Integrand size = 282, antiderivative size = 23 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=\frac {4}{\left (1+x-\frac {x}{4+e^3-5 e^8 x}\right )^2} \] Output:
4/(x-x/(4-5*x*exp(4)^2+exp(3))+1)^2
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.65 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=\frac {4 \left (4+e^3-5 e^8 x\right )^2}{\left (4+3 x+e^3 (1+x)-5 e^8 x (1+x)\right )^2} \] Input:
Integrate[(384 + 320*E^3 + 88*E^6 + 8*E^9 - 1000*E^24*x^3 + E^8*(-1760*x - 920*E^3*x - 120*E^6*x) + E^16*(2400*x^2 + 600*E^3*x^2))/(-64 - 144*x - 10 8*x^2 - 27*x^3 + E^3*(-48 - 120*x - 99*x^2 - 27*x^3) + E^6*(-12 - 33*x - 3 0*x^2 - 9*x^3) + E^9*(-1 - 3*x - 3*x^2 - x^3) + E^24*(125*x^3 + 375*x^4 + 375*x^5 + 125*x^6) + E^8*(240*x + 600*x^2 + 495*x^3 + 135*x^4 + E^6*(15*x + 45*x^2 + 45*x^3 + 15*x^4) + E^3*(120*x + 330*x^2 + 300*x^3 + 90*x^4)) + E^16*(-300*x^2 - 825*x^3 - 750*x^4 - 225*x^5 + E^3*(-75*x^2 - 225*x^3 - 22 5*x^4 - 75*x^5))),x]
Output:
(4*(4 + E^3 - 5*E^8*x)^2)/(4 + 3*x + E^3*(1 + x) - 5*E^8*x*(1 + x))^2
Leaf count is larger than twice the leaf count of optimal. \(248\) vs. \(2(23)=46\).
Time = 1.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 10.78, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, 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.
| \(\displaystyle \int \frac {-1000 e^{24} x^3+e^{16} \left (600 e^3 x^2+2400 x^2\right )+e^8 \left (-120 e^6 x-920 e^3 x-1760 x\right )+8 e^9+88 e^6+320 e^3+384}{-27 x^3-108 x^2+e^3 \left (-27 x^3-99 x^2-120 x-48\right )+e^6 \left (-9 x^3-30 x^2-33 x-12\right )+e^9 \left (-x^3-3 x^2-3 x-1\right )+e^8 \left (135 x^4+495 x^3+600 x^2+e^6 \left (15 x^4+45 x^3+45 x^2+15 x\right )+e^3 \left (90 x^4+300 x^3+330 x^2+120 x\right )+240 x\right )+e^{24} \left (125 x^6+375 x^5+375 x^4+125 x^3\right )+e^{16} \left (-225 x^5-750 x^4-825 x^3-300 x^2+e^3 \left (-75 x^5-225 x^4-225 x^3-75 x^2\right )\right )-144 x-64} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {40 e^8 \left (-5 e^8 x+5 e^8+2 e^3+9\right )}{\left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )^2}+\frac {8 \left (5 e^8 \left (17+8 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}\right ) x-\left (4+e^3\right ) \left (12+7 e^3+e^6+45 e^8+10 e^{11}+25 e^{16}\right )\right )}{\left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {60 e^8 \left (5+e^3+5 e^8\right ) \left (-10 e^8 x-5 e^8+e^3+3\right )}{\left (9+6 e^3+e^6+50 e^8+10 e^{11}+25 e^{16}\right ) \left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )}-\frac {40 e^8 \left (-15 e^8 \left (5+e^3+5 e^8\right ) x-25 e^{16}+5 e^{11}+10 e^8+2 e^6+15 e^3+27\right )}{\left (20 e^8 \left (4+e^3\right )+\left (3+e^3-5 e^8\right )^2\right ) \left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )}+\frac {4 \left (\left (4+e^3\right ) \left (4+e^3+5 e^8\right )-5 e^8 \left (5+e^3+5 e^8\right ) x\right )}{\left (-5 e^8 x^2+\left (3+e^3-5 e^8\right ) x+e^3+4\right )^2}\) |
Input:
Int[(384 + 320*E^3 + 88*E^6 + 8*E^9 - 1000*E^24*x^3 + E^8*(-1760*x - 920*E ^3*x - 120*E^6*x) + E^16*(2400*x^2 + 600*E^3*x^2))/(-64 - 144*x - 108*x^2 - 27*x^3 + E^3*(-48 - 120*x - 99*x^2 - 27*x^3) + E^6*(-12 - 33*x - 30*x^2 - 9*x^3) + E^9*(-1 - 3*x - 3*x^2 - x^3) + E^24*(125*x^3 + 375*x^4 + 375*x^ 5 + 125*x^6) + E^8*(240*x + 600*x^2 + 495*x^3 + 135*x^4 + E^6*(15*x + 45*x ^2 + 45*x^3 + 15*x^4) + E^3*(120*x + 330*x^2 + 300*x^3 + 90*x^4)) + E^16*( -300*x^2 - 825*x^3 - 750*x^4 - 225*x^5 + E^3*(-75*x^2 - 225*x^3 - 225*x^4 - 75*x^5))),x]
Output:
(4*((4 + E^3)*(4 + E^3 + 5*E^8) - 5*E^8*(5 + E^3 + 5*E^8)*x))/(4 + E^3 + ( 3 + E^3 - 5*E^8)*x - 5*E^8*x^2)^2 + (60*E^8*(5 + E^3 + 5*E^8)*(3 + E^3 - 5 *E^8 - 10*E^8*x))/((9 + 6*E^3 + E^6 + 50*E^8 + 10*E^11 + 25*E^16)*(4 + E^3 + (3 + E^3 - 5*E^8)*x - 5*E^8*x^2)) - (40*E^8*(27 + 15*E^3 + 2*E^6 + 10*E ^8 + 5*E^11 - 25*E^16 - 15*E^8*(5 + E^3 + 5*E^8)*x))/((20*E^8*(4 + E^3) + (3 + E^3 - 5*E^8)^2)*(4 + E^3 + (3 + E^3 - 5*E^8)*x - 5*E^8*x^2))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(23)=46\).
Time = 1.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74
| method | result | size |
| norman | \(\frac {-40 \,{\mathrm e}^{8} \left (4+{\mathrm e}^{3}\right ) x +100 \,{\mathrm e}^{16} x^{2}+64+4 \,{\mathrm e}^{6}+32 \,{\mathrm e}^{3}}{\left (-5 x^{2} {\mathrm e}^{8}-5 x \,{\mathrm e}^{8}+x \,{\mathrm e}^{3}+{\mathrm e}^{3}+3 x +4\right )^{2}}\) | \(63\) |
| risch | \(\frac {\frac {64}{25}+4 \,{\mathrm e}^{16} x^{2}-\frac {8 \,{\mathrm e}^{8} \left (4+{\mathrm e}^{3}\right ) x}{5}+\frac {4 \,{\mathrm e}^{6}}{25}+\frac {32 \,{\mathrm e}^{3}}{25}}{{\mathrm e}^{16} x^{4}+2 \,{\mathrm e}^{16} x^{3}+{\mathrm e}^{16} x^{2}-\frac {2 x^{3} {\mathrm e}^{11}}{5}-\frac {4 x^{2} {\mathrm e}^{11}}{5}-\frac {2 x \,{\mathrm e}^{11}}{5}-\frac {6 \,{\mathrm e}^{8} x^{3}}{5}-\frac {14 x^{2} {\mathrm e}^{8}}{5}-\frac {8 x \,{\mathrm e}^{8}}{5}+\frac {x^{2} {\mathrm e}^{6}}{25}+\frac {2 x \,{\mathrm e}^{6}}{25}+\frac {{\mathrm e}^{6}}{25}+\frac {6 x^{2} {\mathrm e}^{3}}{25}+\frac {14 x \,{\mathrm e}^{3}}{25}+\frac {8 \,{\mathrm e}^{3}}{25}+\frac {9 x^{2}}{25}+\frac {24 x}{25}+\frac {16}{25}}\) | \(129\) |
