Integrand size = 363, antiderivative size = 22 \[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=\log (x)+\log \left (-1+x+\frac {5}{\left (1+(3+x-\log (4))^2\right )^2}\right ) \] Output:
ln(5/(1+(3-2*ln(2)+x)^2)^2+x-1)+ln(x)
Leaf count is larger than twice the leaf count of optimal. \(147\) vs. \(2(22)=44\).
Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.68 \[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=\log (x)-2 \log \left (10+6 x+x^2-6 \log (4)-2 x \log (4)+\log ^2(4)\right )+\log \left (95+20 x-64 x^2-44 x^3-11 x^4-x^5-120 \log (4)+8 x \log (4)+76 x^2 \log (4)+32 x^3 \log (4)+4 x^4 \log (4)+56 \log ^2(4)-20 x \log ^2(4)-30 x^2 \log ^2(4)-6 x^3 \log ^2(4)-12 \log ^3(4)+8 x \log ^3(4)+4 x^2 \log ^3(4)+\log ^4(4)-x \log ^4(4)\right ) \] Input:
Integrate[(-950 + 170*x + 2205*x^2 + 2184*x^3 + 1014*x^4 + 258*x^5 + 35*x^ 6 + 2*x^7 + (1770 - 830*x - 3792*x^2 - 2904*x^3 - 1014*x^4 - 174*x^5 - 12* x^6)*Log[4] + (-1375 + 1032*x + 2628*x^2 + 1476*x^3 + 345*x^4 + 30*x^5)*Lo g[4]^2 + (576 - 600*x - 924*x^2 - 340*x^3 - 40*x^4)*Log[4]^3 + (-138 + 186 *x + 165*x^2 + 30*x^3)*Log[4]^4 + (18 - 30*x - 12*x^2)*Log[4]^5 + (-1 + 2* x)*Log[4]^6)/(-950*x - 770*x^2 + 425*x^3 + 804*x^4 + 438*x^5 + 120*x^6 + 1 7*x^7 + x^8 + (1770*x + 950*x^2 - 1032*x^3 - 1176*x^4 - 462*x^5 - 84*x^6 - 6*x^7)*Log[4] + (-1375*x - 348*x^2 + 900*x^3 + 648*x^4 + 165*x^5 + 15*x^6 )*Log[4]^2 + (576*x - 24*x^2 - 372*x^3 - 160*x^4 - 20*x^5)*Log[4]^3 + (-13 8*x + 48*x^2 + 75*x^3 + 15*x^4)*Log[4]^4 + (18*x - 12*x^2 - 6*x^3)*Log[4]^ 5 + (-x + x^2)*Log[4]^6),x]
Output:
Log[x] - 2*Log[10 + 6*x + x^2 - 6*Log[4] - 2*x*Log[4] + Log[4]^2] + Log[95 + 20*x - 64*x^2 - 44*x^3 - 11*x^4 - x^5 - 120*Log[4] + 8*x*Log[4] + 76*x^ 2*Log[4] + 32*x^3*Log[4] + 4*x^4*Log[4] + 56*Log[4]^2 - 20*x*Log[4]^2 - 30 *x^2*Log[4]^2 - 6*x^3*Log[4]^2 - 12*Log[4]^3 + 8*x*Log[4]^3 + 4*x^2*Log[4] ^3 + Log[4]^4 - x*Log[4]^4]
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(22)=44\).
Time = 1.46 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2026, 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.
