Integrand size = 62, antiderivative size = 19 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {4}{9} (8+x)^2 (-x+\log (2 x))^2 \] Output:
4*(ln(2*x)-x)^2*(1/3*x+8/3)^2
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(19)=38\).
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.05 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {8}{9} \left (32 x^2+8 x^3+\frac {x^4}{2}-64 x \log (2 x)-16 x^2 \log (2 x)-x^3 \log (2 x)+32 \log ^2(2 x)+8 x \log ^2(2 x)+\frac {1}{2} x^2 \log ^2(2 x)\right ) \] Input:
Integrate[(-512*x + 384*x^2 + 184*x^3 + 16*x^4 + (512 - 384*x - 248*x^2 - 24*x^3)*Log[2*x] + (64*x + 8*x^2)*Log[2*x]^2)/(9*x),x]
Output:
(8*(32*x^2 + 8*x^3 + x^4/2 - 64*x*Log[2*x] - 16*x^2*Log[2*x] - x^3*Log[2*x ] + 32*Log[2*x]^2 + 8*x*Log[2*x]^2 + (x^2*Log[2*x]^2)/2))/9
Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(19)=38\).
Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {27, 27, 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.
\(\displaystyle \int \frac {16 x^4+184 x^3+384 x^2+\left (8 x^2+64 x\right ) \log ^2(2 x)+\left (-24 x^3-248 x^2-384 x+512\right ) \log (2 x)-512 x}{9 x} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int -\frac {8 \left (-2 x^4-23 x^3-48 x^2+64 x-\left (x^2+8 x\right ) \log ^2(2 x)-\left (-3 x^3-31 x^2-48 x+64\right ) \log (2 x)\right )}{x}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {8}{9} \int \frac {-2 x^4-23 x^3-48 x^2+64 x-\left (x^2+8 x\right ) \log ^2(2 x)-\left (-3 x^3-31 x^2-48 x+64\right ) \log (2 x)}{x}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {8}{9} \int \left (-2 x^3-23 x^2-48 x-(x+8) \log ^2(2 x)+64+\frac {(x+8) \left (3 x^2+7 x-8\right ) \log (2 x)}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {8}{9} \left (-\frac {x^4}{2}-8 x^3+x^3 \log (2 x)-32 x^2-\frac {1}{2} x^2 \log ^2(2 x)+16 x^2 \log (2 x)-8 x \log ^2(2 x)-32 \log ^2(2 x)+64 x \log (2 x)\right )\) |
Input:
Int[(-512*x + 384*x^2 + 184*x^3 + 16*x^4 + (512 - 384*x - 248*x^2 - 24*x^3 )*Log[2*x] + (64*x + 8*x^2)*Log[2*x]^2)/(9*x),x]
Output:
(-8*(-32*x^2 - 8*x^3 - x^4/2 + 64*x*Log[2*x] + 16*x^2*Log[2*x] + x^3*Log[2 *x] - 32*Log[2*x]^2 - 8*x*Log[2*x]^2 - (x^2*Log[2*x]^2)/2))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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]]
Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(19)=38\).
Time = 1.97 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.89
method | result | size |
risch | \(\frac {\left (4 x^{2}+64 x +256\right ) \ln \left (2 x \right )^{2}}{9}+\frac {\left (-8 x^{3}-128 x^{2}-512 x \right ) \ln \left (2 x \right )}{9}+\frac {4 x^{4}}{9}+\frac {64 x^{3}}{9}+\frac {256 x^{2}}{9}\) | \(55\) |
derivativedivides | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 \ln \left (2 x \right ) x^{3}}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
default | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 \ln \left (2 x \right ) x^{3}}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
norman | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 \ln \left (2 x \right ) x^{3}}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
parallelrisch | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 \ln \left (2 x \right ) x^{3}}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
