Integrand size = 127, antiderivative size = 32 \[ \int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 \left (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15}\right )}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx=5 \left (-e^4+\frac {4}{\frac {3}{4}+x^2+\frac {x^2}{\left (3+x^2\right )^2}}\right )^2 \]
Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 \left (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15}\right )}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx=\frac {160 \left (8 \left (3+x^2\right )^4-e^4 \left (3+x^2\right )^2 \left (27+58 x^2+27 x^4+4 x^6\right )\right )}{\left (27+58 x^2+27 x^4+4 x^6\right )^2} \]
Integrate[(-16588800*x - 30965760*x^3 - 24883200*x^5 - 10936320*x^7 - 2744 320*x^9 - 368640*x^11 - 20480*x^13 + E^4*(3110400*x + 10414080*x^3 + 12960 000*x^5 + 8542720*x^7 + 3310080*x^9 + 760320*x^11 + 96000*x^13 + 5120*x^15 ))/(19683 + 126846*x^2 + 331533*x^4 + 457552*x^6 + 369117*x^8 + 184710*x^1 0 + 58563*x^12 + 11532*x^14 + 1296*x^16 + 64*x^18),x]
(160*(8*(3 + x^2)^4 - E^4*(3 + x^2)^2*(27 + 58*x^2 + 27*x^4 + 4*x^6)))/(27 + 58*x^2 + 27*x^4 + 4*x^6)^2
Leaf count is larger than twice the leaf count of optimal. \(198\) vs. \(2(32)=64\).
Time = 11.47 (sec) , antiderivative size = 198, normalized size of antiderivative = 6.19, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.126, Rules used = {2460, 7239, 27, 25, 2527, 27, 2527, 27, 2527, 27, 2527, 27, 2029, 2527, 27, 2021}
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.
| \(\displaystyle \int \frac {-20480 x^{13}-368640 x^{11}-2744320 x^9-10936320 x^7-24883200 x^5-30965760 x^3+e^4 \left (5120 x^{15}+96000 x^{13}+760320 x^{11}+3310080 x^9+8542720 x^7+12960000 x^5+10414080 x^3+3110400 x\right )-16588800 x}{64 x^{18}+1296 x^{16}+11532 x^{14}+58563 x^{12}+184710 x^{10}+369117 x^8+457552 x^6+331533 x^4+126846 x^2+19683} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {80 x \left (4 e^4 x^2+21 e^4-16\right )}{4 x^6+27 x^4+58 x^2+27}+\frac {80 x \left (-\left (\left (288-49 e^4\right ) x^4\right )-6 \left (310-67 e^4\right ) x^2-3 \left (1351-291 e^4\right )\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {80 x \left (8801 x^4+51666 x^2+86265\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1280 x \left (x^8+12 x^6+53 x^4+108 x^2+90\right ) \left (e^4 \left (4 x^6+27 x^4+58 x^2+27\right )-16 \left (x^2+3\right )^2\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 1280 \int -\frac {x \left (x^8+12 x^6+53 x^4+108 x^2+90\right ) \left (16 \left (x^2+3\right )^2-e^4 \left (4 x^6+27 x^4+58 x^2+27\right )\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -1280 \int \frac {x \left (x^8+12 x^6+53 x^4+108 x^2+90\right ) \left (16 \left (x^2+3\right )^2-e^4 \left (4 x^6+27 x^4+58 x^2+27\right )\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx\) |
| \(\Big \downarrow \) 2527 |
| \(\displaystyle -1280 \left (\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {1}{8} \int -\frac {8 \left (2 \left (8-51 e^4\right ) x^{13}+96 \left (3-8 e^4\right ) x^{11}+\left (2144-2721 e^4\right ) x^9+2 \left (4272-3337 e^4\right ) x^7+405 \left (48-25 e^4\right ) x^5+72 \left (336-113 e^4\right ) x^3+810 \left (16-3 e^4\right ) x\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -1280 \left (\int \frac {2 \left (8-51 e^4\right ) x^{13}+96 \left (3-8 e^4\right ) x^{11}+\left (2144-2721 e^4\right ) x^9+2 \left (4272-3337 e^4\right ) x^7+405 \left (48-25 e^4\right ) x^5+72 \left (336-113 e^4\right ) x^3+810 \left (16-3 e^4\right ) x}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 2527 |
