Integrand size = 83, antiderivative size = 20 \[ \int \frac {1280 e^6 x^4+1024 x^5+2048 e^3 x^5+768 x^6}{\left (e^{12}+4 e^9 x+x^2+2 x^3+x^4+e^6 \left (2 x+6 x^2\right )+e^3 \left (4 x^2+4 x^3\right )\right ) \log ^4(2)} \, dx=\frac {256 x^5}{\left (x+\left (e^3+x\right )^2\right ) \log ^4(2)} \]
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(20)=40\).
Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 4.25 \[ \int \frac {1280 e^6 x^4+1024 x^5+2048 e^3 x^5+768 x^6}{\left (e^{12}+4 e^9 x+x^2+2 x^3+x^4+e^6 \left (2 x+6 x^2\right )+e^3 \left (4 x^2+4 x^3\right )\right ) \log ^4(2)} \, dx=\frac {256 \left (4 e^{15}+2 e^3 x (4+3 x)+2 e^{12} (5+4 x)+e^9 \left (6+24 x+4 x^2\right )+e^6 \left (1+22 x+10 x^2\right )+x \left (1+x+x^4\right )\right )}{\left (e^6+x+2 e^3 x+x^2\right ) \log ^4(2)} \]
Integrate[(1280*E^6*x^4 + 1024*x^5 + 2048*E^3*x^5 + 768*x^6)/((E^12 + 4*E^ 9*x + x^2 + 2*x^3 + x^4 + E^6*(2*x + 6*x^2) + E^3*(4*x^2 + 4*x^3))*Log[2]^ 4),x]
(256*(4*E^15 + 2*E^3*x*(4 + 3*x) + 2*E^12*(5 + 4*x) + E^9*(6 + 24*x + 4*x^ 2) + E^6*(1 + 22*x + 10*x^2) + x*(1 + x + x^4)))/((E^6 + x + 2*E^3*x + x^2 )*Log[2]^4)
Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(20)=40\).
Time = 0.96 (sec) , antiderivative size = 217, normalized size of antiderivative = 10.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6, 27, 27, 2028, 2459, 1380, 27, 2345, 27, 2019, 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.
| \(\displaystyle \int \frac {768 x^6+2048 e^3 x^5+1024 x^5+1280 e^6 x^4}{\left (x^4+2 x^3+x^2+e^6 \left (6 x^2+2 x\right )+e^3 \left (4 x^3+4 x^2\right )+4 e^9 x+e^{12}\right ) \log ^4(2)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {768 x^6+\left (1024+2048 e^3\right ) x^5+1280 e^6 x^4}{\left (x^4+2 x^3+x^2+e^6 \left (6 x^2+2 x\right )+e^3 \left (4 x^3+4 x^2\right )+4 e^9 x+e^{12}\right ) \log ^4(2)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {256 \left (3 x^6+4 \left (1+2 e^3\right ) x^5+5 e^6 x^4\right )}{x^4+2 x^3+x^2+4 e^9 x+2 e^6 \left (3 x^2+x\right )+4 e^3 \left (x^3+x^2\right )+e^{12}}dx}{\log ^4(2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {256 \int \frac {3 x^6+4 \left (1+2 e^3\right ) x^5+5 e^6 x^4}{x^4+2 x^3+x^2+4 e^9 x+2 e^6 \left (3 x^2+x\right )+4 e^3 \left (x^3+x^2\right )+e^{12}}dx}{\log ^4(2)}\) |
| \(\Big \downarrow \) 2028 |
| \(\displaystyle \frac {256 \int \frac {x^4 \left (3 x^2+4 \left (1+2 e^3\right ) x+5 e^6\right )}{x^4+2 x^3+x^2+4 e^9 x+2 e^6 \left (3 x^2+x\right )+4 e^3 \left (x^3+x^2\right )+e^{12}}dx}{\log ^4(2)}\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \frac {256 \int \frac {3 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^6-5 \left (1+2 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^5+\frac {5}{4} \left (1+4 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^4+\frac {5}{2} \left (1+6 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^3-\frac {5}{16} \left (1+2 e^3\right )^2 \left (7+28 e^3+4 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+\frac {1}{16} \left (1+2 e^3\right )^3 \left (11+44 e^3+4 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )-\frac {5}{64} \left (1+2 e^3\right )^4 \left (1+4 e^3\right )}{\left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^4-\frac {1}{2} \left (1+4 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+\frac {1}{16} \left (1+4 e^3\right )^2}d\left (x+\frac {1}{4} \left (2+4 e^3\right )\right )}{\log ^4(2)}\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle \frac {256 \int -\frac {-192 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^6+320 \left (1+2 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^5-80 \left (1+4 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^4-160 \left (1+6 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^3+20 \left (1+2 e^3\right )^2 \left (7+28 e^3+4 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2-4 \left (1+2 e^3\right )^3 \left (11+44 e^3+4 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )+5 \left (1+2 e^3\right )^4 \left (1+4 e^3\right )}{4 \left (-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1\right )^2}d\left (x+\frac {1}{4} \left (2+4 e^3\right )\right )}{\log ^4(2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {64 \int \frac {-192 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^6+320 \left (1+2 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^5-80 \left (1+4 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^4-160 \left (1+6 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^3+20 \left (1+2 e^3\right )^2 \left (7+28 e^3+4 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2-4 \left (1+2 e^3\right )^3 \left (11+44 e^3+4 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )+5 \left (1+2 e^3\right )^4 \left (1+4 e^3\right )}{\left (-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1\right )^2}d\left (x+\frac {1}{4} \left (2+4 e^3\right )\right )}{\log ^4(2)}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {64 \left (\frac {\int \frac {2 \left (-48 \left (1+4 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^4+80 \left (1+6 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^3-32 \left (1+8 e^3+21 e^6+20 e^9\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2-20 \left (1+2 e^3\right ) \left (1+4 e^3\right )^2 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )+\left (1+4 e^3\right )^2 \left (11+44 e^3+40 e^6\right )\right )}{-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1}d\left (x+\frac {1}{4} \left (2+4 e^3\right )\right )}{2 \left (1+4 e^3\right )}+\frac {8 \left (\left (1+4 e^3\right ) \left (1+10 e^3+35 e^6+50 e^9+25 e^{12}+2 e^{15}\right )-2 \left (1+12 e^3+53 e^6+104 e^9+85 e^{12}+20 e^{15}\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )\right )}{\left (1+4 e^3\right ) \left (-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1\right )}\right )}{\log ^4(2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {64 \left (\frac {\int \frac {-48 \left (1+4 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^4+80 \left (1+6 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^3-32 \left (1+8 e^3+21 e^6+20 e^9\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2-20 \left (1+2 e^3\right ) \left (1+4 e^3\right )^2 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )+\left (1+4 e^3\right )^2 \left (11+44 e^3+40 e^6\right )}{-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1}d\left (x+\frac {1}{4} \left (2+4 e^3\right )\right )}{1+4 e^3}+\frac {8 \left (\left (1+4 e^3\right ) \left (1+10 e^3+35 e^6+50 e^9+25 e^{12}+2 e^{15}\right )-2 \left (1+12 e^3+53 e^6+104 e^9+85 e^{12}+20 e^{15}\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )\right )}{\left (1+4 e^3\right ) \left (-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1\right )}\right )}{\log ^4(2)}\) |
| \(\Big \downarrow \) 2019 |
| \(\displaystyle \frac {64 \left (\frac {\int \left (\left (12+48 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+\left (-20-120 e^3-160 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )+160 e^9+216 e^6+88 e^3+11\right )d\left (x+\frac {1}{4} \left (2+4 e^3\right )\right )}{1+4 e^3}+\frac {8 \left (\left (1+4 e^3\right ) \left (1+10 e^3+35 e^6+50 e^9+25 e^{12}+2 e^{15}\right )-2 \left (1+12 e^3+53 e^6+104 e^9+85 e^{12}+20 e^{15}\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )\right )}{\left (1+4 e^3\right ) \left (-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1\right )}\right )}{\log ^4(2)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {64 \left (\frac {8 \left (\left (1+4 e^3\right ) \left (1+10 e^3+35 e^6+50 e^9+25 e^{12}+2 e^{15}\right )-2 \left (1+12 e^3+53 e^6+104 e^9+85 e^{12}+20 e^{15}\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )\right )}{\left (1+4 e^3\right ) \left (-4 \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+4 e^3+1\right )}+\frac {4 \left (1+4 e^3\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^3-10 \left (1+6 e^3+8 e^6\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )^2+\left (11+88 e^3+216 e^6+160 e^9\right ) \left (x+\frac {1}{4} \left (2+4 e^3\right )\right )}{1+4 e^3}\right )}{\log ^4(2)}\) |
Int[(1280*E^6*x^4 + 1024*x^5 + 2048*E^3*x^5 + 768*x^6)/((E^12 + 4*E^9*x + x^2 + 2*x^3 + x^4 + E^6*(2*x + 6*x^2) + E^3*(4*x^2 + 4*x^3))*Log[2]^4),x]
(64*((8*((1 + 4*E^3)*(1 + 10*E^3 + 35*E^6 + 50*E^9 + 25*E^12 + 2*E^15) - 2 *(1 + 12*E^3 + 53*E^6 + 104*E^9 + 85*E^12 + 20*E^15)*((2 + 4*E^3)/4 + x))) /((1 + 4*E^3)*(1 + 4*E^3 - 4*((2 + 4*E^3)/4 + x)^2)) + ((11 + 88*E^3 + 216 *E^6 + 160*E^9)*((2 + 4*E^3)/4 + x) - 10*(1 + 6*E^3 + 8*E^6)*((2 + 4*E^3)/ 4 + x)^2 + 4*(1 + 4*E^3)*((2 + 4*E^3)/4 + x)^3)/(1 + 4*E^3)))/Log[2]^4
3.2.87.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px , Qx, x]^p*Qx^(p + q), x] /; FreeQ[q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]
Int[(Fx_.)*((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))^p*Fx, x] /; FreeQ[ {a, b, c, r, s, t}, x] && IntegerQ[p] && PosQ[s - r] && PosQ[t - r] && !(E qQ[p, 1] && EqQ[u, 1])
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Time = 1.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30
