Integrand size = 106, antiderivative size = 26 \[ \int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+\left (108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6\right ) \log (3)+\left (108 x^3+1728 x^4+9216 x^5+16384 x^6\right ) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx=\left (1+x-x^2-\frac {x}{3+16 x}+x^2 \log (3)\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(26)=52\).
Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.38 \[ \int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+\left (108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6\right ) \log (3)+\left (108 x^3+1728 x^4+9216 x^5+16384 x^6\right ) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx=\frac {16777216 x^6 (-1+\log (3))^2-6912 x^2 \left (-2493+370 \log (3)+3 \log ^2(3)-220 \log (9)\right )+65536 x^4 \left (-407+714 \log (3)+9 \log ^2(3)-30 \log (9)\right )+98304 x^3 (248+106 \log (3)+15 \log (9))-288 x \left (-12709+1170 \log (3)+27 \log ^2(3)-900 \log (9)+512 \log (27)\right )-9 \left (-29935+270 \log (3)+81 \log ^2(3)-1080 \log (9)+1536 \log (27)\right )+2097152 x^5 \left (-13+3 \log ^2(3)+\log (59049)\right )}{65536 (3+16 x)^2} \]
Integrate[(36 + 684*x + 3348*x^2 + 1068*x^3 - 19264*x^4 - 15360*x^5 + 1638 4*x^6 + (108*x + 1836*x^2 + 11112*x^3 + 25728*x^4 + 6144*x^5 - 32768*x^6)* Log[3] + (108*x^3 + 1728*x^4 + 9216*x^5 + 16384*x^6)*Log[3]^2)/(27 + 432*x + 2304*x^2 + 4096*x^3),x]
(16777216*x^6*(-1 + Log[3])^2 - 6912*x^2*(-2493 + 370*Log[3] + 3*Log[3]^2 - 220*Log[9]) + 65536*x^4*(-407 + 714*Log[3] + 9*Log[3]^2 - 30*Log[9]) + 9 8304*x^3*(248 + 106*Log[3] + 15*Log[9]) - 288*x*(-12709 + 1170*Log[3] + 27 *Log[3]^2 - 900*Log[9] + 512*Log[27]) - 9*(-29935 + 270*Log[3] + 81*Log[3] ^2 - 1080*Log[9] + 1536*Log[27]) + 2097152*x^5*(-13 + 3*Log[3]^2 + Log[590 49]))/(65536*(3 + 16*x)^2)
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).
Time = 0.37 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.77, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2007, 2389, 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.
\(\displaystyle \int \frac {16384 x^6-15360 x^5-19264 x^4+1068 x^3+3348 x^2+\left (16384 x^6+9216 x^5+1728 x^4+108 x^3\right ) \log ^2(3)+\left (-32768 x^6+6144 x^5+25728 x^4+11112 x^3+1836 x^2+108 x\right ) \log (3)+684 x+36}{4096 x^3+2304 x^2+432 x+27} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {16384 x^6-15360 x^5-19264 x^4+1068 x^3+3348 x^2+\left (16384 x^6+9216 x^5+1728 x^4+108 x^3\right ) \log ^2(3)+\left (-32768 x^6+6144 x^5+25728 x^4+11112 x^3+1836 x^2+108 x\right ) \log (3)+684 x+36}{(16 x+3)^3}dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (4 x^3 (\log (3)-1)^2+6 x^2 (\log (3)-1)-\frac {9}{8 (16 x+3)^3}+\frac {1}{4} x (15 \log (3)-7)-\frac {9 (61+\log (27))}{128 (16 x+3)^2}+\frac {3}{128} (79+\log (3))\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^4 (1-\log (3))^2-2 x^3 (1-\log (3))-\frac {1}{8} x^2 (7-15 \log (3))+\frac {9}{256 (16 x+3)^2}+\frac {3}{128} x (79+\log (3))+\frac {9 (61+\log (27))}{2048 (16 x+3)}\) |
Int[(36 + 684*x + 3348*x^2 + 1068*x^3 - 19264*x^4 - 15360*x^5 + 16384*x^6 + (108*x + 1836*x^2 + 11112*x^3 + 25728*x^4 + 6144*x^5 - 32768*x^6)*Log[3] + (108*x^3 + 1728*x^4 + 9216*x^5 + 16384*x^6)*Log[3]^2)/(27 + 432*x + 230 4*x^2 + 4096*x^3),x]
9/(256*(3 + 16*x)^2) - (x^2*(7 - 15*Log[3]))/8 - 2*x^3*(1 - Log[3]) + x^4* (1 - Log[3])^2 + (3*x*(79 + Log[3]))/128 + (9*(61 + Log[27]))/(2048*(3 + 1 6*x))
3.28.25.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(26)=52\).