| \(\displaystyle -1280 \left (-\frac {1}{16} \int -\frac {16 \left (96 \left (3-8 e^4\right ) x^{11}+24 \left (99-175 e^4\right ) x^9+\left (8760-8051 e^4\right ) x^7+405 \left (48-25 e^4\right ) x^5+72 \left (336-113 e^4\right ) x^3+810 \left (16-3 e^4\right ) x\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -1280 \left (\int \frac {96 \left (3-8 e^4\right ) x^{11}+24 \left (99-175 e^4\right ) x^9+\left (8760-8051 e^4\right ) x^7+405 \left (48-25 e^4\right ) x^5+72 \left (336-113 e^4\right ) x^3+810 \left (16-3 e^4\right ) x}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 2527 |
| \(\displaystyle -1280 \left (-\frac {1}{24} \int -\frac {72 \left (8 \left (72-103 e^4\right ) x^9+3 \left (1128-1307 e^4\right ) x^7+81 \left (88-63 e^4\right ) x^5+24 \left (336-113 e^4\right ) x^3+270 \left (16-3 e^4\right ) x\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -1280 \left (3 \int \frac {8 \left (72-103 e^4\right ) x^9+3 \left (1128-1307 e^4\right ) x^7+81 \left (88-63 e^4\right ) x^5+24 \left (336-113 e^4\right ) x^3+270 \left (16-3 e^4\right ) x}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 2527 |
| \(\displaystyle -1280 \left (3 \left (-\frac {1}{32} \int -\frac {96 \left (20 \left (24-19 e^4\right ) x^7+27 \left (88-63 e^4\right ) x^5+\left (3336-1831 e^4\right ) x^3+90 \left (16-3 e^4\right ) x\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx-\frac {\left (72-103 e^4\right ) x^4}{4 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -1280 \left (3 \left (3 \int \frac {20 \left (24-19 e^4\right ) x^7+27 \left (88-63 e^4\right ) x^5+\left (3336-1831 e^4\right ) x^3+90 \left (16-3 e^4\right ) x}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx-\frac {\left (72-103 e^4\right ) x^4}{4 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 2029 |
| \(\displaystyle -1280 \left (3 \left (3 \int \frac {x \left (20 \left (24-19 e^4\right ) x^6+27 \left (88-63 e^4\right ) x^4+\left (3336-1831 e^4\right ) x^2+90 \left (16-3 e^4\right )\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx-\frac {\left (72-103 e^4\right ) x^4}{4 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 2527 |
| \(\displaystyle -1280 \left (3 \left (3 \left (-\frac {1}{40} \int -\frac {360 \left (6 \left (8-3 e^4\right ) x^5+27 \left (8-3 e^4\right ) x^3+29 \left (8-3 e^4\right ) x\right )}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx-\frac {\left (24-19 e^4\right ) x^2}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {\left (72-103 e^4\right ) x^4}{4 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -1280 \left (3 \left (3 \left (9 \int \frac {6 \left (8-3 e^4\right ) x^5+27 \left (8-3 e^4\right ) x^3+29 \left (8-3 e^4\right ) x}{\left (4 x^6+27 x^4+58 x^2+27\right )^3}dx-\frac {\left (24-19 e^4\right ) x^2}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {\left (72-103 e^4\right ) x^4}{4 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
| \(\Big \downarrow \) 2021 |
| \(\displaystyle -1280 \left (-\frac {4 \left (3-8 e^4\right ) x^6}{\left (4 x^6+27 x^4+58 x^2+27\right )^2}+3 \left (3 \left (-\frac {\left (24-19 e^4\right ) x^2}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {9 \left (8-3 e^4\right )}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )-\frac {\left (72-103 e^4\right ) x^4}{4 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )+\frac {e^4 x^{10}}{2 \left (4 x^6+27 x^4+58 x^2+27\right )^2}-\frac {\left (8-51 e^4\right ) x^8}{8 \left (4 x^6+27 x^4+58 x^2+27\right )^2}\right )\) |
