Integrand size = 290, antiderivative size = 26 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x \log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{4+x} \]
Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\log \left (x^3-\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )-\frac {4 \log \left (-x^3+\log \left (x+\frac {2 x}{\log (2+x)}\right )\right )}{4+x} \]
Integrate[(-8*x - 2*x^2 + (16 + 12*x + 2*x^2 - 48*x^3 - 36*x^4 - 6*x^5)*Lo g[2 + x] + (8 + 6*x + x^2 - 24*x^3 - 18*x^4 - 3*x^5)*Log[2 + x]^2 + ((-16* x^3 - 8*x^4)*Log[2 + x] + (-8*x^3 - 4*x^4)*Log[2 + x]^2 + ((16 + 8*x)*Log[ 2 + x] + (8 + 4*x)*Log[2 + x]^2)*Log[(2*x + x*Log[2 + x])/Log[2 + x]])*Log [-x^3 + Log[(2*x + x*Log[2 + x])/Log[2 + x]]])/((-64*x^3 - 64*x^4 - 20*x^5 - 2*x^6)*Log[2 + x] + (-32*x^3 - 32*x^4 - 10*x^5 - x^6)*Log[2 + x]^2 + (( 64 + 64*x + 20*x^2 + 2*x^3)*Log[2 + x] + (32 + 32*x + 10*x^2 + x^3)*Log[2 + x]^2)*Log[(2*x + x*Log[2 + x])/Log[2 + x]]),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.
| \(\displaystyle \int \frac {-2 x^2+\left (\left (-4 x^4-8 x^3\right ) \log ^2(x+2)+\left (-8 x^4-16 x^3\right ) \log (x+2)+\left ((4 x+8) \log ^2(x+2)+(8 x+16) \log (x+2)\right ) \log \left (\frac {2 x+x \log (x+2)}{\log (x+2)}\right )\right ) \log \left (\log \left (\frac {2 x+x \log (x+2)}{\log (x+2)}\right )-x^3\right )+\left (-3 x^5-18 x^4-24 x^3+x^2+6 x+8\right ) \log ^2(x+2)+\left (-6 x^5-36 x^4-48 x^3+2 x^2+12 x+16\right ) \log (x+2)-8 x}{\left (\left (x^3+10 x^2+32 x+32\right ) \log ^2(x+2)+\left (2 x^3+20 x^2+64 x+64\right ) \log (x+2)\right ) \log \left (\frac {2 x+x \log (x+2)}{\log (x+2)}\right )+\left (-x^6-10 x^5-32 x^4-32 x^3\right ) \log ^2(x+2)+\left (-2 x^6-20 x^5-64 x^4-64 x^3\right ) \log (x+2)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 x^2-\left (\left (-4 x^4-8 x^3\right ) \log ^2(x+2)+\left (-8 x^4-16 x^3\right ) \log (x+2)+\left ((4 x+8) \log ^2(x+2)+(8 x+16) \log (x+2)\right ) \log \left (\frac {2 x+x \log (x+2)}{\log (x+2)}\right )\right ) \log \left (\log \left (\frac {2 x+x \log (x+2)}{\log (x+2)}\right )-x^3\right )-\left (-3 x^5-18 x^4-24 x^3+x^2+6 x+8\right ) \log ^2(x+2)-\left (-6 x^5-36 x^4-48 x^3+2 x^2+12 x+16\right ) \log (x+2)+8 x}{(x+2) (x+4)^2 \log (x+2) (\log (x+2)+2) \left (x^3-\log \left (x+\frac {2 x}{\log (x+2)}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {8 x}{(x+2) (x+4)^2 \log (x+2) (\log (x+2)+2) \left (x^3-\log \left (x+\frac {2 x}{\log (x+2)}\right )\right )}+\frac {4 \log \left (\log \left (x+\frac {2 x}{\log (x+2)}\right )-x^3\right )}{(x+4)^2}+\frac {2 \left (3 x^3-1\right )}{(x+4) (\log (x+2)+2) \left (x^3-\log \left (x+\frac {2 x}{\log (x+2)}\right )\right )}+\frac {\left (3 x^3-1\right ) \log (x+2)}{(x+4) (\log (x+2)+2) \left (x^3-\log \left (x+\frac {2 x}{\log (x+2)}\right )\right )}+\frac {2 x^2}{(x+2) (x+4)^2 \log (x+2) (\log (x+2)+2) \left (x^3-\log \left (x+\frac {2 x}{\log (x+2)}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 