Integrand size = 124, antiderivative size = 20 \begin {dmath*} \int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx=\left (x+\left (-5+\log \left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )\right )^2 \end {dmath*}
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx=\left (x+\left (-5+\log \left (x^3\right )\right ) \log \left (2 x (4+x)^2\right )\right )^2 \end {dmath*}
Integrate[(-40*x - 22*x^2 + 2*x^3 + (8*x + 6*x^2)*Log[x^3] + (200 + 134*x - 4*x^2 + (-80 - 52*x + 2*x^2)*Log[x^3] + (8 + 6*x)*Log[x^3]^2)*Log[32*x + 16*x^2 + 2*x^3] + (-120 - 30*x + (24 + 6*x)*Log[x^3])*Log[32*x + 16*x^2 + 2*x^3]^2)/(4*x + x^2),x]
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.
\(\displaystyle \int \frac {2 x^3-22 x^2+\left ((6 x+24) \log \left (x^3\right )-30 x-120\right ) \log ^2\left (2 x^3+16 x^2+32 x\right )+\left ((6 x+8) \log ^2\left (x^3\right )-4 x^2+\left (2 x^2-52 x-80\right ) \log \left (x^3\right )+134 x+200\right ) \log \left (2 x^3+16 x^2+32 x\right )+\left (6 x^2+8 x\right ) \log \left (x^3\right )-40 x}{x^2+4 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^3-22 x^2+\left ((6 x+24) \log \left (x^3\right )-30 x-120\right ) \log ^2\left (2 x^3+16 x^2+32 x\right )+\left ((6 x+8) \log ^2\left (x^3\right )-4 x^2+\left (2 x^2-52 x-80\right ) \log \left (x^3\right )+134 x+200\right ) \log \left (2 x^3+16 x^2+32 x\right )+\left (6 x^2+8 x\right ) \log \left (x^3\right )-40 x}{x (x+4)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {6 \left (\log \left (x^3\right )-5\right ) \log ^2\left (2 x (x+4)^2\right )}{x}+\frac {2 \left (3 x \log ^2\left (x^3\right )+4 \log ^2\left (x^3\right )-26 x \log \left (x^3\right )-40 \log \left (x^3\right )-2 x^2+x^2 \log \left (x^3\right )+67 x+100\right ) \log \left (2 x (x+4)^2\right )}{x (x+4)}+\frac {2 \left (3 x \log \left (x^3\right )+4 \log \left (x^3\right )+x^2-11 x-20\right )}{x+4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {\log ^2\left (x^3\right ) \log \left (2 x (x+4)^2\right )}{x+4}dx+6 \int \frac {\left (\log \left (x^3\right )-5\right ) \log ^2\left (2 x (x+4)^2\right )}{x}dx-40 \int \frac {\log (x) \log \left (x^3\right )}{x+4}dx-4 \operatorname {PolyLog}\left (2,-\frac {x}{4}\right ) \log ^2\left (x^3\right )+40 \operatorname {PolyLog}\left (2,-\frac {x}{4}\right ) \log \left (x^3\right )+24 \operatorname {PolyLog}\left (3,-\frac {x}{4}\right ) \log \left (x^3\right )-120 \operatorname {PolyLog}\left (3,-\frac {x}{4}\right )-240 \operatorname {PolyLog}\left (3,\frac {x+4}{4}\right )-72 \operatorname {PolyLog}\left (4,-\frac {x}{4}\right )+120 \operatorname {PolyLog}\left (2,-\frac {x}{4}\right ) \left (\log (x)+\log \left ((x+4)^2\right )-\log \left (2 x (x+4)^2\right )\right )+120 \operatorname {PolyLog}\left (2,\frac {x+4}{4}\right ) \log \left ((x+4)^2\right )-\frac {1}{54} \log ^4\left (x^3\right )-\frac {4}{9} \log \left (\frac {x}{4}+1\right ) \log ^3\left (x^3\right )+\frac {2}{9} \log \left (2 x (x+4)^2\right ) \log ^3\left (x^3\right )+\frac {10}{27} \log ^3\left (x^3\right )+\frac {20}{3} \log \left (\frac {x}{4}+1\right ) \log ^2\left (x^3\right )-\frac {10}{3} \log \left (2 x (x+4)^2\right ) \log ^2\left (x^3\right )-10 \log ^2\left ((x+4)^2\right ) \log \left (x^3\right )+40 \log \left (\frac {x}{4}+1\right ) \left (\log (x)+\log \left ((x+4)^2\right )-\log \left (2 x (x+4)^2\right )\right ) \log \left (x^3\right )+2 x \log \left (2 x (x+4)^2\right ) \log \left (x^3\right )+x^2-25 \log ^2(x)-100 \log ^2(x+4)+30 \log \left (-\frac {x}{4}\right ) \log ^2\left ((x+4)^2\right )-100 \log \left (\frac {x}{4}+1\right ) \log (x)-100 \log (4) \log (x)-10 x \log \left (2 x (x+4)^2\right )+50 \log (x) \log \left (2 x (x+4)^2\right )+100 \log (x+4) \log \left (2 x (x+4)^2\right )\) |
Int[(-40*x - 22*x^2 + 2*x^3 + (8*x + 6*x^2)*Log[x^3] + (200 + 134*x - 4*x^ 2 + (-80 - 52*x + 2*x^2)*Log[x^3] + (8 + 6*x)*Log[x^3]^2)*Log[32*x + 16*x^ 2 + 2*x^3] + (-120 - 30*x + (24 + 6*x)*Log[x^3])*Log[32*x + 16*x^2 + 2*x^3 ]^2)/(4*x + x^2),x]
3.12.90.3.1 Defintions of rubi rules used
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])
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 691.84 (sec) , antiderivative size = 17841276, normalized size of antiderivative = 892063.80
int((((24+6*x)*ln(x^3)-30*x-120)*ln(2*x^3+16*x^2+32*x)^2+((6*x+8)*ln(x^3)^ 2+(2*x^2-52*x-80)*ln(x^3)-4*x^2+134*x+200)*ln(2*x^3+16*x^2+32*x)+(6*x^2+8* x)*ln(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.15 \begin {dmath*} \int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx={\left (\log \left (x^{3}\right )^{2} - 10 \, \log \left (x^{3}\right ) + 25\right )} \log \left (2 \, x^{3} + 16 \, x^{2} + 32 \, x\right )^{2} + x^{2} + 2 \, {\left (x \log \left (x^{3}\right ) - 5 \, x\right )} \log \left (2 \, x^{3} + 16 \, x^{2} + 32 \, x\right ) \end {dmath*}
integrate((((24+6*x)*log(x^3)-30*x-120)*log(2*x^3+16*x^2+32*x)^2+((6*x+8)* log(x^3)^2+(2*x^2-52*x-80)*log(x^3)-4*x^2+134*x+200)*log(2*x^3+16*x^2+32*x )+(6*x^2+8*x)*log(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x, algorithm=\
(log(x^3)^2 - 10*log(x^3) + 25)*log(2*x^3 + 16*x^2 + 32*x)^2 + x^2 + 2*(x* log(x^3) - 5*x)*log(2*x^3 + 16*x^2 + 32*x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).
Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.05 \begin {dmath*} \int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx=x^{2} + \left (2 x \log {\left (x^{3} \right )} - 10 x\right ) \log {\left (2 x^{3} + 16 x^{2} + 32 x \right )} + \left (\log {\left (x^{3} \right )}^{2} - 10 \log {\left (x^{3} \right )} + 25\right ) \log {\left (2 x^{3} + 16 x^{2} + 32 x \right )}^{2} \end {dmath*}
integrate((((24+6*x)*ln(x**3)-30*x-120)*ln(2*x**3+16*x**2+32*x)**2+((6*x+8 )*ln(x**3)**2+(2*x**2-52*x-80)*ln(x**3)-4*x**2+134*x+200)*ln(2*x**3+16*x** 2+32*x)+(6*x**2+8*x)*ln(x**3)+2*x**3-22*x**2-40*x)/(x**2+4*x),x)
x**2 + (2*x*log(x**3) - 10*x)*log(2*x**3 + 16*x**2 + 32*x) + (log(x**3)**2 - 10*log(x**3) + 25)*log(2*x**3 + 16*x**2 + 32*x)**2
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (20) = 40\).
