Integrand size = 85, antiderivative size = 19 \[ \int \frac {-1296-864 x-216 x^2-24 x^3-x^4+\left (144+96 x-2 x^3\right ) \log (2)}{1296+864 x+216 x^2+24 x^3+x^4+\left (-288-24 x+16 x^2+2 x^3\right ) \log (2)+\left (16-8 x+x^2\right ) \log ^2(2)} \, dx=\frac {x}{-1+\frac {(4-x) \log (2)}{(6+x)^2}} \]
Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(19)=38\).
Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 6.05 \[ \int \frac {-1296-864 x-216 x^2-24 x^3-x^4+\left (144+96 x-2 x^3\right ) \log (2)}{1296+864 x+216 x^2+24 x^3+x^4+\left (-288-24 x+16 x^2+2 x^3\right ) \log (2)+\left (16-8 x+x^2\right ) \log ^2(2)} \, dx=-\frac {x^3 \log (2) (40+\log (2))+x^2 \log (2) \left (480+52 \log (2)+\log ^2(2)\right )+x \left (64 \log ^3(2)+\log ^4(2)-6 \log ^2(2) (-150+\log (4))-192 \log (2) (-12+\log (4))-432 \log (4)\right )-4 (-9+\log (2)) \left (52 \log ^2(2)+\log ^3(2)-\log (4) \log (64)\right )}{\log (2) (40+\log (2)) \left (x^2-4 (-9+\log (2))+x (12+\log (2))\right )} \]
Integrate[(-1296 - 864*x - 216*x^2 - 24*x^3 - x^4 + (144 + 96*x - 2*x^3)*L og[2])/(1296 + 864*x + 216*x^2 + 24*x^3 + x^4 + (-288 - 24*x + 16*x^2 + 2* x^3)*Log[2] + (16 - 8*x + x^2)*Log[2]^2),x]
-((x^3*Log[2]*(40 + Log[2]) + x^2*Log[2]*(480 + 52*Log[2] + Log[2]^2) + x* (64*Log[2]^3 + Log[2]^4 - 6*Log[2]^2*(-150 + Log[4]) - 192*Log[2]*(-12 + L og[4]) - 432*Log[4]) - 4*(-9 + Log[2])*(52*Log[2]^2 + Log[2]^3 - Log[4]*Lo g[64]))/(Log[2]*(40 + Log[2])*(x^2 - 4*(-9 + Log[2]) + x*(12 + Log[2]))))
Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(19)=38\).
Time = 0.42 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.84, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2459, 1380, 25, 2345, 24}
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.
\(\displaystyle \int \frac {-x^4-24 x^3+\left (-2 x^3+96 x+144\right ) \log (2)-216 x^2-864 x-1296}{x^4+24 x^3+216 x^2+\left (x^2-8 x+16\right ) \log ^2(2)+\left (2 x^3+16 x^2-24 x-288\right ) \log (2)+864 x+1296} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \frac {-\log (2) \left (120+\log ^2(2)+36 \log (2)\right ) \left (x+\frac {1}{4} (24+2 \log (2))\right )-\left (x+\frac {1}{4} (24+2 \log (2))\right )^4+\frac {3}{2} \log (2) (24+\log (2)) \left (x+\frac {1}{4} (24+2 \log (2))\right )^2+\frac {3}{16} \log ^2(2) (8+\log (2)) (40+\log (2))}{\left (x+\frac {1}{4} (24+2 \log (2))\right )^4-\frac {1}{2} \log (2) (40+\log (2)) \left (x+\frac {1}{4} (24+2 \log (2))\right )^2+\frac {1}{16} \log ^2(2) (40+\log (2))^2}d\left (x+\frac {1}{4} (24+2 \log (2))\right )\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle \int -\frac {16 \log (2) \left (120+\log ^2(2)+36 \log (2)\right ) \left (x+\frac {1}{4} (24+2 \log (2))\right )+16 \left (x+\frac {1}{4} (24+2 \log (2))\right )^4-24 \log (2) (24+\log (2)) \left (x+\frac {1}{4} (24+2 \log (2))\right )^2-3 \log ^2(2) (8+\log (2)) (40+\log (2))}{\left (4 \left (x+\frac {1}{4} (24+2 \log (2))\right )^2-\log (2) (40+\log (2))\right )^2}d\left (x+\frac {1}{4} (24+2 \log (2))\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {16 \left (x+\frac {1}{4} (24+2 \log (2))\right )^4-24 \log (2) (24+\log (2)) \left (x+\frac {1}{4} (24+2 \log (2))\right )^2+16 \log (2) \left (120+36 \log (2)+\log ^2(2)\right ) \left (x+\frac {1}{4} (24+2 \log (2))\right )-3 \log ^2(2) (8+\log (2)) (40+\log (2))}{\left (4 \left (x+\frac {1}{4} (24+2 \log (2))\right )^2-\log (2) (40+\log (2))\right )^2}d\left (x+\frac {1}{4} (24+2 \log (2))\right )\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {2 \log (2) \left (-2 (16+\log (2)) \left (x+\frac {1}{4} (24+2 \log (2))\right )+120+\log ^2(2)+36 \log (2)\right )}{4 \left (x+\frac {1}{4} (24+2 \log (2))\right )^2-\log (2) (40+\log (2))}-\frac {\int 2 \log (2) (40+\log (2))d\left (x+\frac {1}{4} (24+2 \log (2))\right )}{2 \log (2) (40+\log (2))}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -x+\frac {2 \log (2) \left (-2 (16+\log (2)) \left (x+\frac {1}{4} (24+2 \log (2))\right )+120+\log ^2(2)+36 \log (2)\right )}{4 \left (x+\frac {1}{4} (24+2 \log (2))\right )^2-\log (2) (40+\log (2))}+\frac {1}{4} (-24-2 \log (2))\) |
Int[(-1296 - 864*x - 216*x^2 - 24*x^3 - x^4 + (144 + 96*x - 2*x^3)*Log[2]) /(1296 + 864*x + 216*x^2 + 24*x^3 + x^4 + (-288 - 24*x + 16*x^2 + 2*x^3)*L og[2] + (16 - 8*x + x^2)*Log[2]^2),x]
-x + (-24 - 2*Log[2])/4 + (2*Log[2]*(120 + 36*Log[2] + Log[2]^2 - 2*(16 + Log[2])*(x + (24 + 2*Log[2])/4)))/(-(Log[2]*(40 + Log[2])) + 4*(x + (24 + 2*Log[2])/4)^2)
3.29.55.3.1 Defintions of rubi rules used
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[(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 = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05
method | result | size |
gosper | \(\frac {-x^{3}+12 x \ln \left (2\right )-48 \ln \left (2\right )+108 x +432}{x \ln \left (2\right )+x^{2}-4 \ln \left (2\right )+12 x +36}\) | \(39\) |
norman | \(\frac {\left (108+12 \ln \left (2\right )\right ) x -x^{3}-48 \ln \left (2\right )+432}{x \ln \left (2\right )+x^{2}-4 \ln \left (2\right )+12 x +36}\) | \(39\) |
parallelrisch | \(\frac {-x^{3}+12 x \ln \left (2\right )-48 \ln \left (2\right )+108 x +432}{x \ln \left (2\right )+x^{2}-4 \ln \left (2\right )+12 x +36}\) | \(39\) |
default | \(-x +\frac {\ln \left (2\right ) \left (\left (-16-\ln \left (2\right )\right ) x +4 \ln \left (2\right )-36\right )}{x \ln \left (2\right )+x^{2}-4 \ln \left (2\right )+12 x +36}\) | \(40\) |
risch | \(-x +\frac {\left (-\ln \left (2\right )^{2}-16 \ln \left (2\right )\right ) x +4 \ln \left (2\right )^{2}-36 \ln \left (2\right )}{x \ln \left (2\right )+x^{2}-4 \ln \left (2\right )+12 x +36}\) | \(48\) |
int(((-2*x^3+96*x+144)*ln(2)-x^4-24*x^3-216*x^2-864*x-1296)/((x^2-8*x+16)* ln(2)^2+(2*x^3+16*x^2-24*x-288)*ln(2)+x^4+24*x^3+216*x^2+864*x+1296),x,met hod=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.58 \[ \int \frac {-1296-864 x-216 x^2-24 x^3-x^4+\left (144+96 x-2 x^3\right ) \log (2)}{1296+864 x+216 x^2+24 x^3+x^4+\left (-288-24 x+16 x^2+2 x^3\right ) \log (2)+\left (16-8 x+x^2\right ) \log ^2(2)} \, dx=-\frac {x^{3} + {\left (x - 4\right )} \log \left (2\right )^{2} + 12 \, x^{2} + {\left (x^{2} + 12 \, x + 36\right )} \log \left (2\right ) + 36 \, x}{x^{2} + {\left (x - 4\right )} \log \left (2\right ) + 12 \, x + 36} \]
integrate(((-2*x^3+96*x+144)*log(2)-x^4-24*x^3-216*x^2-864*x-1296)/((x^2-8 *x+16)*log(2)^2+(2*x^3+16*x^2-24*x-288)*log(2)+x^4+24*x^3+216*x^2+864*x+12 96),x, algorithm=\
-(x^3 + (x - 4)*log(2)^2 + 12*x^2 + (x^2 + 12*x + 36)*log(2) + 36*x)/(x^2 + (x - 4)*log(2) + 12*x + 36)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (14) = 28\).
