Integrand size = 206, antiderivative size = 23 \[ \int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} \left (-320+480 x^4-320 x^6+60 x^8\right )+e^{20} \left (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9\right )+e^{15} \left (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10}\right )+e^{10} \left (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11}\right )+e^5 \left (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12}\right )}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx=4+\frac {10 \left (-2+x^2+\frac {x}{e^5+x}\right )^4}{x^2} \]
Leaf count is larger than twice the leaf count of optimal. \(127\) vs. \(2(23)=46\).
Time = 0.11 (sec) , antiderivative size = 127, normalized size of antiderivative = 5.52 \[ \int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} \left (-320+480 x^4-320 x^6+60 x^8\right )+e^{20} \left (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9\right )+e^{15} \left (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10}\right )+e^{10} \left (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11}\right )+e^5 \left (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12}\right )}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx=10 \left (\frac {16}{x^2}-\frac {32}{e^5 x}-4 e^5 \left (-3+e^{10}\right ) x+2 \left (3+2 e^{10}\right ) x^2-4 e^5 x^3-4 x^4+x^6+\frac {e^{10}}{\left (e^5+x\right )^4}+\frac {6 e^5-4 e^{15}}{\left (e^5+x\right )^3}+\frac {17-12 e^{10}+6 e^{20}}{\left (e^5+x\right )^2}-\frac {4 \left (-8+3 e^{10}+e^{30}\right )}{e^5 \left (e^5+x\right )}\right ) \]
Integrate[(-20*x^5 + 120*x^9 - 160*x^11 + 60*x^13 + E^25*(-320 + 480*x^4 - 320*x^6 + 60*x^8) + E^20*(-1280*x + 480*x^3 + 1680*x^5 - 1400*x^7 + 300*x ^9) + E^15*(-1600*x^2 + 960*x^4 + 2400*x^6 - 2440*x^8 + 600*x^10) + E^10*( -880*x^3 + 600*x^5 + 1800*x^7 - 2120*x^9 + 600*x^11) + E^5*(-220*x^4 + 120 *x^6 + 720*x^8 - 920*x^10 + 300*x^12))/(E^25*x^3 + 5*E^20*x^4 + 10*E^15*x^ 5 + 10*E^10*x^6 + 5*E^5*x^7 + x^8),x]
10*(16/x^2 - 32/(E^5*x) - 4*E^5*(-3 + E^10)*x + 2*(3 + 2*E^10)*x^2 - 4*E^5 *x^3 - 4*x^4 + x^6 + E^10/(E^5 + x)^4 + (6*E^5 - 4*E^15)/(E^5 + x)^3 + (17 - 12*E^10 + 6*E^20)/(E^5 + x)^2 - (4*(-8 + 3*E^10 + E^30))/(E^5*(E^5 + x) ))
Leaf count is larger than twice the leaf count of optimal. \(133\) vs. \(2(23)=46\).
Time = 0.67 (sec) , antiderivative size = 133, normalized size of antiderivative = 5.78, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {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.