Int[(-16588800*x - 30965760*x^3 - 24883200*x^5 - 10936320*x^7 - 2744320*x^ 9 - 368640*x^11 - 20480*x^13 + E^4*(3110400*x + 10414080*x^3 + 12960000*x^ 5 + 8542720*x^7 + 3310080*x^9 + 760320*x^11 + 96000*x^13 + 5120*x^15))/(19 683 + 126846*x^2 + 331533*x^4 + 457552*x^6 + 369117*x^8 + 184710*x^10 + 58 563*x^12 + 11532*x^14 + 1296*x^16 + 64*x^18),x]
-1280*((-4*(3 - 8*E^4)*x^6)/(27 + 58*x^2 + 27*x^4 + 4*x^6)^2 - ((8 - 51*E^ 4)*x^8)/(8*(27 + 58*x^2 + 27*x^4 + 4*x^6)^2) + (E^4*x^10)/(2*(27 + 58*x^2 + 27*x^4 + 4*x^6)^2) + 3*(-1/4*((72 - 103*E^4)*x^4)/(27 + 58*x^2 + 27*x^4 + 4*x^6)^2 + 3*((-9*(8 - 3*E^4))/(8*(27 + 58*x^2 + 27*x^4 + 4*x^6)^2) - (( 24 - 19*E^4)*x^2)/(2*(27 + 58*x^2 + 27*x^4 + 4*x^6)^2))))
3.21.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x ]}, Simp[Coeff[Pp, x, p]*x^(p - q + 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp , Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; Free Q[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
Int[(Fx_.)*((d_.)*(x_)^(q_.) + (a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)* (x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r) + d*x^(q - r))^p*Fx, x] /; FreeQ[{a, b, c, d, r, s, t, q}, x] && IntegerQ[p ] && PosQ[s - r] && PosQ[t - r] && PosQ[q - r] && !(EqQ[p, 1] && EqQ[u, 1] )
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[Q x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
Int[(Pm_)*(Qn_)^(p_.), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x ]}, Simp[Coeff[Pm, x, m]*x^(m - n + 1)*(Qn^(p + 1)/((m + n*p + 1)*Coeff[Qn, x, n])), x] + Simp[1/((m + n*p + 1)*Coeff[Qn, x, n]) Int[ExpandToSum[(m + n*p + 1)*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*x^(m - n)*((m - n + 1)*Qn + (p + 1)*x*D[Qn, x]), x]*Qn^p, x], x] /; LtQ[1, n, m + 1] && m + n*p + 1 < 0] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && LtQ[p, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(29)=58\).
Time = 0.12 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31
| method | result | size |
| norman | \(\frac {-640 \,{\mathrm e}^{4} x^{10}+\left (1280-8160 \,{\mathrm e}^{4}\right ) x^{8}+\left (15360-40960 \,{\mathrm e}^{4}\right ) x^{6}+\left (69120-98880 \,{\mathrm e}^{4}\right ) x^{4}+\left (138240-109440 \,{\mathrm e}^{4}\right ) x^{2}+103680-38880 \,{\mathrm e}^{4}}{\left (4 x^{6}+27 x^{4}+58 x^{2}+27\right )^{2}}\) | \(74\) |
| default | \(\frac {-640 \,{\mathrm e}^{4} x^{10}+10240 \left (-\frac {51 \,{\mathrm e}^{4}}{64}+\frac {1}{8}\right ) x^{8}+10240 \left (-4 \,{\mathrm e}^{4}+\frac {3}{2}\right ) x^{6}+10240 \left (-\frac {309 \,{\mathrm e}^{4}}{32}+\frac {27}{4}\right ) x^{4}+10240 \left (-\frac {171 \,{\mathrm e}^{4}}{16}+\frac {27}{2}\right ) x^{2}-38880 \,{\mathrm e}^{4}+103680}{\left (4 x^{6}+27 x^{4}+58 x^{2}+27\right )^{2}}\) | \(75\) |
| risch | \(\frac {6480-40 \,{\mathrm e}^{4} x^{10}+\left (80-510 \,{\mathrm e}^{4}\right ) x^{8}+\left (-2560 \,{\mathrm e}^{4}+960\right ) x^{6}+\left (-6180 \,{\mathrm e}^{4}+4320\right ) x^{4}+\left (8640-6840 \,{\mathrm e}^{4}\right ) x^{2}-2430 \,{\mathrm e}^{4}}{x^{12}+\frac {27}{2} x^{10}+\frac {1193}{16} x^{8}+\frac {837}{4} x^{6}+\frac {2411}{8} x^{4}+\frac {783}{4} x^{2}+\frac {729}{16}}\) | \(87\) |