Time = 0.51 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21 \[ \int \frac {-1296-864 x-216 x^2-24 x^3-x^4+\left (144+96 x-2 x^3\right ) \log (2)}{1296+864 x+216 x^2+24 x^3+x^4+\left (-288-24 x+16 x^2+2 x^3\right ) \log (2)+\left (16-8 x+x^2\right ) \log ^2(2)} \, dx=- x - \frac {x \left (\log {\left (2 \right )}^{2} + 16 \log {\left (2 \right )}\right ) - 4 \log {\left (2 \right )}^{2} + 36 \log {\left (2 \right )}}{x^{2} + x \left (\log {\left (2 \right )} + 12\right ) - 4 \log {\left (2 \right )} + 36} \]
integrate(((-2*x**3+96*x+144)*ln(2)-x**4-24*x**3-216*x**2-864*x-1296)/((x* *2-8*x+16)*ln(2)**2+(2*x**3+16*x**2-24*x-288)*ln(2)+x**4+24*x**3+216*x**2+ 864*x+1296),x)
-x - (x*(log(2)**2 + 16*log(2)) - 4*log(2)**2 + 36*log(2))/(x**2 + x*(log( 2) + 12) - 4*log(2) + 36)
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (18) = 36\).
Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.37 \[ \int \frac {-1296-864 x-216 x^2-24 x^3-x^4+\left (144+96 x-2 x^3\right ) \log (2)}{1296+864 x+216 x^2+24 x^3+x^4+\left (-288-24 x+16 x^2+2 x^3\right ) \log (2)+\left (16-8 x+x^2\right ) \log ^2(2)} \, dx=-x - \frac {{\left (\log \left (2\right )^{2} + 16 \, \log \left (2\right )\right )} x - 4 \, \log \left (2\right )^{2} + 36 \, \log \left (2\right )}{x^{2} + x {\left (\log \left (2\right ) + 12\right )} - 4 \, \log \left (2\right ) + 36} \]
integrate(((-2*x^3+96*x+144)*log(2)-x^4-24*x^3-216*x^2-864*x-1296)/((x^2-8 *x+16)*log(2)^2+(2*x^3+16*x^2-24*x-288)*log(2)+x^4+24*x^3+216*x^2+864*x+12 96),x, algorithm=\
-x - ((log(2)^2 + 16*log(2))*x - 4*log(2)^2 + 36*log(2))/(x^2 + x*(log(2) + 12) - 4*log(2) + 36)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.42 \[ \int \frac {-1296-864 x-216 x^2-24 x^3-x^4+\left (144+96 x-2 x^3\right ) \log (2)}{1296+864 x+216 x^2+24 x^3+x^4+\left (-288-24 x+16 x^2+2 x^3\right ) \log (2)+\left (16-8 x+x^2\right ) \log ^2(2)} \, dx=-x - \frac {x \log \left (2\right )^{2} + 16 \, x \log \left (2\right ) - 4 \, \log \left (2\right )^{2} + 36 \, \log \left (2\right )}{x^{2} + x \log \left (2\right ) + 12 \, x - 4 \, \log \left (2\right ) + 36} \]
integrate(((-2*x^3+96*x+144)*log(2)-x^4-24*x^3-216*x^2-864*x-1296)/((x^2-8 *x+16)*log(2)^2+(2*x^3+16*x^2-24*x-288)*log(2)+x^4+24*x^3+216*x^2+864*x+12 96),x, algorithm=\
-x - (x*log(2)^2 + 16*x*log(2) - 4*log(2)^2 + 36*log(2))/(x^2 + x*log(2) + 12*x - 4*log(2) + 36)