parts | \(\frac {4 x^{2} \ln \left (2 x \right )^{2}}{9}-\frac {128 x^{2} \ln \left (2 x \right )}{9}+\frac {256 x^{2}}{9}-\frac {8 \ln \left (2 x \right ) x^{3}}{9}+\frac {64 x^{3}}{9}+\frac {4 x^{4}}{9}+\frac {64 x \ln \left (2 x \right )^{2}}{9}-\frac {512 x \ln \left (2 x \right )}{9}+\frac {256 \ln \left (2 x \right )^{2}}{9}\) | \(70\) |
orering | \(\frac {\left (54 x^{9}+1608 x^{8}+17367 x^{7}+79323 x^{6}+153552 x^{5}+372352 x^{4}+1421824 x^{3}-1163264 x^{2}-196608 x -786432\right ) \left (\left (8 x^{2}+64 x \right ) \ln \left (2 x \right )^{2}+\left (-24 x^{3}-248 x^{2}-384 x +512\right ) \ln \left (2 x \right )+16 x^{4}+184 x^{3}+384 x^{2}-512 x \right )}{54 \left (12 x^{5}-25 x^{4}+121 x^{3}-36 x^{2}-8 x -64\right ) \left (x +8\right )^{3} x}-\frac {x \left (18 x^{8}+531 x^{7}+5580 x^{6}+22887 x^{5}+23784 x^{4}-21440 x^{3}+171520 x^{2}-598016 x +294912\right ) \left (\frac {\left (16 x +64\right ) \ln \left (2 x \right )^{2}+\frac {2 \left (8 x^{2}+64 x \right ) \ln \left (2 x \right )}{x}+\left (-72 x^{2}-496 x -384\right ) \ln \left (2 x \right )+\frac {-24 x^{3}-248 x^{2}-384 x +512}{x}+64 x^{3}+552 x^{2}+768 x -512}{9 x}-\frac {\left (8 x^{2}+64 x \right ) \ln \left (2 x \right )^{2}+\left (-24 x^{3}-248 x^{2}-384 x +512\right ) \ln \left (2 x \right )+16 x^{4}+184 x^{3}+384 x^{2}-512 x}{9 x^{2}}\right )}{6 \left (12 x^{5}-25 x^{4}+121 x^{3}-36 x^{2}-8 x -64\right ) \left (x +8\right )^{2}}+\frac {\left (3 x^{7}+87 x^{6}+873 x^{5}+2973 x^{4}-21440 x^{2}+38144 x -12288\right ) x^{2} \left (\frac {16 \ln \left (2 x \right )^{2}+\frac {4 \left (16 x +64\right ) \ln \left (2 x \right )}{x}+\frac {16 x^{2}+128 x}{x^{2}}-\frac {2 \left (8 x^{2}+64 x \right ) \ln \left (2 x \right )}{x^{2}}+\left (-144 x -496\right ) \ln \left (2 x \right )+\frac {-144 x^{2}-992 x -768}{x}-\frac {-24 x^{3}-248 x^{2}-384 x +512}{x^{2}}+192 x^{2}+1104 x +768}{9 x}-\frac {2 \left (\left (16 x +64\right ) \ln \left (2 x \right )^{2}+\frac {2 \left (8 x^{2}+64 x \right ) \ln \left (2 x \right )}{x}+\left (-72 x^{2}-496 x -384\right ) \ln \left (2 x \right )+\frac {-24 x^{3}-248 x^{2}-384 x +512}{x}+64 x^{3}+552 x^{2}+768 x -512\right )}{9 x^{2}}+\frac {\frac {2 \left (8 x^{2}+64 x \right ) \ln \left (2 x \right )^{2}}{9}+\frac {2 \left (-24 x^{3}-248 x^{2}-384 x +512\right ) \ln \left (2 x \right )}{9}+\frac {32 x^{4}}{9}+\frac {368 x^{3}}{9}+\frac {256 x^{2}}{3}-\frac {1024 x}{9}}{x^{3}}\right )}{6 \left (x +8\right ) \left (12 x^{4}-13 x^{3}+108 x^{2}+72 x +64\right ) \left (-1+x \right )}\) | \(685\) |
Input:
int(1/9*((8*x^2+64*x)*ln(2*x)^2+(-24*x^3-248*x^2-384*x+512)*ln(2*x)+16*x^4 +184*x^3+384*x^2-512*x)/x,x,method=_RETURNVERBOSE)
Output:
1/9*(4*x^2+64*x+256)*ln(2*x)^2+1/9*(-8*x^3-128*x^2-512*x)*ln(2*x)+4/9*x^4+ 64/9*x^3+256/9*x^2
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (17) = 34\).
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.63 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {4}{9} \, x^{4} + \frac {64}{9} \, x^{3} + \frac {4}{9} \, {\left (x^{2} + 16 \, x + 64\right )} \log \left (2 \, x\right )^{2} + \frac {256}{9} \, x^{2} - \frac {8}{9} \, {\left (x^{3} + 16 \, x^{2} + 64 \, x\right )} \log \left (2 \, x\right ) \] Input:
integrate(1/9*((8*x^2+64*x)*log(2*x)^2+(-24*x^3-248*x^2-384*x+512)*log(2*x )+16*x^4+184*x^3+384*x^2-512*x)/x,x, algorithm="fricas")
Output:
4/9*x^4 + 64/9*x^3 + 4/9*(x^2 + 16*x + 64)*log(2*x)^2 + 256/9*x^2 - 8/9*(x ^3 + 16*x^2 + 64*x)*log(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (17) = 34\).