Time = 0.33 (sec) , antiderivative size = 146, normalized size of antiderivative = 7.30 \begin {dmath*} \int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx=6 \, {\left (3 \, \log \left (2\right ) - 5\right )} \log \left (x\right )^{3} + 9 \, \log \left (x\right )^{4} + 4 \, {\left (9 \, \log \left (x\right )^{2} - 30 \, \log \left (x\right ) + 25\right )} \log \left (x + 4\right )^{2} + {\left (9 \, \log \left (2\right )^{2} + 6 \, x - 60 \, \log \left (2\right ) + 25\right )} \log \left (x\right )^{2} + x^{2} - 10 \, x {\left (\log \left (2\right ) - 3\right )} + 4 \, {\left (3 \, {\left (3 \, \log \left (2\right ) - 10\right )} \log \left (x\right )^{2} + 9 \, \log \left (x\right )^{3} + {\left (3 \, x - 30 \, \log \left (2\right ) + 25\right )} \log \left (x\right ) - 5 \, x + 25 \, \log \left (2\right ) - 20\right )} \log \left (x + 4\right ) + 2 \, {\left (x {\left (3 \, \log \left (2\right ) - 5\right )} - 15 \, \log \left (2\right )^{2} + 25 \, \log \left (2\right )\right )} \log \left (x\right ) - 30 \, x + 80 \, \log \left (x + 4\right ) \end {dmath*}
integrate((((24+6*x)*log(x^3)-30*x-120)*log(2*x^3+16*x^2+32*x)^2+((6*x+8)* log(x^3)^2+(2*x^2-52*x-80)*log(x^3)-4*x^2+134*x+200)*log(2*x^3+16*x^2+32*x )+(6*x^2+8*x)*log(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x, algorithm=\
6*(3*log(2) - 5)*log(x)^3 + 9*log(x)^4 + 4*(9*log(x)^2 - 30*log(x) + 25)*l og(x + 4)^2 + (9*log(2)^2 + 6*x - 60*log(2) + 25)*log(x)^2 + x^2 - 10*x*(l og(2) - 3) + 4*(3*(3*log(2) - 10)*log(x)^2 + 9*log(x)^3 + (3*x - 30*log(2) + 25)*log(x) - 5*x + 25*log(2) - 20)*log(x + 4) + 2*(x*(3*log(2) - 5) - 1 5*log(2)^2 + 25*log(2))*log(x) - 30*x + 80*log(x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (20) = 40\).
Time = 0.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 5.45 \begin {dmath*} \int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx=9 \, \log \left (x\right )^{4} + 3 \, {\left (3 \, \log \left (x\right )^{2} - 10 \, \log \left (x\right )\right )} \log \left (2 \, x^{2} + 16 \, x + 32\right )^{2} + {\left (6 \, x + 25\right )} \log \left (x\right )^{2} - 30 \, \log \left (x\right )^{3} + x^{2} + 2 \, {\left (9 \, \log \left (x\right )^{3} + 3 \, x \log \left (x\right ) - 30 \, \log \left (x\right )^{2} - 5 \, x + 50 \, \log \left (x + 4\right ) + 25 \, \log \left (x\right )\right )} \log \left (2 \, x^{2} + 16 \, x + 32\right ) - 100 \, \log \left (x + 4\right )^{2} - 10 \, x \log \left (x\right ) \end {dmath*}
integrate((((24+6*x)*log(x^3)-30*x-120)*log(2*x^3+16*x^2+32*x)^2+((6*x+8)* log(x^3)^2+(2*x^2-52*x-80)*log(x^3)-4*x^2+134*x+200)*log(2*x^3+16*x^2+32*x )+(6*x^2+8*x)*log(x^3)+2*x^3-22*x^2-40*x)/(x^2+4*x),x, algorithm=\
9*log(x)^4 + 3*(3*log(x)^2 - 10*log(x))*log(2*x^2 + 16*x + 32)^2 + (6*x + 25)*log(x)^2 - 30*log(x)^3 + x^2 + 2*(9*log(x)^3 + 3*x*log(x) - 30*log(x)^ 2 - 5*x + 50*log(x + 4) + 25*log(x))*log(2*x^2 + 16*x + 32) - 100*log(x + 4)^2 - 10*x*log(x)
Time = 15.44 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05 \begin {dmath*} \int \frac {-40 x-22 x^2+2 x^3+\left (8 x+6 x^2\right ) \log \left (x^3\right )+\left (200+134 x-4 x^2+\left (-80-52 x+2 x^2\right ) \log \left (x^3\right )+(8+6 x) \log ^2\left (x^3\right )\right ) \log \left (32 x+16 x^2+2 x^3\right )+\left (-120-30 x+(24+6 x) \log \left (x^3\right )\right ) \log ^2\left (32 x+16 x^2+2 x^3\right )}{4 x+x^2} \, dx={\left (x-5\,\ln \left (2\,x^3+16\,x^2+32\,x\right )+\ln \left (2\,x^3+16\,x^2+32\,x\right )\,\ln \left (x^3\right )\right )}^2 \end {dmath*}