Integrand size = 135, antiderivative size = 22 \[ \int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx=3 x-\frac {8}{\left (x+\frac {1}{3} x \left (-e^4+x\right )\right )^2} \]
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx=\frac {3 \left (-24+\left (-3+e^4\right )^2 x^3-2 \left (-3+e^4\right ) x^4+x^5\right )}{x^2 \left (3-e^4+x\right )^2} \]
Integrate[(-432 - 288*x - 81*x^3 + 3*E^12*x^3 - 81*x^4 - 27*x^5 - 3*x^6 + E^8*(-27*x^3 - 9*x^4) + E^4*(144 + 81*x^3 + 54*x^4 + 9*x^5))/(-27*x^3 + E^ 12*x^3 - 27*x^4 - 9*x^5 - x^6 + E^8*(-9*x^3 - 3*x^4) + E^4*(27*x^3 + 18*x^ 4 + 3*x^5)),x]
Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(22)=44\).
Time = 0.49 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.36, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {6, 6, 2026, 2007, 2123, 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.
\(\displaystyle \int \frac {-3 x^6-27 x^5-81 x^4+3 e^{12} x^3-81 x^3+e^8 \left (-9 x^4-27 x^3\right )+e^4 \left (9 x^5+54 x^4+81 x^3+144\right )-288 x-432}{-x^6-9 x^5-27 x^4+e^{12} x^3-27 x^3+e^8 \left (-3 x^4-9 x^3\right )+e^4 \left (3 x^5+18 x^4+27 x^3\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-3 x^6-27 x^5-81 x^4+3 e^{12} x^3-81 x^3+e^8 \left (-9 x^4-27 x^3\right )+e^4 \left (9 x^5+54 x^4+81 x^3+144\right )-288 x-432}{-x^6-9 x^5-27 x^4+\left (e^{12}-27\right ) x^3+e^8 \left (-3 x^4-9 x^3\right )+e^4 \left (3 x^5+18 x^4+27 x^3\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-3 x^6-27 x^5-81 x^4+\left (3 e^{12}-81\right ) x^3+e^8 \left (-9 x^4-27 x^3\right )+e^4 \left (9 x^5+54 x^4+81 x^3+144\right )-288 x-432}{-x^6-9 x^5-27 x^4+\left (e^{12}-27\right ) x^3+e^8 \left (-3 x^4-9 x^3\right )+e^4 \left (3 x^5+18 x^4+27 x^3\right )}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-3 x^6-27 x^5-81 x^4+\left (3 e^{12}-81\right ) x^3+e^8 \left (-9 x^4-27 x^3\right )+e^4 \left (9 x^5+54 x^4+81 x^3+144\right )-288 x-432}{x^3 \left (-x^3-3 \left (3-e^4\right ) x^2-3 \left (3-e^4\right )^2 x+\left (e^4-3\right )^3\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-3 x^6-27 x^5-81 x^4+\left (3 e^{12}-81\right ) x^3+e^8 \left (-9 x^4-27 x^3\right )+e^4 \left (9 x^5+54 x^4+81 x^3+144\right )-288 x-432}{\left (-x+e^4-3\right )^3 x^3}dx\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \int \left (\frac {144}{\left (e^4-3\right )^2 x^3}+\frac {144}{\left (e^4-3\right )^3 x^2}-\frac {144}{\left (e^4-3\right )^3 \left (-x+e^4-3\right )^2}-\frac {144}{\left (e^4-3\right )^2 \left (-x+e^4-3\right )^3}+3\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {72}{\left (3-e^4\right )^2 x^2}+3 x-\frac {144}{\left (3-e^4\right )^3 \left (x-e^4+3\right )}-\frac {72}{\left (3-e^4\right )^2 \left (x-e^4+3\right )^2}+\frac {144}{\left (3-e^4\right )^3 x}\) |
