Integrand size = 249, antiderivative size = 27 \[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\frac {x}{\log \left (6 \left (3+\left (2+\frac {9}{x}-x-\log (2 x)\right )^2\right )\right )} \] Output:
x/ln(6*(9/x+2-x-ln(2*x))^2+18)
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\frac {x}{\log \left (\frac {6 \left (81+36 x-11 x^2-4 x^3+x^4+2 x \left (-9-2 x+x^2\right ) \log (2 x)+x^2 \log ^2(2 x)\right )}{x^2}\right )} \] Input:
Integrate[(162 + 54*x + 4*x^2 + 2*x^3 - 2*x^4 + (-18*x - 2*x^2 - 2*x^3)*Lo g[2*x] + (81 + 36*x - 11*x^2 - 4*x^3 + x^4 + (-18*x - 4*x^2 + 2*x^3)*Log[2 *x] + x^2*Log[2*x]^2)*Log[(486 + 216*x - 66*x^2 - 24*x^3 + 6*x^4 + (-108*x - 24*x^2 + 12*x^3)*Log[2*x] + 6*x^2*Log[2*x]^2)/x^2])/((81 + 36*x - 11*x^ 2 - 4*x^3 + x^4 + (-18*x - 4*x^2 + 2*x^3)*Log[2*x] + x^2*Log[2*x]^2)*Log[( 486 + 216*x - 66*x^2 - 24*x^3 + 6*x^4 + (-108*x - 24*x^2 + 12*x^3)*Log[2*x ] + 6*x^2*Log[2*x]^2)/x^2]^2),x]
Output:
x/Log[(6*(81 + 36*x - 11*x^2 - 4*x^3 + x^4 + 2*x*(-9 - 2*x + x^2)*Log[2*x] + x^2*Log[2*x]^2))/x^2]
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.
| \(\displaystyle \int \frac {-2 x^4+2 x^3+4 x^2+\left (-2 x^3-2 x^2-18 x\right ) \log (2 x)+\left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+\left (2 x^3-4 x^2-18 x\right ) \log (2 x)+36 x+81\right ) \log \left (\frac {6 x^4-24 x^3-66 x^2+6 x^2 \log ^2(2 x)+\left (12 x^3-24 x^2-108 x\right ) \log (2 x)+216 x+486}{x^2}\right )+54 x+162}{\left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+\left (2 x^3-4 x^2-18 x\right ) \log (2 x)+36 x+81\right ) \log ^2\left (\frac {6 x^4-24 x^3-66 x^2+6 x^2 \log ^2(2 x)+\left (12 x^3-24 x^2-108 x\right ) \log (2 x)+216 x+486}{x^2}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-2 x^4+2 x^3+4 x^2+\left (-2 x^3-2 x^2-18 x\right ) \log (2 x)+\left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+\left (2 x^3-4 x^2-18 x\right ) \log (2 x)+36 x+81\right ) \log \left (\frac {6 x^4-24 x^3-66 x^2+6 x^2 \log ^2(2 x)+\left (12 x^3-24 x^2-108 x\right ) \log (2 x)+216 x+486}{x^2}\right )+54 x+162}{\left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+\left (2 x^3-4 x^2-18 x\right ) \log (2 x)+36 x+81\right ) \log ^2\left (6 x^2+\frac {486}{x^2}+\frac {\left (12 x^3-24 x^2-108 x\right ) \log (2 x)}{x^2}-24 x+\frac {216}{x}+6 \log ^2(2 x)-66\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {2 x^4}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}+\frac {2 x^3}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}+\frac {4 x^2}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}-\frac {2 \left (x^2+x+9\right ) x \log (2 x)}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}+\frac {54 x}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}+\frac {1}{\log \left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}+\frac {162}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-x^4+x^3+2 x^2+27 x+81\right )+x^2 \log \left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right ) \log ^2(2 x)+2 x \left (-x^2+\left (x^2-2 x-9\right ) \log \left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )-x-9\right ) \log (2 x)+\left (x^4-4 x^3-11 x^2+36 x+81\right ) \log \left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}{\left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{\log \left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}-\frac {2 \left (x^2+x+9\right ) \left (x^2-2 x+x \log (2 x)-9\right )}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {1}{\log \left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}-\frac {2 \left (x^2+x+9\right ) \left (x^2-2 x+x \log (2 x)-9\right )}{\left (x^4-4 x^3+2 x^3 \log (2 x)-11 x^2+x^2 \log ^2(2 x)-4 x^2 \log (2 x)+36 x-18 x \log (2 x)+81\right ) \log ^2\left (\frac {6 \left (x^4-4 x^3-11 x^2+x^2 \log ^2(2 x)+2 \left (x^2-2 x-9\right ) x \log (2 x)+36 x+81\right )}{x^2}\right )}\right )dx\) |
Input:
Int[(162 + 54*x + 4*x^2 + 2*x^3 - 2*x^4 + (-18*x - 2*x^2 - 2*x^3)*Log[2*x] + (81 + 36*x - 11*x^2 - 4*x^3 + x^4 + (-18*x - 4*x^2 + 2*x^3)*Log[2*x] + x^2*Log[2*x]^2)*Log[(486 + 216*x - 66*x^2 - 24*x^3 + 6*x^4 + (-108*x - 24* x^2 + 12*x^3)*Log[2*x] + 6*x^2*Log[2*x]^2)/x^2])/((81 + 36*x - 11*x^2 - 4* x^3 + x^4 + (-18*x - 4*x^2 + 2*x^3)*Log[2*x] + x^2*Log[2*x]^2)*Log[(486 + 216*x - 66*x^2 - 24*x^3 + 6*x^4 + (-108*x - 24*x^2 + 12*x^3)*Log[2*x] + 6* x^2*Log[2*x]^2)/x^2]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(27)=54\).
Time = 6.86 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22
| method | result | size |
| parallelrisch | \(\frac {x}{\ln \left (\frac {6 x^{2} \ln \left (2 x \right )^{2}+\left (12 x^{3}-24 x^{2}-108 x \right ) \ln \left (2 x \right )+6 x^{4}-24 x^{3}-66 x^{2}+216 x +486}{x^{2}}\right )}\) | \(60\) |
| default | \(\frac {2 i x}{\pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (2\right )^{2}+\left (2 x^{2} \ln \left (x \right )+2 x^{3}-4 x^{2}-18 x \right ) \ln \left (2\right )+x^{2} \ln \left (x \right )^{2}+\left (2 x^{3}-4 x^{2}-18 x \right ) \ln \left (x \right )+x^{4}-4 x^{3}-11 x^{2}+36 x +81\right )\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (2\right )^{2}+\left (2 x^{2} \ln \left (x \right )+2 x^{3}-4 x^{2}-18 x \right ) \ln \left (2\right )+x^{2} \ln \left (x \right )^{2}+\left (2 x^{3}-4 x^{2}-18 x \right ) \ln \left (x \right )+x^{4}-4 x^{3}-11 x^{2}+36 x +81\right )}{x^{2}}\right )-\pi \,\operatorname {csgn}\left (i \left (x^{2} \ln \left (2\right )^{2}+\left (2 x^{2} \ln \left (x \right )+2 x^{3}-4 x^{2}-18 x \right ) \ln \left (2\right )+x^{2} \ln \left (x \right )^{2}+\left (2 x^{3}-4 x^{2}-18 x \right ) \ln \left (x \right )+x^{4}-4 x^{3}-11 x^{2}+36 