Integrand size = 124, antiderivative size = 32 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\frac {2}{e^2 \log \left (\frac {75}{x^2}-x+\frac {x^2}{-x+x^2}\right )}} \] Output:
exp(2/exp(2)/ln(x^2/(x^2-x)-x+75/x^2))
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\frac {2}{e^2 \log \left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}} \] Input:
Integrate[(E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]))*( -300 + 600*x - 300*x^2 - 4*x^3 + 4*x^4 - 2*x^5))/((-75*x + 150*x^2 - 75*x^ 3 + 2*x^4 - 3*x^5 + x^6)*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]^2),x ]
Output:
E^(2/(E^2*Log[-((75 - 75*x - 2*x^3 + x^4)/((-1 + x)*x^2))]))
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.
\(\displaystyle \int \frac {\left (-2 x^5+4 x^4-4 x^3-300 x^2+600 x-300\right ) \exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{\left (x^6-3 x^5+2 x^4-75 x^3+150 x^2-75 x\right ) \log ^2\left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (-2 x^5+4 x^4-4 x^3-300 x^2+600 x-300\right ) \exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{x \left (x^5-3 x^4+2 x^3-75 x^2+150 x-75\right ) \log ^2\left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {\left (-2 x^5+4 x^4-4 x^3-300 x^2+600 x-300\right ) \exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{(1-x) x \log ^2\left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}+\frac {\left (x^3-x^2-x-76\right ) \left (-2 x^5+4 x^4-4 x^3-300 x^2+600 x-300\right ) \exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{x \left (x^4-2 x^3-75 x+75\right ) \log ^2\left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{(x-1) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx+4 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{x \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx+150 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{\left (x^4-2 x^3-75 x+75\right ) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx+12 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right ) x^2}{\left (x^4-2 x^3-75 x+75\right ) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx-8 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right ) x^3}{\left (x^4-2 x^3-75 x+75\right ) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx\) |
Input:
Int[(E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]))*(-300 + 600*x - 300*x^2 - 4*x^3 + 4*x^4 - 2*x^5))/((-75*x + 150*x^2 - 75*x^3 + 2* x^4 - 3*x^5 + x^6)*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]^2),x]
Output:
$Aborted
Time = 42.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {2 \,{\mathrm e}^{-2}}{\ln \left (-\frac {x^{4}-2 x^{3}-75 x +75}{x^{2} \left (-1+x \right )}\right )}}\) | \(34\) |
risch | \({\mathrm e}^{\frac {2 \,{\mathrm e}^{-2}}{\ln \left (\frac {-x^{4}+2 x^{3}+75 x -75}{x^{3}-x^{2}}\right )}}\) | \(36\) |
Input:
int((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/ln((-x^4+2*x^3+75* x-75)/(x^3-x^2)))/(x^6-3*x^5+2*x^4-75*x^3+150*x^2-75*x)/exp(2)/ln((-x^4+2* x^3+75*x-75)/(x^3-x^2))^2,x,method=_RETURNVERBOSE)
Output:
exp(2/exp(2)/ln(-(x^4-2*x^3-75*x+75)/x^2/(-1+x)))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\left (-\frac {2 \, {\left (e^{2} \log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right ) - 1\right )} e^{\left (-2\right )}}{\log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )} + 2\right )} \] Input:
integrate((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/log((-x^4+2* x^3+75*x-75)/(x^3-x^2)))/(x^6-3*x^5+2*x^4-75*x^3+150*x^2-75*x)/exp(2)/log( (-x^4+2*x^3+75*x-75)/(x^3-x^2))^2,x, algorithm="fricas")
Output:
e^(-2*(e^2*log(-(x^4 - 2*x^3 - 75*x + 75)/(x^3 - x^2)) - 1)*e^(-2)/log(-(x ^4 - 2*x^3 - 75*x + 75)/(x^3 - x^2)) + 2)
Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\frac {2}{e^{2} \log {\left (\frac {- x^{4} + 2 x^{3} + 75 x - 75}{x^{3} - x^{2}} \right )}}} \] Input:
integrate((-2*x**5+4*x**4-4*x**3-300*x**2+600*x-300)*exp(2/exp(2)/ln((-x** 4+2*x**3+75*x-75)/(x**3-x**2)))/(x**6-3*x**5+2*x**4-75*x**3+150*x**2-75*x) /exp(2)/ln((-x**4+2*x**3+75*x-75)/(x**3-x**2))**2,x)
Output:
exp(2*exp(-2)/log((-x**4 + 2*x**3 + 75*x - 75)/(x**3 - x**2)))
\[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=\int { -\frac {2 \, {\left (x^{5} - 2 \, x^{4} + 2 \, x^{3} + 150 \, x^{2} - 300 \, x + 150\right )} e^{\left (\frac {2 \, e^{\left (-2\right )}}{\log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )} - 2\right )}}{{\left (x^{6} - 3 \, x^{5} + 2 \, x^{4} - 75 \, x^{3} + 150 \, x^{2} - 75 \, x\right )} \log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )^{2}} \,d x } \] Input:
integrate((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/log((-x^4+2* x^3+75*x-75)/(x^3-x^2)))/(x^6-3*x^5+2*x^4-75*x^3+150*x^2-75*x)/exp(2)/log( (-x^4+2*x^3+75*x-75)/(x^3-x^2))^2,x, algorithm="maxima")
Output:
-2*integrate((x^5 - 2*x^4 + 2*x^3 + 150*x^2 - 300*x + 150)*e^(2*e^(-2)/log (-(x^4 - 2*x^3 - 75*x + 75)/(x^3 - x^2)) - 2)/((x^6 - 3*x^5 + 2*x^4 - 75*x ^3 + 150*x^2 - 75*x)*log(-(x^4 - 2*x^3 - 75*x + 75)/(x^3 - x^2))^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\left (\frac {2 \, e^{\left (-2\right )}}{\log \left (-\frac {x^{4}}{x^{3} - x^{2}} + \frac {2 \, x^{3}}{x^{3} - x^{2}} + \frac {75 \, x}{x^{3} - x^{2}} - \frac {75}{x^{3} - x^{2}}\right )}\right )} \] Input:
integrate((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/log((-x^4+2* x^3+75*x-75)/(x^3-x^2)))/(x^6-3*x^5+2*x^4-75*x^3+150*x^2-75*x)/exp(2)/log( (-x^4+2*x^3+75*x-75)/(x^3-x^2))^2,x, algorithm="giac")
Output:
e^(2*e^(-2)/log(-x^4/(x^3 - x^2) + 2*x^3/(x^3 - x^2) + 75*x/(x^3 - x^2) - 75/(x^3 - x^2)))
Time = 4.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx={\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-2}}{\ln \left (-\frac {-x^4+2\,x^3+75\,x-75}{x^2-x^3}\right )}} \] Input:
int((exp(-2)*exp((2*exp(-2))/log(-(75*x + 2*x^3 - x^4 - 75)/(x^2 - x^3)))* (300*x^2 - 600*x + 4*x^3 - 4*x^4 + 2*x^5 + 300))/(log(-(75*x + 2*x^3 - x^4 - 75)/(x^2 - x^3))^2*(75*x - 150*x^2 + 75*x^3 - 2*x^4 + 3*x^5 - x^6)),x)
Output:
exp((2*exp(-2))/log(-(75*x + 2*x^3 - x^4 - 75)/(x^2 - x^3)))
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\frac {2}{\mathrm {log}\left (\frac {-x^{4}+2 x^{3}+75 x -75}{x^{3}-x^{2}}\right ) e^{2}}} \] Input:
int((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/log((-x^4+2*x^3+75 *x-75)/(x^3-x^2)))/(x^6-3*x^5+2*x^4-75*x^3+150*x^2-75*x)/exp(2)/log((-x^4+ 2*x^3+75*x-75)/(x^3-x^2))^2,x)
Output:
e**(2/(log(( - x**4 + 2*x**3 + 75*x - 75)/(x**3 - x**2))*e**2))