| \(\displaystyle \int \frac {60 x^{13}-160 x^{11}+120 x^9-20 x^5+e^{25} \left (60 x^8-320 x^6+480 x^4-320\right )+e^{20} \left (300 x^9-1400 x^7+1680 x^5+480 x^3-1280 x\right )+e^5 \left (300 x^{12}-920 x^{10}+720 x^8+120 x^6-220 x^4\right )+e^{10} \left (600 x^{11}-2120 x^9+1800 x^7+600 x^5-880 x^3\right )+e^{15} \left (600 x^{10}-2440 x^8+2400 x^6+960 x^4-1600 x^2\right )}{x^8+5 e^5 x^7+10 e^{10} x^6+10 e^{15} x^5+5 e^{20} x^4+e^{25} x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {60 x^{13}-160 x^{11}+120 x^9-20 x^5+e^{25} \left (60 x^8-320 x^6+480 x^4-320\right )+e^{20} \left (300 x^9-1400 x^7+1680 x^5+480 x^3-1280 x\right )+e^5 \left (300 x^{12}-920 x^{10}+720 x^8+120 x^6-220 x^4\right )+e^{10} \left (600 x^{11}-2120 x^9+1800 x^7+600 x^5-880 x^3\right )+e^{15} \left (600 x^{10}-2440 x^8+2400 x^6+960 x^4-1600 x^2\right )}{x^3 \left (x^5+5 e^5 x^4+10 e^{10} x^3+10 e^{15} x^2+5 e^{20} x+e^{25}\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {60 x^{13}-160 x^{11}+120 x^9-20 x^5+e^{25} \left (60 x^8-320 x^6+480 x^4-320\right )+e^{20} \left (300 x^9-1400 x^7+1680 x^5+480 x^3-1280 x\right )+e^5 \left (300 x^{12}-920 x^{10}+720 x^8+120 x^6-220 x^4\right )+e^{10} \left (600 x^{11}-2120 x^9+1800 x^7+600 x^5-880 x^3\right )+e^{15} \left (600 x^{10}-2440 x^8+2400 x^6+960 x^4-1600 x^2\right )}{x^3 \left (x+e^5\right )^5}dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (60 x^5-160 x^3-\frac {320}{x^3}-120 e^5 x^2+\frac {320}{e^5 x^2}+40 \left (3+2 e^{10}\right ) x+\frac {40 \left (-8+3 e^{10}+e^{30}\right )}{e^5 \left (x+e^5\right )^2}-\frac {20 \left (17-12 e^{10}+6 e^{20}\right )}{\left (x+e^5\right )^3}+\frac {60 e^5 \left (2 e^{10}-3\right )}{\left (x+e^5\right )^4}-\frac {40 e^{10}}{\left (x+e^5\right )^5}-40 e^5 \left (e^{10}-3\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 10 x^6-40 x^4-40 e^5 x^3+20 \left (3+2 e^{10}\right ) x^2+\frac {160}{x^2}+40 e^5 \left (3-e^{10}\right ) x+\frac {40 \left (8-3 e^{10}-e^{30}\right )}{e^5 \left (x+e^5\right )}+\frac {10 \left (17-12 e^{10}+6 e^{20}\right )}{\left (x+e^5\right )^2}+\frac {20 e^5 \left (3-2 e^{10}\right )}{\left (x+e^5\right )^3}+\frac {10 e^{10}}{\left (x+e^5\right )^4}-\frac {320}{e^5 x}\) |
Int[(-20*x^5 + 120*x^9 - 160*x^11 + 60*x^13 + E^25*(-320 + 480*x^4 - 320*x ^6 + 60*x^8) + E^20*(-1280*x + 480*x^3 + 1680*x^5 - 1400*x^7 + 300*x^9) + E^15*(-1600*x^2 + 960*x^4 + 2400*x^6 - 2440*x^8 + 600*x^10) + E^10*(-880*x ^3 + 600*x^5 + 1800*x^7 - 2120*x^9 + 600*x^11) + E^5*(-220*x^4 + 120*x^6 + 720*x^8 - 920*x^10 + 300*x^12))/(E^25*x^3 + 5*E^20*x^4 + 10*E^15*x^5 + 10 *E^10*x^6 + 5*E^5*x^7 + x^8),x]
160/x^2 - 320/(E^5*x) + 40*E^5*(3 - E^10)*x + 20*(3 + 2*E^10)*x^2 - 40*E^5 *x^3 - 40*x^4 + 10*x^6 + (10*E^10)/(E^5 + x)^4 + (20*E^5*(3 - 2*E^10))/(E^ 5 + x)^3 + (10*(17 - 12*E^10 + 6*E^20))/(E^5 + x)^2 + (40*(8 - 3*E^10 - E^ 30))/(E^5*(E^5 + x))
3.9.2.3.1 Defintions of rubi rules used
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])
Leaf count of result is larger than twice the leaf count of optimal. \(154\) vs. \(2(22)=44\).
Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 6.74
| method | result | size |
| risch | \(10 x^{6}-40 x \,{\mathrm e}^{15}+40 x^{2} {\mathrm e}^{10}-40 x^{3} {\mathrm e}^{5}-40 x^{4}+120 x \,{\mathrm e}^{5}+60 x^{2}+\frac {\left (-40 \,{\mathrm e}^{25}-120 \,{\mathrm e}^{5}\right ) x^{5}+\left (-120 \,{\mathrm e}^{30}+60 \,{\mathrm e}^{20}-480 \,{\mathrm e}^{10}+10\right ) x^{4}+\left (-120 \,{\mathrm e}^{35}+120 \,{\mathrm e}^{25}-640 \,{\mathrm e}^{15}+80 \,{\mathrm e}^{5}\right ) x^{3}+\left (-40 \,{\mathrm e}^{40}+60 \,{\mathrm e}^{30}-280 \,{\mathrm e}^{20}+240 \,{\mathrm e}^{10}\right ) x^{2}+320 x \,{\mathrm e}^{15}+160 \,{\mathrm e}^{20}}{x^{2} \left ({\mathrm e}^{20}+4 x \,{\mathrm e}^{15}+6 x^{2} {\mathrm e}^{10}+4 x^{3} {\mathrm e}^{5}+x^{4}\right )}\) | \(155\) |
| norman | \(\frac {\left (60 \,{\mathrm e}^{10}-40\right ) x^{10}+\left (-280 \,{\mathrm e}^{15}+360 \,{\mathrm e}^{5}\right ) x^{7}+\left (40 \,{\mathrm e}^{15}-200 \,{\mathrm e}^{5}\right ) x^{9}+80 \,{\mathrm e}^{5} \left (4 \,{\mathrm e}^{30}-39 \,{\mathrm e}^{20}-8 \,{\mathrm e}^{10}+1\right ) x^{3}+\left (10 \,{\mathrm e}^{20}-360 \,{\mathrm e}^{10}+60\right ) x^{8}+\left (320 \,{\mathrm e}^{25}-2400 \,{\mathrm e}^{15}-120 \,{\mathrm e}^{5}\right ) x^{5}+\left (480 \,{\mathrm e}^{30}-4440 \,{\mathrm e}^{20}-480 \,{\mathrm e}^{10}+10\right ) x^{4}+\left (80 \,{\mathrm e}^{40}-780 \,{\mathrm e}^{30}-280 \,{\mathrm e}^{20}+240 \,{\mathrm e}^{10}\right ) x^{2}+10 x^{12}+160 \,{\mathrm e}^{20}+320 x \,{\mathrm e}^{15}+40 \,{\mathrm e}^{5} x^{11}}{x^{2} \left ({\mathrm e}^{5}+x \right )^{4}}\) | \(198\) |
| default | \(10 x^{6}-3840 x \,{\mathrm e}^{15}+40 x^{2} {\mathrm e}^{10}-40 x^{3} {\mathrm e}^{5}-40 x^{4}+3800 x \,{\mathrm e}^{5} {\mathrm e}^{10}+120 x \,{\mathrm e}^{5}+60 x^{2}-\frac {320 \,{\mathrm e}^{-50} {\mathrm e}^{45}}{x}+\frac {160 \,{\mathrm e}^{-50} {\mathrm e}^{50}}{x^{2}}+4 \,{\mathrm e}^{-50} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left ({\mathrm e}^{25}+5 \textit {\_Z} \,{\mathrm e}^{20}+10 \textit {\_Z}^{2} {\mathrm e}^{15}+10 \textit {\_Z}^{3} {\mathrm e}^{10}+5 \textit {\_Z}^{4} {\mathrm e}^{5}+\textit {\_Z}^{5}\right )}{\sum }\frac {\left (2 \left (3 \,{\mathrm e}^{55}-8 \,{\mathrm e}^{45}+{\mathrm e}^{75}\right ) \textit {\_R}^{3}+\left (6 \,{\mathrm e}^{80}-6 \,{\mathrm e}^{70}-65 \,{\mathrm e}^{50}+30 \,{\mathrm e}^{60}\right ) \textit {\_R}^{2}+\left (-91 \,{\mathrm e}^{55}-12 \,{\mathrm e}^{75}+6 \,{\mathrm e}^{85}+48 \,{\mathrm e}^{65}\right ) \textit {\_R} -6 \,{\mathrm e}^{80}+24 \,{\mathrm e}^{70}-44 \,{\mathrm e}^{60}+2 \,{\mathrm e}^{90}\right ) \ln \left (x -\textit {\_R} \right )}{{\mathrm e}^{20}+4 \textit {\_R} \,{\mathrm e}^{15}+6 \textit {\_R}^{2} {\mathrm e}^{10}+4 \textit {\_R}^{3} {\mathrm e}^{5}+\textit {\_R}^{4}}\right )\) | \(229\) |
| gosper | \(\frac {320 x \,{\mathrm e}^{15}-280 x^{2} {\mathrm e}^{20}-3120 x^{3} {\mathrm e}^{25}+360 x^{7} {\mathrm e}^{5}-120 x^{5} {\mathrm e}^{5}+80 x^{3} {\mathrm e}^{5}+160 \,{\mathrm e}^{20}+10 x^{12}-40 x^{10}+60 x^{8}+10 x^{4}+80 x^{2} {\mathrm e}^{40}-2400 x^{5} {\mathrm e}^{15}-480 x^{4} {\mathrm e}^{10}-4440 \,{\mathrm e}^{20} x^{4}+40 \,{\mathrm e}^{5} x^{11}-640 x^{3} {\mathrm e}^{15}-360 \,{\mathrm e}^{10} x^{8}+320 x^{5} {\mathrm e}^{25}+480 x^{4} {\mathrm e}^{30}+320 x^{3} {\mathrm e}^{35}-780 x^{2} {\mathrm e}^{30}-200 \,{\mathrm e}^{5} x^{9}+60 \,{\mathrm e}^{10} x^{10}+40 \,{\mathrm e}^{15} x^{9}+10 \,{\mathrm e}^{20} x^{8}+240 x^{2} {\mathrm e}^{10}-280 x^{7} {\mathrm e}^{15}}{x^{2} \left ({\mathrm e}^{20}+4 x \,{\mathrm e}^{15}+6 x^{2} {\mathrm e}^{10}+4 x^{3} {\mathrm e}^{5}+x^{4}\right )}\) | \(256\) |
| parallelrisch | \(\frac {320 x \,{\mathrm e}^{15}-280 x^{2} {\mathrm e}^{20}-3120 x^{3} {\mathrm e}^{25}+360 x^{7} {\mathrm e}^{5}-120 x^{5} {\mathrm e}^{5}+80 x^{3} {\mathrm e}^{5}+160 \,{\mathrm e}^{20}+10 x^{12}-40 x^{10}+60 x^{8}+10 x^{4}+80 x^{2} {\mathrm e}^{40}-2400 x^{5} {\mathrm e}^{15}-480 x^{4} {\mathrm e}^{10}-4440 \,{\mathrm e}^{20} x^{4}+40 \,{\mathrm e}^{5} x^{11}-640 x^{3} {\mathrm e}^{15}-360 \,{\mathrm e}^{10} x^{8}+320 x^{5} {\mathrm e}^{25}+480 x^{4} {\mathrm e}^{30}+320 x^{3} {\mathrm e}^{35}-780 x^{2} {\mathrm e}^{30}-200 \,{\mathrm e}^{5} x^{9}+60 \,{\mathrm e}^{10} x^{10}+40 \,{\mathrm e}^{15} x^{9}+10 \,{\mathrm e}^{20} x^{8}+240 x^{2} {\mathrm e}^{10}-280 x^{7} {\mathrm e}^{15}}{x^{2} \left ({\mathrm e}^{20}+4 x \,{\mathrm e}^{15}+6 x^{2} {\mathrm e}^{10}+4 x^{3} {\mathrm e}^{5}+x^{4}\right )}\) | \(260\) |
int(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^5+480* x^3-1280*x)*exp(5)^4+(600*x^10-2440*x^8+2400*x^6+960*x^4-1600*x^2)*exp(5)^ 3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-920*x^10 +720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x^3*exp (5)^5+5*x^4*exp(5)^4+10*x^5*exp(5)^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^8),x,m ethod=_RETURNVERBOSE)