96 \int \frac {1}{(\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx-24 \int \frac {x}{(\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx-386 \int \frac {1}{(x+4) (\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx-2 \int \frac {1}{(x+2) \log (x+2) (\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx+4 \int \frac {1}{(x+4) \log (x+2) (\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx+48 \int \frac {\log (x+2)}{(\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx-12 \int \frac {x \log (x+2)}{(\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx-193 \int \frac {\log (x+2)}{(x+4) (\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx+4 \int \frac {\log \left (\log \left (\frac {2 x}{\log (x+2)}+x\right )-x^3\right )}{(x+4)^2}dx+6 \int \frac {x^2}{(\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx+3 \int \frac {x^2 \log (x+2)}{(\log (x+2)+2) \left (x^3-\log \left (\frac {2 x}{\log (x+2)}+x\right )\right )}dx\) |
Int[(-8*x - 2*x^2 + (16 + 12*x + 2*x^2 - 48*x^3 - 36*x^4 - 6*x^5)*Log[2 + x] + (8 + 6*x + x^2 - 24*x^3 - 18*x^4 - 3*x^5)*Log[2 + x]^2 + ((-16*x^3 - 8*x^4)*Log[2 + x] + (-8*x^3 - 4*x^4)*Log[2 + x]^2 + ((16 + 8*x)*Log[2 + x] + (8 + 4*x)*Log[2 + x]^2)*Log[(2*x + x*Log[2 + x])/Log[2 + x]])*Log[-x^3 + Log[(2*x + x*Log[2 + x])/Log[2 + x]]])/((-64*x^3 - 64*x^4 - 20*x^5 - 2*x ^6)*Log[2 + x] + (-32*x^3 - 32*x^4 - 10*x^5 - x^6)*Log[2 + x]^2 + ((64 + 6 4*x + 20*x^2 + 2*x^3)*Log[2 + x] + (32 + 32*x + 10*x^2 + x^3)*Log[2 + x]^2 )*Log[(2*x + x*Log[2 + x])/Log[2 + x]]),x]
3.18.21.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.03 (sec) , antiderivative size = 470, normalized size of antiderivative = 18.08
\[-\frac {4 \ln \left (\ln \left (x \right )-\ln \left (\ln \left (2+x \right )\right )+\ln \left (\ln \left (2+x \right )+2\right )-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (\frac {i}{\ln \left (2+x \right )}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (i \left (\ln \left (2+x \right )+2\right )\right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )\right )}{2}-x^{3}\right )}{4+x}+\ln \left (\ln \left (\ln \left (2+x \right )+2\right )+\frac {i \left (2 i x^{3}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (2+x \right )+2\right )\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )+\pi \,\operatorname {csgn}\left (i \left (\ln \left (2+x \right )+2\right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}+\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{3}+\pi \,\operatorname {csgn}\left (\frac {i \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right ) \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{2}-\pi \operatorname {csgn}\left (\frac {i x \left (\ln \left (2+x \right )+2\right )}{\ln \left (2+x \right )}\right )^{3}-2 i \ln \left (x \right )+2 i \ln \left (\ln \left (2+x \right )\right )\right )}{2}\right )\]
