Integrand size = 184, antiderivative size = 29 \[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=9 \left (3+x-\frac {1}{\left (x+\frac {x}{x+5 (i \pi +\log (4))^2}\right )^2}\right ) \] Output:
9*x+27-9/(x+x/(x+5*(2*ln(2)+I*Pi)^2))^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(163\) vs. \(2(29)=58\).
Time = 0.13 (sec) , antiderivative size = 163, normalized size of antiderivative = 5.62 \[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=\frac {9 \left (-x^2+x^5+25 \pi ^4 \left (-1+x^3\right )-100 i \pi ^3 \left (-1+x^3\right ) \log (4)-10 x \log ^2(4)-25 \log ^4(4)+2 x^4 \left (1+5 \log ^2(4)\right )+x^3 \left (1+5 \log ^2(4)\right )^2+20 i \pi \log (4) \left (-x+x^4-5 \log ^2(4)+x^3 \left (1+5 \log ^2(4)\right )\right )-10 \pi ^2 \left (-x+x^4-15 \log ^2(4)+x^3 \left (1+15 \log ^2(4)\right )\right )\right )}{x^2 \left (1-5 \pi ^2+x+10 i \pi \log (4)+5 \log ^2(4)\right )^2} \] Input:
Integrate[(27*x^3 + 27*x^4 + 27*x^5 + 9*x^6 + (90*x + 270*x^2 + 135*x^3 + 270*x^4 + 135*x^5)*(I*Pi + Log[4])^2 + (450 + 1350*x + 675*x^3 + 675*x^4)* (I*Pi + Log[4])^4 + (2250 + 1125*x^3)*(I*Pi + Log[4])^6)/(x^3 + 3*x^4 + 3* x^5 + x^6 + (15*x^3 + 30*x^4 + 15*x^5)*(I*Pi + Log[4])^2 + (75*x^3 + 75*x^ 4)*(I*Pi + Log[4])^4 + 125*x^3*(I*Pi + Log[4])^6),x]
Output:
(9*(-x^2 + x^5 + 25*Pi^4*(-1 + x^3) - (100*I)*Pi^3*(-1 + x^3)*Log[4] - 10* x*Log[4]^2 - 25*Log[4]^4 + 2*x^4*(1 + 5*Log[4]^2) + x^3*(1 + 5*Log[4]^2)^2 + (20*I)*Pi*Log[4]*(-x + x^4 - 5*Log[4]^2 + x^3*(1 + 5*Log[4]^2)) - 10*Pi ^2*(-x + x^4 - 15*Log[4]^2 + x^3*(1 + 15*Log[4]^2))))/(x^2*(1 - 5*Pi^2 + x + (10*I)*Pi*Log[4] + 5*Log[4]^2)^2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(182\) vs. \(2(29)=58\).
Time = 0.97 (sec) , antiderivative size = 182, normalized size of antiderivative = 6.28, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6, 2026, 2007, 2123, 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.
| \(\displaystyle \int \frac {9 x^6+27 x^5+27 x^4+27 x^3+\left (1125 x^3+2250\right ) (\log (4)+i \pi )^6+\left (675 x^4+675 x^3+1350 x+450\right ) (\log (4)+i \pi )^4+\left (135 x^5+270 x^4+135 x^3+270 x^2+90 x\right ) (\log (4)+i \pi )^2}{x^6+3 x^5+3 x^4+x^3+125 x^3 (\log (4)+i \pi )^6+\left (75 x^4+75 x^3\right ) (\log (4)+i \pi )^4+\left (15 x^5+30 x^4+15 x^3\right ) (\log (4)+i \pi )^2} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {9 x^6+27 x^5+27 x^4+27 x^3+\left (1125 x^3+2250\right ) (\log (4)+i \pi )^6+\left (675 x^4+675 x^3+1350 x+450\right ) (\log (4)+i \pi )^4+\left (135 x^5+270 x^4+135 x^3+270 x^2+90 x\right ) (\log (4)+i \pi )^2}{x^6+3 x^5+3 x^4+x^3 \left (1+125 (\log (4)+i \pi )^6\right )+\left (75 x^4+75 x^3\right ) (\log (4)+i \pi )^4+\left (15 x^5+30 x^4+15 x^3\right ) (\log (4)+i \pi )^2}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {9 x^6+27 x^5+27 x^4+27 x^3+\left (1125 x^3+2250\right ) (\log (4)+i \pi )^6+\left (675 x^4+675 x^3+1350 x+450\right ) (\log (4)+i \pi )^4+\left (135 x^5+270 x^4+135 x^3+270 x^2+90 x\right ) (\log (4)+i \pi )^2}{x^3 \left (x^3+3 x^2 \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )+3 x \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^2+\left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^3\right )}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {9 x^6+27 x^5+27 x^4+27 x^3+\left (1125 x^3+2250\right ) (\log (4)+i \pi )^6+\left (675 x^4+675 x^3+1350 x+450\right ) (\log (4)+i \pi )^4+\left (135 x^5+270 x^4+135 x^3+270 x^2+90 x\right ) (\log (4)+i \pi )^2}{x^3 \left (x-5 \pi ^2+1+5 \log ^2(4)+10 i \pi \log (4)\right )^3}dx\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \int \left (\frac {450 (\pi -i \log (4))^4}{x^3 \left (-1+5 \pi ^2-5 \log ^2(4)-10 i \pi \log (4)\right )^2}+\frac {90 (\pi -i \log (4))^2}{x^2 \left (-1+5 \pi ^2-5 \log ^2(4)-10 i \pi \log (4)\right )^3}-\frac {90 (\pi -i \log (4))^2}{\left (-1+5 \pi ^2-5 \log ^2(4)-10 i \pi \log (4)\right )^3 \left (-x+5 \pi ^2-1-5 \log ^2(4)-10 i \pi \log (4)\right )^2}-\frac {18}{\left (-1+5 \pi ^2-5 \log ^2(4)-10 i \pi \log (4)\right )^2 \left (-x+5 \pi ^2-1-5 \log ^2(4)-10 i \pi \log (4)\right )^3}+9\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {225 (\pi -i \log (4))^4}{x^2 \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^2}+9 x-\frac {90 (\pi -i \log (4))^2}{\left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^3 \left (x-5 \pi ^2+1+5 \log ^2(4)+10 i \pi \log (4)\right )}-\frac {9}{\left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^2 \left (x-5 \pi ^2+1+5 \log ^2(4)+10 i \pi \log (4)\right )^2}+\frac {90 (\pi -i \log (4))^2}{x \left (1-5 \pi ^2+5 \log ^2(4)+10 i \pi \log (4)\right )^3}\) |
Input:
Int[(27*x^3 + 27*x^4 + 27*x^5 + 9*x^6 + (90*x + 270*x^2 + 135*x^3 + 270*x^ 4 + 135*x^5)*(I*Pi + Log[4])^2 + (450 + 1350*x + 675*x^3 + 675*x^4)*(I*Pi + Log[4])^4 + (2250 + 1125*x^3)*(I*Pi + Log[4])^6)/(x^3 + 3*x^4 + 3*x^5 + x^6 + (15*x^3 + 30*x^4 + 15*x^5)*(I*Pi + Log[4])^2 + (75*x^3 + 75*x^4)*(I* Pi + Log[4])^4 + 125*x^3*(I*Pi + Log[4])^6),x]
Output:
9*x + (90*(Pi - I*Log[4])^2)/(x*(1 - 5*Pi^2 + (10*I)*Pi*Log[4] + 5*Log[4]^ 2)^3) - (225*(Pi - I*Log[4])^4)/(x^2*(1 - 5*Pi^2 + (10*I)*Pi*Log[4] + 5*Lo g[4]^2)^2) - 9/((1 - 5*Pi^2 + (10*I)*Pi*Log[4] + 5*Log[4]^2)^2*(1 - 5*Pi^2 + x + (10*I)*Pi*Log[4] + 5*Log[4]^2)^2) - (90*(Pi - I*Log[4])^2)/((1 - 5* Pi^2 + (10*I)*Pi*Log[4] + 5*Log[4]^2)^3*(1 - 5*Pi^2 + x + (10*I)*Pi*Log[4] + 5*Log[4]^2))
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[(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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (30 ) = 60\).
Time = 3.75 (sec) , antiderivative size = 154, normalized size of antiderivative = 5.31
| method | result | size |
| risch | \(9 x +\frac {-\frac {9 x^{2}}{25}+\left (-\frac {72 i \pi \ln \left (2\right )}{5}+\frac {18 \pi ^{2}}{5}-\frac {72 \ln \left (2\right )^{2}}{5}\right ) x +72 i \pi ^{3} \ln \left (2\right )-288 i \pi \ln \left (2\right )^{3}-9 \pi ^{4}+216 \pi ^{2} \ln \left (2\right )^{2}-144 \ln \left (2\right )^{4}}{x^{2} \left (-8 i \pi ^{3} \ln \left (2\right )+32 i \pi \ln \left (2\right )^{3}+\pi ^{4}-24 \pi ^{2} \ln \left (2\right )^{2}+\frac {8 i \pi \ln \left (2\right ) x}{5}+16 \ln \left (2\right )^{4}-\frac {2 x \,\pi ^{2}}{5}+\frac {8 i \pi \ln \left (2\right )}{5}+\frac {8 x \ln \left (2\right )^{2}}{5}-\frac {2 \pi ^{2}}{5}+\frac {8 \ln \left (2\right )^{2}}{5}+\frac {x^{2}}{25}+\frac {2 x}{25}+\frac {1}{25}\right )}\) | \(154\) |
