Integrand size = 95, antiderivative size = 18 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\frac {1}{\log ^4\left ((2-x) x \left (x^4+\log (36)\right )\right )}} \]
Time = 0.50 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} \]
Integrate[(E^Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^(-4)*(40*x^4 - 24*x^5 + (8 - 8*x)*Log[36]))/((-2*x^5 + x^6 + (-2*x + x^2)*Log[36])*Log[2*x^5 - x ^6 + (2*x - x^2)*Log[36]]^5),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.
\(\displaystyle \int \frac {\left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{\left (x^6-2 x^5+\left (x^2-2 x\right ) \log (36)\right ) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{x \left (x^5-2 x^4+x \log (36)-2 \log (36)\right ) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {\left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{(x-2) x (16+\log (36)) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}+\frac {\left (-x^3-2 x^2-4 x-8\right ) \left (-24 x^5+40 x^4+(8-8 x) \log (36)\right ) \exp \left (\frac {1}{\log ^4\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )}{x (16+\log (36)) \left (x^4+\log (36)\right ) \log ^5\left (-x^6+2 x^5+\left (2 x-x^2\right ) \log (36)\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{(x-2) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx-\frac {4 \log (36) \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx}{16+\log (36)}-\frac {64 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx}{16+\log (36)}+4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (\sqrt [4]{-\log (36)}-x\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (x+\sqrt [4]{-\log (36)}\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx+4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (-x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((x-2) x \left (x^4+\log (36)\right )\right )\right )}dx\) |
Int[(E^Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^(-4)*(40*x^4 - 24*x^5 + (8 - 8*x)*Log[36]))/((-2*x^5 + x^6 + (-2*x + x^2)*Log[36])*Log[2*x^5 - x^6 + ( 2*x - x^2)*Log[36]]^5),x]
3.11.84.3.1 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 = 0.49 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.78
\[{\mathrm e}^{\frac {1}{{\ln \left (2 \left (-x^{2}+2 x \right ) \left (\ln \left (2\right )+\ln \left (3\right )\right )-x^{6}+2 x^{5}\right )}^{4}}}\]
int((2*(-8*x+8)*ln(6)-24*x^5+40*x^4)*exp(1/ln(2*(-x^2+2*x)*ln(6)-x^6+2*x^5 )^4)/(2*(x^2-2*x)*ln(6)+x^6-2*x^5)/ln(2*(-x^2+2*x)*ln(6)-x^6+2*x^5)^5,x)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\left (\frac {1}{\log \left (-x^{6} + 2 \, x^{5} - 2 \, {\left (x^{2} - 2 \, x\right )} \log \left (6\right )\right )^{4}}\right )} \]
integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)- x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2 *x^5)^5,x, algorithm=\
Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\frac {1}{\log {\left (- x^{6} + 2 x^{5} + \left (- 2 x^{2} + 4 x\right ) \log {\left (6 \right )} \right )}^{4}}} \]
integrate((2*(-8*x+8)*ln(6)-24*x**5+40*x**4)*exp(1/ln(2*(-x**2+2*x)*ln(6)- x**6+2*x**5)**4)/(2*(x**2-2*x)*ln(6)+x**6-2*x**5)/ln(2*(-x**2+2*x)*ln(6)-x **6+2*x**5)**5,x)
Leaf count of result is larger than twice the leaf count of optimal. 1148 vs. \(2 (18) = 36\).
Time = 0.49 (sec) , antiderivative size = 1148, normalized size of antiderivative = 63.78 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=\text {Too large to display} \]
integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)- x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2 *x^5)^5,x, algorithm=\
3*x^5*e^(1/(log(x^4 + 2*log(3) + 2*log(2))^4 + 4*(log(x^4 + 2*log(3) + 2*l og(2)) + log(-x + 2))*log(x)^3 + log(x)^4 + 4*log(x^4 + 2*log(3) + 2*log(2 ))^3*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2)^2 + 4*lo g(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^3 + log(-x + 2)^4 + 6*(log(x^4 + 2*log(3) + 2*log(2))^2 + 2*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2) + lo g(-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3) + 2*log(2))^3 + 3*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2) + 3*log(x^4 + 2*log(3) + 2*log(2))*log( -x + 2)^2 + log(-x + 2)^3)*log(x)))/(3*x^5 - 5*x^4 + 2*x*(log(3) + log(2)) - 2*log(3) - 2*log(2)) - 5*x^4*e^(1/(log(x^4 + 2*log(3) + 2*log(2))^4 + 4 *(log(x^4 + 2*log(3) + 2*log(2)) + log(-x + 2))*log(x)^3 + log(x)^4 + 4*lo g(x^4 + 2*log(3) + 2*log(2))^3*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log( 2))^2*log(-x + 2)^2 + 4*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^3 + log (-x + 2)^4 + 6*(log(x^4 + 2*log(3) + 2*log(2))^2 + 2*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2) + log(-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3) + 2*log(2))^3 + 3*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2) + 3*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^2 + log(-x + 2)^3)*log(x)))/(3*x^5 - 5*x ^4 + 2*x*(log(3) + log(2)) - 2*log(3) - 2*log(2)) + 2*x*e^(1/(log(x^4 + 2* log(3) + 2*log(2))^4 + 4*(log(x^4 + 2*log(3) + 2*log(2)) + log(-x + 2))*lo g(x)^3 + log(x)^4 + 4*log(x^4 + 2*log(3) + 2*log(2))^3*log(-x + 2) + 6*log (x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2)^2 + 4*log(x^4 + 2*log(3) + 2*...
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx=e^{\left (\frac {1}{\log \left (-x^{6} + 2 \, x^{5} - 2 \, x^{2} \log \left (6\right ) + 4 \, x \log \left (6\right )\right )^{4}}\right )} \]
integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)- x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2 *x^5)^5,x, algorithm=\
Time = 10.71 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx={\mathrm {e}}^{\frac {1}{{\ln \left (-x^6+2\,x^5-2\,\ln \left (6\right )\,x^2+4\,\ln \left (6\right )\,x\right )}^4}} \]