10*x^6-40*x*exp(15)+40*x^2*exp(10)-40*x^3*exp(5)-40*x^4+120*x*exp(5)+60*x^ 2+((-40*exp(25)-120*exp(5))*x^5+(-120*exp(30)+60*exp(20)-480*exp(10)+10)*x ^4+(-120*exp(35)+120*exp(25)-640*exp(15)+80*exp(5))*x^3+(-40*exp(40)+60*ex p(30)-280*exp(20)+240*exp(10))*x^2+320*x*exp(15)+160*exp(20))/x^2/(exp(20) +4*x*exp(15)+6*x^2*exp(10)+4*x^3*exp(5)+x^4)
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 201, normalized size of antiderivative = 8.74 \[ \int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} \left (-320+480 x^4-320 x^6+60 x^8\right )+e^{20} \left (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9\right )+e^{15} \left (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10}\right )+e^{10} \left (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11}\right )+e^5 \left (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12}\right )}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx=\frac {10 \, {\left (x^{12} - 4 \, x^{10} + 6 \, x^{8} + x^{4} - 16 \, x^{3} e^{35} - 4 \, x^{2} e^{40} - 6 \, {\left (4 \, x^{4} - x^{2}\right )} e^{30} - 8 \, {\left (2 \, x^{5} - 3 \, x^{3}\right )} e^{25} + {\left (x^{8} - 12 \, x^{6} + 60 \, x^{4} - 28 \, x^{2} + 16\right )} e^{20} + 4 \, {\left (x^{9} - 7 \, x^{7} + 24 \, x^{5} - 16 \, x^{3} + 8 \, x\right )} e^{15} + 6 \, {\left (x^{10} - 6 \, x^{8} + 14 \, x^{6} - 8 \, x^{4} + 4 \, x^{2}\right )} e^{10} + 4 \, {\left (x^{11} - 5 \, x^{9} + 9 \, x^{7} - 3 \, x^{5} + 2 \, x^{3}\right )} e^{5}\right )}}{x^{6} + 4 \, x^{5} e^{5} + 6 \, x^{4} e^{10} + 4 \, x^{3} e^{15} + x^{2} e^{20}} \]
integrate(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^ 5+480*x^3-1280*x)*exp(5)^4+(600*x^10-2440*x^8+2400*x^6+960*x^4-1600*x^2)*e xp(5)^3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-92 0*x^10+720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x ^3*exp(5)^5+5*x^4*exp(5)^4+10*x^5*exp(5)^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^ 8),x, algorithm=\
10*(x^12 - 4*x^10 + 6*x^8 + x^4 - 16*x^3*e^35 - 4*x^2*e^40 - 6*(4*x^4 - x^ 2)*e^30 - 8*(2*x^5 - 3*x^3)*e^25 + (x^8 - 12*x^6 + 60*x^4 - 28*x^2 + 16)*e ^20 + 4*(x^9 - 7*x^7 + 24*x^5 - 16*x^3 + 8*x)*e^15 + 6*(x^10 - 6*x^8 + 14* x^6 - 8*x^4 + 4*x^2)*e^10 + 4*(x^11 - 5*x^9 + 9*x^7 - 3*x^5 + 2*x^3)*e^5)/ (x^6 + 4*x^5*e^5 + 6*x^4*e^10 + 4*x^3*e^15 + x^2*e^20)
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (19) = 38\).