x +81\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (2\right )^{2}+\left (2 x^{2} \ln \left (x \right )+2 x^{3}-4 x^{2}-18 x \right ) \ln \left (2\right )+x^{2} \ln \left (x \right )^{2}+\left (2 x^{3}-4 x^{2}-18 x \right ) \ln \left (x \right )+x^{4}-4 x^{3}-11 x^{2}+36 x +81\right )}{x^{2}}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{x^{2}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (2\right )^{2}+\left (2 x^{2} \ln \left (x \right )+2 x^{3}-4 x^{2}-18 x \right ) \ln \left (2\right )+x^{2} \ln \left (x \right )^{2}+\left (2 x^{3}-4 x^{2}-18 x \right ) \ln \left (x \right )+x^{4}-4 x^{3}-11 x^{2}+36 x +81\right )}{x^{2}}\right )}^{2}-\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )+2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}-\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2} \ln \left (2\right )^{2}+\left (2 x^{2} \ln \left (x \right )+2 x^{3}-4 x^{2}-18 x \right ) \ln \left (2\right )+x^{2} \ln \left (x \right )^{2}+\left (2 x^{3}-4 x^{2}-18 x \right ) \ln \left (x \right )+x^{4}-4 x^{3}-11 x^{2}+36 x +81\right )}{x^{2}}\right )}^{3}+2 i \ln \left (2\right )+2 i \ln \left (3\right )+2 i \ln \left (x^{2} \ln \left (2\right )^{2}+\left (2 x^{2} \ln \left (x \right )+2 x^{3}-4 x^{2}-18 x \right ) \ln \left (2\right )+x^{2} \ln \left (x \right )^{2}+\left (2 x^{3}-4 x^{2}-18 x \right ) \ln \left (x \right )+x^{4}-4 x^{3}-11 x^{2}+36 x +81\right )-4 i \ln \left (x \right )}\) | \(664\) |
Input:
int(((x^2*ln(2*x)^2+(2*x^3-4*x^2-18*x)*ln(2*x)+x^4-4*x^3-11*x^2+36*x+81)*l n((6*x^2*ln(2*x)^2+(12*x^3-24*x^2-108*x)*ln(2*x)+6*x^4-24*x^3-66*x^2+216*x +486)/x^2)+(-2*x^3-2*x^2-18*x)*ln(2*x)-2*x^4+2*x^3+4*x^2+54*x+162)/(x^2*ln (2*x)^2+(2*x^3-4*x^2-18*x)*ln(2*x)+x^4-4*x^3-11*x^2+36*x+81)/ln((6*x^2*ln( 2*x)^2+(12*x^3-24*x^2-108*x)*ln(2*x)+6*x^4-24*x^3-66*x^2+216*x+486)/x^2)^2 ,x,method=_RETURNVERBOSE)
Output:
x/ln((6*x^2*ln(2*x)^2+(12*x^3-24*x^2-108*x)*ln(2*x)+6*x^4-24*x^3-66*x^2+21 6*x+486)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\frac {x}{\log \left (\frac {6 \, {\left (x^{4} + x^{2} \log \left (2 \, x\right )^{2} - 4 \, x^{3} - 11 \, x^{2} + 2 \, {\left (x^{3} - 2 \, x^{2} - 9 \, x\right )} \log \left (2 \, x\right ) + 36 \, x + 81\right )}}{x^{2}}\right )} \] Input:
integrate(((x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36 *x+81)*log((6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-6 6*x^2+216*x+486)/x^2)+(-2*x^3-2*x^2-18*x)*log(2*x)-2*x^4+2*x^3+4*x^2+54*x+ 162)/(x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36*x+81) /log((6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-66*x^2+ 216*x+486)/x^2)^2,x, algorithm="fricas")
Output:
x/log(6*(x^4 + x^2*log(2*x)^2 - 4*x^3 - 11*x^2 + 2*(x^3 - 2*x^2 - 9*x)*log (2*x) + 36*x + 81)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).
