Integrand size = 129, antiderivative size = 33 \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=\frac {x+\frac {1}{3} \left (3+\frac {x}{4+x^2}\right )}{-\frac {4}{3}-x+\frac {x}{\log (4)}} \] Output:
(x+1+1/3*x/(x^2+4))/(-4/3+1/2*x/ln(2)-x)
Leaf count is larger than twice the leaf count of optimal. \(346\) vs. \(2(33)=66\).
Time = 0.15 (sec) , antiderivative size = 346, normalized size of antiderivative = 10.48 \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=-\frac {\log (4) \left (3 x (-1+\log (4)) \left (1296+1296 \log ^4(4)+576 \log ^3(4) (-9+\log (256))-72 \log ^2(256)-19 \log ^4(256)-72 \log ^2(4) \left (-108+16 \log (256)+\log ^2(256)\right )+16 \log (4) \left (-324+36 \log (256)+9 \log ^2(256)+5 \log ^3(256)\right )\right )-3 \left (15552+1296 \log (256)+864 \log ^2(256)+168 \log ^3(256)+12 \log ^4(256)+\log ^5(256)+1296 \log ^4(4) (12+\log (256))-192 \log ^3(4) \left (324+27 \log (256)+2 \log ^2(256)\right )-48 \log (4) \left (1296+108 \log (256)+44 \log ^2(256)+7 \log ^3(256)\right )+24 \log ^2(4) \left (3888+324 \log (256)+68 \log ^2(256)+7 \log ^3(256)\right )\right )+x^2 \left (-11664+1296 \log ^5(4)-1296 \log (256)-648 \log ^2(256)-180 \log ^3(256)-9 \log ^4(256)-\log ^5(256)-1296 \log ^4(4) (13+\log (256))+72 \log ^3(4) \left (756+72 \log (256)+7 \log ^2(256)\right )-36 \log ^2(4) \left (2088+216 \log (256)+46 \log ^2(256)+5 \log ^3(256)\right )+\log (4) \left (47952+5184 \log (256)+1800 \log ^2(256)+360 \log ^3(256)+\log ^4(256)\right )\right )\right )}{3 \left (4+x^2\right ) (-1+\log (4)) (3 x (-1+\log (4))+\log (256)) \left (36-72 \log (4)+36 \log ^2(4)+\log ^2(256)\right )^2} \] Input:
Integrate[((-144 - 72*x^2 - 6*x^3 - 9*x^4)*Log[4] + (-64 - 20*x^2 + 6*x^3 - 3*x^4)*Log[4]^2)/(144*x^2 + 72*x^4 + 9*x^6 + (-384*x - 288*x^2 - 192*x^3 - 144*x^4 - 24*x^5 - 18*x^6)*Log[4] + (256 + 384*x + 272*x^2 + 192*x^3 + 88*x^4 + 24*x^5 + 9*x^6)*Log[4]^2),x]
Output:
-1/3*(Log[4]*(3*x*(-1 + Log[4])*(1296 + 1296*Log[4]^4 + 576*Log[4]^3*(-9 + Log[256]) - 72*Log[256]^2 - 19*Log[256]^4 - 72*Log[4]^2*(-108 + 16*Log[25 6] + Log[256]^2) + 16*Log[4]*(-324 + 36*Log[256] + 9*Log[256]^2 + 5*Log[25 6]^3)) - 3*(15552 + 1296*Log[256] + 864*Log[256]^2 + 168*Log[256]^3 + 12*L og[256]^4 + Log[256]^5 + 1296*Log[4]^4*(12 + Log[256]) - 192*Log[4]^3*(324 + 27*Log[256] + 2*Log[256]^2) - 48*Log[4]*(1296 + 108*Log[256] + 44*Log[2 56]^2 + 7*Log[256]^3) + 24*Log[4]^2*(3888 + 324*Log[256] + 68*Log[256]^2 + 7*Log[256]^3)) + x^2*(-11664 + 1296*Log[4]^5 - 1296*Log[256] - 648*Log[25 6]^2 - 180*Log[256]^3 - 9*Log[256]^4 - Log[256]^5 - 1296*Log[4]^4*(13 + Lo g[256]) + 72*Log[4]^3*(756 + 72*Log[256] + 7*Log[256]^2) - 36*Log[4]^2*(20 88 + 216*Log[256] + 46*Log[256]^2 + 5*Log[256]^3) + Log[4]*(47952 + 5184*L og[256] + 1800*Log[256]^2 + 360*Log[256]^3 + Log[256]^4))))/((4 + x^2)*(-1 + Log[4])*(3*x*(-1 + Log[4]) + Log[256])*(36 - 72*Log[4] + 36*Log[4]^2 + Log[256]^2)^2)
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(33)=66\).