| method | result | size |
| gosper | \(\frac {256 x^{5}}{\ln \left (2\right )^{4} \left ({\mathrm e}^{6}+2 x \,{\mathrm e}^{3}+x^{2}+x \right )}\) | \(26\) |
| norman | \(\frac {256 x^{5}}{\ln \left (2\right )^{4} \left ({\mathrm e}^{6}+2 x \,{\mathrm e}^{3}+x^{2}+x \right )}\) | \(26\) |
| parallelrisch | \(\frac {256 x^{5}}{\ln \left (2\right )^{4} \left ({\mathrm e}^{6}+2 x \,{\mathrm e}^{3}+x^{2}+x \right )}\) | \(26\) |
| risch | \(-\frac {256 \left (-x^{5}-10 x^{2} {\mathrm e}^{6}-4 x^{2} {\mathrm e}^{9}-24 x \,{\mathrm e}^{9}-6 x^{2} {\mathrm e}^{3}-8 x \,{\mathrm e}^{12}-8 x \,{\mathrm e}^{3}-22 x \,{\mathrm e}^{6}-x^{2}-{\mathrm e}^{6}-6 \,{\mathrm e}^{9}-4 \,{\mathrm e}^{15}-10 \,{\mathrm e}^{12}-x \right )}{\ln \left (2\right )^{4} \left ({\mathrm e}^{6}+2 x \,{\mathrm e}^{3}+x^{2}+x \right )}\) | \(92\) |
| default | \(\frac {768 x \,{\mathrm e}^{6}-512 x^{2} {\mathrm e}^{3}+256 x^{3}+1024 x \,{\mathrm e}^{3}-256 x^{2}+256 x +128 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\left (4 \,{\mathrm e}^{3}+2\right ) \textit {\_Z}^{3}+\left (4 \,{\mathrm e}^{3}+6 \,{\mathrm e}^{6}+1\right ) \textit {\_Z}^{2}+\left (2 \,{\mathrm e}^{6}+4 \,{\mathrm e}^{9}\right ) \textit {\_Z} +{\mathrm e}^{12}\right )}{\sum }\frac {\left (\left (-20 \,{\mathrm e}^{9}-5 \,{\mathrm e}^{12}-21 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}-1\right ) \textit {\_R}^{2}+2 \left (-10 \,{\mathrm e}^{12}-4 \,{\mathrm e}^{15}-6 \,{\mathrm e}^{9}-{\mathrm e}^{6}\right ) \textit {\_R} -4 \,{\mathrm e}^{12} {\mathrm e}^{3}-3 \,{\mathrm e}^{12} {\mathrm e}^{6}-{\mathrm e}^{12}\right ) \ln \left (x -\textit {\_R} \right )}{2 \,{\mathrm e}^{9}+6 \textit {\_R} \,{\mathrm e}^{6}+6 \textit {\_R}^{2} {\mathrm e}^{3}+2 \textit {\_R}^{3}+{\mathrm e}^{6}+4 \textit {\_R} \,{\mathrm e}^{3}+3 \textit {\_R}^{2}+\textit {\_R}}\right )}{\ln \left (2\right )^{4}}\) | \(191\) |
int((1280*x^4*exp(3)^2+2048*x^5*exp(3)+768*x^6+1024*x^5)/(exp(3)^4+4*x*exp (3)^3+(6*x^2+2*x)*exp(3)^2+(4*x^3+4*x^2)*exp(3)+x^4+2*x^3+x^2)/ln(2)^4,x,m ethod=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 4.05 \[ \int \frac {1280 e^6 x^4+1024 x^5+2048 e^3 x^5+768 x^6}{\left (e^{12}+4 e^9 x+x^2+2 x^3+x^4+e^6 \left (2 x+6 x^2\right )+e^3 \left (4 x^2+4 x^3\right )\right ) \log ^4(2)} \, dx=\frac {256 \, {\left (x^{5} + x^{2} + 2 \, {\left (4 \, x + 5\right )} e^{12} + 2 \, {\left (2 \, x^{2} + 12 \, x + 3\right )} e^{9} + {\left (10 \, x^{2} + 22 \, x + 1\right )} e^{6} + 2 \, {\left (3 \, x^{2} + 4 \, x\right )} e^{3} + x + 4 \, e^{15}\right )}}{{\left (x^{2} + 2 \, x e^{3} + x + e^{6}\right )} \log \left (2\right )^{4}} \]
integrate((1280*x^4*exp(3)^2+2048*x^5*exp(3)+768*x^6+1024*x^5)/(exp(3)^4+4 *x*exp(3)^3+(6*x^2+2*x)*exp(3)^2+(4*x^3+4*x^2)*exp(3)+x^4+2*x^3+x^2)/log(2 )^4,x, algorithm=\
256*(x^5 + x^2 + 2*(4*x + 5)*e^12 + 2*(2*x^2 + 12*x + 3)*e^9 + (10*x^2 + 2 2*x + 1)*e^6 + 2*(3*x^2 + 4*x)*e^3 + x + 4*e^15)/((x^2 + 2*x*e^3 + x + e^6 )*log(2)^4)
Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (17) = 34\).