| \(\displaystyle \int \frac {2 x^7+35 x^6+258 x^5+1014 x^4+2184 x^3+2205 x^2+\left (-12 x^2-30 x+18\right ) \log ^5(4)+\left (30 x^3+165 x^2+186 x-138\right ) \log ^4(4)+\left (-40 x^4-340 x^3-924 x^2-600 x+576\right ) \log ^3(4)+\left (30 x^5+345 x^4+1476 x^3+2628 x^2+1032 x-1375\right ) \log ^2(4)+\left (-12 x^6-174 x^5-1014 x^4-2904 x^3-3792 x^2-830 x+1770\right ) \log (4)+170 x+(2 x-1) \log ^6(4)-950}{x^8+17 x^7+120 x^6+438 x^5+804 x^4+425 x^3-770 x^2+\left (x^2-x\right ) \log ^6(4)+\left (-6 x^3-12 x^2+18 x\right ) \log ^5(4)+\left (15 x^4+75 x^3+48 x^2-138 x\right ) \log ^4(4)+\left (-20 x^5-160 x^4-372 x^3-24 x^2+576 x\right ) \log ^3(4)+\left (15 x^6+165 x^5+648 x^4+900 x^3-348 x^2-1375 x\right ) \log ^2(4)+\left (-6 x^7-84 x^6-462 x^5-1176 x^4-1032 x^3+950 x^2+1770 x\right ) \log (4)-950 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^7+35 x^6+258 x^5+1014 x^4+2184 x^3+2205 x^2+\left (-12 x^2-30 x+18\right ) \log ^5(4)+\left (30 x^3+165 x^2+186 x-138\right ) \log ^4(4)+\left (-40 x^4-340 x^3-924 x^2-600 x+576\right ) \log ^3(4)+\left (30 x^5+345 x^4+1476 x^3+2628 x^2+1032 x-1375\right ) \log ^2(4)+\left (-12 x^6-174 x^5-1014 x^4-2904 x^3-3792 x^2-830 x+1770\right ) \log (4)+170 x+(2 x-1) \log ^6(4)-950}{x \left (x^7+x^6 (17-6 \log (4))+3 x^5 \left (40+5 \log ^2(4)-28 \log (4)\right )+x^4 \left (438-20 \log ^3(4)+165 \log ^2(4)-462 \log (4)\right )+x^3 \left (804+15 \log ^4(4)-160 \log ^3(4)+648 \log ^2(4)-1176 \log (4)\right )+x^2 \left (425-6 \log ^5(4)+75 \log ^4(4)-372 \log ^3(4)+900 \log ^2(4)-1032 \log (4)\right )-x \left (770-\log ^6(4)+12 \log ^5(4)-48 \log ^4(4)+24 \log ^3(4)+348 \log ^2(4)-950 \log (4)\right )-\left (10+\log ^2(4)-6 \log (4)\right ) \left (95+\log ^4(4)-12 \log ^3(4)+56 \log ^2(4)-120 \log (4)\right )\right )}dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {4 (-x-3+\log (4))}{x^2+2 x (3-\log (4))+10+\log ^2(4)-6 \log (4)}+\frac {5 x^4+4 x^3 (11-4 \log (4))+6 x^2 \left (22+3 \log ^2(4)-16 \log (4)\right )+4 x (2-\log (4)) \left (16+2 \log ^2(4)-11 \log (4)\right )-\left (2-\log ^2(4)+2 \log (4)\right ) \left (10+\log ^2(4)-6 \log (4)\right )}{x^5+11 x^4 \left (1-\frac {8 \log (2)}{11}\right )+44 x^3 \left (1+\frac {1}{11} \log (4) (\log (8)-8)\right )+64 x^2 \left (1-\frac {1}{32} \log (4) (38+\log (4) (\log (16)-15))\right )-20 x \left (1+\frac {2}{5} \log (4) \left (1-\frac {1}{4} \log (4) (10+(\log (2)-4) \log (4))\right )\right )-95 \left (1+\frac {2}{95} \log (4) \left (-60+(\log (2)-6) \log ^2(4)+28 \log (4)\right )\right )}+\frac {1}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \log \left (x^2+2 x (3-\log (4))+10+\log ^2(4)-6 \log (4)\right )+\log \left (x^5+x^4 (11-8 \log (2))+4 x^3 (11-\log (4) (8-\log (8)))+2 x^2 \left (32+\log ^2(4) (15-\log (16))-38 \log (4)\right )-2 x \left (10+\log (4) \left (4+(4-\log (2)) \log ^2(4)-10 \log (4)\right )\right )-95+2 \log (4) \left (60+(6-\log (2)) \log ^2(4)-28 \log (4)\right )\right )+\log (x)\) |
Input:
Int[(-950 + 170*x + 2205*x^2 + 2184*x^3 + 1014*x^4 + 258*x^5 + 35*x^6 + 2* x^7 + (1770 - 830*x - 3792*x^2 - 2904*x^3 - 1014*x^4 - 174*x^5 - 12*x^6)*L og[4] + (-1375 + 1032*x + 2628*x^2 + 1476*x^3 + 345*x^4 + 30*x^5)*Log[4]^2 + (576 - 600*x - 924*x^2 - 340*x^3 - 40*x^4)*Log[4]^3 + (-138 + 186*x + 1 65*x^2 + 30*x^3)*Log[4]^4 + (18 - 30*x - 12*x^2)*Log[4]^5 + (-1 + 2*x)*Log [4]^6)/(-950*x - 770*x^2 + 425*x^3 + 804*x^4 + 438*x^5 + 120*x^6 + 17*x^7 + x^8 + (1770*x + 950*x^2 - 1032*x^3 - 1176*x^4 - 462*x^5 - 84*x^6 - 6*x^7 )*Log[4] + (-1375*x - 348*x^2 + 900*x^3 + 648*x^4 + 165*x^5 + 15*x^6)*Log[ 4]^2 + (576*x - 24*x^2 - 372*x^3 - 160*x^4 - 20*x^5)*Log[4]^3 + (-138*x + 48*x^2 + 75*x^3 + 15*x^4)*Log[4]^4 + (18*x - 12*x^2 - 6*x^3)*Log[4]^5 + (- x + x^2)*Log[4]^6),x]
Output:
Log[x] - 2*Log[10 + x^2 + 2*x*(3 - Log[4]) - 6*Log[4] + Log[4]^2] + Log[-9 5 + x^5 + x^4*(11 - 8*Log[2]) + 2*Log[4]*(60 - 28*Log[4] + (6 - Log[2])*Lo g[4]^2) - 2*x*(10 + Log[4]*(4 - 10*Log[4] + (4 - Log[2])*Log[4]^2)) + 4*x^ 3*(11 - Log[4]*(8 - Log[8])) + 2*x^2*(32 - 38*Log[4] + Log[4]^2*(15 - Log[ 16]))]
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[(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. \(136\) vs. \(2(22)=44\).