Time = 0.63 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, 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.
| \(\displaystyle \int \frac {\left (-3 x^4+6 x^3-20 x^2-64\right ) \log ^2(4)+\left (-9 x^4-6 x^3-72 x^2-144\right ) \log (4)}{9 x^6+72 x^4+144 x^2+\left (9 x^6+24 x^5+88 x^4+192 x^3+272 x^2+384 x+256\right ) \log ^2(4)+\left (-18 x^6-24 x^5-144 x^4-192 x^3-288 x^2-384 x\right ) \log (4)} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {2 \log (4) (-3 x (1-\log (4))-4 \log (4))}{\left (x^2+4\right )^2 \left (9+13 \log ^2(4)-18 \log (4)\right )}+\frac {\log ^2(4)}{\left (x^2+4\right ) \left (9+13 \log ^2(4)-18 \log (4)\right )}+\frac {3 \log (4) \left (-27-16 \log ^3(4)-15 \log ^2(4)+42 \log (4)\right )}{\left (9+13 \log ^2(4)-18 \log (4)\right ) (3 x (1-\log (4))-\log (256))^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (4) (3 (1-\log (4))-x \log (4))}{\left (x^2+4\right ) \left (9+13 \log ^2(4)-18 \log (4)\right )}+\frac {\log (4) \left (27+16 \log ^3(4)+15 \log ^2(4)-42 \log (4)\right )}{(1-\log (4)) \left (9+13 \log ^2(4)-18 \log (4)\right ) (3 x (1-\log (4))-\log (256))}\) |
Input:
Int[((-144 - 72*x^2 - 6*x^3 - 9*x^4)*Log[4] + (-64 - 20*x^2 + 6*x^3 - 3*x^ 4)*Log[4]^2)/(144*x^2 + 72*x^4 + 9*x^6 + (-384*x - 288*x^2 - 192*x^3 - 144 *x^4 - 24*x^5 - 18*x^6)*Log[4] + (256 + 384*x + 272*x^2 + 192*x^3 + 88*x^4 + 24*x^5 + 9*x^6)*Log[4]^2),x]
Output:
(Log[4]*(3*(1 - Log[4]) - x*Log[4]))/((4 + x^2)*(9 - 18*Log[4] + 13*Log[4] ^2)) + (Log[4]*(27 - 42*Log[4] + 15*Log[4]^2 + 16*Log[4]^3))/((1 - Log[4]) *(9 - 18*Log[4] + 13*Log[4]^2)*(3*x*(1 - Log[4]) - Log[256]))
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. \(2(28)=56\).
Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91
| method | result | size |
| norman | \(\frac {-\frac {3 \left (4 \ln \left (2\right )^{2}+6 \ln \left (2\right )\right ) x^{3}}{8 \ln \left (2\right )}-\frac {\left (16 \ln \left (2\right )^{2}+18 \ln \left (2\right )\right ) x}{2 \ln \left (2\right )}}{\left (x^{2}+4\right ) \left (6 x \ln \left (2\right )+8 \ln \left (2\right )-3 x \right )}\) | \(63\) |
| parallelrisch | \(-\frac {12 x^{3} \ln \left (2\right )^{2}+18 x^{3} \ln \left (2\right )+64 x \ln \left (2\right )^{2}+72 x \ln \left (2\right )}{8 \ln \left (2\right ) \left (6 x^{3} \ln \left (2\right )+8 x^{2} \ln \left (2\right )-3 x^{3}+24 x \ln \left (2\right )+32 \ln \left (2\right )-12 x \right )}\) | \(70\) |
| gosper | \(\frac {2 \left (2 x^{2} \ln \left (2\right )-2 x \ln \left (2\right )+3 x^{2}+8 \ln \left (2\right )+x +12\right ) \ln \left (2\right )}{\left (6 x^{3} \ln \left (2\right )+8 x^{2} \ln \left (2\right )-3 x^{3}+24 x \ln \left (2\right )+32 \ln \left (2\right )-12 x \right ) \left (2 \ln \left (2\right )-1\right )}\) | \(71\) |
| risch | \(\frac {\frac {\ln \left (2\right ) \left (2 \ln \left (2\right )+3\right ) x^{2}}{6 \ln \left (2\right )-3}-\frac {x \ln \left (2\right )}{3}+\frac {4 \ln \left (2\right ) \left (2 \ln \left (2\right )+3\right )}{3 \left (2 \ln \left (2\right )-1\right )}}{x^{3} \ln \left (2\right )+\frac {4 x^{2} \ln \left (2\right )}{3}-\frac {x^{3}}{2}+4 x \ln \left (2\right )+\frac {16 \ln \left (2\right )}{3}-2 x}\) | \(80\) |
| default | \(2 \ln \left (2\right ) \left (\frac {-2 x \ln \left (2\right )+3-6 \ln \left (2\right )}{\left (x^{2}+4\right ) \left (52 \ln \left (2\right )^{2}-36 \ln \left (2\right )+9\right )}-\frac {-384 \ln \left (2\right )^{3}-180 \ln \left (2\right )^{2}+252 \ln \left (2\right )-81}{\left (52 \ln \left (2\right )^{2}-36 \ln \left (2\right )+9\right ) \left (6 \ln \left (2\right )-3\right ) \left (6 x \ln \left (2\right )+8 \ln \left (2\right )-3 x \right )}\right )\) | \(97\) |
Input:
int((4*(-3*x^4+6*x^3-20*x^2-64)*ln(2)^2+2*(-9*x^4-6*x^3-72*x^2-144)*ln(2)) /(4*(9*x^6+24*x^5+88*x^4+192*x^3+272*x^2+384*x+256)*ln(2)^2+2*(-18*x^6-24* x^5-144*x^4-192*x^3-288*x^2-384*x)*ln(2)+9*x^6+72*x^4+144*x^2),x,method=_R ETURNVERBOSE)
Output:
(-3/8*(4*ln(2)^2+6*ln(2))/ln(2)*x^3-1/2*(16*ln(2)^2+18*ln(2))/ln(2)*x)/(x^ 2+4)/(6*x*ln(2)+8*ln(2)-3*x)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (30) = 60\).
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.39 \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=\frac {2 \, {\left (2 \, {\left (x^{2} - x + 4\right )} \log \left (2\right )^{2} + {\left (3 \, x^{2} + x + 12\right )} \log \left (2\right )\right )}}{3 \, x^{3} + 4 \, {\left (3 \, x^{3} + 4 \, x^{2} + 12 \, x + 16\right )} \log \left (2\right )^{2} - 4 \, {\left (3 \, x^{3} + 2 \, x^{2} + 12 \, x + 8\right )} \log \left (2\right ) + 12 \, x} \] Input:
integrate((4*(-3*x^4+6*x^3-20*x^2-64)*log(2)^2+2*(-9*x^4-6*x^3-72*x^2-144) *log(2))/(4*(9*x^6+24*x^5+88*x^4+192*x^3+272*x^2+384*x+256)*log(2)^2+2*(-1 8*x^6-24*x^5-144*x^4-192*x^3-288*x^2-384*x)*log(2)+9*x^6+72*x^4+144*x^2),x , algorithm="fricas")
Output:
2*(2*(x^2 - x + 4)*log(2)^2 + (3*x^2 + x + 12)*log(2))/(3*x^3 + 4*(3*x^3 + 4*x^2 + 12*x + 16)*log(2)^2 - 4*(3*x^3 + 2*x^2 + 12*x + 8)*log(2) + 12*x)
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (22) = 44\).