Time = 0.91 (sec) , antiderivative size = 138, normalized size of antiderivative = 6.90 \[ \int \frac {1280 e^6 x^4+1024 x^5+2048 e^3 x^5+768 x^6}{\left (e^{12}+4 e^9 x+x^2+2 x^3+x^4+e^6 \left (2 x+6 x^2\right )+e^3 \left (4 x^2+4 x^3\right )\right ) \log ^4(2)} \, dx=\frac {256 x^{3}}{\log {\left (2 \right )}^{4}} + x^{2} \left (- \frac {512 e^{3}}{\log {\left (2 \right )}^{4}} - \frac {256}{\log {\left (2 \right )}^{4}}\right ) + x \left (\frac {256}{\log {\left (2 \right )}^{4}} + \frac {1024 e^{3}}{\log {\left (2 \right )}^{4}} + \frac {768 e^{6}}{\log {\left (2 \right )}^{4}}\right ) + \frac {x \left (256 + 2048 e^{3} + 5376 e^{6} + 5120 e^{9} + 1280 e^{12}\right ) + 256 e^{6} + 1536 e^{9} + 2560 e^{12} + 1024 e^{15}}{x^{2} \log {\left (2 \right )}^{4} + x \left (\log {\left (2 \right )}^{4} + 2 e^{3} \log {\left (2 \right )}^{4}\right ) + e^{6} \log {\left (2 \right )}^{4}} \]
integrate((1280*x**4*exp(3)**2+2048*x**5*exp(3)+768*x**6+1024*x**5)/(exp(3 )**4+4*x*exp(3)**3+(6*x**2+2*x)*exp(3)**2+(4*x**3+4*x**2)*exp(3)+x**4+2*x* *3+x**2)/ln(2)**4,x)
256*x**3/log(2)**4 + x**2*(-512*exp(3)/log(2)**4 - 256/log(2)**4) + x*(256 /log(2)**4 + 1024*exp(3)/log(2)**4 + 768*exp(6)/log(2)**4) + (x*(256 + 204 8*exp(3) + 5376*exp(6) + 5120*exp(9) + 1280*exp(12)) + 256*exp(6) + 1536*e xp(9) + 2560*exp(12) + 1024*exp(15))/(x**2*log(2)**4 + x*(log(2)**4 + 2*ex p(3)*log(2)**4) + exp(6)*log(2)**4)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (19) = 38\).
Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 4.25 \[ \int \frac {1280 e^6 x^4+1024 x^5+2048 e^3 x^5+768 x^6}{\left (e^{12}+4 e^9 x+x^2+2 x^3+x^4+e^6 \left (2 x+6 x^2\right )+e^3 \left (4 x^2+4 x^3\right )\right ) \log ^4(2)} \, dx=\frac {256 \, {\left (x^{3} - x^{2} {\left (2 \, e^{3} + 1\right )} + x {\left (3 \, e^{6} + 4 \, e^{3} + 1\right )} + \frac {x {\left (5 \, e^{12} + 20 \, e^{9} + 21 \, e^{6} + 8 \, e^{3} + 1\right )} + 4 \, e^{15} + 10 \, e^{12} + 6 \, e^{9} + e^{6}}{x^{2} + x {\left (2 \, e^{3} + 1\right )} + e^{6}}\right )}}{\log \left (2\right )^{4}} \]
integrate((1280*x^4*exp(3)^2+2048*x^5*exp(3)+768*x^6+1024*x^5)/(exp(3)^4+4 *x*exp(3)^3+(6*x^2+2*x)*exp(3)^2+(4*x^3+4*x^2)*exp(3)+x^4+2*x^3+x^2)/log(2 )^4,x, algorithm=\
256*(x^3 - x^2*(2*e^3 + 1) + x*(3*e^6 + 4*e^3 + 1) + (x*(5*e^12 + 20*e^9 + 21*e^6 + 8*e^3 + 1) + 4*e^15 + 10*e^12 + 6*e^9 + e^6)/(x^2 + x*(2*e^3 + 1 ) + e^6))/log(2)^4