Time = 0.79 (sec) , antiderivative size = 137, normalized size of antiderivative = 6.23
| method | result | size |
| risch | \(-2 \ln \left (-x^{2}+\left (4 \ln \left (2\right )-6\right ) x -4 \ln \left (2\right )^{2}+12 \ln \left (2\right )-10\right )+\ln \left (x^{6}+\left (-8 \ln \left (2\right )+11\right ) x^{5}+\left (24 \ln \left (2\right )^{2}-64 \ln \left (2\right )+44\right ) x^{4}+\left (-32 \ln \left (2\right )^{3}+120 \ln \left (2\right )^{2}-152 \ln \left (2\right )+64\right ) x^{3}+\left (16 \ln \left (2\right )^{4}-64 \ln \left (2\right )^{3}+80 \ln \left (2\right )^{2}-16 \ln \left (2\right )-20\right ) x^{2}+\left (-16 \ln \left (2\right )^{4}+96 \ln \left (2\right )^{3}-224 \ln \left (2\right )^{2}+240 \ln \left (2\right )-95\right ) x \right )\) | \(137\) |
| default | \(-2 \ln \left (4 \ln \left (2\right )^{2}-4 x \ln \left (2\right )+x^{2}-12 \ln \left (2\right )+6 x +10\right )+\ln \left (16 x \ln \left (2\right )^{4}-32 x^{2} \ln \left (2\right )^{3}+24 x^{3} \ln \left (2\right )^{2}-8 x^{4} \ln \left (2\right )+x^{5}-16 \ln \left (2\right )^{4}-64 x \ln \left (2\right )^{3}+120 x^{2} \ln \left (2\right )^{2}-64 x^{3} \ln \left (2\right )+11 x^{4}+96 \ln \left (2\right )^{3}+80 x \ln \left (2\right )^{2}-152 x^{2} \ln \left (2\right )+44 x^{3}-224 \ln \left (2\right )^{2}-16 x \ln \left (2\right )+64 x^{2}+240 \ln \left (2\right )-20 x -95\right )+\ln \left (x \right )\) | \(150\) |
| norman | \(-2 \ln \left (4 \ln \left (2\right )^{2}-4 x \ln \left (2\right )+x^{2}-12 \ln \left (2\right )+6 x +10\right )+\ln \left (16 x \ln \left (2\right )^{4}-32 x^{2} \ln \left (2\right )^{3}+24 x^{3} \ln \left (2\right )^{2}-8 x^{4} \ln \left (2\right )+x^{5}-16 \ln \left (2\right )^{4}-64 x \ln \left (2\right )^{3}+120 x^{2} \ln \left (2\right )^{2}-64 x^{3} \ln \left (2\right )+11 x^{4}+96 \ln \left (2\right )^{3}+80 x \ln \left (2\right )^{2}-152 x^{2} \ln \left (2\right )+44 x^{3}-224 \ln \left (2\right )^{2}-16 x \ln \left (2\right )+64 x^{2}+240 \ln \left (2\right )-20 x -95\right )+\ln \left (x \right )\) | \(150\) |
| parallelrisch | \(-2 \ln \left (4 \ln \left (2\right )^{2}-4 x \ln \left (2\right )+x^{2}-12 \ln \left (2\right )+6 x +10\right )+\ln \left (16 x \ln \left (2\right )^{4}-32 x^{2} \ln \left (2\right )^{3}+24 x^{3} \ln \left (2\right )^{2}-8 x^{4} \ln \left (2\right )+x^{5}-16 \ln \left (2\right )^{4}-64 x \ln \left (2\right )^{3}+120 x^{2} \ln \left (2\right )^{2}-64 x^{3} \ln \left (2\right )+11 x^{4}+96 \ln \left (2\right )^{3}+80 x \ln \left (2\right )^{2}-152 x^{2} \ln \left (2\right )+44 x^{3}-224 \ln \left (2\right )^{2}-16 x \ln \left (2\right )+64 x^{2}+240 \ln \left (2\right )-20 x -95\right )+\ln \left (x \right )\) | \(150\) |