Time = 2.37 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.09 \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=- \frac {x^{2} \left (- 6 \log {\left (2 \right )} - 4 \log {\left (2 \right )}^{2}\right ) + x \left (- 2 \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2}\right ) - 24 \log {\left (2 \right )} - 16 \log {\left (2 \right )}^{2}}{x^{3} \left (- 12 \log {\left (2 \right )} + 3 + 12 \log {\left (2 \right )}^{2}\right ) + x^{2} \left (- 8 \log {\left (2 \right )} + 16 \log {\left (2 \right )}^{2}\right ) + x \left (- 48 \log {\left (2 \right )} + 12 + 48 \log {\left (2 \right )}^{2}\right ) - 32 \log {\left (2 \right )} + 64 \log {\left (2 \right )}^{2}} \] Input:
integrate((4*(-3*x**4+6*x**3-20*x**2-64)*ln(2)**2+2*(-9*x**4-6*x**3-72*x** 2-144)*ln(2))/(4*(9*x**6+24*x**5+88*x**4+192*x**3+272*x**2+384*x+256)*ln(2 )**2+2*(-18*x**6-24*x**5-144*x**4-192*x**3-288*x**2-384*x)*ln(2)+9*x**6+72 *x**4+144*x**2),x)
Output:
-(x**2*(-6*log(2) - 4*log(2)**2) + x*(-2*log(2) + 4*log(2)**2) - 24*log(2) - 16*log(2)**2)/(x**3*(-12*log(2) + 3 + 12*log(2)**2) + x**2*(-8*log(2) + 16*log(2)**2) + x*(-48*log(2) + 12 + 48*log(2)**2) - 32*log(2) + 64*log(2 )**2)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (30) = 60\).
Time = 0.03 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.12 \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=\frac {2 \, {\left ({\left (2 \, \log \left (2\right )^{2} + 3 \, \log \left (2\right )\right )} x^{2} - {\left (2 \, \log \left (2\right )^{2} - \log \left (2\right )\right )} x + 8 \, \log \left (2\right )^{2} + 12 \, \log \left (2\right )\right )}}{3 \, {\left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 1\right )} x^{3} + 8 \, {\left (2 \, \log \left (2\right )^{2} - \log \left (2\right )\right )} x^{2} + 12 \, {\left (4 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) + 1\right )} x + 64 \, \log \left (2\right )^{2} - 32 \, \log \left (2\right )} \] Input:
integrate((4*(-3*x^4+6*x^3-20*x^2-64)*log(2)^2+2*(-9*x^4-6*x^3-72*x^2-144) *log(2))/(4*(9*x^6+24*x^5+88*x^4+192*x^3+272*x^2+384*x+256)*log(2)^2+2*(-1 8*x^6-24*x^5-144*x^4-192*x^3-288*x^2-384*x)*log(2)+9*x^6+72*x^4+144*x^2),x , algorithm="maxima")
Output:
2*((2*log(2)^2 + 3*log(2))*x^2 - (2*log(2)^2 - log(2))*x + 8*log(2)^2 + 12 *log(2))/(3*(4*log(2)^2 - 4*log(2) + 1)*x^3 + 8*(2*log(2)^2 - log(2))*x^2 + 12*(4*log(2)^2 - 4*log(2) + 1)*x + 64*log(2)^2 - 32*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (30) = 60\).
