Integrand size = 164, antiderivative size = 31 \begin {dmath*} \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=-8+x+\frac {x}{1-\frac {\log ^2\left (\log \left (3 e^3\right )\right )}{(2 x-x \log (3))^2}} \end {dmath*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \begin {dmath*} \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=x \left (2+\frac {\log ^2(3+\log (3))}{x^2 (-2+\log (3))^2-\log ^2(3+\log (3))}\right ) \end {dmath*}
Integrate[(32*x^4 - 64*x^4*Log[3] + 48*x^4*Log[3]^2 - 16*x^4*Log[3]^3 + 2* x^4*Log[3]^4 + (-20*x^2 + 20*x^2*Log[3] - 5*x^2*Log[3]^2)*Log[Log[3*E^3]]^ 2 + Log[Log[3*E^3]]^4)/(16*x^4 - 32*x^4*Log[3] + 24*x^4*Log[3]^2 - 8*x^4*L og[3]^3 + x^4*Log[3]^4 + (-8*x^2 + 8*x^2*Log[3] - 2*x^2*Log[3]^2)*Log[Log[ 3*E^3]]^2 + Log[Log[3*E^3]]^4),x]
Time = 0.42 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.085, Rules used = {6, 6, 6, 6, 6, 6, 6, 6, 2454, 1380, 27, 2087, 1471, 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.
| \(\displaystyle \int \frac {32 x^4+2 x^4 \log ^4(3)-16 x^4 \log ^3(3)+48 x^4 \log ^2(3)-64 x^4 \log (3)+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4+x^4 \log ^4(3)-8 x^4 \log ^3(3)+24 x^4 \log ^2(3)-32 x^4 \log (3)+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 x^4 \log ^4(3)-16 x^4 \log ^3(3)+48 x^4 \log ^2(3)+x^4 (32-64 \log (3))+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4+x^4 \log ^4(3)-8 x^4 \log ^3(3)+24 x^4 \log ^2(3)-32 x^4 \log (3)+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {2 x^4 \log ^4(3)-16 x^4 \log ^3(3)+x^4 \left (32+48 \log ^2(3)-64 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4+x^4 \log ^4(3)-8 x^4 \log ^3(3)+24 x^4 \log ^2(3)-32 x^4 \log (3)+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (32+48 \log ^2(3)-64 \log (3)\right )+x^4 \left (2 \log ^4(3)-16 \log ^3(3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4+x^4 \log ^4(3)-8 x^4 \log ^3(3)+24 x^4 \log ^2(3)-32 x^4 \log (3)+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (32+2 \log ^4(3)-16 \log ^3(3)+48 \log ^2(3)-64 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4+x^4 \log ^4(3)-8 x^4 \log ^3(3)+24 x^4 \log ^2(3)-32 x^4 \log (3)+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (32+2 \log ^4(3)-16 \log ^3(3)+48 \log ^2(3)-64 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 \log ^4(3)-8 x^4 \log ^3(3)+24 x^4 \log ^2(3)+x^4 (16-32 \log (3))+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (32+2 \log ^4(3)-16 \log ^3(3)+48 \log ^2(3)-64 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 \log ^4(3)-8 x^4 \log ^3(3)+x^4 \left (16+24 \log ^2(3)-32 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (32+2 \log ^4(3)-16 \log ^3(3)+48 \log ^2(3)-64 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 \left (16+24 \log ^2(3)-32 \log (3)\right )+x^4 \left (\log ^4(3)-8 \log ^3(3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^4 \left (32+2 \log ^4(3)-16 \log ^3(3)+48 \log ^2(3)-64 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 \left (16+\log ^4(3)-8 \log ^3(3)+24 \log ^2(3)-32 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-8 x^2-2 x^2 \log ^2(3)+8 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}dx\) |