Input:
int((64*(-1+2*x)*ln(2)^6+32*(-12*x^2-30*x+18)*ln(2)^5+16*(30*x^3+165*x^2+1 86*x-138)*ln(2)^4+8*(-40*x^4-340*x^3-924*x^2-600*x+576)*ln(2)^3+4*(30*x^5+ 345*x^4+1476*x^3+2628*x^2+1032*x-1375)*ln(2)^2+2*(-12*x^6-174*x^5-1014*x^4 -2904*x^3-3792*x^2-830*x+1770)*ln(2)+2*x^7+35*x^6+258*x^5+1014*x^4+2184*x^ 3+2205*x^2+170*x-950)/(64*(x^2-x)*ln(2)^6+32*(-6*x^3-12*x^2+18*x)*ln(2)^5+ 16*(15*x^4+75*x^3+48*x^2-138*x)*ln(2)^4+8*(-20*x^5-160*x^4-372*x^3-24*x^2+ 576*x)*ln(2)^3+4*(15*x^6+165*x^5+648*x^4+900*x^3-348*x^2-1375*x)*ln(2)^2+2 *(-6*x^7-84*x^6-462*x^5-1176*x^4-1032*x^3+950*x^2+1770*x)*ln(2)+x^8+17*x^7 +120*x^6+438*x^5+804*x^4+425*x^3-770*x^2-950*x),x,method=_RETURNVERBOSE)
Output:
-2*ln(-x^2+(4*ln(2)-6)*x-4*ln(2)^2+12*ln(2)-10)+ln(x^6+(-8*ln(2)+11)*x^5+( 24*ln(2)^2-64*ln(2)+44)*x^4+(-32*ln(2)^3+120*ln(2)^2-152*ln(2)+64)*x^3+(16 *ln(2)^4-64*ln(2)^3+80*ln(2)^2-16*ln(2)-20)*x^2+(-16*ln(2)^4+96*ln(2)^3-22 4*ln(2)^2+240*ln(2)-95)*x)
Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 135, normalized size of antiderivative = 6.14 \[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=\log \left (x^{6} + 11 \, x^{5} + 16 \, {\left (x^{2} - x\right )} \log \left (2\right )^{4} + 44 \, x^{4} - 32 \, {\left (x^{3} + 2 \, x^{2} - 3 \, x\right )} \log \left (2\right )^{3} + 64 \, x^{3} + 8 \, {\left (3 \, x^{4} + 15 \, x^{3} + 10 \, x^{2} - 28 \, x\right )} \log \left (2\right )^{2} - 20 \, x^{2} - 8 \, {\left (x^{5} + 8 \, x^{4} + 19 \, x^{3} + 2 \, x^{2} - 30 \, x\right )} \log \left (2\right ) - 95 \, x\right ) - 2 \, \log \left (x^{2} - 4 \, {\left (x + 3\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + 6 \, x + 10\right ) \] Input:
integrate((64*(-1+2*x)*log(2)^6+32*(-12*x^2-30*x+18)*log(2)^5+16*(30*x^3+1 65*x^2+186*x-138)*log(2)^4+8*(-40*x^4-340*x^3-924*x^2-600*x+576)*log(2)^3+ 4*(30*x^5+345*x^4+1476*x^3+2628*x^2+1032*x-1375)*log(2)^2+2*(-12*x^6-174*x ^5-1014*x^4-2904*x^3-3792*x^2-830*x+1770)*log(2)+2*x^7+35*x^6+258*x^5+1014 *x^4+2184*x^3+2205*x^2+170*x-950)/(64*(x^2-x)*log(2)^6+32*(-6*x^3-12*x^2+1 8*x)*log(2)^5+16*(15*x^4+75*x^3+48*x^2-138*x)*log(2)^4+8*(-20*x^5-160*x^4- 372*x^3-24*x^2+576*x)*log(2)^3+4*(15*x^6+165*x^5+648*x^4+900*x^3-348*x^2-1 375*x)*log(2)^2+2*(-6*x^7-84*x^6-462*x^5-1176*x^4-1032*x^3+950*x^2+1770*x) *log(2)+x^8+17*x^7+120*x^6+438*x^5+804*x^4+425*x^3-770*x^2-950*x),x, algor ithm="fricas")
Output:
log(x^6 + 11*x^5 + 16*(x^2 - x)*log(2)^4 + 44*x^4 - 32*(x^3 + 2*x^2 - 3*x) *log(2)^3 + 64*x^3 + 8*(3*x^4 + 15*x^3 + 10*x^2 - 28*x)*log(2)^2 - 20*x^2 - 8*(x^5 + 8*x^4 + 19*x^3 + 2*x^2 - 30*x)*log(2) - 95*x) - 2*log(x^2 - 4*( x + 3)*log(2) + 4*log(2)^2 + 6*x + 10)
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (22) = 44\).
