Integrand size = 124, antiderivative size = 32 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\frac {2}{e^2 \log \left (\frac {75}{x^2}-x+\frac {x^2}{-x+x^2}\right )}} \]
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\frac {2}{e^2 \log \left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}} \]
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 ]
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 |
\(\displaystyle 2 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{(x-1) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx+4 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{x \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx+150 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right )}{\left (x^4-2 x^3-75 x+75\right ) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx+12 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right ) x^2}{\left (x^4-2 x^3-75 x+75\right ) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx-8 \int \frac {\exp \left (\frac {2}{e^2 \log \left (\frac {-x^4+2 x^3+75 x-75}{x^3-x^2}\right )}-2\right ) x^3}{\left (x^4-2 x^3-75 x+75\right ) \log ^2\left (-\frac {x^4-2 x^3-75 x+75}{(x-1) x^2}\right )}dx\) |
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]
3.16.36.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 44.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {2 \,{\mathrm e}^{-2}}{\ln \left (-\frac {x^{4}-2 x^{3}-75 x +75}{\left (-1+x \right ) x^{2}}\right )}}\) | \(34\) |
risch | \({\mathrm e}^{\frac {2 \,{\mathrm e}^{-2}}{\ln \left (\frac {-x^{4}+2 x^{3}+75 x -75}{x^{3}-x^{2}}\right )}}\) | \(36\) |
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)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (30) = 60\).
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\left (-\frac {2 \, {\left (e^{2} \log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right ) - 1\right )} e^{\left (-2\right )}}{\log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )} + 2\right )} \]
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=\
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 = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\frac {2}{e^{2} \log {\left (\frac {- x^{4} + 2 x^{3} + 75 x - 75}{x^{3} - x^{2}} \right )}}} \]
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)
\[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=\int { -\frac {2 \, {\left (x^{5} - 2 \, x^{4} + 2 \, x^{3} + 150 \, x^{2} - 300 \, x + 150\right )} e^{\left (\frac {2 \, e^{\left (-2\right )}}{\log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )} - 2\right )}}{{\left (x^{6} - 3 \, x^{5} + 2 \, x^{4} - 75 \, x^{3} + 150 \, x^{2} - 75 \, x\right )} \log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )^{2}} \,d x } \]
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=\
-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 = 0.38 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx=e^{\left (\frac {2 \, e^{\left (-2\right )}}{\log \left (-\frac {x^{4}}{x^{3} - x^{2}} + \frac {2 \, x^{3}}{x^{3} - x^{2}} + \frac {75 \, x}{x^{3} - x^{2}} - \frac {75}{x^{3} - x^{2}}\right )}\right )} \]
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=\
Time = 14.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}} \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx={\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-2}}{\ln \left (-\frac {-x^4+2\,x^3+75\,x-75}{x^2-x^3}\right )}} \]