Integrand size = 140, antiderivative size = 33 \[ \int \frac {360+600 x+262 x^2+48 x^3-34 x^4-30 x^5+e^4 \left (-900-900 x-15 x^2+360 x^3+114 x^4-36 x^5-15 x^6\right )}{36 x^2+60 x^3+25 x^4+e^4 \left (-360 x-600 x^2-178 x^3+120 x^4+50 x^5\right )+e^8 \left (900+1500 x+265 x^2-600 x^3-214 x^4+60 x^5+25 x^6\right )} \, dx=\frac {2-x+\frac {2 x}{5+\frac {6}{x}}}{e^4+\frac {x}{-5+x^2}} \]
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(33)=66\).
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.39 \[ \int \frac {360+600 x+262 x^2+48 x^3-34 x^4-30 x^5+e^4 \left (-900-900 x-15 x^2+360 x^3+114 x^4-36 x^5-15 x^6\right )}{36 x^2+60 x^3+25 x^4+e^4 \left (-360 x-600 x^2-178 x^3+120 x^4+50 x^5\right )+e^8 \left (900+1500 x+265 x^2-600 x^3-214 x^4+60 x^5+25 x^6\right )} \, dx=\frac {-15 x (6+5 x)+e^4 \left (450+39 x-370 x^2-75 x^3\right )+e^8 \left (180+900 x+339 x^2-180 x^3-75 x^4\right )}{25 e^8 (6+5 x) \left (x+e^4 \left (-5+x^2\right )\right )} \]
Integrate[(360 + 600*x + 262*x^2 + 48*x^3 - 34*x^4 - 30*x^5 + E^4*(-900 - 900*x - 15*x^2 + 360*x^3 + 114*x^4 - 36*x^5 - 15*x^6))/(36*x^2 + 60*x^3 + 25*x^4 + E^4*(-360*x - 600*x^2 - 178*x^3 + 120*x^4 + 50*x^5) + E^8*(900 + 1500*x + 265*x^2 - 600*x^3 - 214*x^4 + 60*x^5 + 25*x^6)),x]
(-15*x*(6 + 5*x) + E^4*(450 + 39*x - 370*x^2 - 75*x^3) + E^8*(180 + 900*x + 339*x^2 - 180*x^3 - 75*x^4))/(25*E^8*(6 + 5*x)*(x + E^4*(-5 + x^2)))
Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(33)=66\).
Time = 0.48 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.73, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {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.
\(\displaystyle \int \frac {-30 x^5-34 x^4+48 x^3+262 x^2+e^4 \left (-15 x^6-36 x^5+114 x^4+360 x^3-15 x^2-900 x-900\right )+600 x+360}{25 x^4+60 x^3+36 x^2+e^4 \left (50 x^5+120 x^4-178 x^3-600 x^2-360 x\right )+e^8 \left (25 x^6+60 x^5-214 x^4-600 x^3+265 x^2+1500 x+900\right )} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {5 e^4 \left (18+159 e^4+236 e^8\right )-\left (18+99 e^4+298 e^8+390 e^{12}\right ) x}{e^8 \left (30+89 e^4\right ) \left (-e^4 x^2-x+5 e^4\right )^2}+\frac {18+99 e^4+118 e^8}{e^8 \left (30+89 e^4\right ) \left (e^4 x^2+x-5 e^4\right )}-\frac {6408}{5 \left (30+89 e^4\right ) (5 x+6)^2}-\frac {3}{5 e^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {15 e^4 \left (6+13 e^4\right )-\left (18+99 e^4+118 e^8\right ) x}{e^8 \left (30+89 e^4\right ) \left (-e^4 x^2-x+5 e^4\right )}-\frac {3 x}{5 e^4}+\frac {6408}{25 \left (30+89 e^4\right ) (5 x+6)}\) |
Int[(360 + 600*x + 262*x^2 + 48*x^3 - 34*x^4 - 30*x^5 + E^4*(-900 - 900*x - 15*x^2 + 360*x^3 + 114*x^4 - 36*x^5 - 15*x^6))/(36*x^2 + 60*x^3 + 25*x^4 + E^4*(-360*x - 600*x^2 - 178*x^3 + 120*x^4 + 50*x^5) + E^8*(900 + 1500*x + 265*x^2 - 600*x^3 - 214*x^4 + 60*x^5 + 25*x^6)),x]
(-3*x)/(5*E^4) + 6408/(25*(30 + 89*E^4)*(6 + 5*x)) - (15*E^4*(6 + 13*E^4) - (18 + 99*E^4 + 118*E^8)*x)/(E^8*(30 + 89*E^4)*(5*E^4 - x - E^4*x^2))