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 3.47 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {4 x^{4}}{9} + \frac {64 x^{3}}{9} + \frac {256 x^{2}}{9} + \left (\frac {4 x^{2}}{9} + \frac {64 x}{9} + \frac {256}{9}\right ) \log {\left (2 x \right )}^{2} + \left (- \frac {8 x^{3}}{9} - \frac {128 x^{2}}{9} - \frac {512 x}{9}\right ) \log {\left (2 x \right )} \] Input:
integrate(1/9*((8*x**2+64*x)*ln(2*x)**2+(-24*x**3-248*x**2-384*x+512)*ln(2 *x)+16*x**4+184*x**3+384*x**2-512*x)/x,x)
Output:
4*x**4/9 + 64*x**3/9 + 256*x**2/9 + (4*x**2/9 + 64*x/9 + 256/9)*log(2*x)** 2 + (-8*x**3/9 - 128*x**2/9 - 512*x/9)*log(2*x)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.74 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {4}{9} \, x^{4} - \frac {8}{9} \, x^{3} \log \left (2 \, x\right ) + \frac {2}{9} \, {\left (2 \, \log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 1\right )} x^{2} + \frac {64}{9} \, x^{3} - \frac {124}{9} \, x^{2} \log \left (2 \, x\right ) + \frac {64}{9} \, {\left (\log \left (2 \, x\right )^{2} - 2 \, \log \left (2 \, x\right ) + 2\right )} x + \frac {254}{9} \, x^{2} - \frac {128}{3} \, x \log \left (2 \, x\right ) + \frac {256}{9} \, \log \left (2 \, x\right )^{2} - \frac {128}{9} \, x \] Input:
integrate(1/9*((8*x^2+64*x)*log(2*x)^2+(-24*x^3-248*x^2-384*x+512)*log(2*x )+16*x^4+184*x^3+384*x^2-512*x)/x,x, algorithm="maxima")
Output:
4/9*x^4 - 8/9*x^3*log(2*x) + 2/9*(2*log(2*x)^2 - 2*log(2*x) + 1)*x^2 + 64/ 9*x^3 - 124/9*x^2*log(2*x) + 64/9*(log(2*x)^2 - 2*log(2*x) + 2)*x + 254/9* x^2 - 128/3*x*log(2*x) + 256/9*log(2*x)^2 - 128/9*x
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (17) = 34\).
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.63 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {4}{9} \, x^{4} + \frac {64}{9} \, x^{3} + \frac {4}{9} \, {\left (x^{2} + 16 \, x + 64\right )} \log \left (2 \, x\right )^{2} + \frac {256}{9} \, x^{2} - \frac {8}{9} \, {\left (x^{3} + 16 \, x^{2} + 64 \, x\right )} \log \left (2 \, x\right ) \] Input:
integrate(1/9*((8*x^2+64*x)*log(2*x)^2+(-24*x^3-248*x^2-384*x+512)*log(2*x )+16*x^4+184*x^3+384*x^2-512*x)/x,x, algorithm="giac")
Output:
4/9*x^4 + 64/9*x^3 + 4/9*(x^2 + 16*x + 64)*log(2*x)^2 + 256/9*x^2 - 8/9*(x ^3 + 16*x^2 + 64*x)*log(2*x)
Time = 3.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {4\,{\left (x-\ln \left (2\,x\right )\right )}^2\,{\left (x+8\right )}^2}{9} \] Input:
int(((log(2*x)^2*(64*x + 8*x^2))/9 - (log(2*x)*(384*x + 248*x^2 + 24*x^3 - 512))/9 - (512*x)/9 + (128*x^2)/3 + (184*x^3)/9 + (16*x^4)/9)/x,x)
Output:
(4*(x - log(2*x))^2*(x + 8)^2)/9
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.63 \[ \int \frac {-512 x+384 x^2+184 x^3+16 x^4+\left (512-384 x-248 x^2-24 x^3\right ) \log (2 x)+\left (64 x+8 x^2\right ) \log ^2(2 x)}{9 x} \, dx=\frac {4 \mathrm {log}\left (2 x \right )^{2} x^{2}}{9}+\frac {64 \mathrm {log}\left (2 x \right )^{2} x}{9}+\frac {256 \mathrm {log}\left (2 x \right )^{2}}{9}-\frac {8 \,\mathrm {log}\left (2 x \right ) x^{3}}{9}-\frac {128 \,\mathrm {log}\left (2 x \right ) x^{2}}{9}-\frac {512 \,\mathrm {log}\left (2 x \right ) x}{9}+\frac {4 x^{4}}{9}+\frac {64 x^{3}}{9}+\frac {256 x^{2}}{9} \] Input:
int(1/9*((8*x^2+64*x)*log(2*x)^2+(-24*x^3-248*x^2-384*x+512)*log(2*x)+16*x ^4+184*x^3+384*x^2-512*x)/x,x)
Output:
(4*(log(2*x)**2*x**2 + 16*log(2*x)**2*x + 64*log(2*x)**2 - 2*log(2*x)*x**3 - 32*log(2*x)*x**2 - 128*log(2*x)*x + x**4 + 16*x**3 + 64*x**2))/9