| gosper | \(\frac {100 \,{\mathrm e}^{16} x^{2}-40 \,{\mathrm e}^{3} {\mathrm e}^{8} x -160 x \,{\mathrm e}^{8}+4 \,{\mathrm e}^{6}+32 \,{\mathrm e}^{3}+64}{25 \,{\mathrm e}^{16} x^{4}+50 \,{\mathrm e}^{16} x^{3}-10 \,{\mathrm e}^{3} {\mathrm e}^{8} x^{3}+25 \,{\mathrm e}^{16} x^{2}-20 x^{2} {\mathrm e}^{3} {\mathrm e}^{8}-30 \,{\mathrm e}^{8} x^{3}+x^{2} {\mathrm e}^{6}-10 \,{\mathrm e}^{3} {\mathrm e}^{8} x -70 x^{2} {\mathrm e}^{8}+2 x \,{\mathrm e}^{6}+6 x^{2} {\mathrm e}^{3}-40 x \,{\mathrm e}^{8}+{\mathrm e}^{6}+14 x \,{\mathrm e}^{3}+9 x^{2}+8 \,{\mathrm e}^{3}+24 x +16}\) | \(168\) |
| parallelrisch | \(\frac {\left (2500 \,{\mathrm e}^{32} x^{2}-1000 \,{\mathrm e}^{3} {\mathrm e}^{24} x -4000 \,{\mathrm e}^{24} x +100 \,{\mathrm e}^{16} {\mathrm e}^{6}+800 \,{\mathrm e}^{3} {\mathrm e}^{16}+1600 \,{\mathrm e}^{16}\right ) {\mathrm e}^{-16}}{625 \,{\mathrm e}^{16} x^{4}+1250 \,{\mathrm e}^{16} x^{3}-250 \,{\mathrm e}^{3} {\mathrm e}^{8} x^{3}+625 \,{\mathrm e}^{16} x^{2}-500 x^{2} {\mathrm e}^{3} {\mathrm e}^{8}-750 \,{\mathrm e}^{8} x^{3}+25 x^{2} {\mathrm e}^{6}-250 \,{\mathrm e}^{3} {\mathrm e}^{8} x -1750 x^{2} {\mathrm e}^{8}+50 x \,{\mathrm e}^{6}+150 x^{2} {\mathrm e}^{3}-1000 x \,{\mathrm e}^{8}+25 \,{\mathrm e}^{6}+350 x \,{\mathrm e}^{3}+225 x^{2}+200 \,{\mathrm e}^{3}+600 x +400}\) | \(187\) |
| default | \(-\frac {8 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (125 \,{\mathrm e}^{24} \textit {\_Z}^{6}-\left (75 \,{\mathrm e}^{19}-375 \,{\mathrm e}^{24}+225 \,{\mathrm e}^{16}\right ) \textit {\_Z}^{5}-\left (-90 \,{\mathrm e}^{11}+225 \,{\mathrm e}^{19}-15 \,{\mathrm e}^{14}-375 \,{\mathrm e}^{24}+750 \,{\mathrm e}^{16}-135 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{4}-\left (27 \,{\mathrm e}^{3}+9 \,{\mathrm e}^{6}+{\mathrm e}^{9}-300 \,{\mathrm e}^{11}+225 \,{\mathrm e}^{19}-45 \,{\mathrm e}^{14}-125 \,{\mathrm e}^{24}+825 \,{\mathrm e}^{16}-495 \,{\mathrm e}^{8}+27\right ) \textit {\_Z}^{3}-\left (99 \,{\mathrm e}^{3}+30 \,{\mathrm e}^{6}+3 \,{\mathrm e}^{9}-330 \,{\mathrm e}^{11}+75 \,{\mathrm e}^{19}-45 \,{\mathrm e}^{14}+300 \,{\mathrm e}^{16}-600 \,{\mathrm e}^{8}+108\right ) \textit {\_Z}^{2}-\left (120 \,{\mathrm e}^{3}+33 \,{\mathrm e}^{6}+3 \,{\mathrm e}^{9}-120 \,{\mathrm e}^{11}-15 \,{\mathrm e}^{14}-240 \,{\mathrm e}^{8}+144\right ) \textit {\_Z} -64-12 \,{\mathrm e}^{6}-{\mathrm e}^{9}-48 \,{\mathrm e}^{3}\right )}{\sum }\frac {\left (48-125 \textit {\_R}^{3} {\mathrm e}^{24}+75 \left ({\mathrm e}^{19}+4 \,{\mathrm e}^{16}\right ) \textit {\_R}^{2}+5 \left (-23 \,{\mathrm e}^{11}-3 \,{\mathrm e}^{14}-44 \,{\mathrm e}^{8}\right ) \textit {\_R} +{\mathrm e}^{9}+11 \,{\mathrm e}^{6}+40 \,{\mathrm e}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{48+72 \textit {\_R} -80 \,{\mathrm e}^{8}+11 \,{\mathrm e}^{6}+{\mathrm e}^{9}+40 \,{\mathrm e}^{3}-40 \,{\mathrm e}^{11}+27 \textit {\_R}^{2}-5 \,{\mathrm e}^{14}-500 \textit {\_R}^{3} {\mathrm e}^{24}-495 \textit {\_R}^{2} {\mathrm e}^{8}+200 \textit {\_R} \,{\mathrm e}^{16}+9 \textit {\_R}^{2} {\mathrm e}^{6}+66 \textit {\_R} \,{\mathrm e}^{3}+27 \textit {\_R}^{2} {\mathrm e}^{3}+2 \textit {\_R} \,{\mathrm e}^{9}+20 \textit {\_R} \,{\mathrm e}^{6}+\textit {\_R}^{2} {\mathrm e}^{9}+125 \textit {\_R}^{4} {\mathrm e}^{19}+300 \textit {\_R}^{3} {\mathrm e}^{19}-20 \textit {\_R}^{3} {\mathrm e}^{14}-45 \textit {\_R}^{2} {\mathrm e}^{14}-120 \textit {\_R}^{3} {\mathrm e}^{11}-250 \,{\mathrm e}^{24} \textit {\_R}^{5}-625 \,{\mathrm e}^{24} \textit {\_R}^{4}-300 \textit {\_R}^{2} {\mathrm e}^{11}-220 \textit {\_R} \,{\mathrm e}^{11}+225 \textit {\_R}^{2} {\mathrm e}^{19}-30 \textit {\_R} \,{\mathrm e}^{14}-180 \,{\mathrm e}^{8} \textit {\_R}^{3}-400 \textit {\_R} \,{\mathrm e}^{8}-125 \textit {\_R}^{2} {\mathrm e}^{24}+50 \textit {\_R} \,{\mathrm e}^{19}+825 \,{\mathrm e}^{16} \textit {\_R}^{2}+1000 \,{\mathrm e}^{16} \textit {\_R}^{3}+375 \,{\mathrm e}^{16} \textit {\_R}^{4}}\right )}{3}\) | \(445\) |
Input:
int((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*exp(3)^ 2-920*x*exp(3)-1760*x)*exp(4)^2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+384)/((1 25*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3-75*x^2) *exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45*x^2+1 5*x)*exp(3)^2+(90*x^4+300*x^3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+600*x^ 2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*x-12)*exp( 3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x,method=_R ETURNVERBOSE)
Output:
(-40*exp(4)^2*(4+exp(3))*x+100*x^2*exp(4)^4+64+4*exp(3)^2+32*exp(3))/(-5*x ^2*exp(4)^2-5*x*exp(4)^2+x*exp(3)+exp(3)+3*x+4)^2
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (22) = 44\).