3.8.64.3.1 Defintions of rubi rules used
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 = 0.21 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58
method | result | size |
norman | \(\frac {-3 x^{4}-\frac {24 x \,{\mathrm e}^{-4}}{5}+\frac {{\mathrm e}^{-4} \left (-20+111 \,{\mathrm e}^{4}\right ) x^{2}}{5}-36}{\left (5 x +6\right ) \left (x^{2} {\mathrm e}^{4}-5 \,{\mathrm e}^{4}+x \right )}\) | \(52\) |
gosper | \(-\frac {\left (15 x^{4} {\mathrm e}^{4}-111 x^{2} {\mathrm e}^{4}+20 x^{2}+180 \,{\mathrm e}^{4}+24 x \right ) {\mathrm e}^{-4}}{5 \left (5 x^{3} {\mathrm e}^{4}+6 x^{2} {\mathrm e}^{4}-25 x \,{\mathrm e}^{4}+5 x^{2}-30 \,{\mathrm e}^{4}+6 x \right )}\) | \(68\) |
parallelrisch | \(-\frac {\left (15 x^{4} {\mathrm e}^{4}-111 x^{2} {\mathrm e}^{4}+20 x^{2}+180 \,{\mathrm e}^{4}+24 x \right ) {\mathrm e}^{-4}}{5 \left (5 x^{3} {\mathrm e}^{4}+6 x^{2} {\mathrm e}^{4}-25 x \,{\mathrm e}^{4}+5 x^{2}-30 \,{\mathrm e}^{4}+6 x \right )}\) | \(68\) |
risch | \(-\frac {3 x \,{\mathrm e}^{-4}}{5}+\frac {\left (-\frac {38 x^{2}}{5}+\frac {72 x^{2} {\mathrm e}^{4}}{25}-3 \,{\mathrm e}^{-4} x^{2}+\frac {147 x}{25}-\frac {18 x \,{\mathrm e}^{-4}}{5}-\frac {72 \,{\mathrm e}^{4}}{5}+18\right ) {\mathrm e}^{-4}}{5 x^{3} {\mathrm e}^{4}+6 x^{2} {\mathrm e}^{4}-25 x \,{\mathrm e}^{4}+5 x^{2}-30 \,{\mathrm e}^{4}+6 x}\) | \(75\) |
default | \(-\frac {3 x \,{\mathrm e}^{4} {\mathrm e}^{-8}}{5}-\frac {-1539842400 \,{\mathrm e}^{4}-4568199120 \,{\mathrm e}^{8}-4517441352 \,{\mathrm e}^{12}-173016000}{25 \left (7921 \,{\mathrm e}^{8}+5340 \,{\mathrm e}^{4}+900\right )^{2} \left (5 x +6\right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} {\mathrm e}^{8}+2 \textit {\_Z}^{3} {\mathrm e}^{4}+\left (-10 \,{\mathrm e}^{8}+1\right ) \textit {\_Z}^{2}-10 \textit {\_Z} \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}\right )}{\sum }\frac {\left (8100000+\left (39807720 \,{\mathrm e}^{4}+83186342 \,{\mathrm e}^{16}+486000 \,{\mathrm e}^{-4}+111620952 \,{\mathrm e}^{8}+153912951 \,{\mathrm e}^{12}+6998400\right ) \textit {\_R}^{2}+30 \left (-162000-1792800 \,{\mathrm e}^{4}-9164597 \,{\mathrm e}^{16}-7401240 \,{\mathrm e}^{8}-13497384 \,{\mathrm e}^{12}\right ) \textit {\_R} +88020000 \,{\mathrm e}^{4}+415931710 \,{\mathrm e}^{16}+355644000 \,{\mathrm e}^{8}+632095800 \,{\mathrm e}^{12}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} {\mathrm e}^{8}+3 \textit {\_R}^{2} {\mathrm e}^{4}-10 \textit {\_R} \,{\mathrm e}^{8}-5 \,{\mathrm e}^{4}+\textit {\_R}}}{2 \left (7921 \,{\mathrm e}^{8}+5340 \,{\mathrm e}^{4}+900\right )^{2}}\) | \(196\) |
int(((-15*x^6-36*x^5+114*x^4+360*x^3-15*x^2-900*x-900)*exp(4)-30*x^5-34*x^ 4+48*x^3+262*x^2+600*x+360)/((25*x^6+60*x^5-214*x^4-600*x^3+265*x^2+1500*x +900)*exp(4)^2+(50*x^5+120*x^4-178*x^3-600*x^2-360*x)*exp(4)+25*x^4+60*x^3 +36*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31) = 62\).
Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61 \[ \int \frac {360+600 x+262 x^2+48 x^3-34 x^4-30 x^5+e^4 \left (-900-900 x-15 x^2+360 x^3+114 x^4-36 x^5-15 x^6\right )}{36 x^2+60 x^3+25 x^4+e^4 \left (-360 x-600 x^2-178 x^3+120 x^4+50 x^5\right )+e^8 \left (900+1500 x+265 x^2-600 x^3-214 x^4+60 x^5+25 x^6\right )} \, dx=-\frac {75 \, x^{2} + 3 \, {\left (25 \, x^{4} + 30 \, x^{3} - 149 \, x^{2} - 150 \, x + 120\right )} e^{8} + {\left (75 \, x^{3} + 280 \, x^{2} - 147 \, x - 450\right )} e^{4} + 90 \, x}{25 \, {\left ({\left (5 \, x^{3} + 6 \, x^{2} - 25 \, x - 30\right )} e^{12} + {\left (5 \, x^{2} + 6 \, x\right )} e^{8}\right )}} \]
integrate(((-15*x^6-36*x^5+114*x^4+360*x^3-15*x^2-900*x-900)*exp(4)-30*x^5 -34*x^4+48*x^3+262*x^2+600*x+360)/((25*x^6+60*x^5-214*x^4-600*x^3+265*x^2+ 1500*x+900)*exp(4)^2+(50*x^5+120*x^4-178*x^3-600*x^2-360*x)*exp(4)+25*x^4+ 60*x^3+36*x^2),x, algorithm=\
-1/25*(75*x^2 + 3*(25*x^4 + 30*x^3 - 149*x^2 - 150*x + 120)*e^8 + (75*x^3 + 280*x^2 - 147*x - 450)*e^4 + 90*x)/((5*x^3 + 6*x^2 - 25*x - 30)*e^12 + ( 5*x^2 + 6*x)*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (20) = 40\).
Time = 2.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.48 \[ \int \frac {360+600 x+262 x^2+48 x^3-34 x^4-30 x^5+e^4 \left (-900-900 x-15 x^2+360 x^3+114 x^4-36 x^5-15 x^6\right )}{36 x^2+60 x^3+25 x^4+e^4 \left (-360 x-600 x^2-178 x^3+120 x^4+50 x^5\right )+e^8 \left (900+1500 x+265 x^2-600 x^3-214 x^4+60 x^5+25 x^6\right )} \, dx=- \frac {3 x}{5 e^{4}} - \frac {x^{2} \left (- 72 e^{8} + 75 + 190 e^{4}\right ) + x \left (90 - 147 e^{4}\right ) - 450 e^{4} + 360 e^{8}}{125 x^{3} e^{12} + x^{2} \cdot \left (125 e^{8} + 150 e^{12}\right ) + x \left (- 625 e^{12} + 150 e^{8}\right ) - 750 e^{12}} \]
integrate(((-15*x**6-36*x**5+114*x**4+360*x**3-15*x**2-900*x-900)*exp(4)-3 0*x**5-34*x**4+48*x**3+262*x**2+600*x+360)/((25*x**6+60*x**5-214*x**4-600* x**3+265*x**2+1500*x+900)*exp(4)**2+(50*x**5+120*x**4-178*x**3-600*x**2-36 0*x)*exp(4)+25*x**4+60*x**3+36*x**2),x)
-3*x*exp(-4)/5 - (x**2*(-72*exp(8) + 75 + 190*exp(4)) + x*(90 - 147*exp(4) ) - 450*exp(4) + 360*exp(8))/(125*x**3*exp(12) + x**2*(125*exp(8) + 150*ex p(12)) + x*(-625*exp(12) + 150*exp(8)) - 750*exp(12))
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (31) = 62\).
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.39 \[ \int \frac {360+600 x+262 x^2+48 x^3-34 x^4-30 x^5+e^4 \left (-900-900 x-15 x^2+360 x^3+114 x^4-36 x^5-15 x^6\right )}{36 x^2+60 x^3+25 x^4+e^4 \left (-360 x-600 x^2-178 x^3+120 x^4+50 x^5\right )+e^8 \left (900+1500 x+265 x^2-600 x^3-214 x^4+60 x^5+25 x^6\right )} \, dx=-\frac {3}{5} \, x e^{\left (-4\right )} + \frac {x^{2} {\left (72 \, e^{8} - 190 \, e^{4} - 75\right )} + 3 \, x {\left (49 \, e^{4} - 30\right )} - 360 \, e^{8} + 450 \, e^{4}}{25 \, {\left (5 \, x^{3} e^{12} + x^{2} {\left (6 \, e^{12} + 5 \, e^{8}\right )} - x {\left (25 \, e^{12} - 6 \, e^{8}\right )} - 30 \, e^{12}\right )}} \]
integrate(((-15*x^6-36*x^5+114*x^4+360*x^3-15*x^2-900*x-900)*exp(4)-30*x^5 -34*x^4+48*x^3+262*x^2+600*x+360)/((25*x^6+60*x^5-214*x^4-600*x^3+265*x^2+ 1500*x+900)*exp(4)^2+(50*x^5+120*x^4-178*x^3-600*x^2-360*x)*exp(4)+25*x^4+ 60*x^3+36*x^2),x, algorithm=\
-3/5*x*e^(-4) + 1/25*(x^2*(72*e^8 - 190*e^4 - 75) + 3*x*(49*e^4 - 30) - 36 0*e^8 + 450*e^4)/(5*x^3*e^12 + x^2*(6*e^12 + 5*e^8) - x*(25*e^12 - 6*e^8) - 30*e^12)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (31) = 62\).