int(((((4*x+8)*ln(2+x)^2+(8*x+16)*ln(2+x))*ln((x*ln(2+x)+2*x)/ln(2+x))+(-4 *x^4-8*x^3)*ln(2+x)^2+(-8*x^4-16*x^3)*ln(2+x))*ln(ln((x*ln(2+x)+2*x)/ln(2+ x))-x^3)+(-3*x^5-18*x^4-24*x^3+x^2+6*x+8)*ln(2+x)^2+(-6*x^5-36*x^4-48*x^3+ 2*x^2+12*x+16)*ln(2+x)-2*x^2-8*x)/(((x^3+10*x^2+32*x+32)*ln(2+x)^2+(2*x^3+ 20*x^2+64*x+64)*ln(2+x))*ln((x*ln(2+x)+2*x)/ln(2+x))+(-x^6-10*x^5-32*x^4-3 2*x^3)*ln(2+x)^2+(-2*x^6-20*x^5-64*x^4-64*x^3)*ln(2+x)),x)
-4/(4+x)*ln(ln(x)-ln(ln(2+x))+ln(ln(2+x)+2)-1/2*I*Pi*csgn(I/ln(2+x)*(ln(2+ x)+2))*(-csgn(I/ln(2+x)*(ln(2+x)+2))+csgn(I/ln(2+x)))*(-csgn(I/ln(2+x)*(ln (2+x)+2))+csgn(I*(ln(2+x)+2)))-1/2*I*Pi*csgn(I*x/ln(2+x)*(ln(2+x)+2))*(-cs gn(I*x/ln(2+x)*(ln(2+x)+2))+csgn(I*x))*(-csgn(I*x/ln(2+x)*(ln(2+x)+2))+csg n(I/ln(2+x)*(ln(2+x)+2)))-x^3)+ln(ln(ln(2+x)+2)+1/2*I*(2*I*x^3-Pi*csgn(I*x )*csgn(I/ln(2+x)*(ln(2+x)+2))*csgn(I*x/ln(2+x)*(ln(2+x)+2))+Pi*csgn(I*x)*c sgn(I*x/ln(2+x)*(ln(2+x)+2))^2-Pi*csgn(I*(ln(2+x)+2))*csgn(I/ln(2+x))*csgn (I/ln(2+x)*(ln(2+x)+2))+Pi*csgn(I*(ln(2+x)+2))*csgn(I/ln(2+x)*(ln(2+x)+2)) ^2+Pi*csgn(I/ln(2+x))*csgn(I/ln(2+x)*(ln(2+x)+2))^2-Pi*csgn(I/ln(2+x)*(ln( 2+x)+2))^3+Pi*csgn(I/ln(2+x)*(ln(2+x)+2))*csgn(I*x/ln(2+x)*(ln(2+x)+2))^2- Pi*csgn(I*x/ln(2+x)*(ln(2+x)+2))^3-2*I*ln(x)+2*I*ln(ln(2+x))))
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x \log \left (-x^{3} + \log \left (\frac {x \log \left (x + 2\right ) + 2 \, x}{\log \left (x + 2\right )}\right )\right )}{x + 4} \]
integrate(((((4*x+8)*log(2+x)^2+(8*x+16)*log(2+x))*log((x*log(2+x)+2*x)/lo g(2+x))+(-4*x^4-8*x^3)*log(2+x)^2+(-8*x^4-16*x^3)*log(2+x))*log(log((x*log (2+x)+2*x)/log(2+x))-x^3)+(-3*x^5-18*x^4-24*x^3+x^2+6*x+8)*log(2+x)^2+(-6* x^5-36*x^4-48*x^3+2*x^2+12*x+16)*log(2+x)-2*x^2-8*x)/(((x^3+10*x^2+32*x+32 )*log(2+x)^2+(2*x^3+20*x^2+64*x+64)*log(2+x))*log((x*log(2+x)+2*x)/log(2+x ))+(-x^6-10*x^5-32*x^4-32*x^3)*log(2+x)^2+(-2*x^6-20*x^5-64*x^4-64*x^3)*lo g(2+x)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 19.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\log {\left (- x^{3} + \log {\left (\frac {x \log {\left (x + 2 \right )} + 2 x}{\log {\left (x + 2 \right )}} \right )} \right )} - \frac {4 \log {\left (- x^{3} + \log {\left (\frac {x \log {\left (x + 2 \right )} + 2 x}{\log {\left (x + 2 \right )}} \right )} \right )}}{x + 4} \]
integrate(((((4*x+8)*ln(2+x)**2+(8*x+16)*ln(2+x))*ln((x*ln(2+x)+2*x)/ln(2+ x))+(-4*x**4-8*x**3)*ln(2+x)**2+(-8*x**4-16*x**3)*ln(2+x))*ln(ln((x*ln(2+x )+2*x)/ln(2+x))-x**3)+(-3*x**5-18*x**4-24*x**3+x**2+6*x+8)*ln(2+x)**2+(-6* x**5-36*x**4-48*x**3+2*x**2+12*x+16)*ln(2+x)-2*x**2-8*x)/(((x**3+10*x**2+3 2*x+32)*ln(2+x)**2+(2*x**3+20*x**2+64*x+64)*ln(2+x))*ln((x*ln(2+x)+2*x)/ln (2+x))+(-x**6-10*x**5-32*x**4-32*x**3)*ln(2+x)**2+(-2*x**6-20*x**5-64*x**4 -64*x**3)*ln(2+x)),x)
log(-x**3 + log((x*log(x + 2) + 2*x)/log(x + 2))) - 4*log(-x**3 + log((x*l og(x + 2) + 2*x)/log(x + 2)))/(x + 4)
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x \log \left (-x^{3} + \log \left (x\right ) + \log \left (\log \left (x + 2\right ) + 2\right ) - \log \left (\log \left (x + 2\right )\right )\right )}{x + 4} \]