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.87 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=\frac {4 \, {\left (25 \, x^{2} e^{16} - 10 \, x e^{11} - 40 \, x e^{8} + e^{6} + 8 \, e^{3} + 16\right )}}{9 \, x^{2} + 25 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{16} - 10 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{11} - 10 \, {\left (3 \, x^{3} + 7 \, x^{2} + 4 \, x\right )} e^{8} + {\left (x^{2} + 2 \, x + 1\right )} e^{6} + 2 \, {\left (3 \, x^{2} + 7 \, x + 4\right )} e^{3} + 24 \, x + 16} \] Input:
integrate((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*e xp(3)^2-920*x*exp(3)-1760*x)*exp(4)^2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+38 4)/((125*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3-7 5*x^2)*exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45 *x^2+15*x)*exp(3)^2+(90*x^4+300*x^3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+ 600*x^2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*x-12 )*exp(3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x, al gorithm="fricas")
Output:
4*(25*x^2*e^16 - 10*x*e^11 - 40*x*e^8 + e^6 + 8*e^3 + 16)/(9*x^2 + 25*(x^4 + 2*x^3 + x^2)*e^16 - 10*(x^3 + 2*x^2 + x)*e^11 - 10*(3*x^3 + 7*x^2 + 4*x )*e^8 + (x^2 + 2*x + 1)*e^6 + 2*(3*x^2 + 7*x + 4)*e^3 + 24*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (19) = 38\).
Time = 6.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 5.30 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=- \frac {- 100 x^{2} e^{16} + x \left (160 e^{8} + 40 e^{11}\right ) - 4 e^{6} - 32 e^{3} - 64}{25 x^{4} e^{16} + x^{3} \left (- 10 e^{11} - 30 e^{8} + 50 e^{16}\right ) + x^{2} \left (- 20 e^{11} - 70 e^{8} + 9 + 6 e^{3} + e^{6} + 25 e^{16}\right ) + x \left (- 10 e^{11} - 40 e^{8} + 24 + 14 e^{3} + 2 e^{6}\right ) + 16 + 8 e^{3} + e^{6}} \] Input:
integrate((-1000*x**3*exp(4)**6+(600*x**2*exp(3)+2400*x**2)*exp(4)**4+(-12 0*x*exp(3)**2-920*x*exp(3)-1760*x)*exp(4)**2+8*exp(3)**3+88*exp(3)**2+320* exp(3)+384)/((125*x**6+375*x**5+375*x**4+125*x**3)*exp(4)**6+((-75*x**5-22 5*x**4-225*x**3-75*x**2)*exp(3)-225*x**5-750*x**4-825*x**3-300*x**2)*exp(4 )**4+((15*x**4+45*x**3+45*x**2+15*x)*exp(3)**2+(90*x**4+300*x**3+330*x**2+ 120*x)*exp(3)+135*x**4+495*x**3+600*x**2+240*x)*exp(4)**2+(-x**3-3*x**2-3* x-1)*exp(3)**3+(-9*x**3-30*x**2-33*x-12)*exp(3)**2+(-27*x**3-99*x**2-120*x -48)*exp(3)-27*x**3-108*x**2-144*x-64),x)
Output:
-(-100*x**2*exp(16) + x*(160*exp(8) + 40*exp(11)) - 4*exp(6) - 32*exp(3) - 64)/(25*x**4*exp(16) + x**3*(-10*exp(11) - 30*exp(8) + 50*exp(16)) + x**2 *(-20*exp(11) - 70*exp(8) + 9 + 6*exp(3) + exp(6) + 25*exp(16)) + x*(-10*e xp(11) - 40*exp(8) + 24 + 14*exp(3) + 2*exp(6)) + 16 + 8*exp(3) + exp(6))
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (22) = 44\).