Time = 2.38 (sec) , antiderivative size = 170, normalized size of antiderivative = 7.39 \[ \int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} \left (-320+480 x^4-320 x^6+60 x^8\right )+e^{20} \left (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9\right )+e^{15} \left (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10}\right )+e^{10} \left (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11}\right )+e^5 \left (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12}\right )}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx=10 x^{6} - 40 x^{4} - 40 x^{3} e^{5} + x^{2} \cdot \left (60 + 40 e^{10}\right ) + x \left (- 40 e^{15} + 120 e^{5}\right ) + \frac {x^{5} \left (- 40 e^{25} - 120 e^{5}\right ) + x^{4} \left (- 120 e^{30} - 480 e^{10} + 10 + 60 e^{20}\right ) + x^{3} \left (- 120 e^{35} - 640 e^{15} + 80 e^{5} + 120 e^{25}\right ) + x^{2} \left (- 40 e^{40} - 280 e^{20} + 240 e^{10} + 60 e^{30}\right ) + 320 x e^{15} + 160 e^{20}}{x^{6} + 4 x^{5} e^{5} + 6 x^{4} e^{10} + 4 x^{3} e^{15} + x^{2} e^{20}} \]
integrate(((60*x**8-320*x**6+480*x**4-320)*exp(5)**5+(300*x**9-1400*x**7+1 680*x**5+480*x**3-1280*x)*exp(5)**4+(600*x**10-2440*x**8+2400*x**6+960*x** 4-1600*x**2)*exp(5)**3+(600*x**11-2120*x**9+1800*x**7+600*x**5-880*x**3)*e xp(5)**2+(300*x**12-920*x**10+720*x**8+120*x**6-220*x**4)*exp(5)+60*x**13- 160*x**11+120*x**9-20*x**5)/(x**3*exp(5)**5+5*x**4*exp(5)**4+10*x**5*exp(5 )**3+10*x**6*exp(5)**2+5*x**7*exp(5)+x**8),x)
10*x**6 - 40*x**4 - 40*x**3*exp(5) + x**2*(60 + 40*exp(10)) + x*(-40*exp(1 5) + 120*exp(5)) + (x**5*(-40*exp(25) - 120*exp(5)) + x**4*(-120*exp(30) - 480*exp(10) + 10 + 60*exp(20)) + x**3*(-120*exp(35) - 640*exp(15) + 80*ex p(5) + 120*exp(25)) + x**2*(-40*exp(40) - 280*exp(20) + 240*exp(10) + 60*e xp(30)) + 320*x*exp(15) + 160*exp(20))/(x**6 + 4*x**5*exp(5) + 6*x**4*exp( 10) + 4*x**3*exp(15) + x**2*exp(20))
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 158, normalized size of antiderivative = 6.87 \[ \int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} \left (-320+480 x^4-320 x^6+60 x^8\right )+e^{20} \left (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9\right )+e^{15} \left (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10}\right )+e^{10} \left (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11}\right )+e^5 \left (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12}\right )}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx=10 \, x^{6} - 40 \, x^{4} - 40 \, x^{3} e^{5} + 20 \, x^{2} {\left (2 \, e^{10} + 3\right )} - 40 \, x {\left (e^{15} - 3 \, e^{5}\right )} - \frac {10 \, {\left (4 \, x^{5} {\left (e^{25} + 3 \, e^{5}\right )} + x^{4} {\left (12 \, e^{30} - 6 \, e^{20} + 48 \, e^{10} - 1\right )} + 4 \, x^{3} {\left (3 \, e^{35} - 3 \, e^{25} + 16 \, e^{15} - 2 \, e^{5}\right )} + 2 \, x^{2} {\left (2 \, e^{40} - 3 \, e^{30} + 14 \, e^{20} - 12 \, e^{10}\right )} - 32 \, x e^{15} - 16 \, e^{20}\right )}}{x^{6} + 4 \, x^{5} e^{5} + 6 \, x^{4} e^{10} + 4 \, x^{3} e^{15} + x^{2} e^{20}} \]
integrate(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^ 5+480*x^3-1280*x)*exp(5)^4+(600*x^10-2440*x^8+2400*x^6+960*x^4-1600*x^2)*e xp(5)^3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-92 0*x^10+720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x ^3*exp(5)^5+5*x^4*exp(5)^4+10*x^5*exp(5)^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^ 8),x, algorithm=\
10*x^6 - 40*x^4 - 40*x^3*e^5 + 20*x^2*(2*e^10 + 3) - 40*x*(e^15 - 3*e^5) - 10*(4*x^5*(e^25 + 3*e^5) + x^4*(12*e^30 - 6*e^20 + 48*e^10 - 1) + 4*x^3*( 3*e^35 - 3*e^25 + 16*e^15 - 2*e^5) + 2*x^2*(2*e^40 - 3*e^30 + 14*e^20 - 12 *e^10) - 32*x*e^15 - 16*e^20)/(x^6 + 4*x^5*e^5 + 6*x^4*e^10 + 4*x^3*e^15 + x^2*e^20)
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (22) = 44\).