| gosper | \(-\frac {160 \left (4 x^{8} {\mathrm e}^{4}+39 x^{6} {\mathrm e}^{4}-8 x^{6}+139 x^{4} {\mathrm e}^{4}-72 x^{4}+201 x^{2} {\mathrm e}^{4}-216 x^{2}+81 \,{\mathrm e}^{4}-216\right ) \left (x^{2}+3\right )}{16 x^{12}+216 x^{10}+1193 x^{8}+3348 x^{6}+4822 x^{4}+3132 x^{2}+729}\) | \(91\) |
| parallelrisch | \(-\frac {-1658880+10240 \,{\mathrm e}^{4} x^{10}+130560 x^{8} {\mathrm e}^{4}-20480 x^{8}+655360 x^{6} {\mathrm e}^{4}-245760 x^{6}+1582080 x^{4} {\mathrm e}^{4}-1105920 x^{4}+1751040 x^{2} {\mathrm e}^{4}-2211840 x^{2}+622080 \,{\mathrm e}^{4}}{16 \left (16 x^{12}+216 x^{10}+1193 x^{8}+3348 x^{6}+4822 x^{4}+3132 x^{2}+729\right )}\) | \(98\) |
int(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960000*x^ 5+10414080*x^3+3110400*x)*exp(4)-20480*x^13-368640*x^11-2744320*x^9-109363 20*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+11532*x^14 +58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+19683) ,x,method=_RETURNVERBOSE)
(-640*exp(4)*x^10+(1280-8160*exp(4))*x^8+(15360-40960*exp(4))*x^6+(69120-9 8880*exp(4))*x^4+(138240-109440*exp(4))*x^2+103680-38880*exp(4))/(4*x^6+27 *x^4+58*x^2+27)^2
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (32) = 64\).
Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 \left (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15}\right )}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx=\frac {160 \, {\left (8 \, x^{8} + 96 \, x^{6} + 432 \, x^{4} + 864 \, x^{2} - {\left (4 \, x^{10} + 51 \, x^{8} + 256 \, x^{6} + 618 \, x^{4} + 684 \, x^{2} + 243\right )} e^{4} + 648\right )}}{16 \, x^{12} + 216 \, x^{10} + 1193 \, x^{8} + 3348 \, x^{6} + 4822 \, x^{4} + 3132 \, x^{2} + 729} \]
integrate(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960 000*x^5+10414080*x^3+3110400*x)*exp(4)-20480*x^13-368640*x^11-2744320*x^9- 10936320*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+1153 2*x^14+58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+ 19683),x, algorithm=\
160*(8*x^8 + 96*x^6 + 432*x^4 + 864*x^2 - (4*x^10 + 51*x^8 + 256*x^6 + 618 *x^4 + 684*x^2 + 243)*e^4 + 648)/(16*x^12 + 216*x^10 + 1193*x^8 + 3348*x^6 + 4822*x^4 + 3132*x^2 + 729)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).
Time = 5.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.72 \[ \int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 \left (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15}\right )}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx=\frac {- 640 x^{10} e^{4} + x^{8} \cdot \left (1280 - 8160 e^{4}\right ) + x^{6} \cdot \left (15360 - 40960 e^{4}\right ) + x^{4} \cdot \left (69120 - 98880 e^{4}\right ) + x^{2} \cdot \left (138240 - 109440 e^{4}\right ) - 38880 e^{4} + 103680}{16 x^{12} + 216 x^{10} + 1193 x^{8} + 3348 x^{6} + 4822 x^{4} + 3132 x^{2} + 729} \]
integrate(((5120*x**15+96000*x**13+760320*x**11+3310080*x**9+8542720*x**7+ 12960000*x**5+10414080*x**3+3110400*x)*exp(4)-20480*x**13-368640*x**11-274 4320*x**9-10936320*x**7-24883200*x**5-30965760*x**3-16588800*x)/(64*x**18+ 1296*x**16+11532*x**14+58563*x**12+184710*x**10+369117*x**8+457552*x**6+33 1533*x**4+126846*x**2+19683),x)
(-640*x**10*exp(4) + x**8*(1280 - 8160*exp(4)) + x**6*(15360 - 40960*exp(4 )) + x**4*(69120 - 98880*exp(4)) + x**2*(138240 - 109440*exp(4)) - 38880*e xp(4) + 103680)/(16*x**12 + 216*x**10 + 1193*x**8 + 3348*x**6 + 4822*x**4 + 3132*x**2 + 729)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (32) = 64\).
