )\) |
Int[(4095 - 2047*x + 384*x^2 - 32*x^3 + x^4 + E^x*(4095 - 2047*x + 384*x^2 - 32*x^3 + x^4) + (2047 - 2815*x + 864*x^2 - 100*x^3 + 4*x^4 + E^x*(-2047 + 768*x - 96*x^2 + 4*x^3))*Log[(-1 + E^x + x)/Log[3]])/(-1 + E^x + x),x]
3.12.6.3.1 Defintions of rubi rules used
Time = 0.80 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70
method | result | size |
risch | \(\left (x^{4}-32 x^{3}+384 x^{2}-2047 x \right ) \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right )+4095 \ln \left (x +{\mathrm e}^{x}-1\right )\) | \(39\) |
norman | \(4095 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right )+\ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x^{4}-2047 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x +384 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x^{2}-32 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x^{3}\) | \(76\) |
parallelrisch | \(4095 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right )+\ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x^{4}-2047 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x +384 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x^{2}-32 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \left (3\right )}\right ) x^{3}\) | \(76\) |
int((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+2047)* ln((x+exp(x)-1)/ln(3))+(x^4-32*x^3+384*x^2-2047*x+4095)*exp(x)+x^4-32*x^3+ 384*x^2-2047*x+4095)/(x+exp(x)-1),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx={\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2047 \, x + 4095\right )} \log \left (\frac {x + e^{x} - 1}{\log \left (3\right )}\right ) \]
integrate((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+ 2047)*log((x+exp(x)-1)/log(3))+(x^4-32*x^3+384*x^2-2047*x+4095)*exp(x)+x^4 -32*x^3+384*x^2-2047*x+4095)/(x+exp(x)-1),x, algorithm=\
Time = 0.16 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx=\left (x^{4} - 32 x^{3} + 384 x^{2} - 2047 x\right ) \log {\left (\frac {x + e^{x} - 1}{\log {\left (3 \right )}} \right )} + 4095 \log {\left (x + e^{x} - 1 \right )} \]
integrate((((4*x**3-96*x**2+768*x-2047)*exp(x)+4*x**4-100*x**3+864*x**2-28 15*x+2047)*ln((x+exp(x)-1)/ln(3))+(x**4-32*x**3+384*x**2-2047*x+4095)*exp( x)+x**4-32*x**3+384*x**2-2047*x+4095)/(x+exp(x)-1),x)
(x**4 - 32*x**3 + 384*x**2 - 2047*x)*log((x + exp(x) - 1)/log(3)) + 4095*l og(x + exp(x) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx=-x^{4} \log \left (\log \left (3\right )\right ) + 32 \, x^{3} \log \left (\log \left (3\right )\right ) - 384 \, x^{2} \log \left (\log \left (3\right )\right ) + {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2047 \, x + 4095\right )} \log \left (x + e^{x} - 1\right ) + 2047 \, x \log \left (\log \left (3\right )\right ) \]
integrate((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+ 2047)*log((x+exp(x)-1)/log(3))+(x^4-32*x^3+384*x^2-2047*x+4095)*exp(x)+x^4 -32*x^3+384*x^2-2047*x+4095)/(x+exp(x)-1),x, algorithm=\
-x^4*log(log(3)) + 32*x^3*log(log(3)) - 384*x^2*log(log(3)) + (x^4 - 32*x^ 3 + 384*x^2 - 2047*x + 4095)*log(x + e^x - 1) + 2047*x*log(log(3))
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (20) = 40\).
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx=x^{4} \log \left (x + e^{x} - 1\right ) - x^{4} \log \left (\log \left (3\right )\right ) - 32 \, x^{3} \log \left (x + e^{x} - 1\right ) + 32 \, x^{3} \log \left (\log \left (3\right )\right ) + 384 \, x^{2} \log \left (x + e^{x} - 1\right ) - 384 \, x^{2} \log \left (\log \left (3\right )\right ) - 2047 \, x \log \left (x + e^{x} - 1\right ) + 2047 \, x \log \left (\log \left (3\right )\right ) + 4095 \, \log \left (x + e^{x} - 1\right ) \]
integrate((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+ 2047)*log((x+exp(x)-1)/log(3))+(x^4-32*x^3+384*x^2-2047*x+4095)*exp(x)+x^4 -32*x^3+384*x^2-2047*x+4095)/(x+exp(x)-1),x, algorithm=\
x^4*log(x + e^x - 1) - x^4*log(log(3)) - 32*x^3*log(x + e^x - 1) + 32*x^3* log(log(3)) + 384*x^2*log(x + e^x - 1) - 384*x^2*log(log(3)) - 2047*x*log( x + e^x - 1) + 2047*x*log(log(3)) + 4095*log(x + e^x - 1)
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx=4095\,\ln \left (x+{\mathrm {e}}^x-1\right )-\ln \left (\frac {x+{\mathrm {e}}^x-1}{\ln \left (3\right )}\right )\,\left (-x^4+32\,x^3-384\,x^2+2047\,x\right ) \]