| parallelrisch | \(\frac {1080 x^{3} \ln \left (2\right )^{2}-2250 \pi ^{6} x^{2}+1350 \pi ^{4} x^{2}-32400 \pi ^{2} \ln \left (2\right )^{2} x^{2}-16200 \pi ^{2} \ln \left (2\right )^{2} x^{3}-90 x \,\pi ^{2}+360 x \ln \left (2\right )^{2}+1080 x^{2} \ln \left (2\right )^{2}+10800 \ln \left (2\right )^{4} x^{3}+21600 \ln \left (2\right )^{4} x^{2}+144000 \ln \left (2\right )^{6} x^{2}+3600 \ln \left (2\right )^{4}+27 x^{3}+27 x^{2}-9 x^{5}-270 \pi ^{2} x^{3}-5400 \pi ^{2} \ln \left (2\right )^{2}-270 \pi ^{2} x^{2}-1800 i \pi ^{3} \ln \left (2\right )+7200 i \pi \ln \left (2\right )^{3}+675 \pi ^{4} x^{3}+27000 i \pi ^{5} \ln \left (2\right ) x^{2}-360000 i \pi ^{3} \ln \left (2\right )^{3} x^{2}+432000 i \pi \ln \left (2\right )^{5} x^{2}-10800 i \pi ^{3} \ln \left (2\right ) x^{2}+43200 i \pi \ln \left (2\right )^{3} x^{2}+1080 i \pi \ln \left (2\right ) x^{3}+1080 i \pi \ln \left (2\right ) x^{2}-5400 i \pi ^{3} \ln \left (2\right ) x^{3}+21600 i \pi \ln \left (2\right )^{3} x^{3}+225 \pi ^{4}+360 i \pi \ln \left (2\right ) x -540000 \pi ^{2} \ln \left (2\right )^{4} x^{2}+135000 \pi ^{4} \ln \left (2\right )^{2} x^{2}}{x^{2} \left (200 i \pi ^{3} \ln \left (2\right )-800 i \pi \ln \left (2\right )^{3}-40 i \pi \ln \left (2\right ) x -25 \pi ^{4}+600 \pi ^{2} \ln \left (2\right )^{2}-400 \ln \left (2\right )^{4}-40 i \pi \ln \left (2\right )+10 x \,\pi ^{2}-40 x \ln \left (2\right )^{2}+10 \pi ^{2}-40 \ln \left (2\right )^{2}-x^{2}-2 x -1\right )}\) | \(392\) |
| default | \(9 x -\frac {9 \left (-400 i \pi ^{3} \ln \left (2\right )+1600 i \pi \ln \left (2\right )^{3}+50 \pi ^{4}-1200 \pi ^{2} \ln \left (2\right )^{2}+800 \ln \left (2\right )^{4}+40 i \pi \ln \left (2\right )-10 \pi ^{2}+40 \ln \left (2\right )^{2}\right )}{\left (20 i \pi \ln \left (2\right )-5 \pi ^{2}+20 \ln \left (2\right )^{2}+1\right )^{4} x}-\frac {9 \left (-400 i \pi ^{3} \ln \left (2\right )+6000 i \pi ^{5} \ln \left (2\right )+560000 i \pi ^{5} \ln \left (2\right )^{3}-2240000 i \pi ^{3} \ln \left (2\right )^{5}+1250 \pi ^{8}-140000 \pi ^{6} \ln \left (2\right )^{2}+1400000 \pi ^{4} \ln \left (2\right )^{4}-2240000 \pi ^{2} \ln \left (2\right )^{6}+320000 \ln \left (2\right )^{8}+96000 i \pi \ln \left (2\right )^{5}+1280000 i \pi \ln \left (2\right )^{7}+1600 i \pi \ln \left (2\right )^{3}-500 \pi ^{6}+30000 \pi ^{4} \ln \left (2\right )^{2}-120000 \pi ^{2} \ln \left (2\right )^{4}+32000 \ln \left (2\right )^{6}-80000 i \pi ^{3} \ln \left (2\right )^{3}-20000 i \pi ^{7} \ln \left (2\right )+50 \pi ^{4}-1200 \pi ^{2} \ln \left (2\right )^{2}+800 \ln \left (2\right )^{4}\right )}{2 \left (20 i \pi \ln \left (2\right )-5 \pi ^{2}+20 \ln \left (2\right )^{2}+1\right )^{4} x^{2}}-\frac {9 \left (400 i \pi ^{3} \ln \left (2\right )-1600 i \pi \ln \left (2\right )^{3}-50 \pi ^{4}+1200 \pi ^{2} \ln \left (2\right )^{2}-800 \ln \left (2\right )^{4}-40 i \pi \ln \left (2\right )+10 \pi ^{2}-40 \ln \left (2\right )^{2}\right )}{\left (20 i \pi \ln \left (2\right )-5 \pi ^{2}+20 \ln \left (2\right )^{2}+1\right )^{4} \left (-5 \pi ^{2}+20 i \pi \ln \left (2\right )+20 \ln \left (2\right )^{2}+x +1\right )}-\frac {9 \left (-1200 \pi ^{2} \ln \left (2\right )^{2}-400 i \pi ^{3} \ln \left (2\right )+1600 i \pi \ln \left (2\right )^{3}+50 \pi ^{4}+800 \ln \left (2\right )^{4}+80 i \pi \ln \left (2\right )-20 \pi ^{2}+80 \ln \left (2\right )^{2}+2\right )}{2 \left (20 i \pi \ln \left (2\right )-5 \pi ^{2}+20 \ln \left (2\right )^{2}+1\right )^{4} \left (-5 \pi ^{2}+20 i \pi \ln \left (2\right )+20 \ln \left (2\right )^{2}+x +1\right )^{2}}\) | \(476\) |