Time = 12.85 (sec) , antiderivative size = 322, normalized size of antiderivative = 16.95 \[ \int \frac {-1296-864 x-216 x^2-24 x^3-x^4+\left (144+96 x-2 x^3\right ) \log (2)}{1296+864 x+216 x^2+24 x^3+x^4+\left (-288-24 x+16 x^2+2 x^3\right ) \log (2)+\left (16-8 x+x^2\right ) \log ^2(2)} \, dx=\left (\sum _{k=1}^4\ln \left (-{\ln \left (2\right )}^3\,\left (1728000\,\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )+864000\,x+259200\,\ln \left (2\right )+\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,\ln \left (2\right )\,916800+\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,x\,288000+163200\,x\,\ln \left (2\right )-\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,{\ln \left (2\right )}^2\,15920-\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,{\ln \left (2\right )}^3\,8464-\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,{\ln \left (2\right )}^4\,348-\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,{\ln \left (2\right )}^5\,4-21600\,x\,{\ln \left (2\right )}^2-800\,x\,{\ln \left (2\right )}^3-20800\,{\ln \left (2\right )}^2-6200\,{\ln \left (2\right )}^3-200\,{\ln \left (2\right )}^4+\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,x\,\ln \left (2\right )\,384800+\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,x\,{\ln \left (2\right )}^2\,70880+\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,x\,{\ln \left (2\right )}^3\,4416+\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,x\,{\ln \left (2\right )}^4\,112+\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\,x\,{\ln \left (2\right )}^5+5184000\right )\,2\right )\,\mathrm {root}\left (625\,{\ln \left (2\right )}^6\,{\left (\ln \left (2\right )+40\right )}^2,z,k\right )\right )-x \]
int(-(864*x - log(2)*(96*x - 2*x^3 + 144) + 216*x^2 + 24*x^3 + x^4 + 1296) /(864*x + log(2)^2*(x^2 - 8*x + 16) - log(2)*(24*x - 16*x^2 - 2*x^3 + 288) + 216*x^2 + 24*x^3 + x^4 + 1296),x)
symsum(log(-2*log(2)^3*(1728000*root(625*log(2)^6*(log(2) + 40)^2, z, k) + 864000*x + 259200*log(2) + 916800*root(625*log(2)^6*(log(2) + 40)^2, z, k )*log(2) + 288000*root(625*log(2)^6*(log(2) + 40)^2, z, k)*x + 163200*x*lo g(2) - 15920*root(625*log(2)^6*(log(2) + 40)^2, z, k)*log(2)^2 - 8464*root (625*log(2)^6*(log(2) + 40)^2, z, k)*log(2)^3 - 348*root(625*log(2)^6*(log (2) + 40)^2, z, k)*log(2)^4 - 4*root(625*log(2)^6*(log(2) + 40)^2, z, k)*l og(2)^5 - 21600*x*log(2)^2 - 800*x*log(2)^3 - 20800*log(2)^2 - 6200*log(2) ^3 - 200*log(2)^4 + 384800*root(625*log(2)^6*(log(2) + 40)^2, z, k)*x*log( 2) + 70880*root(625*log(2)^6*(log(2) + 40)^2, z, k)*x*log(2)^2 + 4416*root (625*log(2)^6*(log(2) + 40)^2, z, k)*x*log(2)^3 + 112*root(625*log(2)^6*(l og(2) + 40)^2, z, k)*x*log(2)^4 + root(625*log(2)^6*(log(2) + 40)^2, z, k) *x*log(2)^5 + 5184000))*root(625*log(2)^6*(log(2) + 40)^2, z, k), k, 1, 4) - x