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.48 \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=\frac {2 \, {\left (2 \, x^{2} \log \left (2\right )^{2} + 3 \, x^{2} \log \left (2\right ) - 2 \, x \log \left (2\right )^{2} + x \log \left (2\right ) + 8 \, \log \left (2\right )^{2} + 12 \, \log \left (2\right )\right )}}{{\left (6 \, x^{3} \log \left (2\right ) - 3 \, x^{3} + 8 \, x^{2} \log \left (2\right ) + 24 \, x \log \left (2\right ) - 12 \, x + 32 \, \log \left (2\right )\right )} {\left (2 \, \log \left (2\right ) - 1\right )}} \] Input:
integrate((4*(-3*x^4+6*x^3-20*x^2-64)*log(2)^2+2*(-9*x^4-6*x^3-72*x^2-144) *log(2))/(4*(9*x^6+24*x^5+88*x^4+192*x^3+272*x^2+384*x+256)*log(2)^2+2*(-1 8*x^6-24*x^5-144*x^4-192*x^3-288*x^2-384*x)*log(2)+9*x^6+72*x^4+144*x^2),x , algorithm="giac")
Output:
2*(2*x^2*log(2)^2 + 3*x^2*log(2) - 2*x*log(2)^2 + x*log(2) + 8*log(2)^2 + 12*log(2))/((6*x^3*log(2) - 3*x^3 + 8*x^2*log(2) + 24*x*log(2) - 12*x + 32 *log(2))*(2*log(2) - 1))
Timed out. \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=\text {Hanged} \] Input:
int(-(4*log(2)^2*(20*x^2 - 6*x^3 + 3*x^4 + 64) + 2*log(2)*(72*x^2 + 6*x^3 + 9*x^4 + 144))/(4*log(2)^2*(384*x + 272*x^2 + 192*x^3 + 88*x^4 + 24*x^5 + 9*x^6 + 256) - 2*log(2)*(384*x + 288*x^2 + 192*x^3 + 144*x^4 + 24*x^5 + 1 8*x^6) + 144*x^2 + 72*x^4 + 9*x^6),x)
Output:
\text{Hanged}
Time = 0.15 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64 \[ \int \frac {\left (-144-72 x^2-6 x^3-9 x^4\right ) \log (4)+\left (-64-20 x^2+6 x^3-3 x^4\right ) \log ^2(4)}{144 x^2+72 x^4+9 x^6+\left (-384 x-288 x^2-192 x^3-144 x^4-24 x^5-18 x^6\right ) \log (4)+\left (256+384 x+272 x^2+192 x^3+88 x^4+24 x^5+9 x^6\right ) \log ^2(4)} \, dx=\frac {x \left (-6 \,\mathrm {log}\left (2\right ) x^{2}-32 \,\mathrm {log}\left (2\right )-9 x^{2}-36\right )}{24 \,\mathrm {log}\left (2\right ) x^{3}+32 \,\mathrm {log}\left (2\right ) x^{2}+96 \,\mathrm {log}\left (2\right ) x +128 \,\mathrm {log}\left (2\right )-12 x^{3}-48 x} \] Input:
int((4*(-3*x^4+6*x^3-20*x^2-64)*log(2)^2+2*(-9*x^4-6*x^3-72*x^2-144)*log(2 ))/(4*(9*x^6+24*x^5+88*x^4+192*x^3+272*x^2+384*x+256)*log(2)^2+2*(-18*x^6- 24*x^5-144*x^4-192*x^3-288*x^2-384*x)*log(2)+9*x^6+72*x^4+144*x^2),x)
Output:
(x*( - 6*log(2)*x**2 - 32*log(2) - 9*x**2 - 36))/(4*(6*log(2)*x**3 + 8*log (2)*x**2 + 24*log(2)*x + 32*log(2) - 3*x**3 - 12*x))