Integrand size = 80, antiderivative size = 22 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx=\log (2)+\left (4+\frac {(-8-x)^4}{x}+x\right )^2 \log (x) \] Output:
ln(2)+((-x-8)^4/x+x+4)^2*ln(x)
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(22)=44\).
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx=7364624 \log (x)+\frac {16777216 \log (x)}{x^2}+\frac {16809984 \log (x)}{x}+1842184 x \log (x)+287745 x^2 \log (x)+28744 x^3 \log (x)+1794 x^4 \log (x)+64 x^5 \log (x)+x^6 \log (x) \] Input:
Integrate[(16777216 + 16809984*x + 7364624*x^2 + 1842184*x^3 + 287745*x^4 + 28744*x^5 + 1794*x^6 + 64*x^7 + x^8 + (-33554432 - 16809984*x + 1842184* x^3 + 575490*x^4 + 86232*x^5 + 7176*x^6 + 320*x^7 + 6*x^8)*Log[x])/x^3,x]
Output:
7364624*Log[x] + (16777216*Log[x])/x^2 + (16809984*Log[x])/x + 1842184*x*L og[x] + 287745*x^2*Log[x] + 28744*x^3*Log[x] + 1794*x^4*Log[x] + 64*x^5*Lo g[x] + x^6*Log[x]
Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(22)=44\).
Time = 0.58 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2010, 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.
\(\displaystyle \int \frac {x^8+64 x^7+1794 x^6+28744 x^5+287745 x^4+1842184 x^3+7364624 x^2+\left (6 x^8+320 x^7+7176 x^6+86232 x^5+575490 x^4+1842184 x^3-16809984 x-33554432\right ) \log (x)+16809984 x+16777216}{x^3} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (\frac {\left (x^4+32 x^3+385 x^2+2052 x+4096\right )^2}{x^3}+\frac {2 \left (3 x^4+64 x^3+385 x^2-4096\right ) \left (x^4+32 x^3+385 x^2+2052 x+4096\right ) \log (x)}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^6 \log (x)+64 x^5 \log (x)+1794 x^4 \log (x)+28744 x^3 \log (x)+287745 x^2 \log (x)+\frac {16777216 \log (x)}{x^2}+1842184 x \log (x)+7364624 \log (x)+\frac {16809984 \log (x)}{x}\) |
Input:
Int[(16777216 + 16809984*x + 7364624*x^2 + 1842184*x^3 + 287745*x^4 + 2874 4*x^5 + 1794*x^6 + 64*x^7 + x^8 + (-33554432 - 16809984*x + 1842184*x^3 + 575490*x^4 + 86232*x^5 + 7176*x^6 + 320*x^7 + 6*x^8)*Log[x])/x^3,x]
Output:
7364624*Log[x] + (16777216*Log[x])/x^2 + (16809984*Log[x])/x + 1842184*x*L og[x] + 287745*x^2*Log[x] + 28744*x^3*Log[x] + 1794*x^4*Log[x] + 64*x^5*Lo g[x] + x^6*Log[x]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 3.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05
method | result | size |
risch | \(\frac {\left (x^{8}+64 x^{7}+1794 x^{6}+28744 x^{5}+287745 x^{4}+1842184 x^{3}+16809984 x +16777216\right ) \ln \left (x \right )}{x^{2}}+7364624 \ln \left (x \right )\) | \(45\) |
default | \(1794 x^{4} \ln \left (x \right )+x^{6} \ln \left (x \right )+28744 x^{3} \ln \left (x \right )+\frac {16809984 \ln \left (x \right )}{x}+\frac {16777216 \ln \left (x \right )}{x^{2}}+64 x^{5} \ln \left (x \right )+287745 x^{2} \ln \left (x \right )+1842184 x \ln \left (x \right )+7364624 \ln \left (x \right )\) | \(59\) |
parts | \(1794 x^{4} \ln \left (x \right )+x^{6} \ln \left (x \right )+28744 x^{3} \ln \left (x \right )+\frac {16809984 \ln \left (x \right )}{x}+\frac {16777216 \ln \left (x \right )}{x^{2}}+64 x^{5} \ln \left (x \right )+287745 x^{2} \ln \left (x \right )+1842184 x \ln \left (x \right )+7364624 \ln \left (x \right )\) | \(59\) |
norman | \(\frac {x^{8} \ln \left (x \right )+7364624 x^{2} \ln \left (x \right )+16809984 x \ln \left (x \right )+1842184 x^{3} \ln \left (x \right )+287745 x^{4} \ln \left (x \right )+28744 x^{5} \ln \left (x \right )+1794 x^{6} \ln \left (x \right )+64 x^{7} \ln \left (x \right )+16777216 \ln \left (x \right )}{x^{2}}\) | \(63\) |