Time = 10.66 (sec) , antiderivative size = 143, normalized size of antiderivative = 6.50 \[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=- 2 \log {\left (x^{2} + x \left (6 - 4 \log {\left (2 \right )}\right ) - 12 \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2} + 10 \right )} + \log {\left (x^{6} + x^{5} \cdot \left (11 - 8 \log {\left (2 \right )}\right ) + x^{4} \left (- 64 \log {\left (2 \right )} + 24 \log {\left (2 \right )}^{2} + 44\right ) + x^{3} \left (- 152 \log {\left (2 \right )} - 32 \log {\left (2 \right )}^{3} + 120 \log {\left (2 \right )}^{2} + 64\right ) + x^{2} \left (- 64 \log {\left (2 \right )}^{3} - 20 - 16 \log {\left (2 \right )} + 16 \log {\left (2 \right )}^{4} + 80 \log {\left (2 \right )}^{2}\right ) + x \left (- 224 \log {\left (2 \right )}^{2} - 95 - 16 \log {\left (2 \right )}^{4} + 96 \log {\left (2 \right )}^{3} + 240 \log {\left (2 \right )}\right ) \right )} \] Input:
integrate((64*(-1+2*x)*ln(2)**6+32*(-12*x**2-30*x+18)*ln(2)**5+16*(30*x**3 +165*x**2+186*x-138)*ln(2)**4+8*(-40*x**4-340*x**3-924*x**2-600*x+576)*ln( 2)**3+4*(30*x**5+345*x**4+1476*x**3+2628*x**2+1032*x-1375)*ln(2)**2+2*(-12 *x**6-174*x**5-1014*x**4-2904*x**3-3792*x**2-830*x+1770)*ln(2)+2*x**7+35*x **6+258*x**5+1014*x**4+2184*x**3+2205*x**2+170*x-950)/(64*(x**2-x)*ln(2)** 6+32*(-6*x**3-12*x**2+18*x)*ln(2)**5+16*(15*x**4+75*x**3+48*x**2-138*x)*ln (2)**4+8*(-20*x**5-160*x**4-372*x**3-24*x**2+576*x)*ln(2)**3+4*(15*x**6+16 5*x**5+648*x**4+900*x**3-348*x**2-1375*x)*ln(2)**2+2*(-6*x**7-84*x**6-462* x**5-1176*x**4-1032*x**3+950*x**2+1770*x)*ln(2)+x**8+17*x**7+120*x**6+438* x**5+804*x**4+425*x**3-770*x**2-950*x),x)
Output:
-2*log(x**2 + x*(6 - 4*log(2)) - 12*log(2) + 4*log(2)**2 + 10) + log(x**6 + x**5*(11 - 8*log(2)) + x**4*(-64*log(2) + 24*log(2)**2 + 44) + x**3*(-15 2*log(2) - 32*log(2)**3 + 120*log(2)**2 + 64) + x**2*(-64*log(2)**3 - 20 - 16*log(2) + 16*log(2)**4 + 80*log(2)**2) + x*(-224*log(2)**2 - 95 - 16*lo g(2)**4 + 96*log(2)**3 + 240*log(2)))
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (22) = 44\).