Time = 0.36 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\frac {x}{\log {\left (\frac {6 x^{4} - 24 x^{3} + 6 x^{2} \log {\left (2 x \right )}^{2} - 66 x^{2} + 216 x + \left (12 x^{3} - 24 x^{2} - 108 x\right ) \log {\left (2 x \right )} + 486}{x^{2}} \right )}} \] Input:
integrate(((x**2*ln(2*x)**2+(2*x**3-4*x**2-18*x)*ln(2*x)+x**4-4*x**3-11*x* *2+36*x+81)*ln((6*x**2*ln(2*x)**2+(12*x**3-24*x**2-108*x)*ln(2*x)+6*x**4-2 4*x**3-66*x**2+216*x+486)/x**2)+(-2*x**3-2*x**2-18*x)*ln(2*x)-2*x**4+2*x** 3+4*x**2+54*x+162)/(x**2*ln(2*x)**2+(2*x**3-4*x**2-18*x)*ln(2*x)+x**4-4*x* *3-11*x**2+36*x+81)/ln((6*x**2*ln(2*x)**2+(12*x**3-24*x**2-108*x)*ln(2*x)+ 6*x**4-24*x**3-66*x**2+216*x+486)/x**2)**2,x)
Output:
x/log((6*x**4 - 24*x**3 + 6*x**2*log(2*x)**2 - 66*x**2 + 216*x + (12*x**3 - 24*x**2 - 108*x)*log(2*x) + 486)/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (23) = 46\).
Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\frac {x}{\log \left (3\right ) + \log \left (2\right ) + \log \left (x^{4} + 2 \, x^{3} {\left (\log \left (2\right ) - 2\right )} + x^{2} \log \left (x\right )^{2} + {\left (\log \left (2\right )^{2} - 4 \, \log \left (2\right ) - 11\right )} x^{2} - 18 \, x {\left (\log \left (2\right ) - 2\right )} + 2 \, {\left (x^{3} + x^{2} {\left (\log \left (2\right ) - 2\right )} - 9 \, x\right )} \log \left (x\right ) + 81\right ) - 2 \, \log \left (x\right )} \] Input:
integrate(((x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36 *x+81)*log((6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-6 6*x^2+216*x+486)/x^2)+(-2*x^3-2*x^2-18*x)*log(2*x)-2*x^4+2*x^3+4*x^2+54*x+ 162)/(x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36*x+81) /log((6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-66*x^2+ 216*x+486)/x^2)^2,x, algorithm="maxima")
Output:
x/(log(3) + log(2) + log(x^4 + 2*x^3*(log(2) - 2) + x^2*log(x)^2 + (log(2) ^2 - 4*log(2) - 11)*x^2 - 18*x*(log(2) - 2) + 2*(x^3 + x^2*(log(2) - 2) - 9*x)*log(x) + 81) - 2*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 3434 vs. \(2 (23) = 46\).
Time = 1.00 (sec) , antiderivative size = 3434, normalized size of antiderivative = 127.19 \[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\text {Too large to display} \] Input:
integrate(((x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36 *x+81)*log((6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-6 6*x^2+216*x+486)/x^2)+(-2*x^3-2*x^2-18*x)*log(2*x)-2*x^4+2*x^3+4*x^2+54*x+ 162)/(x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36*x+81) /log((6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-66*x^2+ 216*x+486)/x^2)^2,x, algorithm="giac")
Output:
(x^7 + x^6*log(2) + 2*x^6*log(2*x) + 2*x^5*log(2)*log(2*x) + x^5*log(2*x)^ 2 + x^4*log(2)*log(2*x)^2 + x^6*log(x) + 2*x^5*log(2*x)*log(x) + x^4*log(2 *x)^2*log(x) - 6*x^6 - 4*x^5*log(2) - 8*x^5*log(2*x) - 4*x^4*log(2)*log(2* x) - 2*x^4*log(2*x)^2 - 4*x^5*log(x) - 4*x^4*log(2*x)*log(x) - 12*x^5 - 11 *x^4*log(2) - 28*x^4*log(2*x) - 18*x^3*log(2)*log(2*x) - 9*x^3*log(2*x)^2 - 11*x^4*log(x) - 18*x^3*log(2*x)*log(x) + 94*x^4 + 36*x^3*log(2) + 72*x^3 *log(2*x) + 36*x^3*log(x) + 108*x^3 + 81*x^2*log(2) + 162*x^2*log(2*x) + 8 1*x^2*log(x) - 486*x^2 - 729*x)/(x^6*log(2) + 2*x^5*log(2)^2 + x^4*log(2)^ 3 + x^6*log(3*x^4 + 6*x^3*log(2*x) + 3*x^2*log(2*x)^2 - 12*x^3 - 12*x^2*lo g(2*x) - 33*x^2 - 54*x*log(2*x) + 108*x + 243) + 2*x^5*log(2)*log(3*x^4 + 6*x^3*log(2*x) + 3*x^2*log(2*x)^2 - 12*x^3 - 12*x^2*log(2*x) - 33*x^2 - 54 *x*log(2*x) + 108*x + 243) + x^4*log(2)^2*log(3*x^4 + 6*x^3*log(2*x) + 3*x ^2*log(2*x)^2 - 12*x^3 - 12*x^2*log(2*x) - 33*x^2 - 54*x*log(2*x) + 108*x + 243) + x^5*log(2)*log(2*x) + 2*x^4*log(2)^2*log(2*x) + x^3*log(2)^3*log( 2*x) + x^5*log(3*x^4 + 6*x^3*log(2*x) + 3*x^2*log(2*x)^2 - 12*x^3 - 12*x^2 *log(2*x) - 33*x^2 - 54*x*log(2*x) + 108*x + 243)*log(2*x) + 2*x^4*log(2)* log(3*x^4 + 6*x^3*log(2*x) + 3*x^2*log(2*x)^2 - 12*x^3 - 12*x^2*log(2*x) - 33*x^2 - 54*x*log(2*x) + 108*x + 243)*log(2*x) + x^3*log(2)^2*log(3*x^4 + 6*x^3*log(2*x) + 3*x^2*log(2*x)^2 - 12*x^3 - 12*x^2*log(2*x) - 33*x^2 - 5 4*x*log(2*x) + 108*x + 243)*log(2*x) - 2*x^6*log(x) - 2*x^5*log(2)*log(...
Time = 0.96 (sec) , antiderivative size = 295, normalized size of antiderivative = 10.93 \[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\frac {x}{2}+\frac {\ln \left (x\right )}{2}+\frac {x+\frac {x\,\ln \left (\frac {216\,x-\ln \left (2\,x\right )\,\left (-12\,x^3+24\,x^2+108\,x\right )-66\,x^2-24\,x^3+6\,x^4+6\,x^2\,{\ln \left (2\,x\right )}^2+486}{x^2}\right )\,\left (x^4+2\,x^3\,\ln \left (2\,x\right )-4\,x^3+x^2\,{\ln \left (2\,x\right )}^2-4\,x^2\,\ln \left (2\,x\right )-11\,x^2-18\,x\,\ln \left (2\,x\right )+36\,x+81\right )}{2\,\left (x^2+x+9\right )\,\left (2\,x-x\,\ln \left (2\,x\right )-x^2+9\right )}}{\ln \left (\frac {216\,x-\ln \left (2\,x\right )\,\left (-12\,x^3+24\,x^2+108\,x\right )-66\,x^2-24\,x^3+6\,x^4+6\,x^2\,{\ln \left (2\,x\right )}^2+486}{x^2}\right )}-\frac {\frac {15\,x}{2}-\frac {27}{2}}{x^2+x+9}-\frac {3\,\left (x^7+2\,x^6+19\,x^5+18\,x^4+81\,x^3\right )}{2\,{\left (x^2+x+9\right )}^3\,\left (2\,x-x\,\ln \left (2\,x\right )-x^2+9\right )}-\frac {\ln \left (2\,x\right )\,\left (\frac {x}{2}+\frac {9}{2}\right )}{x^2+x+9} \] Input:
int((54*x - log(2*x)*(18*x + 2*x^2 + 2*x^3) + log((216*x - log(2*x)*(108*x + 24*x^2 - 12*x^3) - 66*x^2 - 24*x^3 + 6*x^4 + 6*x^2*log(2*x)^2 + 486)/x^ 2)*(36*x - log(2*x)*(18*x + 4*x^2 - 2*x^3) - 11*x^2 - 4*x^3 + x^4 + x^2*lo g(2*x)^2 + 81) + 4*x^2 + 2*x^3 - 2*x^4 + 162)/(log((216*x - log(2*x)*(108* x + 24*x^2 - 12*x^3) - 66*x^2 - 24*x^3 + 6*x^4 + 6*x^2*log(2*x)^2 + 486)/x ^2)^2*(36*x - log(2*x)*(18*x + 4*x^2 - 2*x^3) - 11*x^2 - 4*x^3 + x^4 + x^2 *log(2*x)^2 + 81)),x)
Output:
x/2 + log(x)/2 + (x + (x*log((216*x - log(2*x)*(108*x + 24*x^2 - 12*x^3) - 66*x^2 - 24*x^3 + 6*x^4 + 6*x^2*log(2*x)^2 + 486)/x^2)*(36*x - 18*x*log(2 *x) - 4*x^2*log(2*x) + 2*x^3*log(2*x) - 11*x^2 - 4*x^3 + x^4 + x^2*log(2*x )^2 + 81))/(2*(x + x^2 + 9)*(2*x - x*log(2*x) - x^2 + 9)))/log((216*x - lo g(2*x)*(108*x + 24*x^2 - 12*x^3) - 66*x^2 - 24*x^3 + 6*x^4 + 6*x^2*log(2*x )^2 + 486)/x^2) - ((15*x)/2 - 27/2)/(x + x^2 + 9) - (3*(81*x^3 + 18*x^4 + 19*x^5 + 2*x^6 + x^7))/(2*(x + x^2 + 9)^3*(2*x - x*log(2*x) - x^2 + 9)) - (log(2*x)*(x/2 + 9/2))/(x + x^2 + 9)
\[ \int \frac {162+54 x+4 x^2+2 x^3-2 x^4+\left (-18 x-2 x^2-2 x^3\right ) \log (2 x)+\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log \left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )}{\left (81+36 x-11 x^2-4 x^3+x^4+\left (-18 x-4 x^2+2 x^3\right ) \log (2 x)+x^2 \log ^2(2 x)\right ) \log ^2\left (\frac {486+216 x-66 x^2-24 x^3+6 x^4+\left (-108 x-24 x^2+12 x^3\right ) \log (2 x)+6 x^2 \log ^2(2 x)}{x^2}\right )} \, dx=\text {too large to display} \] Input:
int(((x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36*x+81) *log((6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-66*x^2+ 216*x+486)/x^2)+(-2*x^3-2*x^2-18*x)*log(2*x)-2*x^4+2*x^3+4*x^2+54*x+162)/( x^2*log(2*x)^2+(2*x^3-4*x^2-18*x)*log(2*x)+x^4-4*x^3-11*x^2+36*x+81)/log(( 6*x^2*log(2*x)^2+(12*x^3-24*x^2-108*x)*log(2*x)+6*x^4-24*x^3-66*x^2+216*x+ 486)/x^2)^2,x)
Output:
- 2*int(x**4/(log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log(2*x)*x* *2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x**2)**2*l og(2*x)**2*x**2 + 2*log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log(2* x)*x**2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x**2) **2*log(2*x)*x**3 - 4*log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log( 2*x)*x**2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x** 2)**2*log(2*x)*x**2 - 18*log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*l og(2*x)*x**2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/ x**2)**2*log(2*x)*x + log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log( 2*x)*x**2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x** 2)**2*x**4 - 4*log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log(2*x)*x* *2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x**2)**2*x **3 - 11*log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log(2*x)*x**2 - 1 08*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x**2)**2*x**2 + 36*log((6*log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log(2*x)*x**2 - 108*log (2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x**2)**2*x + 81*log((6 *log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log(2*x)*x**2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x**2)**2),x) + int(x**4/(log((6 *log(2*x)**2*x**2 + 12*log(2*x)*x**3 - 24*log(2*x)*x**2 - 108*log(2*x)*x + 6*x**4 - 24*x**3 - 66*x**2 + 216*x + 486)/x**2)*log(2*x)**2*x**2 + 2*l...