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92
method | result | size |
default | \(x^{4} \ln \left (3\right )^{2}-2 x^{4} \ln \left (3\right )+x^{4}+2 x^{3} \ln \left (3\right )-2 x^{3}+\frac {15 x^{2} \ln \left (3\right )}{8}-\frac {7 x^{2}}{8}+\frac {3 x \ln \left (3\right )}{128}+\frac {237 x}{128}-\frac {4 \left (-\frac {27 \ln \left (3\right )}{8192}-\frac {549}{8192}\right )}{16 x +3}+\frac {9}{256 \left (16 x +3\right )^{2}}\) | \(76\) |
risch | \(x^{4} \ln \left (3\right )^{2}-2 x^{4} \ln \left (3\right )+2 x^{3} \ln \left (3\right )+x^{4}+\frac {15 x^{2} \ln \left (3\right )}{8}-2 x^{3}+\frac {3 x \ln \left (3\right )}{128}-\frac {7 x^{2}}{8}+\frac {237 x}{128}+\frac {\frac {\left (\frac {549}{256}+\frac {27 \ln \left (3\right )}{256}\right ) x}{128}+\frac {1719}{524288}+\frac {81 \ln \left (3\right )}{524288}}{x^{2}+\frac {3}{8} x +\frac {9}{256}}\) | \(78\) |
norman | \(\frac {\left (372+204 \ln \left (3\right )\right ) x^{3}+12 x +\left (-416+96 \ln \left (3\right )^{2}+320 \ln \left (3\right )\right ) x^{5}+\left (-407+654 \ln \left (3\right )+9 \ln \left (3\right )^{2}\right ) x^{4}+\left (146+18 \ln \left (3\right )\right ) x^{2}+\left (256 \ln \left (3\right )^{2}-512 \ln \left (3\right )+256\right ) x^{6}}{\left (16 x +3\right )^{2}}\) | \(81\) |
gosper | \(\frac {x \left (256 x^{5} \ln \left (3\right )^{2}+96 x^{4} \ln \left (3\right )^{2}-512 x^{5} \ln \left (3\right )+9 x^{3} \ln \left (3\right )^{2}+320 x^{4} \ln \left (3\right )+256 x^{5}+654 x^{3} \ln \left (3\right )-416 x^{4}+204 x^{2} \ln \left (3\right )-407 x^{3}+18 x \ln \left (3\right )+372 x^{2}+146 x +12\right )}{256 x^{2}+96 x +9}\) | \(100\) |
parallelrisch | \(\frac {2304 x^{6} \ln \left (3\right )^{2}+864 x^{5} \ln \left (3\right )^{2}-4608 x^{6} \ln \left (3\right )+81 x^{4} \ln \left (3\right )^{2}+2880 x^{5} \ln \left (3\right )+2304 x^{6}+5886 x^{4} \ln \left (3\right )-3744 x^{5}+1836 x^{3} \ln \left (3\right )-3663 x^{4}+162 x^{2} \ln \left (3\right )+3348 x^{3}+1314 x^{2}+108 x}{2304 x^{2}+864 x +81}\) | \(106\) |
int(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*ln(3)^2+(-32768*x^6+6144*x^5+25 728*x^4+11112*x^3+1836*x^2+108*x)*ln(3)+16384*x^6-15360*x^5-19264*x^4+1068 *x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x,method=_RETURNVERBO SE)
x^4*ln(3)^2-2*x^4*ln(3)+x^4+2*x^3*ln(3)-2*x^3+15/8*x^2*ln(3)-7/8*x^2+3/128 *x*ln(3)+237/128*x-4*(-27/8192*ln(3)-549/8192)/(16*x+3)+9/256/(16*x+3)^2
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (26) = 52\).
