Integrand size = 143, antiderivative size = 32 \begin {dmath*} \int \frac {3 x^2+3 x^3-x^4+e^{e^5} \left (-3-3 x-6 x^3-6 x^4+2 x^5\right )+e^{2 e^5} \left (6 x+3 x^2+3 x^4+3 x^5-x^6\right )}{-4 x^2-x^3+x^4+e^{e^5} \left (3 x+8 x^3+2 x^4-2 x^5\right )+e^{2 e^5} \left (-3 x^2-4 x^4-x^5+x^6\right )} \, dx=-x+\log \left (4+x-x^2-\frac {3}{e^{-e^5} x-x^2}\right ) \end {dmath*}
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(32)=64\).
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \begin {dmath*} \int \frac {3 x^2+3 x^3-x^4+e^{e^5} \left (-3-3 x-6 x^3-6 x^4+2 x^5\right )+e^{2 e^5} \left (6 x+3 x^2+3 x^4+3 x^5-x^6\right )}{-4 x^2-x^3+x^4+e^{e^5} \left (3 x+8 x^3+2 x^4-2 x^5\right )+e^{2 e^5} \left (-3 x^2-4 x^4-x^5+x^6\right )} \, dx=-x-\log (x)-\log \left (1-e^{e^5} x\right )+\log \left (-3 e^{e^5}+4 x+x^2-4 e^{e^5} x^2-x^3-e^{e^5} x^3+e^{e^5} x^4\right ) \end {dmath*}
Integrate[(3*x^2 + 3*x^3 - x^4 + E^E^5*(-3 - 3*x - 6*x^3 - 6*x^4 + 2*x^5) + E^(2*E^5)*(6*x + 3*x^2 + 3*x^4 + 3*x^5 - x^6))/(-4*x^2 - x^3 + x^4 + E^E ^5*(3*x + 8*x^3 + 2*x^4 - 2*x^5) + E^(2*E^5)*(-3*x^2 - 4*x^4 - x^5 + x^6)) ,x]
-x - Log[x] - Log[1 - E^E^5*x] + Log[-3*E^E^5 + 4*x + x^2 - 4*E^E^5*x^2 - x^3 - E^E^5*x^3 + E^E^5*x^4]
Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(32)=64\).
Time = 0.75 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2026, 2462, 2009}
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.
\(\displaystyle \int \frac {-x^4+3 x^3+3 x^2+e^{e^5} \left (2 x^5-6 x^4-6 x^3-3 x-3\right )+e^{2 e^5} \left (-x^6+3 x^5+3 x^4+3 x^2+6 x\right )}{x^4-x^3-4 x^2+e^{e^5} \left (-2 x^5+2 x^4+8 x^3+3 x\right )+e^{2 e^5} \left (x^6-x^5-4 x^4-3 x^2\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-x^4+3 x^3+3 x^2+e^{e^5} \left (2 x^5-6 x^4-6 x^3-3 x-3\right )+e^{2 e^5} \left (-x^6+3 x^5+3 x^4+3 x^2+6 x\right )}{x \left (e^{2 e^5} x^5-e^{e^5} \left (2+e^{e^5}\right ) x^4+\left (1+2 e^{e^5}-4 e^{2 e^5}\right ) x^3-\left (1-8 e^{e^5}\right ) x^2-\left (4+3 e^{2 e^5}\right ) x+3 e^{e^5}\right )}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {-4 e^{e^5} x^3+3 \left (1+e^{e^5}\right ) x^2-2 \left (1-4 e^{e^5}\right ) x-4}{-e^{e^5} x^4+\left (1+e^{e^5}\right ) x^3-\left (1-4 e^{e^5}\right ) x^2-4 x+3 e^{e^5}}-\frac {1}{x}-\frac {e^{e^5}}{e^{e^5} x-1}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \log \left (-e^{e^5} x^4+\left (1+e^{e^5}\right ) x^3-\left (1-4 e^{e^5}\right ) x^2-4 x+3 e^{e^5}\right )-x-\log (x)-\log \left (1-e^{e^5} x\right )\) |