| gosper | \(-\frac {9 \left (-5 \pi ^{2}+20 i \pi \ln \left (2\right )+20 \ln \left (2\right )^{2}+x +1\right ) \left (120 x^{3} \ln \left (2\right )^{2}-250 \pi ^{6} x^{2}+150 \pi ^{4} x^{2}-3600 \pi ^{2} \ln \left (2\right )^{2} x^{2}-1800 \pi ^{2} \ln \left (2\right )^{2} x^{3}-10 x \,\pi ^{2}+40 x \ln \left (2\right )^{2}+120 x^{2} \ln \left (2\right )^{2}+1200 \ln \left (2\right )^{4} x^{3}+2400 \ln \left (2\right )^{4} x^{2}+16000 \ln \left (2\right )^{6} x^{2}+400 \ln \left (2\right )^{4}+3 x^{3}+3 x^{2}-x^{5}-30 \pi ^{2} x^{3}-600 \pi ^{2} \ln \left (2\right )^{2}-30 \pi ^{2} x^{2}-200 i \pi ^{3} \ln \left (2\right )+800 i \pi \ln \left (2\right )^{3}+75 \pi ^{4} x^{3}+3000 i \pi ^{5} \ln \left (2\right ) x^{2}-40000 i \pi ^{3} \ln \left (2\right )^{3} x^{2}+48000 i \pi \ln \left (2\right )^{5} x^{2}-1200 i \pi ^{3} \ln \left (2\right ) x^{2}+4800 i \pi \ln \left (2\right )^{3} x^{2}+120 i \pi \ln \left (2\right ) x^{3}+120 i \pi \ln \left (2\right ) x^{2}-600 i \pi ^{3} \ln \left (2\right ) x^{3}+2400 i \pi \ln \left (2\right )^{3} x^{3}+25 \pi ^{4}+40 i \pi \ln \left (2\right ) x -60000 \pi ^{2} \ln \left (2\right )^{4} x^{2}+15000 \pi ^{4} \ln \left (2\right )^{2} x^{2}\right )}{x^{2} \left (1+3 x +7500 \pi ^{4} \ln \left (2\right )^{2}+75 \pi ^{4} x -1800 \pi ^{2} \ln \left (2\right )^{2} x -15 \pi ^{2}+1200 x \ln \left (2\right )^{4}-30 x \,\pi ^{2}+120 x \ln \left (2\right )^{2}+60 x^{2} \ln \left (2\right )^{2}+1200 \ln \left (2\right )^{4}+60 \ln \left (2\right )^{2}+x^{3}+3 x^{2}+8000 \ln \left (2\right )^{6}-30000 \pi ^{2} \ln \left (2\right )^{4}-1800 \pi ^{2} \ln \left (2\right )^{2}-15 \pi ^{2} x^{2}-125 \pi ^{6}+60 i \pi \ln \left (2\right )-600 i \pi ^{3} \ln \left (2\right )+2400 i \pi \ln \left (2\right )^{3}+1500 i \pi ^{5} \ln \left (2\right )+24000 i \pi \ln \left (2\right )^{5}-20000 i \pi ^{3} \ln \left (2\right )^{3}-600 i \pi ^{3} \ln \left (2\right ) x +2400 i \pi \ln \left (2\right )^{3} x +60 i \pi \ln \left (2\right ) x^{2}+75 \pi ^{4}+120 i \pi \ln \left (2\right ) x \right )}\) | \(538\) |
| norman | \(\frac {\left (-2250 \pi ^{4}+25200 \pi ^{2} \ln \left (2\right )^{2}-36000 \ln \left (2\right )^{4}+900 \pi ^{2}-3600 \ln \left (2\right )^{2}-90\right ) x^{5}+\left (4500 \pi ^{6}+18000 \pi ^{4} \ln \left (2\right )^{2}-72000 \pi ^{2} \ln \left (2\right )^{4}-288000 \ln \left (2\right )^{6}+3600 i \pi ^{3} \ln \left (2\right )-14400 i \pi \ln \left (2\right )^{3}-1350 \pi ^{4}+3600 \pi ^{2} \ln \left (2\right )^{2}-21600 \ln \left (2\right )^{4}-360 i \pi \ln \left (2\right )+90 \pi ^{2}-360 \ln \left (2\right )^{2}\right ) x +\left (22500 \pi ^{6}-198000 \pi ^{4} \ln \left (2\right )^{2}+792000 \pi ^{2} \ln \left (2\right )^{4}-1440000 \ln \left (2\right )^{6}-13500 \pi ^{4}+93600 \pi ^{2} \ln \left (2\right )^{2}-216000 \ln \left (2\right )^{4}+2700 \pi ^{2}-10800 \ln \left (2\right )^{2}-189\right ) x^{4}+\left (-84375 \pi ^{8}+90000 \pi ^{6} \ln \left (2\right )^{2}+3420000 \pi ^{4} \ln \left (2\right )^{4}+1440000 \pi ^{2} \ln \left (2\right )^{6}-21600000 \ln \left (2\right )^{8}+67500 \pi ^{6}-306000 \pi ^{4} \ln \left (2\right )^{2}+1224000 \pi ^{2} \ln \left (2\right )^{4}-4320000 \ln \left (2\right )^{6}-20250 \pi ^{4}+111600 \pi ^{2} \ln \left (2\right )^{2}-324000 \ln \left (2\right )^{4}+2880 \pi ^{2}-11520 \ln \left (2\right )^{2}-153\right ) x^{3}+\left (112500 \pi ^{10}+1350000 \pi ^{8} \ln \left (2\right )^{2}+3600000 \pi ^{6} \ln \left (2\right )^{4}-14400000 \pi ^{4} \ln \left (2\right )^{6}-86400000 \pi ^{2} \ln \left (2\right )^{8}-115200000 \ln \left (2\right )^{10}-112500 \pi ^{8}-360000 \pi ^{6} \ln \left (2\right )^{2}+720000 \pi ^{4} \ln \left (2\right )^{4}-5760000 \pi ^{2} \ln \left (2\right )^{6}-28800000 \ln \left (2\right )^{8}+45000 \pi ^{6}-108000 \pi ^{4} \ln \left (2\right )^{2}+432000 \pi ^{2} \ln \left (2\right )^{4}-2880000 \ln \left (2\right )^{6}-10350 \pi ^{4}+46800 \pi ^{2} \ln \left (2\right )^{2}-165600 \ln \left (2\right )^{4}-360 i \pi \ln \left (2\right )+1170 \pi ^{2}-4680 \ln \left (2\right )^{2}-45\right ) x^{2}+9 x^{7}-5625 \pi ^{8}-90000 \pi ^{6} \ln \left (2\right )^{2}-540000 \pi ^{4} \ln \left (2\right )^{4}-1440000 \pi ^{2} \ln \left (2\right )^{6}-1440000 \ln \left (2\right )^{8}+1800 i \pi ^{3} \ln \left (2\right )-9000 i \pi ^{5} \ln \left (2\right )-144000 i \pi \ln \left (2\right )^{5}+2250 \pi ^{6}+9000 \pi ^{4} \ln \left (2\right )^{2}-36000 \pi ^{2} \ln \left (2\right )^{4}-144000 \ln \left (2\right )^{6}-7200 i \pi \ln \left (2\right )^{3}-72000 i \pi ^{3} \ln \left (2\right )^{3}-225 \pi ^{4}+5400 \pi ^{2} \ln \left (2\right )^{2}-3600 \ln \left (2\right )^{4}}{x^{2} \left (25 \pi ^{4}+200 \pi ^{2} \ln \left (2\right )^{2}+400 \ln \left (2\right )^{4}-10 x \,\pi ^{2}+40 x \ln \left (2\right )^{2}-10 \pi ^{2}+40 \ln \left (2\right )^{2}+x^{2}+2 x +1\right )^{2}}\) | \(643\) |
Input:
int(((1125*x^3+2250)*(2*ln(2)+I*Pi)^6+(675*x^4+675*x^3+1350*x+450)*(2*ln(2 )+I*Pi)^4+(135*x^5+270*x^4+135*x^3+270*x^2+90*x)*(2*ln(2)+I*Pi)^2+9*x^6+27 *x^5+27*x^4+27*x^3)/(125*x^3*(2*ln(2)+I*Pi)^6+(75*x^4+75*x^3)*(2*ln(2)+I*P i)^4+(15*x^5+30*x^4+15*x^3)*(2*ln(2)+I*Pi)^2+x^6+3*x^5+3*x^4+x^3),x,method =_RETURNVERBOSE)
Output:
9*x+(-9/25*x^2+(-72/5*I*Pi*ln(2)+18/5*Pi^2-72/5*ln(2)^2)*x+72*I*Pi^3*ln(2) -288*I*Pi*ln(2)^3-9*Pi^4+216*Pi^2*ln(2)^2-144*ln(2)^4)/x^2/(-8*I*Pi^3*ln(2 )+32*I*Pi*ln(2)^3+Pi^4-24*Pi^2*ln(2)^2+8/5*I*Pi*ln(2)*x+16*ln(2)^4-2/5*x*P i^2+8/5*I*Pi*ln(2)+8/5*x*ln(2)^2-2/5*Pi^2+8/5*ln(2)^2+1/25*x^2+2/25*x+1/25 )
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 240, normalized size of antiderivative = 8.28 \[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=-\frac {9 \, {\left (2 \, {\left (5 \, \pi ^{2} - 1\right )} x^{4} - x^{5} - 400 \, {\left (x^{3} - 1\right )} \log \left (2\right )^{4} + 25 \, \pi ^{4} - {\left (25 \, \pi ^{4} - 10 \, \pi ^{2} + 1\right )} x^{3} + 800 \, {\left (i \, \pi - i \, \pi x^{3}\right )} \log \left (2\right )^{3} - 10 \, \pi ^{2} x + 40 \, {\left ({\left (15 \, \pi ^{2} - 1\right )} x^{3} - x^{4} - 15 \, \pi ^{2} + x\right )} \log \left (2\right )^{2} + x^{2} + 40 \, {\left (-i \, \pi x^{4} + {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} x^{3} - 5 i \, \pi ^{3} + i \, \pi x\right )} \log \left (2\right )\right )}}{800 i \, \pi x^{2} \log \left (2\right )^{3} + 400 \, x^{2} \log \left (2\right )^{4} - 2 \, {\left (5 \, \pi ^{2} - 1\right )} x^{3} + x^{4} + {\left (25 \, \pi ^{4} - 10 \, \pi ^{2} + 1\right )} x^{2} - 40 \, {\left ({\left (15 \, \pi ^{2} - 1\right )} x^{2} - x^{3}\right )} \log \left (2\right )^{2} - 40 \, {\left (-i \, \pi x^{3} + {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} x^{2}\right )} \log \left (2\right )} \] Input:
integrate(((1125*x^3+2250)*(2*log(2)+I*pi)^6+(675*x^4+675*x^3+1350*x+450)* (2*log(2)+I*pi)^4+(135*x^5+270*x^4+135*x^3+270*x^2+90*x)*(2*log(2)+I*pi)^2 +9*x^6+27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*pi)^6+(75*x^4+75*x^3)*(2 *log(2)+I*pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*pi)^2+x^6+3*x^5+3*x^4+x ^3),x, algorithm="fricas")
Output:
-9*(2*(5*pi^2 - 1)*x^4 - x^5 - 400*(x^3 - 1)*log(2)^4 + 25*pi^4 - (25*pi^4 - 10*pi^2 + 1)*x^3 + 800*(I*pi - I*pi*x^3)*log(2)^3 - 10*pi^2*x + 40*((15 *pi^2 - 1)*x^3 - x^4 - 15*pi^2 + x)*log(2)^2 + x^2 + 40*(-I*pi*x^4 + (-I*p i + 5*I*pi^3)*x^3 - 5*I*pi^3 + I*pi*x)*log(2))/(800*I*pi*x^2*log(2)^3 + 40 0*x^2*log(2)^4 - 2*(5*pi^2 - 1)*x^3 + x^4 + (25*pi^4 - 10*pi^2 + 1)*x^2 - 40*((15*pi^2 - 1)*x^2 - x^3)*log(2)^2 - 40*(-I*pi*x^3 + (-I*pi + 5*I*pi^3) *x^2)*log(2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (24) = 48\).
Time = 18.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 5.79 \[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=9 x + \frac {- 9 x^{2} + x \left (- 360 \log {\left (2 \right )}^{2} + 90 \pi ^{2} - 360 i \pi \log {\left (2 \right )}\right ) - 225 \pi ^{4} - 3600 \log {\left (2 \right )}^{4} + 5400 \pi ^{2} \log {\left (2 \right )}^{2} - 7200 i \pi \log {\left (2 \right )}^{3} + 1800 i \pi ^{3} \log {\left (2 \right )}}{x^{4} + x^{3} \left (- 10 \pi ^{2} + 2 + 40 \log {\left (2 \right )}^{2} + 40 i \pi \log {\left (2 \right )}\right ) + x^{2} \left (- 600 \pi ^{2} \log {\left (2 \right )}^{2} - 10 \pi ^{2} + 1 + 40 \log {\left (2 \right )}^{2} + 400 \log {\left (2 \right )}^{4} + 25 \pi ^{4} - 200 i \pi ^{3} \log {\left (2 \right )} + 40 i \pi \log {\left (2 \right )} + 800 i \pi \log {\left (2 \right )}^{3}\right )} \] Input:
integrate(((1125*x**3+2250)*(2*ln(2)+I*pi)**6+(675*x**4+675*x**3+1350*x+45 0)*(2*ln(2)+I*pi)**4+(135*x**5+270*x**4+135*x**3+270*x**2+90*x)*(2*ln(2)+I *pi)**2+9*x**6+27*x**5+27*x**4+27*x**3)/(125*x**3*(2*ln(2)+I*pi)**6+(75*x* *4+75*x**3)*(2*ln(2)+I*pi)**4+(15*x**5+30*x**4+15*x**3)*(2*ln(2)+I*pi)**2+ x**6+3*x**5+3*x**4+x**3),x)
Output:
9*x + (-9*x**2 + x*(-360*log(2)**2 + 90*pi**2 - 360*I*pi*log(2)) - 225*pi* *4 - 3600*log(2)**4 + 5400*pi**2*log(2)**2 - 7200*I*pi*log(2)**3 + 1800*I* pi**3*log(2))/(x**4 + x**3*(-10*pi**2 + 2 + 40*log(2)**2 + 40*I*pi*log(2)) + x**2*(-600*pi**2*log(2)**2 - 10*pi**2 + 1 + 40*log(2)**2 + 400*log(2)** 4 + 25*pi**4 - 200*I*pi**3*log(2) + 40*I*pi*log(2) + 800*I*pi*log(2)**3))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (29) = 58\).