| \(\Big \downarrow \) 2454 |
| \(\displaystyle \int \frac {x^4 \left (32+2 \log ^4(3)-16 \log ^3(3)+48 \log ^2(3)-64 \log (3)\right )+\log ^2\left (\log \left (3 e^3\right )\right ) \left (-20 x^2-5 x^2 \log ^2(3)+20 x^2 \log (3)\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{x^4 (2-\log (3))^4-2 x^2 (2-\log (3))^2 \log ^2(3+\log (3))+\log ^4(3+\log (3))}dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle (2-\log (3))^4 \int \frac {2 (2-\log (3))^4 x^4+\log ^4(3+\log (3))-5 \left (\log ^2(3) x^2-4 \log (3) x^2+4 x^2\right ) \log ^2(3+\log (3))}{(2-\log (3))^4 \left (x^2 (2-\log (3))^2-\log ^2(3+\log (3))\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {2 x^4 (2-\log (3))^4-5 \log ^2(3+\log (3)) \left (4 x^2+x^2 \log ^2(3)-4 x^2 \log (3)\right )+\log ^4(3+\log (3))}{\left (x^2 (2-\log (3))^2-\log ^2(3+\log (3))\right )^2}dx\) |
| \(\Big \downarrow \) 2087 |
| \(\displaystyle \int \frac {2 x^4 (2-\log (3))^4-5 x^2 (2-\log (3))^2 \log ^2(3+\log (3))+\log ^4(3+\log (3))}{\left (x^2 (2-\log (3))^2-\log ^2(3+\log (3))\right )^2}dx\) |
| \(\Big \downarrow \) 1471 |
| \(\displaystyle \frac {\int 4 \log ^2(3+\log (3))dx}{2 \log ^2(3+\log (3))}+\frac {x \log ^2(3+\log (3))}{x^2 (2-\log (3))^2-\log ^2(3+\log (3))}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {x \log ^2(3+\log (3))}{x^2 (2-\log (3))^2-\log ^2(3+\log (3))}+2 x\) |
Int[(32*x^4 - 64*x^4*Log[3] + 48*x^4*Log[3]^2 - 16*x^4*Log[3]^3 + 2*x^4*Lo g[3]^4 + (-20*x^2 + 20*x^2*Log[3] - 5*x^2*Log[3]^2)*Log[Log[3*E^3]]^2 + Lo g[Log[3*E^3]]^4)/(16*x^4 - 32*x^4*Log[3] + 24*x^4*Log[3]^2 - 8*x^4*Log[3]^ 3 + x^4*Log[3]^4 + (-8*x^2 + 8*x^2*Log[3] - 2*x^2*Log[3]^2)*Log[Log[3*E^3] ]^2 + Log[Log[3*E^3]]^4),x]
3.9.88.3.1 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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 , x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x , 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q + 1)) Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[(u_)^(q_.)*(v_)^(p_.), x_Symbol] :> Int[ExpandToSum[u, x]^q*ExpandToSum [v, x]^p, x] /; FreeQ[{p, q}, x] && BinomialQ[u, x] && TrinomialQ[v, x] && !(BinomialMatchQ[u, x] && TrinomialMatchQ[v, x])
Int[(Pq_)*(u_)^(p_.), x_Symbol] :> Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[ p, x] && PolyQ[Pq, x] && TrinomialQ[u, x] && !TrinomialMatchQ[u, x]
Time = 1.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48
| method | result | size |
| risch | \(2 x +\frac {\ln \left (3+\ln \left (3\right )\right )^{2} x}{x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-\ln \left (3+\ln \left (3\right )\right )^{2}+4 x^{2}}\) | \(46\) |
| norman | \(\frac {\left (2 \ln \left (3\right )^{2}-8 \ln \left (3\right )+8\right ) x^{3}-\ln \left (3+\ln \left (3\right )\right )^{2} x}{x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-\ln \left (3+\ln \left (3\right )\right )^{2}+4 x^{2}}\) | \(61\) |