Int[(3*x^2 + 3*x^3 - x^4 + E^E^5*(-3 - 3*x - 6*x^3 - 6*x^4 + 2*x^5) + E^(2 *E^5)*(6*x + 3*x^2 + 3*x^4 + 3*x^5 - x^6))/(-4*x^2 - x^3 + x^4 + E^E^5*(3* x + 8*x^3 + 2*x^4 - 2*x^5) + E^(2*E^5)*(-3*x^2 - 4*x^4 - x^5 + x^6)),x]
-x - Log[x] - Log[1 - E^E^5*x] + Log[3*E^E^5 - 4*x - (1 - 4*E^E^5)*x^2 + ( 1 + E^E^5)*x^3 - E^E^5*x^4]
3.6.15.3.1 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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.92 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.75
method | result | size |
risch | \(-x -\ln \left (-x^{2} {\mathrm e}^{{\mathrm e}^{5}}+x \right )+\ln \left (-{\mathrm e}^{{\mathrm e}^{5}} x^{4}+\left ({\mathrm e}^{{\mathrm e}^{5}}+1\right ) x^{3}+\left (4 \,{\mathrm e}^{{\mathrm e}^{5}}-1\right ) x^{2}-4 x +3 \,{\mathrm e}^{{\mathrm e}^{5}}\right )\) | \(56\) |
norman | \(-x -\ln \left (x \right )-\ln \left (x \,{\mathrm e}^{{\mathrm e}^{5}}-1\right )+\ln \left ({\mathrm e}^{{\mathrm e}^{5}} x^{4}-{\mathrm e}^{{\mathrm e}^{5}} x^{3}-4 x^{2} {\mathrm e}^{{\mathrm e}^{5}}-x^{3}+x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{5}}+4 x \right )\) | \(60\) |
parallelrisch | \(-x -\ln \left (x \right )-\ln \left (\left (x \,{\mathrm e}^{{\mathrm e}^{5}}-1\right ) {\mathrm e}^{-{\mathrm e}^{5}}\right )+\ln \left (\left ({\mathrm e}^{{\mathrm e}^{5}} x^{4}-{\mathrm e}^{{\mathrm e}^{5}} x^{3}-4 x^{2} {\mathrm e}^{{\mathrm e}^{5}}-x^{3}+x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{5}}+4 x \right ) {\mathrm e}^{-{\mathrm e}^{5}}\right )\) | \(72\) |
default | \(-x -\ln \left (x \right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left ({\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_Z}^{5}+\left (-{\mathrm e}^{2 \,{\mathrm e}^{5}}-2 \,{\mathrm e}^{{\mathrm e}^{5}}\right ) \textit {\_Z}^{4}+\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}+2 \,{\mathrm e}^{{\mathrm e}^{5}}+1\right ) \textit {\_Z}^{3}+\left (8 \,{\mathrm e}^{{\mathrm e}^{5}}-1\right ) \textit {\_Z}^{2}+\left (-3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}-4\right ) \textit {\_Z} +3 \,{\mathrm e}^{{\mathrm e}^{5}}\right )}{\sum }\frac {\left (-4+3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{4}+2 \left (-{\mathrm e}^{2 \,{\mathrm e}^{5}}-3 \,{\mathrm e}^{{\mathrm e}^{5}}\right ) \textit {\_R}^{3}+\left (-4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}+4 \,{\mathrm e}^{{\mathrm e}^{5}}+3\right ) \textit {\_R}^{2}+2 \left (4 \,{\mathrm e}^{{\mathrm e}^{5}}-1\right ) \textit {\_R} +3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}\right ) \ln \left (x -\textit {\_R} \right )}{5 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{4}-4 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{3}-12 \,{\mathrm e}^{2 \,{\mathrm e}^{5}} \textit {\_R}^{2}-8 \,{\mathrm e}^{{\mathrm e}^{5}} \textit {\_R}^{3}+6 \textit {\_R}^{2} {\mathrm e}^{{\mathrm e}^{5}}-3 \,{\mathrm e}^{2 \,{\mathrm e}^{5}}+16 \textit {\_R} \,{\mathrm e}^{{\mathrm e}^{5}}+3 \textit {\_R}^{2}-2 \textit {\_R} -4}\right )\) | \(228\) |
int(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3*x-3)* exp(exp(5))-x^4+3*x^3+3*x^2)/((x^6-x^5-4*x^4-3*x^2)*exp(exp(5))^2+(-2*x^5+ 2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x,method=_RETURNVERBOSE)
-x-ln(-x^2*exp(exp(5))+x)+ln(-exp(exp(5))*x^4+(exp(exp(5))+1)*x^3+(4*exp(e xp(5))-1)*x^2-4*x+3*exp(exp(5)))
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \begin {dmath*} \int \frac {3 x^2+3 x^3-x^4+e^{e^5} \left (-3-3 x-6 x^3-6 x^4+2 x^5\right )+e^{2 e^5} \left (6 x+3 x^2+3 x^4+3 x^5-x^6\right )}{-4 x^2-x^3+x^4+e^{e^5} \left (3 x+8 x^3+2 x^4-2 x^5\right )+e^{2 e^5} \left (-3 x^2-4 x^4-x^5+x^6\right )} \, dx=-x + \log \left (-x^{3} + x^{2} + {\left (x^{4} - x^{3} - 4 \, x^{2} - 3\right )} e^{\left (e^{5}\right )} + 4 \, x\right ) - \log \left (x^{2} e^{\left (e^{5}\right )} - x\right ) \end {dmath*}
integrate(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3 *x-3)*exp(exp(5))-x^4+3*x^3+3*x^2)/((x^6-x^5-4*x^4-3*x^2)*exp(exp(5))^2+(- 2*x^5+2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (20) = 40\).
Time = 4.51 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.91 \begin {dmath*} \int \frac {3 x^2+3 x^3-x^4+e^{e^5} \left (-3-3 x-6 x^3-6 x^4+2 x^5\right )+e^{2 e^5} \left (6 x+3 x^2+3 x^4+3 x^5-x^6\right )}{-4 x^2-x^3+x^4+e^{e^5} \left (3 x+8 x^3+2 x^4-2 x^5\right )+e^{2 e^5} \left (-3 x^2-4 x^4-x^5+x^6\right )} \, dx=- x - \log {\left (x^{2} - \frac {x}{e^{e^{5}}} \right )} + \log {\left (x^{4} + \frac {x^{3} \left (- e^{e^{5}} - 1\right )}{e^{e^{5}}} + \frac {x^{2} \cdot \left (1 - 4 e^{e^{5}}\right )}{e^{e^{5}}} + \frac {4 x}{e^{e^{5}}} - 3 \right )} \end {dmath*}
integrate(((-x**6+3*x**5+3*x**4+3*x**2+6*x)*exp(exp(5))**2+(2*x**5-6*x**4- 6*x**3-3*x-3)*exp(exp(5))-x**4+3*x**3+3*x**2)/((x**6-x**5-4*x**4-3*x**2)*e xp(exp(5))**2+(-2*x**5+2*x**4+8*x**3+3*x)*exp(exp(5))+x**4-x**3-4*x**2),x)
-x - log(x**2 - x*exp(-exp(5))) + log(x**4 + x**3*(-exp(exp(5)) - 1)*exp(- exp(5)) + x**2*(1 - 4*exp(exp(5)))*exp(-exp(5)) + 4*x*exp(-exp(5)) - 3)
Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.78 \begin {dmath*} \int \frac {3 x^2+3 x^3-x^4+e^{e^5} \left (-3-3 x-6 x^3-6 x^4+2 x^5\right )+e^{2 e^5} \left (6 x+3 x^2+3 x^4+3 x^5-x^6\right )}{-4 x^2-x^3+x^4+e^{e^5} \left (3 x+8 x^3+2 x^4-2 x^5\right )+e^{2 e^5} \left (-3 x^2-4 x^4-x^5+x^6\right )} \, dx=-x + \log \left (x^{4} e^{\left (e^{5}\right )} - x^{3} {\left (e^{\left (e^{5}\right )} + 1\right )} - x^{2} {\left (4 \, e^{\left (e^{5}\right )} - 1\right )} + 4 \, x - 3 \, e^{\left (e^{5}\right )}\right ) - \log \left (x e^{\left (e^{5}\right )} - 1\right ) - \log \left (x\right ) \end {dmath*}
integrate(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3 *x-3)*exp(exp(5))-x^4+3*x^3+3*x^2)/((x^6-x^5-4*x^4-3*x^2)*exp(exp(5))^2+(- 2*x^5+2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x, algorithm=\
-x + log(x^4*e^(e^5) - x^3*(e^(e^5) + 1) - x^2*(4*e^(e^5) - 1) + 4*x - 3*e ^(e^5)) - log(x*e^(e^5) - 1) - log(x)
Exception generated. \begin {dmath*} \int \frac {3 x^2+3 x^3-x^4+e^{e^5} \left (-3-3 x-6 x^3-6 x^4+2 x^5\right )+e^{2 e^5} \left (6 x+3 x^2+3 x^4+3 x^5-x^6\right )}{-4 x^2-x^3+x^4+e^{e^5} \left (3 x+8 x^3+2 x^4-2 x^5\right )+e^{2 e^5} \left (-3 x^2-4 x^4-x^5+x^6\right )} \, dx=\text {Exception raised: TypeError} \end {dmath*}
integrate(((-x^6+3*x^5+3*x^4+3*x^2+6*x)*exp(exp(5))^2+(2*x^5-6*x^4-6*x^3-3 *x-3)*exp(exp(5))-x^4+3*x^3+3*x^2)/((x^6-x^5-4*x^4-3*x^2)*exp(exp(5))^2+(- 2*x^5+2*x^4+8*x^3+3*x)*exp(exp(5))+x^4-x^3-4*x^2),x, algorithm=\
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Francis algorithm failure for[undef ,0.0,undef,undef,undef,undef]proot error [undef,0.0,undef,undef,undef,unde f]proot e
Time = 15.55 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94 \begin {dmath*} \int \frac {3 x^2+3 x^3-x^4+e^{e^5} \left (-3-3 x-6 x^3-6 x^4+2 x^5\right )+e^{2 e^5} \left (6 x+3 x^2+3 x^4+3 x^5-x^6\right )}{-4 x^2-x^3+x^4+e^{e^5} \left (3 x+8 x^3+2 x^4-2 x^5\right )+e^{2 e^5} \left (-3 x^2-4 x^4-x^5+x^6\right )} \, dx=\ln \left (4\,x\,{\mathrm {e}}^{-{\mathrm {e}}^5}+x^2\,{\mathrm {e}}^{-{\mathrm {e}}^5}-x^3\,{\mathrm {e}}^{-{\mathrm {e}}^5}-4\,x^2-x^3+x^4-3\right )-\ln \left (x^2-x\,{\mathrm {e}}^{-{\mathrm {e}}^5}\right )-x \end {dmath*}
int(-(exp(2*exp(5))*(6*x + 3*x^2 + 3*x^4 + 3*x^5 - x^6) - exp(exp(5))*(3*x + 6*x^3 + 6*x^4 - 2*x^5 + 3) + 3*x^2 + 3*x^3 - x^4)/(exp(2*exp(5))*(3*x^2 + 4*x^4 + x^5 - x^6) - exp(exp(5))*(3*x + 8*x^3 + 2*x^4 - 2*x^5) + 4*x^2 + x^3 - x^4),x)