Integrand size = 95, antiderivative size = 28 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=-\log \left (-e^x+x\right )+\log \left (\left (2-4 \left (-5-2 x+x^2\right )^2\right )^2\right ) \] Output:
ln((2-2*(x^2-2*x-5)*(2*x^2-4*x-10))^2)-ln(x-exp(x))
Time = 2.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=-\log \left (e^x-x\right )+2 \log \left (49+40 x-12 x^2-8 x^3+2 x^4\right ) \] Input:
Integrate[(49 - 40*x + 36*x^2 + 40*x^3 - 14*x^4 + E^x*(31 - 88*x - 36*x^2 + 24*x^3 - 2*x^4))/(-49*x - 40*x^2 + 12*x^3 + 8*x^4 - 2*x^5 + E^x*(49 + 40 *x - 12*x^2 - 8*x^3 + 2*x^4)),x]
Output:
-Log[E^x - x] + 2*Log[49 + 40*x - 12*x^2 - 8*x^3 + 2*x^4]
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.
\(\displaystyle \int \frac {-14 x^4+40 x^3+36 x^2+e^x \left (-2 x^4+24 x^3-36 x^2-88 x+31\right )-40 x+49}{-2 x^5+8 x^4+12 x^3-40 x^2+e^x \left (2 x^4-8 x^3-12 x^2+40 x+49\right )-49 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-14 x^4+40 x^3+36 x^2+e^x \left (-2 x^4+24 x^3-36 x^2-88 x+31\right )-40 x+49}{\left (e^x-x\right ) \left (2 x^4-8 x^3-12 x^2+40 x+49\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-2 x^4+24 x^3-36 x^2-88 x+31}{2 x^4-8 x^3-12 x^2+40 x+49}+\frac {x-1}{x-e^x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{e^x-x}dx+\int \frac {x}{x-e^x}dx-x+2 \log \left (2 (1-x)^4-24 (1-x)^2+71\right )\) |
Input:
Int[(49 - 40*x + 36*x^2 + 40*x^3 - 14*x^4 + E^x*(31 - 88*x - 36*x^2 + 24*x ^3 - 2*x^4))/(-49*x - 40*x^2 + 12*x^3 + 8*x^4 - 2*x^5 + E^x*(49 + 40*x - 1 2*x^2 - 8*x^3 + 2*x^4)),x]
Output:
$Aborted
Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(-\ln \left (x -{\mathrm e}^{x}\right )+2 \ln \left (x^{4}-4 x^{3}-6 x^{2}+20 x +\frac {49}{2}\right )\) | \(32\) |
norman | \(-\ln \left (x -{\mathrm e}^{x}\right )+2 \ln \left (2 x^{4}-8 x^{3}-12 x^{2}+40 x +49\right )\) | \(34\) |
risch | \(2 \ln \left (2 x^{4}-8 x^{3}-12 x^{2}+40 x +49\right )-\ln \left ({\mathrm e}^{x}-x \right )\) | \(34\) |
Input:
int(((-2*x^4+24*x^3-36*x^2-88*x+31)*exp(x)-14*x^4+40*x^3+36*x^2-40*x+49)/( (2*x^4-8*x^3-12*x^2+40*x+49)*exp(x)-2*x^5+8*x^4+12*x^3-40*x^2-49*x),x,meth od=_RETURNVERBOSE)
Output:
-ln(x-exp(x))+2*ln(x^4-4*x^3-6*x^2+20*x+49/2)
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=2 \, \log \left (2 \, x^{4} - 8 \, x^{3} - 12 \, x^{2} + 40 \, x + 49\right ) - \log \left (-x + e^{x}\right ) \] Input:
integrate(((-2*x^4+24*x^3-36*x^2-88*x+31)*exp(x)-14*x^4+40*x^3+36*x^2-40*x +49)/((2*x^4-8*x^3-12*x^2+40*x+49)*exp(x)-2*x^5+8*x^4+12*x^3-40*x^2-49*x), x, algorithm="fricas")