Time = 0.05 (sec) , antiderivative size = 149, normalized size of antiderivative = 5.14 \[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=9 \, x + \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \left (2\right ) - 600 \, \pi ^{2} \log \left (2\right )^{2} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 10 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (2\right ) - 4 \, \log \left (2\right )^{2}\right )} x + x^{2}\right )}}{2 \, {\left (5 \, \pi ^{2} - 20 i \, \pi \log \left (2\right ) - 20 \, \log \left (2\right )^{2} - 1\right )} x^{3} - x^{4} - {\left (25 \, \pi ^{4} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 40 \, {\left (15 \, \pi ^{2} - 1\right )} \log \left (2\right )^{2} - 10 \, \pi ^{2} - 40 \, {\left (-i \, \pi + 5 i \, \pi ^{3}\right )} \log \left (2\right ) + 1\right )} x^{2}} \] Input:
integrate(((1125*x^3+2250)*(2*log(2)+I*pi)^6+(675*x^4+675*x^3+1350*x+450)* (2*log(2)+I*pi)^4+(135*x^5+270*x^4+135*x^3+270*x^2+90*x)*(2*log(2)+I*pi)^2 +9*x^6+27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*pi)^6+(75*x^4+75*x^3)*(2 *log(2)+I*pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*pi)^2+x^6+3*x^5+3*x^4+x ^3),x, algorithm="maxima")
Output:
9*x + 9*(25*pi^4 - 200*I*pi^3*log(2) - 600*pi^2*log(2)^2 + 800*I*pi*log(2) ^3 + 400*log(2)^4 - 10*(pi^2 - 4*I*pi*log(2) - 4*log(2)^2)*x + x^2)/(2*(5* pi^2 - 20*I*pi*log(2) - 20*log(2)^2 - 1)*x^3 - x^4 - (25*pi^4 + 800*I*pi*l og(2)^3 + 400*log(2)^4 - 40*(15*pi^2 - 1)*log(2)^2 - 10*pi^2 - 40*(-I*pi + 5*I*pi^3)*log(2) + 1)*x^2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (29) = 58\).
Time = 0.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.21 \[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=9 \, x - \frac {9 \, {\left (25 \, \pi ^{4} - 200 i \, \pi ^{3} \log \left (2\right ) - 600 \, \pi ^{2} \log \left (2\right )^{2} + 800 i \, \pi \log \left (2\right )^{3} + 400 \, \log \left (2\right )^{4} - 10 \, \pi ^{2} x + 40 i \, \pi x \log \left (2\right ) + 40 \, x \log \left (2\right )^{2} + x^{2}\right )}}{{\left (5 \, \pi ^{2} x - 20 i \, \pi x \log \left (2\right ) - 20 \, x \log \left (2\right )^{2} - x^{2} - x\right )}^{2}} \] Input:
integrate(((1125*x^3+2250)*(2*log(2)+I*pi)^6+(675*x^4+675*x^3+1350*x+450)* (2*log(2)+I*pi)^4+(135*x^5+270*x^4+135*x^3+270*x^2+90*x)*(2*log(2)+I*pi)^2 +9*x^6+27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*pi)^6+(75*x^4+75*x^3)*(2 *log(2)+I*pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*pi)^2+x^6+3*x^5+3*x^4+x ^3),x, algorithm="giac")
Output:
9*x - 9*(25*pi^4 - 200*I*pi^3*log(2) - 600*pi^2*log(2)^2 + 800*I*pi*log(2) ^3 + 400*log(2)^4 - 10*pi^2*x + 40*I*pi*x*log(2) + 40*x*log(2)^2 + x^2)/(5 *pi^2*x - 20*I*pi*x*log(2) - 20*x*log(2)^2 - x^2 - x)^2
Time = 20.96 (sec) , antiderivative size = 640, normalized size of antiderivative = 22.07 \[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=\text {Too large to display} \] Input:
int(((1125*x^3 + 2250)*(Pi*1i + 2*log(2))^6 + (Pi*1i + 2*log(2))^4*(1350*x + 675*x^3 + 675*x^4 + 450) + (Pi*1i + 2*log(2))^2*(90*x + 270*x^2 + 135*x ^3 + 270*x^4 + 135*x^5) + 27*x^3 + 27*x^4 + 27*x^5 + 9*x^6)/((75*x^3 + 75* x^4)*(Pi*1i + 2*log(2))^4 + 125*x^3*(Pi*1i + 2*log(2))^6 + (Pi*1i + 2*log( 2))^2*(15*x^3 + 30*x^4 + 15*x^5) + x^3 + 3*x^4 + 3*x^5 + x^6),x)