Time = 0.24 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+\left (108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6\right ) \log (3)+\left (108 x^3+1728 x^4+9216 x^5+16384 x^6\right ) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx=\frac {524288 \, x^{6} - 851968 \, x^{5} - 833536 \, x^{4} + 761856 \, x^{3} + 2048 \, {\left (256 \, x^{6} + 96 \, x^{5} + 9 \, x^{4}\right )} \log \left (3\right )^{2} + 347904 \, x^{2} - {\left (1048576 \, x^{6} - 655360 \, x^{5} - 1339392 \, x^{4} - 417792 \, x^{3} - 39168 \, x^{2} - 864 \, x - 81\right )} \log \left (3\right ) + 42912 \, x + 1719}{2048 \, {\left (256 \, x^{2} + 96 \, x + 9\right )}} \]
integrate(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*log(3)^2+(-32768*x^6+6144 *x^5+25728*x^4+11112*x^3+1836*x^2+108*x)*log(3)+16384*x^6-15360*x^5-19264* x^4+1068*x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x, algorithm= \
1/2048*(524288*x^6 - 851968*x^5 - 833536*x^4 + 761856*x^3 + 2048*(256*x^6 + 96*x^5 + 9*x^4)*log(3)^2 + 347904*x^2 - (1048576*x^6 - 655360*x^5 - 1339 392*x^4 - 417792*x^3 - 39168*x^2 - 864*x - 81)*log(3) + 42912*x + 1719)/(2 56*x^2 + 96*x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (20) = 40\).
Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.88 \[ \int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+\left (108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6\right ) \log (3)+\left (108 x^3+1728 x^4+9216 x^5+16384 x^6\right ) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx=x^{4} \left (- 2 \log {\left (3 \right )} + 1 + \log {\left (3 \right )}^{2}\right ) + x^{3} \left (-2 + 2 \log {\left (3 \right )}\right ) + x^{2} \left (- \frac {7}{8} + \frac {15 \log {\left (3 \right )}}{8}\right ) + x \left (\frac {3 \log {\left (3 \right )}}{128} + \frac {237}{128}\right ) + \frac {x \left (432 \log {\left (3 \right )} + 8784\right ) + 81 \log {\left (3 \right )} + 1719}{524288 x^{2} + 196608 x + 18432} \]
integrate(((16384*x**6+9216*x**5+1728*x**4+108*x**3)*ln(3)**2+(-32768*x**6 +6144*x**5+25728*x**4+11112*x**3+1836*x**2+108*x)*ln(3)+16384*x**6-15360*x **5-19264*x**4+1068*x**3+3348*x**2+684*x+36)/(4096*x**3+2304*x**2+432*x+27 ),x)
x**4*(-2*log(3) + 1 + log(3)**2) + x**3*(-2 + 2*log(3)) + x**2*(-7/8 + 15* log(3)/8) + x*(3*log(3)/128 + 237/128) + (x*(432*log(3) + 8784) + 81*log(3 ) + 1719)/(524288*x**2 + 196608*x + 18432)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+\left (108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6\right ) \log (3)+\left (108 x^3+1728 x^4+9216 x^5+16384 x^6\right ) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx={\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) + 1\right )} x^{4} + 2 \, x^{3} {\left (\log \left (3\right ) - 1\right )} + \frac {1}{8} \, x^{2} {\left (15 \, \log \left (3\right ) - 7\right )} + \frac {3}{128} \, x {\left (\log \left (3\right ) + 79\right )} + \frac {9 \, {\left (16 \, x {\left (3 \, \log \left (3\right ) + 61\right )} + 9 \, \log \left (3\right ) + 191\right )}}{2048 \, {\left (256 \, x^{2} + 96 \, x + 9\right )}} \]