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 6.48 \[ \int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} \left (-320+480 x^4-320 x^6+60 x^8\right )+e^{20} \left (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9\right )+e^{15} \left (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10}\right )+e^{10} \left (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11}\right )+e^5 \left (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12}\right )}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx=10 \, x^{6} - 40 \, x^{4} - 40 \, x^{3} e^{5} + 40 \, x^{2} e^{10} + 60 \, x^{2} - 40 \, x e^{15} + 120 \, x e^{5} - \frac {160 \, {\left (2 \, x - e^{5}\right )} e^{\left (-5\right )}}{x^{2}} - \frac {10 \, {\left (4 \, x^{3} e^{30} + 12 \, x^{3} e^{10} - 32 \, x^{3} + 12 \, x^{2} e^{35} - 6 \, x^{2} e^{25} + 48 \, x^{2} e^{15} - 113 \, x^{2} e^{5} + 12 \, x e^{40} - 12 \, x e^{30} + 64 \, x e^{20} - 136 \, x e^{10} + 4 \, e^{45} - 6 \, e^{35} + 28 \, e^{25} - 56 \, e^{15}\right )} e^{\left (-5\right )}}{{\left (x + e^{5}\right )}^{4}} \]
integrate(((60*x^8-320*x^6+480*x^4-320)*exp(5)^5+(300*x^9-1400*x^7+1680*x^ 5+480*x^3-1280*x)*exp(5)^4+(600*x^10-2440*x^8+2400*x^6+960*x^4-1600*x^2)*e xp(5)^3+(600*x^11-2120*x^9+1800*x^7+600*x^5-880*x^3)*exp(5)^2+(300*x^12-92 0*x^10+720*x^8+120*x^6-220*x^4)*exp(5)+60*x^13-160*x^11+120*x^9-20*x^5)/(x ^3*exp(5)^5+5*x^4*exp(5)^4+10*x^5*exp(5)^3+10*x^6*exp(5)^2+5*x^7*exp(5)+x^ 8),x, algorithm=\
10*x^6 - 40*x^4 - 40*x^3*e^5 + 40*x^2*e^10 + 60*x^2 - 40*x*e^15 + 120*x*e^ 5 - 160*(2*x - e^5)*e^(-5)/x^2 - 10*(4*x^3*e^30 + 12*x^3*e^10 - 32*x^3 + 1 2*x^2*e^35 - 6*x^2*e^25 + 48*x^2*e^15 - 113*x^2*e^5 + 12*x*e^40 - 12*x*e^3 0 + 64*x*e^20 - 136*x*e^10 + 4*e^45 - 6*e^35 + 28*e^25 - 56*e^15)*e^(-5)/( x + e^5)^4
Time = 8.11 (sec) , antiderivative size = 268, normalized size of antiderivative = 11.65 \[ \int \frac {-20 x^5+120 x^9-160 x^{11}+60 x^{13}+e^{25} \left (-320+480 x^4-320 x^6+60 x^8\right )+e^{20} \left (-1280 x+480 x^3+1680 x^5-1400 x^7+300 x^9\right )+e^{15} \left (-1600 x^2+960 x^4+2400 x^6-2440 x^8+600 x^{10}\right )+e^{10} \left (-880 x^3+600 x^5+1800 x^7-2120 x^9+600 x^{11}\right )+e^5 \left (-220 x^4+120 x^6+720 x^8-920 x^{10}+300 x^{12}\right )}{e^{25} x^3+5 e^{20} x^4+10 e^{15} x^5+10 e^{10} x^6+5 e^5 x^7+x^8} \, dx=x^3\,\left (\frac {800\,{\mathrm {e}}^5}{3}-200\,{\mathrm {e}}^{15}+\frac {40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )}{3}\right )+\frac {\left (-120\,{\mathrm {e}}^5-40\,{\mathrm {e}}^{25}\right )\,x^5+\left (60\,{\mathrm {e}}^{20}-480\,{\mathrm {e}}^{10}-120\,{\mathrm {e}}^{30}+10\right )\,x^4+\left (80\,{\mathrm {e}}^5-640\,{\mathrm {e}}^{15}+120\,{\mathrm {e}}^{25}-120\,{\mathrm {e}}^{35}\right )\,x^3+\left (240\,{\mathrm {e}}^{10}-280\,{\mathrm {e}}^{20}+60\,{\mathrm {e}}^{30}-40\,{\mathrm {e}}^{40}\right )\,x^2+320\,{\mathrm {e}}^{15}\,x+160\,{\mathrm {e}}^{20}}{x^6+4\,{\mathrm {e}}^5\,x^5+6\,{\mathrm {e}}^{10}\,x^4+4\,{\mathrm {e}}^{15}\,x^3+{\mathrm {e}}^{20}\,x^2}+x\,\left (1600\,{\mathrm {e}}^{15}-60\,{\mathrm {e}}^{25}+5\,{\mathrm {e}}^5\,\left (520\,{\mathrm {e}}^{10}+5\,{\mathrm {e}}^5\,\left (800\,{\mathrm {e}}^5-600\,{\mathrm {e}}^{15}+40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )\right )-120\right )-10\,{\mathrm {e}}^{10}\,\left (800\,{\mathrm {e}}^5-600\,{\mathrm {e}}^{15}+40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )\right )+20\,{\mathrm {e}}^5\,\left (3\,{\mathrm {e}}^{20}-122\,{\mathrm {e}}^{10}+36\right )\right )-40\,x^4+10\,x^6-x^2\,\left (260\,{\mathrm {e}}^{10}+\frac {5\,{\mathrm {e}}^5\,\left (800\,{\mathrm {e}}^5-600\,{\mathrm {e}}^{15}+40\,{\mathrm {e}}^5\,\left (15\,{\mathrm {e}}^{10}-23\right )\right )}{2}-60\right ) \]
int((exp(5)*(120*x^6 - 220*x^4 + 720*x^8 - 920*x^10 + 300*x^12) + exp(10)* (600*x^5 - 880*x^3 + 1800*x^7 - 2120*x^9 + 600*x^11) + exp(15)*(960*x^4 - 1600*x^2 + 2400*x^6 - 2440*x^8 + 600*x^10) + exp(25)*(480*x^4 - 320*x^6 + 60*x^8 - 320) + exp(20)*(480*x^3 - 1280*x + 1680*x^5 - 1400*x^7 + 300*x^9) - 20*x^5 + 120*x^9 - 160*x^11 + 60*x^13)/(5*x^7*exp(5) + 10*x^6*exp(10) + 10*x^5*exp(15) + 5*x^4*exp(20) + x^3*exp(25) + x^8),x)
x^3*((800*exp(5))/3 - 200*exp(15) + (40*exp(5)*(15*exp(10) - 23))/3) + (16 0*exp(20) - x^5*(120*exp(5) + 40*exp(25)) + 320*x*exp(15) + x^2*(240*exp(1 0) - 280*exp(20) + 60*exp(30) - 40*exp(40)) + x^3*(80*exp(5) - 640*exp(15) + 120*exp(25) - 120*exp(35)) - x^4*(480*exp(10) - 60*exp(20) + 120*exp(30 ) - 10))/(4*x^5*exp(5) + 6*x^4*exp(10) + 4*x^3*exp(15) + x^2*exp(20) + x^6 ) + x*(1600*exp(15) - 60*exp(25) + 5*exp(5)*(520*exp(10) + 5*exp(5)*(800*e xp(5) - 600*exp(15) + 40*exp(5)*(15*exp(10) - 23)) - 120) - 10*exp(10)*(80 0*exp(5) - 600*exp(15) + 40*exp(5)*(15*exp(10) - 23)) + 20*exp(5)*(3*exp(2 0) - 122*exp(10) + 36)) - 40*x^4 + 10*x^6 - x^2*(260*exp(10) + (5*exp(5)*( 800*exp(5) - 600*exp(15) + 40*exp(5)*(15*exp(10) - 23)))/2 - 60)