Output:
9*x + (900*Pi*log(4)^3 - 900*Pi^3*log(4) - Pi^4*225i - log(4)^4*225i + Pi^ 2*log(4)^2*1350i - (90*x*(Pi*log(4)*2i - Pi^3*log(2)*60i - Pi*log(4)^3*40i + Pi^3*log(4)*10i - Pi^2 + 5*Pi^4 + log(4)^2 - 10*log(4)^4 - 60*Pi^2*log( 2)^2 + 45*Pi^2*log(4)^2 + 60*log(2)^2*log(4)^2 + Pi*log(2)*log(4)^2*60i + Pi*log(2)^2*log(4)*120i - 120*Pi^2*log(2)*log(4)))/(20*Pi*log(2) + Pi^2*5i - log(2)^2*20i - 1i) + (9*x^2*(Pi*log(2)*40i + Pi*log(2)^3*800i - Pi^3*lo g(2)*2000i - Pi*log(4)^3*1800i + Pi^3*log(4)*900i - 10*Pi^2 + 25*Pi^4 + 40 *log(2)^2 + 400*log(2)^4 - 450*log(4)^4 - 2400*Pi^2*log(2)^2 + 2250*Pi^2*l og(4)^2 + 1800*log(2)^2*log(4)^2 + Pi*log(2)*log(4)^2*1800i + Pi*log(2)^2* log(4)*3600i - 3600*Pi^2*log(2)*log(4) + 1))/(200*Pi^3*log(2) - 800*Pi*log (2)^3 - 40*Pi*log(2) - Pi^2*10i + Pi^4*25i + log(2)^2*40i + log(2)^4*400i - Pi^2*log(2)^2*600i + 1i) + (2700*x^3*(2*Pi^3*log(4) - 4*Pi*log(4)^3 - 4* Pi^3*log(2) + log(4)^4*1i + Pi^2*log(2)^2*4i - Pi^2*log(4)^2*5i - log(2)^2 *log(4)^2*4i + 4*Pi*log(2)*log(4)^2 + 8*Pi*log(2)^2*log(4) + Pi^2*log(2)*l og(4)*8i))/((20*Pi*log(2) + Pi^2*5i - log(2)^2*20i - 1i)*(200*Pi^3*log(2) - 800*Pi*log(2)^3 - 40*Pi*log(2) - Pi^2*10i + Pi^4*25i + log(2)^2*40i + lo g(2)^4*400i - Pi^2*log(2)^2*600i + 1i)))/(x^4*1i - x^3*(40*Pi*log(2) + Pi^ 2*10i - log(2)^2*40i - 2i) + x^2*(200*Pi^3*log(2) - 800*Pi*log(2)^3 - 40*P i*log(2) - Pi^2*10i + Pi^4*25i + log(2)^2*40i + log(2)^4*400i - Pi^2*log(2 )^2*600i + 1i))
\[ \int \frac {27 x^3+27 x^4+27 x^5+9 x^6+\left (90 x+270 x^2+135 x^3+270 x^4+135 x^5\right ) (i \pi +\log (4))^2+\left (450+1350 x+675 x^3+675 x^4\right ) (i \pi +\log (4))^4+\left (2250+1125 x^3\right ) (i \pi +\log (4))^6}{x^3+3 x^4+3 x^5+x^6+\left (15 x^3+30 x^4+15 x^5\right ) (i \pi +\log (4))^2+\left (75 x^3+75 x^4\right ) (i \pi +\log (4))^4+125 x^3 (i \pi +\log (4))^6} \, dx=\int \frac {\left (1125 x^{3}+2250\right ) \left (2 \,\mathrm {log}\left (2\right )+i \pi \right )^{6}+\left (675 x^{4}+675 x^{3}+1350 x +450\right ) \left (2 \,\mathrm {log}\left (2\right )+i \pi \right )^{4}+\left (135 x^{5}+270 x^{4}+135 x^{3}+270 x^{2}+90 x \right ) \left (2 \,\mathrm {log}\left (2\right )+i \pi \right )^{2}+9 x^{6}+27 x^{5}+27 x^{4}+27 x^{3}}{125 x^{3} \left (2 \,\mathrm {log}\left (2\right )+i \pi \right )^{6}+\left (75 x^{4}+75 x^{3}\right ) \left (2 \,\mathrm {log}\left (2\right )+i \pi \right )^{4}+\left (15 x^{5}+30 x^{4}+15 x^{3}\right ) \left (2 \,\mathrm {log}\left (2\right )+i \pi \right )^{2}+x^{6}+3 x^{5}+3 x^{4}+x^{3}}d x \] Input:
int(((1125*x^3+2250)*(2*log(2)+I*Pi)^6+(675*x^4+675*x^3+1350*x+450)*(2*log (2)+I*Pi)^4+(135*x^5+270*x^4+135*x^3+270*x^2+90*x)*(2*log(2)+I*Pi)^2+9*x^6 +27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*Pi)^6+(75*x^4+75*x^3)*(2*log(2 )+I*Pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*Pi)^2+x^6+3*x^5+3*x^4+x^3),x)
Output:
int(((1125*x^3+2250)*(2*log(2)+I*Pi)^6+(675*x^4+675*x^3+1350*x+450)*(2*log (2)+I*Pi)^4+(135*x^5+270*x^4+135*x^3+270*x^2+90*x)*(2*log(2)+I*Pi)^2+9*x^6 +27*x^5+27*x^4+27*x^3)/(125*x^3*(2*log(2)+I*Pi)^6+(75*x^4+75*x^3)*(2*log(2 )+I*Pi)^4+(15*x^5+30*x^4+15*x^3)*(2*log(2)+I*Pi)^2+x^6+3*x^5+3*x^4+x^3),x)