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.20 \[ \int \frac {1280 e^6 x^4+1024 x^5+2048 e^3 x^5+768 x^6}{\left (e^{12}+4 e^9 x+x^2+2 x^3+x^4+e^6 \left (2 x+6 x^2\right )+e^3 \left (4 x^2+4 x^3\right )\right ) \log ^4(2)} \, dx=\frac {256 \, {\left (x^{3} - 2 \, x^{2} e^{3} - x^{2} + 3 \, x e^{6} + 4 \, x e^{3} + x + \frac {5 \, x e^{12} + 20 \, x e^{9} + 21 \, x e^{6} + 8 \, x e^{3} + x + 4 \, e^{15} + 10 \, e^{12} + 6 \, e^{9} + e^{6}}{x^{2} + 2 \, x e^{3} + x + e^{6}}\right )}}{\log \left (2\right )^{4}} \]
integrate((1280*x^4*exp(3)^2+2048*x^5*exp(3)+768*x^6+1024*x^5)/(exp(3)^4+4 *x*exp(3)^3+(6*x^2+2*x)*exp(3)^2+(4*x^3+4*x^2)*exp(3)+x^4+2*x^3+x^2)/log(2 )^4,x, algorithm=\
256*(x^3 - 2*x^2*e^3 - x^2 + 3*x*e^6 + 4*x*e^3 + x + (5*x*e^12 + 20*x*e^9 + 21*x*e^6 + 8*x*e^3 + x + 4*e^15 + 10*e^12 + 6*e^9 + e^6)/(x^2 + 2*x*e^3 + x + e^6))/log(2)^4
Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 8.50 \[ \int \frac {1280 e^6 x^4+1024 x^5+2048 e^3 x^5+768 x^6}{\left (e^{12}+4 e^9 x+x^2+2 x^3+x^4+e^6 \left (2 x+6 x^2\right )+e^3 \left (4 x^2+4 x^3\right )\right ) \log ^4(2)} \, dx=\frac {256\,x^3}{{\ln \left (2\right )}^4}+\frac {256\,{\mathrm {e}}^6+1536\,{\mathrm {e}}^9+2560\,{\mathrm {e}}^{12}+1024\,{\mathrm {e}}^{15}+x\,\left (2048\,{\mathrm {e}}^3+5376\,{\mathrm {e}}^6+5120\,{\mathrm {e}}^9+1280\,{\mathrm {e}}^{12}+256\right )}{{\ln \left (2\right )}^4\,x^2+\left (2\,{\mathrm {e}}^3\,{\ln \left (2\right )}^4+{\ln \left (2\right )}^4\right )\,x+{\mathrm {e}}^6\,{\ln \left (2\right )}^4}-x^2\,\left (\frac {384\,\left (4\,{\mathrm {e}}^3+2\right )}{{\ln \left (2\right )}^4}-\frac {2048\,{\mathrm {e}}^3+1024}{2\,{\ln \left (2\right )}^4}\right )+x\,\left (\left (4\,{\mathrm {e}}^3+2\right )\,\left (\frac {768\,\left (4\,{\mathrm {e}}^3+2\right )}{{\ln \left (2\right )}^4}-\frac {2048\,{\mathrm {e}}^3+1024}{{\ln \left (2\right )}^4}\right )-\frac {768\,\left (4\,{\mathrm {e}}^3+6\,{\mathrm {e}}^6+1\right )}{{\ln \left (2\right )}^4}+\frac {1280\,{\mathrm {e}}^6}{{\ln \left (2\right )}^4}\right ) \]
int((2048*x^5*exp(3) + 1280*x^4*exp(6) + 1024*x^5 + 768*x^6)/(log(2)^4*(ex p(12) + exp(6)*(2*x + 6*x^2) + 4*x*exp(9) + exp(3)*(4*x^2 + 4*x^3) + x^2 + 2*x^3 + x^4)),x)
(256*x^3)/log(2)^4 + (256*exp(6) + 1536*exp(9) + 2560*exp(12) + 1024*exp(1 5) + x*(2048*exp(3) + 5376*exp(6) + 5120*exp(9) + 1280*exp(12) + 256))/(x^ 2*log(2)^4 + exp(6)*log(2)^4 + x*(2*exp(3)*log(2)^4 + log(2)^4)) - x^2*((3 84*(4*exp(3) + 2))/log(2)^4 - (2048*exp(3) + 1024)/(2*log(2)^4)) + x*((4*e xp(3) + 2)*((768*(4*exp(3) + 2))/log(2)^4 - (2048*exp(3) + 1024)/log(2)^4) - (768*(4*exp(3) + 6*exp(6) + 1))/log(2)^4 + (1280*exp(6))/log(2)^4)