| gosper | \(\frac {x \left (2 x^{2} \ln \left (3\right )^{2}-8 x^{2} \ln \left (3\right )-{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}+8 x^{2}\right )}{x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}+4 x^{2}}\) | \(68\) |
| default | \(2 x +\frac {{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}}{2 \left (2-\ln \left (3\right )\right ) \left (-x \ln \left (3\right )+\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )+2 x \right )}+\frac {{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}}{2 \left (2-\ln \left (3\right )\right ) \left (-x \ln \left (3\right )-\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )+2 x \right )}\) | \(77\) |
| parallelrisch | \(\frac {2 x^{3} \ln \left (3\right )^{4}-16 x^{3} \ln \left (3\right )^{3}-\ln \left (3\right )^{2} {\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2} x +48 x^{3} \ln \left (3\right )^{2}+4 \ln \left (3\right ) {\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2} x -64 x^{3} \ln \left (3\right )-4 {\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2} x +32 x^{3}}{\left (\ln \left (3\right )^{2}-4 \ln \left (3\right )+4\right ) \left (x^{2} \ln \left (3\right )^{2}-4 x^{2} \ln \left (3\right )-{\ln \left (\ln \left (3 \,{\mathrm e}^{3}\right )\right )}^{2}+4 x^{2}\right )}\) | \(126\) |
int((ln(ln(3*exp(3)))^4+(-5*x^2*ln(3)^2+20*x^2*ln(3)-20*x^2)*ln(ln(3*exp(3 )))^2+2*x^4*ln(3)^4-16*x^4*ln(3)^3+48*x^4*ln(3)^2-64*x^4*ln(3)+32*x^4)/(ln (ln(3*exp(3)))^4+(-2*x^2*ln(3)^2+8*x^2*ln(3)-8*x^2)*ln(ln(3*exp(3)))^2+x^4 *ln(3)^4-8*x^4*ln(3)^3+24*x^4*ln(3)^2-32*x^4*ln(3)+16*x^4),x,method=_RETUR NVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \begin {dmath*} \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {2 \, x^{3} \log \left (3\right )^{2} - 8 \, x^{3} \log \left (3\right ) + 8 \, x^{3} - x \log \left (\log \left (3\right ) + 3\right )^{2}}{x^{2} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right ) + 4 \, x^{2} - \log \left (\log \left (3\right ) + 3\right )^{2}} \end {dmath*}
integrate((log(log(3*exp(3)))^4+(-5*x^2*log(3)^2+20*x^2*log(3)-20*x^2)*log (log(3*exp(3)))^2+2*x^4*log(3)^4-16*x^4*log(3)^3+48*x^4*log(3)^2-64*x^4*lo g(3)+32*x^4)/(log(log(3*exp(3)))^4+(-2*x^2*log(3)^2+8*x^2*log(3)-8*x^2)*lo g(log(3*exp(3)))^2+x^4*log(3)^4-8*x^4*log(3)^3+24*x^4*log(3)^2-32*x^4*log( 3)+16*x^4),x, algorithm=\
(2*x^3*log(3)^2 - 8*x^3*log(3) + 8*x^3 - x*log(log(3) + 3)^2)/(x^2*log(3)^ 2 - 4*x^2*log(3) + 4*x^2 - log(log(3) + 3)^2)
Time = 1.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \begin {dmath*} \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=2 x + \frac {x \log {\left (\log {\left (3 \right )} + 3 \right )}^{2}}{x^{2} \left (- 4 \log {\left (3 \right )} + \log {\left (3 \right )}^{2} + 4\right ) - \log {\left (\log {\left (3 \right )} + 3 \right )}^{2}} \end {dmath*}
integrate((ln(ln(3*exp(3)))**4+(-5*x**2*ln(3)**2+20*x**2*ln(3)-20*x**2)*ln (ln(3*exp(3)))**2+2*x**4*ln(3)**4-16*x**4*ln(3)**3+48*x**4*ln(3)**2-64*x** 4*ln(3)+32*x**4)/(ln(ln(3*exp(3)))**4+(-2*x**2*ln(3)**2+8*x**2*ln(3)-8*x** 2)*ln(ln(3*exp(3)))**2+x**4*ln(3)**4-8*x**4*ln(3)**3+24*x**4*ln(3)**2-32*x **4*ln(3)+16*x**4),x)
Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \begin {dmath*} \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {x \log \left (\log \left (3 \, e^{3}\right )\right )^{2}}{{\left (\log \left (3\right )^{2} - 4 \, \log \left (3\right ) + 4\right )} x^{2} - \log \left (\log \left (3 \, e^{3}\right )\right )^{2}} + 2 \, x \end {dmath*}
integrate((log(log(3*exp(3)))^4+(-5*x^2*log(3)^2+20*x^2*log(3)-20*x^2)*log (log(3*exp(3)))^2+2*x^4*log(3)^4-16*x^4*log(3)^3+48*x^4*log(3)^2-64*x^4*lo g(3)+32*x^4)/(log(log(3*exp(3)))^4+(-2*x^2*log(3)^2+8*x^2*log(3)-8*x^2)*lo g(log(3*exp(3)))^2+x^4*log(3)^4-8*x^4*log(3)^3+24*x^4*log(3)^2-32*x^4*log( 3)+16*x^4),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.19 \begin {dmath*} \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {x \log \left (\log \left (3 \, e^{3}\right )\right )^{2}}{x^{2} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right ) + 4 \, x^{2} - \log \left (\log \left (3 \, e^{3}\right )\right )^{2}} + \frac {2 \, {\left (x \log \left (3\right )^{4} - 8 \, x \log \left (3\right )^{3} + 24 \, x \log \left (3\right )^{2} - 32 \, x \log \left (3\right ) + 16 \, x\right )}}{\log \left (3\right )^{4} - 8 \, \log \left (3\right )^{3} + 24 \, \log \left (3\right )^{2} - 32 \, \log \left (3\right ) + 16} \end {dmath*}
integrate((log(log(3*exp(3)))^4+(-5*x^2*log(3)^2+20*x^2*log(3)-20*x^2)*log (log(3*exp(3)))^2+2*x^4*log(3)^4-16*x^4*log(3)^3+48*x^4*log(3)^2-64*x^4*lo g(3)+32*x^4)/(log(log(3*exp(3)))^4+(-2*x^2*log(3)^2+8*x^2*log(3)-8*x^2)*lo g(log(3*exp(3)))^2+x^4*log(3)^4-8*x^4*log(3)^3+24*x^4*log(3)^2-32*x^4*log( 3)+16*x^4),x, algorithm=\
x*log(log(3*e^3))^2/(x^2*log(3)^2 - 4*x^2*log(3) + 4*x^2 - log(log(3*e^3)) ^2) + 2*(x*log(3)^4 - 8*x*log(3)^3 + 24*x*log(3)^2 - 32*x*log(3) + 16*x)/( log(3)^4 - 8*log(3)^3 + 24*log(3)^2 - 32*log(3) + 16)
Time = 16.79 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \begin {dmath*} \int \frac {32 x^4-64 x^4 \log (3)+48 x^4 \log ^2(3)-16 x^4 \log ^3(3)+2 x^4 \log ^4(3)+\left (-20 x^2+20 x^2 \log (3)-5 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )}{16 x^4-32 x^4 \log (3)+24 x^4 \log ^2(3)-8 x^4 \log ^3(3)+x^4 \log ^4(3)+\left (-8 x^2+8 x^2 \log (3)-2 x^2 \log ^2(3)\right ) \log ^2\left (\log \left (3 e^3\right )\right )+\log ^4\left (\log \left (3 e^3\right )\right )} \, dx=\frac {x\,\left (2\,x^2\,{\ln \left (3\right )}^2-{\ln \left (\ln \left (3\right )+3\right )}^2-8\,x^2\,\ln \left (3\right )+8\,x^2\right )}{x^2\,{\ln \left (3\right )}^2-{\ln \left (\ln \left (3\right )+3\right )}^2-4\,x^2\,\ln \left (3\right )+4\,x^2} \end {dmath*}
int((48*x^4*log(3)^2 - 16*x^4*log(3)^3 + 2*x^4*log(3)^4 - log(log(3*exp(3) ))^2*(5*x^2*log(3)^2 - 20*x^2*log(3) + 20*x^2) - 64*x^4*log(3) + 32*x^4 + log(log(3*exp(3)))^4)/(24*x^4*log(3)^2 - 8*x^4*log(3)^3 + x^4*log(3)^4 - l og(log(3*exp(3)))^2*(2*x^2*log(3)^2 - 8*x^2*log(3) + 8*x^2) - 32*x^4*log(3 ) + 16*x^4 + log(log(3*exp(3)))^4),x)