integrate(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*log(3)^2+(-32768*x^6+6144 *x^5+25728*x^4+11112*x^3+1836*x^2+108*x)*log(3)+16384*x^6-15360*x^5-19264* x^4+1068*x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x, algorithm= \
(log(3)^2 - 2*log(3) + 1)*x^4 + 2*x^3*(log(3) - 1) + 1/8*x^2*(15*log(3) - 7) + 3/128*x*(log(3) + 79) + 9/2048*(16*x*(3*log(3) + 61) + 9*log(3) + 191 )/(256*x^2 + 96*x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+\left (108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6\right ) \log (3)+\left (108 x^3+1728 x^4+9216 x^5+16384 x^6\right ) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx=x^{4} \log \left (3\right )^{2} - 2 \, x^{4} \log \left (3\right ) + x^{4} + 2 \, x^{3} \log \left (3\right ) - 2 \, x^{3} + \frac {15}{8} \, x^{2} \log \left (3\right ) - \frac {7}{8} \, x^{2} + \frac {3}{128} \, x \log \left (3\right ) + \frac {237}{128} \, x + \frac {9 \, {\left (48 \, x \log \left (3\right ) + 976 \, x + 9 \, \log \left (3\right ) + 191\right )}}{2048 \, {\left (16 \, x + 3\right )}^{2}} \]
integrate(((16384*x^6+9216*x^5+1728*x^4+108*x^3)*log(3)^2+(-32768*x^6+6144 *x^5+25728*x^4+11112*x^3+1836*x^2+108*x)*log(3)+16384*x^6-15360*x^5-19264* x^4+1068*x^3+3348*x^2+684*x+36)/(4096*x^3+2304*x^2+432*x+27),x, algorithm= \
x^4*log(3)^2 - 2*x^4*log(3) + x^4 + 2*x^3*log(3) - 2*x^3 + 15/8*x^2*log(3) - 7/8*x^2 + 3/128*x*log(3) + 237/128*x + 9/2048*(48*x*log(3) + 976*x + 9* log(3) + 191)/(16*x + 3)^2
Time = 13.82 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.19 \[ \int \frac {36+684 x+3348 x^2+1068 x^3-19264 x^4-15360 x^5+16384 x^6+\left (108 x+1836 x^2+11112 x^3+25728 x^4+6144 x^5-32768 x^6\right ) \log (3)+\left (108 x^3+1728 x^4+9216 x^5+16384 x^6\right ) \log ^2(3)}{27+432 x+2304 x^2+4096 x^3} \, dx=x^3\,\left (\frac {\ln \left (3\right )}{2}-\frac {3\,{\left (\ln \left (3\right )-1\right )}^2}{4}+\frac {3\,{\ln \left (3\right )}^2}{4}-\frac {5}{4}\right )+x^2\,\left (\frac {87\,\ln \left (3\right )}{32}+\frac {27\,{\left (\ln \left (3\right )-1\right )}^2}{64}-\frac {27\,{\ln \left (3\right )}^2}{64}-\frac {83}{64}\right )+\frac {\frac {81\,\ln \left (3\right )}{16}+x\,\left (27\,\ln \left (3\right )+549\right )+\frac {1719}{16}}{32768\,x^2+12288\,x+1152}+x^4\,{\left (\ln \left (3\right )-1\right )}^2-x\,\left (\frac {129\,\ln \left (3\right )}{256}+\frac {135\,{\left (\ln \left (3\right )-1\right )}^2}{512}-\frac {135\,{\ln \left (3\right )}^2}{512}-\frac {1083}{512}\right ) \]
int((684*x + log(3)*(108*x + 1836*x^2 + 11112*x^3 + 25728*x^4 + 6144*x^5 - 32768*x^6) + log(3)^2*(108*x^3 + 1728*x^4 + 9216*x^5 + 16384*x^6) + 3348* x^2 + 1068*x^3 - 19264*x^4 - 15360*x^5 + 16384*x^6 + 36)/(432*x + 2304*x^2 + 4096*x^3 + 27),x)
x^3*(log(3)/2 - (3*(log(3) - 1)^2)/4 + (3*log(3)^2)/4 - 5/4) + x^2*((87*lo g(3))/32 + (27*(log(3) - 1)^2)/64 - (27*log(3)^2)/64 - 83/64) + ((81*log(3 ))/16 + x*(27*log(3) + 549) + 1719/16)/(12288*x + 32768*x^2 + 1152) + x^4* (log(3) - 1)^2 - x*((129*log(3))/256 + (135*(log(3) - 1)^2)/512 - (135*log (3)^2)/512 - 1083/512)