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.88 \[ \int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 \left (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15}\right )}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx=-\frac {160 \, {\left (4 \, x^{10} e^{4} + x^{8} {\left (51 \, e^{4} - 8\right )} + 32 \, x^{6} {\left (8 \, e^{4} - 3\right )} + 6 \, x^{4} {\left (103 \, e^{4} - 72\right )} + 36 \, x^{2} {\left (19 \, e^{4} - 24\right )} + 243 \, e^{4} - 648\right )}}{16 \, x^{12} + 216 \, x^{10} + 1193 \, x^{8} + 3348 \, x^{6} + 4822 \, x^{4} + 3132 \, x^{2} + 729} \]
integrate(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960 000*x^5+10414080*x^3+3110400*x)*exp(4)-20480*x^13-368640*x^11-2744320*x^9- 10936320*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+1153 2*x^14+58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+ 19683),x, algorithm=\
-160*(4*x^10*e^4 + x^8*(51*e^4 - 8) + 32*x^6*(8*e^4 - 3) + 6*x^4*(103*e^4 - 72) + 36*x^2*(19*e^4 - 24) + 243*e^4 - 648)/(16*x^12 + 216*x^10 + 1193*x ^8 + 3348*x^6 + 4822*x^4 + 3132*x^2 + 729)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (32) = 64\).
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 \left (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15}\right )}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx=-\frac {160 \, {\left (4 \, x^{10} e^{4} + 51 \, x^{8} e^{4} - 8 \, x^{8} + 256 \, x^{6} e^{4} - 96 \, x^{6} + 618 \, x^{4} e^{4} - 432 \, x^{4} + 684 \, x^{2} e^{4} - 864 \, x^{2} + 243 \, e^{4} - 648\right )}}{{\left (4 \, x^{6} + 27 \, x^{4} + 58 \, x^{2} + 27\right )}^{2}} \]
integrate(((5120*x^15+96000*x^13+760320*x^11+3310080*x^9+8542720*x^7+12960 000*x^5+10414080*x^3+3110400*x)*exp(4)-20480*x^13-368640*x^11-2744320*x^9- 10936320*x^7-24883200*x^5-30965760*x^3-16588800*x)/(64*x^18+1296*x^16+1153 2*x^14+58563*x^12+184710*x^10+369117*x^8+457552*x^6+331533*x^4+126846*x^2+ 19683),x, algorithm=\
-160*(4*x^10*e^4 + 51*x^8*e^4 - 8*x^8 + 256*x^6*e^4 - 96*x^6 + 618*x^4*e^4 - 432*x^4 + 684*x^2*e^4 - 864*x^2 + 243*e^4 - 648)/(4*x^6 + 27*x^4 + 58*x ^2 + 27)^2
Time = 8.52 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {-16588800 x-30965760 x^3-24883200 x^5-10936320 x^7-2744320 x^9-368640 x^{11}-20480 x^{13}+e^4 \left (3110400 x+10414080 x^3+12960000 x^5+8542720 x^7+3310080 x^9+760320 x^{11}+96000 x^{13}+5120 x^{15}\right )}{19683+126846 x^2+331533 x^4+457552 x^6+369117 x^8+184710 x^{10}+58563 x^{12}+11532 x^{14}+1296 x^{16}+64 x^{18}} \, dx=-\frac {160\,{\left (x^2+3\right )}^2\,\left (27\,{\mathrm {e}}^4+58\,x^2\,{\mathrm {e}}^4+27\,x^4\,{\mathrm {e}}^4+4\,x^6\,{\mathrm {e}}^4-48\,x^2-8\,x^4-72\right )}{{\left (4\,x^6+27\,x^4+58\,x^2+27\right )}^2} \]
int(-(16588800*x - exp(4)*(3110400*x + 10414080*x^3 + 12960000*x^5 + 85427 20*x^7 + 3310080*x^9 + 760320*x^11 + 96000*x^13 + 5120*x^15) + 30965760*x^ 3 + 24883200*x^5 + 10936320*x^7 + 2744320*x^9 + 368640*x^11 + 20480*x^13)/ (126846*x^2 + 331533*x^4 + 457552*x^6 + 369117*x^8 + 184710*x^10 + 58563*x ^12 + 11532*x^14 + 1296*x^16 + 64*x^18 + 19683),x)