parallelrisch | \(-\frac {-x^{8} \ln \left (x \right )-64 x^{7} \ln \left (x \right )-1794 x^{6} \ln \left (x \right )-28744 x^{5} \ln \left (x \right )-287745 x^{4} \ln \left (x \right )-1842184 x^{3} \ln \left (x \right )-7364624 x^{2} \ln \left (x \right )-16809984 x \ln \left (x \right )-16777216 \ln \left (x \right )}{x^{2}}\) | \(65\) |
orering | \(\frac {\left (33 x^{16}+3840 x^{15}+207468 x^{14}+6890880 x^{13}+156936318 x^{12}+2585685600 x^{11}+27986222205 x^{10}+133741751744 x^{9}-664459449219 x^{8}-11715232450296 x^{7}-44069538663057 x^{6}-377480921481984 x^{4}-5948479326683136 x^{3}-17019316645920768 x^{2}-6768593580589056 x -4222124650659840\right ) \left (\left (6 x^{8}+320 x^{7}+7176 x^{6}+86232 x^{5}+575490 x^{4}+1842184 x^{3}-16809984 x -33554432\right ) \ln \left (x \right )+x^{8}+64 x^{7}+1794 x^{6}+28744 x^{5}+287745 x^{4}+1842184 x^{3}+7364624 x^{2}+16809984 x +16777216\right )}{12 x^{2} \left (x^{4}+32 x^{3}+385 x^{2}+2052 x +4096\right )^{2} \left (9 x^{8}+368 x^{7}+5636 x^{6}+33886 x^{5}-40447 x^{4}-1509122 x^{3}-6307840 x^{2}-4202496 x +16777216\right )}-\frac {\left (3 x^{12}+288 x^{11}+12681 x^{10}+338532 x^{9}+6101001 x^{8}+78179688 x^{7}-10569623068 x^{5}-68672964769 x^{4}+88999526592 x^{3}+1210434457600 x^{2}+309841625088 x +206158430208\right ) x^{2} \left (\frac {\left (48 x^{7}+2240 x^{6}+43056 x^{5}+431160 x^{4}+2301960 x^{3}+5526552 x^{2}-16809984\right ) \ln \left (x \right )+\frac {6 x^{8}+320 x^{7}+7176 x^{6}+86232 x^{5}+575490 x^{4}+1842184 x^{3}-16809984 x -33554432}{x}+8 x^{7}+448 x^{6}+10764 x^{5}+143720 x^{4}+1150980 x^{3}+5526552 x^{2}+14729248 x +16809984}{x^{3}}-\frac {3 \left (\left (6 x^{8}+320 x^{7}+7176 x^{6}+86232 x^{5}+575490 x^{4}+1842184 x^{3}-16809984 x -33554432\right ) \ln \left (x \right )+x^{8}+64 x^{7}+1794 x^{6}+28744 x^{5}+287745 x^{4}+1842184 x^{3}+7364624 x^{2}+16809984 x +16777216\right )}{x^{4}}\right )}{12 \left (x^{4}+32 x^{3}+385 x^{2}+2052 x +4096\right ) \left (9 x^{8}+368 x^{7}+5636 x^{6}+33886 x^{5}-40447 x^{4}-1509122 x^{3}-6307840 x^{2}-4202496 x +16777216\right )}\) | \(537\) |
Input:
int(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-16809984*x-3 3554432)*ln(x)+x^8+64*x^7+1794*x^6+28744*x^5+287745*x^4+1842184*x^3+736462 4*x^2+16809984*x+16777216)/x^3,x,method=_RETURNVERBOSE)
Output:
(x^8+64*x^7+1794*x^6+28744*x^5+287745*x^4+1842184*x^3+16809984*x+16777216) /x^2*ln(x)+7364624*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx=\frac {{\left (x^{8} + 64 \, x^{7} + 1794 \, x^{6} + 28744 \, x^{5} + 287745 \, x^{4} + 1842184 \, x^{3} + 7364624 \, x^{2} + 16809984 \, x + 16777216\right )} \log \left (x\right )}{x^{2}} \] Input:
integrate(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-168099 84*x-33554432)*log(x)+x^8+64*x^7+1794*x^6+28744*x^5+287745*x^4+1842184*x^3 +7364624*x^2+16809984*x+16777216)/x^3,x, algorithm="fricas")
Output:
(x^8 + 64*x^7 + 1794*x^6 + 28744*x^5 + 287745*x^4 + 1842184*x^3 + 7364624* x^2 + 16809984*x + 16777216)*log(x)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx=7364624 \log {\left (x \right )} + \frac {\left (x^{8} + 64 x^{7} + 1794 x^{6} + 28744 x^{5} + 287745 x^{4} + 1842184 x^{3} + 16809984 x + 16777216\right ) \log {\left (x \right )}}{x^{2}} \] Input:
integrate(((6*x**8+320*x**7+7176*x**6+86232*x**5+575490*x**4+1842184*x**3- 16809984*x-33554432)*ln(x)+x**8+64*x**7+1794*x**6+28744*x**5+287745*x**4+1 842184*x**3+7364624*x**2+16809984*x+16777216)/x**3,x)
Output:
7364624*log(x) + (x**8 + 64*x**7 + 1794*x**6 + 28744*x**5 + 287745*x**4 + 1842184*x**3 + 16809984*x + 16777216)*log(x)/x**2
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (20) = 40\).