Int[(-432 - 288*x - 81*x^3 + 3*E^12*x^3 - 81*x^4 - 27*x^5 - 3*x^6 + E^8*(- 27*x^3 - 9*x^4) + E^4*(144 + 81*x^3 + 54*x^4 + 9*x^5))/(-27*x^3 + E^12*x^3 - 27*x^4 - 9*x^5 - x^6 + E^8*(-9*x^3 - 3*x^4) + E^4*(27*x^3 + 18*x^4 + 3* x^5)),x]
-72/((3 - E^4)^2*x^2) + 144/((3 - E^4)^3*x) + 3*x - 72/((3 - E^4)^2*(3 - E ^4 + x)^2) - 144/((3 - E^4)^3*(3 - E^4 + x))
3.17.80.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
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[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 0.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41
method | result | size |
risch | \(3 x -\frac {72}{x^{2} \left ({\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+x^{2}-6 \,{\mathrm e}^{4}+6 x +9\right )}\) | \(31\) |
norman | \(\frac {-72+\left (-9 \,{\mathrm e}^{8}+54 \,{\mathrm e}^{4}-81\right ) x^{3}+\left (6 \,{\mathrm e}^{12}-54 \,{\mathrm e}^{8}+162 \,{\mathrm e}^{4}-162\right ) x^{2}+3 x^{5}}{x^{2} \left ({\mathrm e}^{4}-x -3\right )^{2}}\) | \(59\) |
gosper | \(\frac {6 x^{2} {\mathrm e}^{12}-9 \,{\mathrm e}^{8} x^{3}+3 x^{5}-54 x^{2} {\mathrm e}^{8}+54 x^{3} {\mathrm e}^{4}+162 x^{2} {\mathrm e}^{4}-81 x^{3}-162 x^{2}-72}{x^{2} \left ({\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+x^{2}-6 \,{\mathrm e}^{4}+6 x +9\right )}\) | \(85\) |
parallelrisch | \(\frac {6 x^{2} {\mathrm e}^{12}-9 \,{\mathrm e}^{8} x^{3}+3 x^{5}-54 x^{2} {\mathrm e}^{8}+54 x^{3} {\mathrm e}^{4}+162 x^{2} {\mathrm e}^{4}-81 x^{3}-162 x^{2}-72}{x^{2} \left ({\mathrm e}^{8}-2 x \,{\mathrm e}^{4}+x^{2}-6 \,{\mathrm e}^{4}+6 x +9\right )}\) | \(86\) |
default | \(3 x -\frac {48 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{4}+9\right ) \textit {\_Z}^{2}+\left (-18 \,{\mathrm e}^{4}+3 \,{\mathrm e}^{8}+27\right ) \textit {\_Z} -27 \,{\mathrm e}^{4}+9 \,{\mathrm e}^{8}-{\mathrm e}^{12}+27\right )}{\sum }\frac {\left (4374-1458 \textit {\_R} \,{\mathrm e}^{4}+135 \textit {\_R} \,{\mathrm e}^{16}-18 \textit {\_R} \,{\mathrm e}^{20}+{\mathrm e}^{24} \textit {\_R} +1215 \textit {\_R} \,{\mathrm e}^{8}-540 \textit {\_R} \,{\mathrm e}^{12}-10206 \,{\mathrm e}^{4}+1890 \,{\mathrm e}^{16}-2 \,{\mathrm e}^{28}-378 \,{\mathrm e}^{20}+42 \,{\mathrm e}^{24}+10206 \,{\mathrm e}^{8}-5670 \,{\mathrm e}^{12}+729 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{9+{\mathrm e}^{8}-2 \textit {\_R} \,{\mathrm e}^{4}+\textit {\_R}^{2}-6 \,{\mathrm e}^{4}+6 \textit {\_R}}\right )}{\left (27 \,{\mathrm e}^{4}+{\mathrm e}^{12}-9 \,{\mathrm e}^{8}-27\right )^{3}}-\frac {3 \left (-69984 \,{\mathrm e}^{4}+6480 \,{\mathrm e}^{16}-864 \,{\mathrm e}^{20}+48 \,{\mathrm e}^{24}+58320 \,{\mathrm e}^{8}-25920 \,{\mathrm e}^{12}+34992\right )}{\left (27 \,{\mathrm e}^{4}+{\mathrm e}^{12}-9 \,{\mathrm e}^{8}-27\right )^{3} x}-\frac {3 \left (244944 \,{\mathrm e}^{4}-45360 \,{\mathrm e}^{16}+9072 \,{\mathrm e}^{20}-1008 \,{\mathrm e}^{24}+48 \,{\mathrm e}^{28}-244944 \,{\mathrm e}^{8}+136080 \,{\mathrm e}^{12}-104976\right )}{2 \left (27 \,{\mathrm e}^{4}+{\mathrm e}^{12}-9 \,{\mathrm e}^{8}-27\right )^{3} x^{2}}\) | \(248\) |