Output:
2*log(2*x^4 - 8*x^3 - 12*x^2 + 40*x + 49) - log(-x + e^x)
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=- \log {\left (- x + e^{x} \right )} + 2 \log {\left (2 x^{4} - 8 x^{3} - 12 x^{2} + 40 x + 49 \right )} \] Input:
integrate(((-2*x**4+24*x**3-36*x**2-88*x+31)*exp(x)-14*x**4+40*x**3+36*x** 2-40*x+49)/((2*x**4-8*x**3-12*x**2+40*x+49)*exp(x)-2*x**5+8*x**4+12*x**3-4 0*x**2-49*x),x)
Output:
-log(-x + exp(x)) + 2*log(2*x**4 - 8*x**3 - 12*x**2 + 40*x + 49)
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=2 \, \log \left (2 \, x^{4} - 8 \, x^{3} - 12 \, x^{2} + 40 \, x + 49\right ) - \log \left (-x + e^{x}\right ) \] Input:
integrate(((-2*x^4+24*x^3-36*x^2-88*x+31)*exp(x)-14*x^4+40*x^3+36*x^2-40*x +49)/((2*x^4-8*x^3-12*x^2+40*x+49)*exp(x)-2*x^5+8*x^4+12*x^3-40*x^2-49*x), x, algorithm="maxima")
Output:
2*log(2*x^4 - 8*x^3 - 12*x^2 + 40*x + 49) - log(-x + e^x)
Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=2 \, \log \left (2 \, x^{4} - 8 \, x^{3} - 12 \, x^{2} + 40 \, x + 49\right ) - \log \left (x - e^{x}\right ) \] Input:
integrate(((-2*x^4+24*x^3-36*x^2-88*x+31)*exp(x)-14*x^4+40*x^3+36*x^2-40*x +49)/((2*x^4-8*x^3-12*x^2+40*x+49)*exp(x)-2*x^5+8*x^4+12*x^3-40*x^2-49*x), x, algorithm="giac")
Output:
2*log(2*x^4 - 8*x^3 - 12*x^2 + 40*x + 49) - log(x - e^x)
Time = 3.42 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=2\,\ln \left (x^4-4\,x^3-6\,x^2+20\,x+\frac {49}{2}\right )-\ln \left ({\mathrm {e}}^x-x\right ) \] Input:
int((40*x + exp(x)*(88*x + 36*x^2 - 24*x^3 + 2*x^4 - 31) - 36*x^2 - 40*x^3 + 14*x^4 - 49)/(49*x - exp(x)*(40*x - 12*x^2 - 8*x^3 + 2*x^4 + 49) + 40*x ^2 - 12*x^3 - 8*x^4 + 2*x^5),x)
Output:
2*log(20*x - 6*x^2 - 4*x^3 + x^4 + 49/2) - log(exp(x) - x)
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {49-40 x+36 x^2+40 x^3-14 x^4+e^x \left (31-88 x-36 x^2+24 x^3-2 x^4\right )}{-49 x-40 x^2+12 x^3+8 x^4-2 x^5+e^x \left (49+40 x-12 x^2-8 x^3+2 x^4\right )} \, dx=-\mathrm {log}\left (e^{x}-x \right )+2 \,\mathrm {log}\left (-\sqrt {\sqrt {2}+12}\, \sqrt {2}+2 x -2\right )+2 \,\mathrm {log}\left (\sqrt {2}+2 x^{2}-4 x -10\right )+2 \,\mathrm {log}\left (\sqrt {\sqrt {2}+12}\, \sqrt {2}+2 x -2\right ) \] Input:
int(((-2*x^4+24*x^3-36*x^2-88*x+31)*exp(x)-14*x^4+40*x^3+36*x^2-40*x+49)/( (2*x^4-8*x^3-12*x^2+40*x+49)*exp(x)-2*x^5+8*x^4+12*x^3-40*x^2-49*x),x)
Output:
- log(e**x - x) + 2*log( - sqrt(sqrt(2) + 12)*sqrt(2) + 2*x - 2) + 2*log( sqrt(2) + 2*x**2 - 4*x - 10) + 2*log(sqrt(sqrt(2) + 12)*sqrt(2) + 2*x - 2)