Time = 0.05 (sec) , antiderivative size = 136, normalized size of antiderivative = 6.18 \[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=\log \left (x^{5} - x^{4} {\left (8 \, \log \left (2\right ) - 11\right )} + 4 \, {\left (6 \, \log \left (2\right )^{2} - 16 \, \log \left (2\right ) + 11\right )} x^{3} - 16 \, \log \left (2\right )^{4} - 8 \, {\left (4 \, \log \left (2\right )^{3} - 15 \, \log \left (2\right )^{2} + 19 \, \log \left (2\right ) - 8\right )} x^{2} + 96 \, \log \left (2\right )^{3} + 4 \, {\left (4 \, \log \left (2\right )^{4} - 16 \, \log \left (2\right )^{3} + 20 \, \log \left (2\right )^{2} - 4 \, \log \left (2\right ) - 5\right )} x - 224 \, \log \left (2\right )^{2} + 240 \, \log \left (2\right ) - 95\right ) - 2 \, \log \left (x^{2} - 2 \, x {\left (2 \, \log \left (2\right ) - 3\right )} + 4 \, \log \left (2\right )^{2} - 12 \, \log \left (2\right ) + 10\right ) + \log \left (x\right ) \] Input:
integrate((64*(-1+2*x)*log(2)^6+32*(-12*x^2-30*x+18)*log(2)^5+16*(30*x^3+1 65*x^2+186*x-138)*log(2)^4+8*(-40*x^4-340*x^3-924*x^2-600*x+576)*log(2)^3+ 4*(30*x^5+345*x^4+1476*x^3+2628*x^2+1032*x-1375)*log(2)^2+2*(-12*x^6-174*x ^5-1014*x^4-2904*x^3-3792*x^2-830*x+1770)*log(2)+2*x^7+35*x^6+258*x^5+1014 *x^4+2184*x^3+2205*x^2+170*x-950)/(64*(x^2-x)*log(2)^6+32*(-6*x^3-12*x^2+1 8*x)*log(2)^5+16*(15*x^4+75*x^3+48*x^2-138*x)*log(2)^4+8*(-20*x^5-160*x^4- 372*x^3-24*x^2+576*x)*log(2)^3+4*(15*x^6+165*x^5+648*x^4+900*x^3-348*x^2-1 375*x)*log(2)^2+2*(-6*x^7-84*x^6-462*x^5-1176*x^4-1032*x^3+950*x^2+1770*x) *log(2)+x^8+17*x^7+120*x^6+438*x^5+804*x^4+425*x^3-770*x^2-950*x),x, algor ithm="maxima")
Output:
log(x^5 - x^4*(8*log(2) - 11) + 4*(6*log(2)^2 - 16*log(2) + 11)*x^3 - 16*l og(2)^4 - 8*(4*log(2)^3 - 15*log(2)^2 + 19*log(2) - 8)*x^2 + 96*log(2)^3 + 4*(4*log(2)^4 - 16*log(2)^3 + 20*log(2)^2 - 4*log(2) - 5)*x - 224*log(2)^ 2 + 240*log(2) - 95) - 2*log(x^2 - 2*x*(2*log(2) - 3) + 4*log(2)^2 - 12*lo g(2) + 10) + log(x)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.86 \[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=-2 \, \log \left (x^{2} - 4 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2} + 6 \, x - 12 \, \log \left (2\right ) + 10\right ) + \log \left ({\left | x^{5} - 8 \, x^{4} \log \left (2\right ) + 24 \, x^{3} \log \left (2\right )^{2} - 32 \, x^{2} \log \left (2\right )^{3} + 16 \, x \log \left (2\right )^{4} + 11 \, x^{4} - 64 \, x^{3} \log \left (2\right ) + 120 \, x^{2} \log \left (2\right )^{2} - 64 \, x \log \left (2\right )^{3} - 16 \, \log \left (2\right )^{4} + 44 \, x^{3} - 152 \, x^{2} \log \left (2\right ) + 80 \, x \log \left (2\right )^{2} + 96 \, \log \left (2\right )^{3} + 64 \, x^{2} - 16 \, x \log \left (2\right ) - 224 \, \log \left (2\right )^{2} - 20 \, x + 240 \, \log \left (2\right ) - 95 \right |}\right ) + \log \left ({\left | x \right |}\right ) \] Input:
integrate((64*(-1+2*x)*log(2)^6+32*(-12*x^2-30*x+18)*log(2)^5+16*(30*x^3+1 65*x^2+186*x-138)*log(2)^4+8*(-40*x^4-340*x^3-924*x^2-600*x+576)*log(2)^3+ 4*(30*x^5+345*x^4+1476*x^3+2628*x^2+1032*x-1375)*log(2)^2+2*(-12*x^6-174*x ^5-1014*x^4-2904*x^3-3792*x^2-830*x+1770)*log(2)+2*x^7+35*x^6+258*x^5+1014 *x^4+2184*x^3+2205*x^2+170*x-950)/(64*(x^2-x)*log(2)^6+32*(-6*x^3-12*x^2+1 8*x)*log(2)^5+16*(15*x^4+75*x^3+48*x^2-138*x)*log(2)^4+8*(-20*x^5-160*x^4- 372*x^3-24*x^2+576*x)*log(2)^3+4*(15*x^6+165*x^5+648*x^4+900*x^3-348*x^2-1 375*x)*log(2)^2+2*(-6*x^7-84*x^6-462*x^5-1176*x^4-1032*x^3+950*x^2+1770*x) *log(2)+x^8+17*x^7+120*x^6+438*x^5+804*x^4+425*x^3-770*x^2-950*x),x, algor ithm="giac")
Output:
-2*log(x^2 - 4*x*log(2) + 4*log(2)^2 + 6*x - 12*log(2) + 10) + log(abs(x^5 - 8*x^4*log(2) + 24*x^3*log(2)^2 - 32*x^2*log(2)^3 + 16*x*log(2)^4 + 11*x ^4 - 64*x^3*log(2) + 120*x^2*log(2)^2 - 64*x*log(2)^3 - 16*log(2)^4 + 44*x ^3 - 152*x^2*log(2) + 80*x*log(2)^2 + 96*log(2)^3 + 64*x^2 - 16*x*log(2) - 224*log(2)^2 - 20*x + 240*log(2) - 95)) + log(abs(x))
Time = 3.52 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.86 \[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=\ln \left (x\,\left (20\,x-240\,\ln \left (2\right )-120\,x^2\,{\ln \left (2\right )}^2+32\,x^2\,{\ln \left (2\right )}^3-24\,x^3\,{\ln \left (2\right )}^2+16\,x\,\ln \left (2\right )-80\,x\,{\ln \left (2\right )}^2+152\,x^2\,\ln \left (2\right )+64\,x\,{\ln \left (2\right )}^3+64\,x^3\,\ln \left (2\right )-16\,x\,{\ln \left (2\right )}^4+8\,x^4\,\ln \left (2\right )+224\,{\ln \left (2\right )}^2-96\,{\ln \left (2\right )}^3+16\,{\ln \left (2\right )}^4-64\,x^2-44\,x^3-11\,x^4-x^5+95\right )\right )-2\,\ln \left (6\,x-12\,\ln \left (2\right )-4\,x\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+10\right ) \] Input:
int((170*x - 8*log(2)^3*(600*x + 924*x^2 + 340*x^3 + 40*x^4 - 576) - 2*log (2)*(830*x + 3792*x^2 + 2904*x^3 + 1014*x^4 + 174*x^5 + 12*x^6 - 1770) + 4 *log(2)^2*(1032*x + 2628*x^2 + 1476*x^3 + 345*x^4 + 30*x^5 - 1375) + 64*lo g(2)^6*(2*x - 1) - 32*log(2)^5*(30*x + 12*x^2 - 18) + 16*log(2)^4*(186*x + 165*x^2 + 30*x^3 - 138) + 2205*x^2 + 2184*x^3 + 1014*x^4 + 258*x^5 + 35*x ^6 + 2*x^7 - 950)/(16*log(2)^4*(48*x^2 - 138*x + 75*x^3 + 15*x^4) - 950*x - 64*log(2)^6*(x - x^2) - 8*log(2)^3*(24*x^2 - 576*x + 372*x^3 + 160*x^4 + 20*x^5) - 2*log(2)*(1032*x^3 - 950*x^2 - 1770*x + 1176*x^4 + 462*x^5 + 84 *x^6 + 6*x^7) + 4*log(2)^2*(900*x^3 - 348*x^2 - 1375*x + 648*x^4 + 165*x^5 + 15*x^6) - 32*log(2)^5*(12*x^2 - 18*x + 6*x^3) - 770*x^2 + 425*x^3 + 804 *x^4 + 438*x^5 + 120*x^6 + 17*x^7 + x^8),x)
Output:
log(x*(20*x - 240*log(2) - 120*x^2*log(2)^2 + 32*x^2*log(2)^3 - 24*x^3*log (2)^2 + 16*x*log(2) - 80*x*log(2)^2 + 152*x^2*log(2) + 64*x*log(2)^3 + 64* x^3*log(2) - 16*x*log(2)^4 + 8*x^4*log(2) + 224*log(2)^2 - 96*log(2)^3 + 1 6*log(2)^4 - 64*x^2 - 44*x^3 - 11*x^4 - x^5 + 95)) - 2*log(6*x - 12*log(2) - 4*x*log(2) + 4*log(2)^2 + x^2 + 10)