int((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+144)*exp (4)-3*x^6-27*x^5-81*x^4-81*x^3-288*x-432)/(x^3*exp(4)^3+(-3*x^4-9*x^3)*exp (4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x,method=_RETU RNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.23 \[ \int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx=\frac {3 \, {\left (x^{5} + 6 \, x^{4} + x^{3} e^{8} + 9 \, x^{3} - 2 \, {\left (x^{4} + 3 \, x^{3}\right )} e^{4} - 24\right )}}{x^{4} + 6 \, x^{3} + x^{2} e^{8} + 9 \, x^{2} - 2 \, {\left (x^{3} + 3 \, x^{2}\right )} e^{4}} \]
integrate((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+14 4)*exp(4)-3*x^6-27*x^5-81*x^4-81*x^3-288*x-432)/(x^3*exp(4)^3+(-3*x^4-9*x^ 3)*exp(4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x, algor ithm=\
3*(x^5 + 6*x^4 + x^3*e^8 + 9*x^3 - 2*(x^4 + 3*x^3)*e^4 - 24)/(x^4 + 6*x^3 + x^2*e^8 + 9*x^2 - 2*(x^3 + 3*x^2)*e^4)
Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx=3 x - \frac {72}{x^{4} + x^{3} \cdot \left (6 - 2 e^{4}\right ) + x^{2} \left (- 6 e^{4} + 9 + e^{8}\right )} \]
integrate((3*x**3*exp(4)**3+(-9*x**4-27*x**3)*exp(4)**2+(9*x**5+54*x**4+81 *x**3+144)*exp(4)-3*x**6-27*x**5-81*x**4-81*x**3-288*x-432)/(x**3*exp(4)** 3+(-3*x**4-9*x**3)*exp(4)**2+(3*x**5+18*x**4+27*x**3)*exp(4)-x**6-9*x**5-2 7*x**4-27*x**3),x)
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx=3 \, x - \frac {72}{x^{4} - 2 \, x^{3} {\left (e^{4} - 3\right )} + x^{2} {\left (e^{8} - 6 \, e^{4} + 9\right )}} \]
integrate((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+14 4)*exp(4)-3*x^6-27*x^5-81*x^4-81*x^3-288*x-432)/(x^3*exp(4)^3+(-3*x^4-9*x^ 3)*exp(4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x, algor ithm=\
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx=3 \, x - \frac {72}{{\left (x^{2} - x e^{4} + 3 \, x\right )}^{2}} \]
integrate((3*x^3*exp(4)^3+(-9*x^4-27*x^3)*exp(4)^2+(9*x^5+54*x^4+81*x^3+14 4)*exp(4)-3*x^6-27*x^5-81*x^4-81*x^3-288*x-432)/(x^3*exp(4)^3+(-3*x^4-9*x^ 3)*exp(4)^2+(3*x^5+18*x^4+27*x^3)*exp(4)-x^6-9*x^5-27*x^4-27*x^3),x, algor ithm=\
Time = 8.65 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {-432-288 x-81 x^3+3 e^{12} x^3-81 x^4-27 x^5-3 x^6+e^8 \left (-27 x^3-9 x^4\right )+e^4 \left (144+81 x^3+54 x^4+9 x^5\right )}{-27 x^3+e^{12} x^3-27 x^4-9 x^5-x^6+e^8 \left (-9 x^3-3 x^4\right )+e^4 \left (27 x^3+18 x^4+3 x^5\right )} \, dx=3\,x-\frac {\left (18\,{\mathrm {e}}^4-3\,{\mathrm {e}}^8+3\,{\left ({\mathrm {e}}^4-3\right )}^2-27\right )\,x^3+72}{x^2\,{\left (x-{\mathrm {e}}^4+3\right )}^2} \]