Time = 0.30 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {360+600 x+262 x^2+48 x^3-34 x^4-30 x^5+e^4 \left (-900-900 x-15 x^2+360 x^3+114 x^4-36 x^5-15 x^6\right )}{36 x^2+60 x^3+25 x^4+e^4 \left (-360 x-600 x^2-178 x^3+120 x^4+50 x^5\right )+e^8 \left (900+1500 x+265 x^2-600 x^3-214 x^4+60 x^5+25 x^6\right )} \, dx=-\frac {3}{5} \, x e^{\left (-4\right )} + \frac {{\left (72 \, x^{2} e^{8} - 190 \, x^{2} e^{4} - 75 \, x^{2} + 147 \, x e^{4} - 90 \, x - 360 \, e^{8} + 450 \, e^{4}\right )} e^{\left (-8\right )}}{25 \, {\left (5 \, x^{3} e^{4} + 6 \, x^{2} e^{4} + 5 \, x^{2} - 25 \, x e^{4} + 6 \, x - 30 \, e^{4}\right )}} \]
integrate(((-15*x^6-36*x^5+114*x^4+360*x^3-15*x^2-900*x-900)*exp(4)-30*x^5 -34*x^4+48*x^3+262*x^2+600*x+360)/((25*x^6+60*x^5-214*x^4-600*x^3+265*x^2+ 1500*x+900)*exp(4)^2+(50*x^5+120*x^4-178*x^3-600*x^2-360*x)*exp(4)+25*x^4+ 60*x^3+36*x^2),x, algorithm=\
-3/5*x*e^(-4) + 1/25*(72*x^2*e^8 - 190*x^2*e^4 - 75*x^2 + 147*x*e^4 - 90*x - 360*e^8 + 450*e^4)*e^(-8)/(5*x^3*e^4 + 6*x^2*e^4 + 5*x^2 - 25*x*e^4 + 6 *x - 30*e^4)
Time = 0.43 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {360+600 x+262 x^2+48 x^3-34 x^4-30 x^5+e^4 \left (-900-900 x-15 x^2+360 x^3+114 x^4-36 x^5-15 x^6\right )}{36 x^2+60 x^3+25 x^4+e^4 \left (-360 x-600 x^2-178 x^3+120 x^4+50 x^5\right )+e^8 \left (900+1500 x+265 x^2-600 x^3-214 x^4+60 x^5+25 x^6\right )} \, dx=-\frac {3\,x\,{\mathrm {e}}^{-4}}{5}-\frac {\frac {{\mathrm {e}}^{-4}\,\left (190\,{\mathrm {e}}^4-72\,{\mathrm {e}}^8+75\right )\,x^2}{5}-\frac {3\,{\mathrm {e}}^{-4}\,\left (49\,{\mathrm {e}}^4-30\right )\,x}{5}+72\,{\mathrm {e}}^4-90}{25\,{\mathrm {e}}^8\,x^3+\left (25\,{\mathrm {e}}^4+30\,{\mathrm {e}}^8\right )\,x^2+\left (30\,{\mathrm {e}}^4-125\,{\mathrm {e}}^8\right )\,x-150\,{\mathrm {e}}^8} \]
int((600*x - exp(4)*(900*x + 15*x^2 - 360*x^3 - 114*x^4 + 36*x^5 + 15*x^6 + 900) + 262*x^2 + 48*x^3 - 34*x^4 - 30*x^5 + 360)/(exp(8)*(1500*x + 265*x ^2 - 600*x^3 - 214*x^4 + 60*x^5 + 25*x^6 + 900) - exp(4)*(360*x + 600*x^2 + 178*x^3 - 120*x^4 - 50*x^5) + 36*x^2 + 60*x^3 + 25*x^4),x)