\[ \int \frac {-950+170 x+2205 x^2+2184 x^3+1014 x^4+258 x^5+35 x^6+2 x^7+\left (1770-830 x-3792 x^2-2904 x^3-1014 x^4-174 x^5-12 x^6\right ) \log (4)+\left (-1375+1032 x+2628 x^2+1476 x^3+345 x^4+30 x^5\right ) \log ^2(4)+\left (576-600 x-924 x^2-340 x^3-40 x^4\right ) \log ^3(4)+\left (-138+186 x+165 x^2+30 x^3\right ) \log ^4(4)+\left (18-30 x-12 x^2\right ) \log ^5(4)+(-1+2 x) \log ^6(4)}{-950 x-770 x^2+425 x^3+804 x^4+438 x^5+120 x^6+17 x^7+x^8+\left (1770 x+950 x^2-1032 x^3-1176 x^4-462 x^5-84 x^6-6 x^7\right ) \log (4)+\left (-1375 x-348 x^2+900 x^3+648 x^4+165 x^5+15 x^6\right ) \log ^2(4)+\left (576 x-24 x^2-372 x^3-160 x^4-20 x^5\right ) \log ^3(4)+\left (-138 x+48 x^2+75 x^3+15 x^4\right ) \log ^4(4)+\left (18 x-12 x^2-6 x^3\right ) \log ^5(4)+\left (-x+x^2\right ) \log ^6(4)} \, dx=\text {too large to display} \] Input:
int((64*(-1+2*x)*log(2)^6+32*(-12*x^2-30*x+18)*log(2)^5+16*(30*x^3+165*x^2 +186*x-138)*log(2)^4+8*(-40*x^4-340*x^3-924*x^2-600*x+576)*log(2)^3+4*(30* x^5+345*x^4+1476*x^3+2628*x^2+1032*x-1375)*log(2)^2+2*(-12*x^6-174*x^5-101 4*x^4-2904*x^3-3792*x^2-830*x+1770)*log(2)+2*x^7+35*x^6+258*x^5+1014*x^4+2 184*x^3+2205*x^2+170*x-950)/(64*(x^2-x)*log(2)^6+32*(-6*x^3-12*x^2+18*x)*l og(2)^5+16*(15*x^4+75*x^3+48*x^2-138*x)*log(2)^4+8*(-20*x^5-160*x^4-372*x^ 3-24*x^2+576*x)*log(2)^3+4*(15*x^6+165*x^5+648*x^4+900*x^3-348*x^2-1375*x) *log(2)^2+2*(-6*x^7-84*x^6-462*x^5-1176*x^4-1032*x^3+950*x^2+1770*x)*log(2 )+x^8+17*x^7+120*x^6+438*x^5+804*x^4+425*x^3-770*x^2-950*x),x)
Output:
( - 288*int(x**4/(768*log(2)**7*x - 768*log(2)**7 - 2304*log(2)**6*x**2 - 5696*log(2)**6*x + 8000*log(2)**6 + 2880*log(2)**5*x**3 + 17664*log(2)**5* x**2 + 15744*log(2)**5*x - 36288*log(2)**5 - 1920*log(2)**4*x**4 - 19440*l og(2)**4*x**3 - 56112*log(2)**4*x**2 - 15360*log(2)**4*x + 92832*log(2)**4 + 720*log(2)**3*x**5 + 10640*log(2)**3*x**4 + 52864*log(2)**3*x**3 + 9379 2*log(2)**3*x**2 - 13440*log(2)**3*x - 144336*log(2)**3 - 144*log(2)**2*x* *6 - 3036*log(2)**2*x**5 - 22308*log(2)**2*x**4 - 72288*log(2)**2*x**3 - 8 5968*log(2)**2*x**2 + 46464*log(2)**2*x + 135980*log(2)**2 + 12*log(2)*x** 7 + 408*log(2)*x**6 + 4296*log(2)*x**5 + 20964*log(2)*x**4 + 49632*log(2)* x**3 + 40188*log(2)*x**2 - 41540*log(2)*x - 71580*log(2) - 17*x**7 - 289*x **6 - 2040*x**5 - 7446*x**4 - 13668*x**3 - 7225*x**2 + 13090*x + 16150),x) *log(2)**3 + 1416*int(x**4/(768*log(2)**7*x - 768*log(2)**7 - 2304*log(2)* *6*x**2 - 5696*log(2)**6*x + 8000*log(2)**6 + 2880*log(2)**5*x**3 + 17664* log(2)**5*x**2 + 15744*log(2)**5*x - 36288*log(2)**5 - 1920*log(2)**4*x**4 - 19440*log(2)**4*x**3 - 56112*log(2)**4*x**2 - 15360*log(2)**4*x + 92832 *log(2)**4 + 720*log(2)**3*x**5 + 10640*log(2)**3*x**4 + 52864*log(2)**3*x **3 + 93792*log(2)**3*x**2 - 13440*log(2)**3*x - 144336*log(2)**3 - 144*lo g(2)**2*x**6 - 3036*log(2)**2*x**5 - 22308*log(2)**2*x**4 - 72288*log(2)** 2*x**3 - 85968*log(2)**2*x**2 + 46464*log(2)**2*x + 135980*log(2)**2 + 12* log(2)*x**7 + 408*log(2)*x**6 + 4296*log(2)*x**5 + 20964*log(2)*x**4 + ...