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.64 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx=x^{6} \log \left (x\right ) + 64 \, x^{5} \log \left (x\right ) + 1794 \, x^{4} \log \left (x\right ) + 28744 \, x^{3} \log \left (x\right ) + 287745 \, x^{2} \log \left (x\right ) + 1842184 \, x \log \left (x\right ) + \frac {16809984 \, \log \left (x\right )}{x} + \frac {16777216 \, \log \left (x\right )}{x^{2}} + 7364624 \, \log \left (x\right ) \] Input:
integrate(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-168099 84*x-33554432)*log(x)+x^8+64*x^7+1794*x^6+28744*x^5+287745*x^4+1842184*x^3 +7364624*x^2+16809984*x+16777216)/x^3,x, algorithm="maxima")
Output:
x^6*log(x) + 64*x^5*log(x) + 1794*x^4*log(x) + 28744*x^3*log(x) + 287745*x ^2*log(x) + 1842184*x*log(x) + 16809984*log(x)/x + 16777216*log(x)/x^2 + 7 364624*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (20) = 40\).
Time = 0.12 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.05 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx={\left (x^{6} + 64 \, x^{5} + 1794 \, x^{4} + 28744 \, x^{3} + 287745 \, x^{2} + 1842184 \, x + \frac {32768 \, {\left (513 \, x + 512\right )}}{x^{2}}\right )} \log \left (x\right ) + 7364624 \, \log \left (x\right ) \] Input:
integrate(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-168099 84*x-33554432)*log(x)+x^8+64*x^7+1794*x^6+28744*x^5+287745*x^4+1842184*x^3 +7364624*x^2+16809984*x+16777216)/x^3,x, algorithm="giac")
Output:
(x^6 + 64*x^5 + 1794*x^4 + 28744*x^3 + 287745*x^2 + 1842184*x + 32768*(513 *x + 512)/x^2)*log(x) + 7364624*log(x)
Time = 3.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx=\frac {\ln \left (x\right )\,{\left (x^4+32\,x^3+385\,x^2+2052\,x+4096\right )}^2}{x^2} \] Input:
int((16809984*x + log(x)*(1842184*x^3 - 16809984*x + 575490*x^4 + 86232*x^ 5 + 7176*x^6 + 320*x^7 + 6*x^8 - 33554432) + 7364624*x^2 + 1842184*x^3 + 2 87745*x^4 + 28744*x^5 + 1794*x^6 + 64*x^7 + x^8 + 16777216)/x^3,x)
Output:
(log(x)*(2052*x + 385*x^2 + 32*x^3 + x^4 + 4096)^2)/x^2
Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {16777216+16809984 x+7364624 x^2+1842184 x^3+287745 x^4+28744 x^5+1794 x^6+64 x^7+x^8+\left (-33554432-16809984 x+1842184 x^3+575490 x^4+86232 x^5+7176 x^6+320 x^7+6 x^8\right ) \log (x)}{x^3} \, dx=\frac {\mathrm {log}\left (x \right ) \left (x^{8}+64 x^{7}+1794 x^{6}+28744 x^{5}+287745 x^{4}+1842184 x^{3}+7364624 x^{2}+16809984 x +16777216\right )}{x^{2}} \] Input:
int(((6*x^8+320*x^7+7176*x^6+86232*x^5+575490*x^4+1842184*x^3-16809984*x-3 3554432)*log(x)+x^8+64*x^7+1794*x^6+28744*x^5+287745*x^4+1842184*x^3+73646 24*x^2+16809984*x+16777216)/x^3,x)
Output:
(log(x)*(x**8 + 64*x**7 + 1794*x**6 + 28744*x**5 + 287745*x**4 + 1842184*x **3 + 7364624*x**2 + 16809984*x + 16777216))/x**2