Time = 0.05 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.65 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=\frac {4 \, {\left (25 \, x^{2} e^{16} - 10 \, x {\left (e^{11} + 4 \, e^{8}\right )} + e^{6} + 8 \, e^{3} + 16\right )}}{25 \, x^{4} e^{16} + 10 \, x^{3} {\left (5 \, e^{16} - e^{11} - 3 \, e^{8}\right )} + x^{2} {\left (25 \, e^{16} - 20 \, e^{11} - 70 \, e^{8} + e^{6} + 6 \, e^{3} + 9\right )} - 2 \, x {\left (5 \, e^{11} + 20 \, e^{8} - e^{6} - 7 \, e^{3} - 12\right )} + e^{6} + 8 \, e^{3} + 16} \] Input:
integrate((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*e xp(3)^2-920*x*exp(3)-1760*x)*exp(4)^2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+38 4)/((125*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3-7 5*x^2)*exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45 *x^2+15*x)*exp(3)^2+(90*x^4+300*x^3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+ 600*x^2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*x-12 )*exp(3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x, al gorithm="maxima")
Output:
4*(25*x^2*e^16 - 10*x*(e^11 + 4*e^8) + e^6 + 8*e^3 + 16)/(25*x^4*e^16 + 10 *x^3*(5*e^16 - e^11 - 3*e^8) + x^2*(25*e^16 - 20*e^11 - 70*e^8 + e^6 + 6*e ^3 + 9) - 2*x*(5*e^11 + 20*e^8 - e^6 - 7*e^3 - 12) + e^6 + 8*e^3 + 16)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (22) = 44\).
Time = 0.15 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.39 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=\frac {4 \, {\left (25 \, x^{2} e^{16} - 10 \, x e^{11} - 40 \, x e^{8} + e^{6} + 8 \, e^{3} + 16\right )}}{{\left (5 \, x^{2} e^{8} + 5 \, x e^{8} - x e^{3} - 3 \, x - e^{3} - 4\right )}^{2}} \] Input:
integrate((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*e xp(3)^2-920*x*exp(3)-1760*x)*exp(4)^2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+38 4)/((125*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3-7 5*x^2)*exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45 *x^2+15*x)*exp(3)^2+(90*x^4+300*x^3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+ 600*x^2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*x-12 )*exp(3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x, al gorithm="giac")
Output:
4*(25*x^2*e^16 - 10*x*e^11 - 40*x*e^8 + e^6 + 8*e^3 + 16)/(5*x^2*e^8 + 5*x *e^8 - x*e^3 - 3*x - e^3 - 4)^2
Time = 0.30 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=\frac {4\,{\left ({\mathrm {e}}^3-5\,x\,{\mathrm {e}}^8+4\right )}^2}{25\,{\mathrm {e}}^{16}\,x^4+\left (50\,{\mathrm {e}}^{16}-10\,{\mathrm {e}}^{11}-30\,{\mathrm {e}}^8\right )\,x^3+\left (6\,{\mathrm {e}}^3+{\mathrm {e}}^6-70\,{\mathrm {e}}^8-20\,{\mathrm {e}}^{11}+25\,{\mathrm {e}}^{16}+9\right )\,x^2+\left (14\,{\mathrm {e}}^3+2\,{\mathrm {e}}^6-40\,{\mathrm {e}}^8-10\,{\mathrm {e}}^{11}+24\right )\,x+8\,{\mathrm {e}}^3+{\mathrm {e}}^6+16} \] Input:
int(-(320*exp(3) + 88*exp(6) + 8*exp(9) - exp(8)*(1760*x + 920*x*exp(3) + 120*x*exp(6)) - 1000*x^3*exp(24) + exp(16)*(600*x^2*exp(3) + 2400*x^2) + 3 84)/(144*x + exp(9)*(3*x + 3*x^2 + x^3 + 1) + exp(6)*(33*x + 30*x^2 + 9*x^ 3 + 12) + exp(3)*(120*x + 99*x^2 + 27*x^3 + 48) + 108*x^2 + 27*x^3 - exp(8 )*(240*x + exp(6)*(15*x + 45*x^2 + 45*x^3 + 15*x^4) + exp(3)*(120*x + 330* x^2 + 300*x^3 + 90*x^4) + 600*x^2 + 495*x^3 + 135*x^4) + exp(16)*(300*x^2 + 825*x^3 + 750*x^4 + 225*x^5 + exp(3)*(75*x^2 + 225*x^3 + 225*x^4 + 75*x^ 5)) - exp(24)*(125*x^3 + 375*x^4 + 375*x^5 + 125*x^6) + 64),x)
Output:
(4*(exp(3) - 5*x*exp(8) + 4)^2)/(8*exp(3) + exp(6) + x^2*(6*exp(3) + exp(6 ) - 70*exp(8) - 20*exp(11) + 25*exp(16) + 9) + x*(14*exp(3) + 2*exp(6) - 4 0*exp(8) - 10*exp(11) + 24) + 25*x^4*exp(16) - x^3*(30*exp(8) + 10*exp(11) - 50*exp(16)) + 16)
Time = 0.20 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.39 \[ \int \frac {384+320 e^3+88 e^6+8 e^9-1000 e^{24} x^3+e^8 \left (-1760 x-920 e^3 x-120 e^6 x\right )+e^{16} \left (2400 x^2+600 e^3 x^2\right )}{-64-144 x-108 x^2-27 x^3+e^3 \left (-48-120 x-99 x^2-27 x^3\right )+e^6 \left (-12-33 x-30 x^2-9 x^3\right )+e^9 \left (-1-3 x-3 x^2-x^3\right )+e^{24} \left (125 x^3+375 x^4+375 x^5+125 x^6\right )+e^8 \left (240 x+600 x^2+495 x^3+135 x^4+e^6 \left (15 x+45 x^2+45 x^3+15 x^4\right )+e^3 \left (120 x+330 x^2+300 x^3+90 x^4\right )\right )+e^{16} \left (-300 x^2-825 x^3-750 x^4-225 x^5+e^3 \left (-75 x^2-225 x^3-225 x^4-75 x^5\right )\right )} \, dx=\frac {100 e^{16} x^{2}-40 e^{11} x -160 e^{8} x +4 e^{6}+32 e^{3}+64}{25 e^{16} x^{4}+50 e^{16} x^{3}+25 e^{16} x^{2}-10 e^{11} x^{3}-20 e^{11} x^{2}-10 e^{11} x -30 e^{8} x^{3}-70 e^{8} x^{2}-40 e^{8} x +e^{6} x^{2}+2 e^{6} x +e^{6}+6 e^{3} x^{2}+14 e^{3} x +8 e^{3}+9 x^{2}+24 x +16} \] Input:
int((-1000*x^3*exp(4)^6+(600*x^2*exp(3)+2400*x^2)*exp(4)^4+(-120*x*exp(3)^ 2-920*x*exp(3)-1760*x)*exp(4)^2+8*exp(3)^3+88*exp(3)^2+320*exp(3)+384)/((1 25*x^6+375*x^5+375*x^4+125*x^3)*exp(4)^6+((-75*x^5-225*x^4-225*x^3-75*x^2) *exp(3)-225*x^5-750*x^4-825*x^3-300*x^2)*exp(4)^4+((15*x^4+45*x^3+45*x^2+1 5*x)*exp(3)^2+(90*x^4+300*x^3+330*x^2+120*x)*exp(3)+135*x^4+495*x^3+600*x^ 2+240*x)*exp(4)^2+(-x^3-3*x^2-3*x-1)*exp(3)^3+(-9*x^3-30*x^2-33*x-12)*exp( 3)^2+(-27*x^3-99*x^2-120*x-48)*exp(3)-27*x^3-108*x^2-144*x-64),x)
Output:
(4*(25*e**16*x**2 - 10*e**11*x - 40*e**8*x + e**6 + 8*e**3 + 16))/(25*e**1 6*x**4 + 50*e**16*x**3 + 25*e**16*x**2 - 10*e**11*x**3 - 20*e**11*x**2 - 1 0*e**11*x - 30*e**8*x**3 - 70*e**8*x**2 - 40*e**8*x + e**6*x**2 + 2*e**6*x + e**6 + 6*e**3*x**2 + 14*e**3*x + 8*e**3 + 9*x**2 + 24*x + 16)