integrate(((((4*x+8)*log(2+x)^2+(8*x+16)*log(2+x))*log((x*log(2+x)+2*x)/lo g(2+x))+(-4*x^4-8*x^3)*log(2+x)^2+(-8*x^4-16*x^3)*log(2+x))*log(log((x*log (2+x)+2*x)/log(2+x))-x^3)+(-3*x^5-18*x^4-24*x^3+x^2+6*x+8)*log(2+x)^2+(-6* x^5-36*x^4-48*x^3+2*x^2+12*x+16)*log(2+x)-2*x^2-8*x)/(((x^3+10*x^2+32*x+32 )*log(2+x)^2+(2*x^3+20*x^2+64*x+64)*log(2+x))*log((x*log(2+x)+2*x)/log(2+x ))+(-x^6-10*x^5-32*x^4-32*x^3)*log(2+x)^2+(-2*x^6-20*x^5-64*x^4-64*x^3)*lo g(2+x)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.72 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=-\frac {4 \, \log \left (-x^{3} + \log \left (x \log \left (x + 2\right ) + 2 \, x\right ) - \log \left (\log \left (x + 2\right )\right )\right )}{x + 4} + \log \left (x^{3} - \log \left (x \log \left (x + 2\right ) + 2 \, x\right ) + \log \left (\log \left (x + 2\right )\right )\right ) \]
integrate(((((4*x+8)*log(2+x)^2+(8*x+16)*log(2+x))*log((x*log(2+x)+2*x)/lo g(2+x))+(-4*x^4-8*x^3)*log(2+x)^2+(-8*x^4-16*x^3)*log(2+x))*log(log((x*log (2+x)+2*x)/log(2+x))-x^3)+(-3*x^5-18*x^4-24*x^3+x^2+6*x+8)*log(2+x)^2+(-6* x^5-36*x^4-48*x^3+2*x^2+12*x+16)*log(2+x)-2*x^2-8*x)/(((x^3+10*x^2+32*x+32 )*log(2+x)^2+(2*x^3+20*x^2+64*x+64)*log(2+x))*log((x*log(2+x)+2*x)/log(2+x ))+(-x^6-10*x^5-32*x^4-32*x^3)*log(2+x)^2+(-2*x^6-20*x^5-64*x^4-64*x^3)*lo g(2+x)),x, algorithm=\
-4*log(-x^3 + log(x*log(x + 2) + 2*x) - log(log(x + 2)))/(x + 4) + log(x^3 - log(x*log(x + 2) + 2*x) + log(log(x + 2)))
Time = 11.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-8 x-2 x^2+\left (16+12 x+2 x^2-48 x^3-36 x^4-6 x^5\right ) \log (2+x)+\left (8+6 x+x^2-24 x^3-18 x^4-3 x^5\right ) \log ^2(2+x)+\left (\left (-16 x^3-8 x^4\right ) \log (2+x)+\left (-8 x^3-4 x^4\right ) \log ^2(2+x)+\left ((16+8 x) \log (2+x)+(8+4 x) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right ) \log \left (-x^3+\log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )\right )}{\left (-64 x^3-64 x^4-20 x^5-2 x^6\right ) \log (2+x)+\left (-32 x^3-32 x^4-10 x^5-x^6\right ) \log ^2(2+x)+\left (\left (64+64 x+20 x^2+2 x^3\right ) \log (2+x)+\left (32+32 x+10 x^2+x^3\right ) \log ^2(2+x)\right ) \log \left (\frac {2 x+x \log (2+x)}{\log (2+x)}\right )} \, dx=\frac {x\,\ln \left (\ln \left (\frac {2\,x+x\,\ln \left (x+2\right )}{\ln \left (x+2\right )}\right )-x^3\right )}{x+4} \]
int((8*x + log(log((2*x + x*log(x + 2))/log(x + 2)) - x^3)*(log(x + 2)*(16 *x^3 + 8*x^4) - log((2*x + x*log(x + 2))/log(x + 2))*(log(x + 2)^2*(4*x + 8) + log(x + 2)*(8*x + 16)) + log(x + 2)^2*(8*x^3 + 4*x^4)) - log(x + 2)*( 12*x + 2*x^2 - 48*x^3 - 36*x^4 - 6*x^5 + 16) - log(x + 2)^2*(6*x + x^2 - 2 4*x^3 - 18*x^4 - 3*x^5 + 8) + 2*x^2)/(log(x + 2)*(64*x^3 + 64*x^4 + 20*x^5 + 2*x^6) - log((2*x + x*log(x + 2))/log(x + 2))*(log(x + 2)*(64*x + 20*x^ 2 + 2*x^3 + 64) + log(x + 2)^2*(32*x + 10*x^2 + x^3 + 32)) + log(x + 2)^2* (32*x^3 + 32*x^4 + 10*x^5 + x^6)),x)