Integrand size = 96, antiderivative size = 29 \[ \int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+\left (-4 x-2 x^2+(4+2 x) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2}\right )}{-2 x^2-x^3+\left (2 x+x^2\right ) \log (3)} \, dx=2 \left (1+\log (x)-\log (x) \left (-1-\log \left (\frac {(x-\log (3))^2}{(2+x)^2}\right )\right )\right ) \]
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+\left (-4 x-2 x^2+(4+2 x) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2}\right )}{-2 x^2-x^3+\left (2 x+x^2\right ) \log (3)} \, dx=\frac {\log (x) \left (8+\log (81)+(4+\log (9)) \log \left (\frac {x^2+\log ^2(3)-x \log (9)}{(2+x)^2}\right )\right )}{2+\log (3)} \]
Integrate[(-8*x - 4*x^2 + (8 + 4*x)*Log[3] + (-8*x - 4*x*Log[3])*Log[x] + (-4*x - 2*x^2 + (4 + 2*x)*Log[3])*Log[(x^2 - 2*x*Log[3] + Log[3]^2)/(4 + 4 *x + x^2)])/(-2*x^2 - x^3 + (2*x + x^2)*Log[3]),x]
(Log[x]*(8 + Log[81] + (4 + Log[9])*Log[(x^2 + Log[3]^2 - x*Log[9])/(2 + x )^2]))/(2 + Log[3])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 9.14, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2026, 7279, 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.
| \(\displaystyle \int \frac {-4 x^2+\left (-2 x^2-4 x+(2 x+4) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{x^2+4 x+4}\right )-8 x+(-8 x-4 x \log (3)) \log (x)+(4 x+8) \log (3)}{-x^3-2 x^2+\left (x^2+2 x\right ) \log (3)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-4 x^2+\left (-2 x^2-4 x+(2 x+4) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{x^2+4 x+4}\right )-8 x+(-8 x-4 x \log (3)) \log (x)+(4 x+8) \log (3)}{x \left (-x^2-x (2-\log (3))+\log (9)\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {4 \left (x^2+2 x \left (1+\frac {\log (3)}{2}\right ) \log (x)+2 x \left (1-\frac {\log (3)}{2}\right )-2 \log (3)\right )}{x \left (x^2+x (2-\log (3))-\log (9)\right )}+\frac {2 (x+2) (x-\log (3)) \log \left (\frac {(x-\log (3))^2}{(x+2)^2}\right )}{x \left (x^2+x (2-\log (3))-\log (9)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \operatorname {PolyLog}\left (2,-\frac {x}{2}\right )-\frac {4 (2+\log (3)) \operatorname {PolyLog}\left (2,-\frac {2 x}{2-\log (3)+\sqrt {4+\log ^2(3)+\log (81)}}\right )}{\sqrt {4+\log ^2(3)+\log (81)}}-\frac {4 (2+\log (3)) \operatorname {PolyLog}\left (2,\frac {2 x}{2-\log (3)-\sqrt {4+\log ^2(3)+\log (81)}}+1\right )}{\sqrt {4+\log ^2(3)+\log (81)}}+4 \operatorname {PolyLog}\left (2,1-\frac {x}{\log (3)}\right )-\frac {4 (2+\log (3)) \log \left (\frac {2 x}{2+\sqrt {4+\log ^2(3)+\log (81)}-\log (3)}+1\right ) \log (x)}{\sqrt {4+\log ^2(3)+\log (81)}}+\frac {4 (2+\log (3)) \log \left (\frac {1}{2} \left (-2+\sqrt {4+\log ^2(3)+\log (81)}+\log (3)\right )\right ) \log \left (-2 x-2+\sqrt {4+\log ^2(3)+\log (81)}+\log (3)\right )}{\sqrt {4+\log ^2(3)+\log (81)}}+4 \log \left (\frac {x}{2}+1\right ) \log (x)+2 \log \left (\frac {(x-\log (3))^2}{(x+2)^2}\right ) \log (x)+4 \log (x)-4 \log (\log (3)) \log (x-\log (3))\) |
Int[(-8*x - 4*x^2 + (8 + 4*x)*Log[3] + (-8*x - 4*x*Log[3])*Log[x] + (-4*x - 2*x^2 + (4 + 2*x)*Log[3])*Log[(x^2 - 2*x*Log[3] + Log[3]^2)/(4 + 4*x + x ^2)])/(-2*x^2 - x^3 + (2*x + x^2)*Log[3]),x]
4*Log[x] + 4*Log[1 + x/2]*Log[x] + 2*Log[x]*Log[(x - Log[3])^2/(2 + x)^2] - 4*Log[x - Log[3]]*Log[Log[3]] + (4*(2 + Log[3])*Log[(-2 + Log[3] + Sqrt[ 4 + Log[3]^2 + Log[81]])/2]*Log[-2 - 2*x + Log[3] + Sqrt[4 + Log[3]^2 + Lo g[81]]])/Sqrt[4 + Log[3]^2 + Log[81]] - (4*(2 + Log[3])*Log[x]*Log[1 + (2* x)/(2 - Log[3] + Sqrt[4 + Log[3]^2 + Log[81]])])/Sqrt[4 + Log[3]^2 + Log[8 1]] + 4*PolyLog[2, -1/2*x] + 4*PolyLog[2, 1 - x/Log[3]] - (4*(2 + Log[3])* PolyLog[2, (-2*x)/(2 - Log[3] + Sqrt[4 + Log[3]^2 + Log[81]])])/Sqrt[4 + L og[3]^2 + Log[81]] - (4*(2 + Log[3])*PolyLog[2, 1 + (2*x)/(2 - Log[3] - Sq rt[4 + Log[3]^2 + Log[81]])])/Sqrt[4 + Log[3]^2 + Log[81]]
3.6.74.3.1 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])
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.32 (sec) , antiderivative size = 310, normalized size of antiderivative = 10.69
| method | result | size |
| risch | \(4 \ln \left (x \right ) \ln \left (\ln \left (3\right )-x \right )-i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (\frac {i}{\left (2+x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (\ln \left (3\right )-x \right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right )-x \right )^{2}}{\left (2+x \right )^{2}}\right )+i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (\frac {i}{\left (2+x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right )-x \right )^{2}}{\left (2+x \right )^{2}}\right )^{2}+i \ln \left (x \right ) \pi \operatorname {csgn}\left (i \left (2+x \right )\right )^{2} \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )-2 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left (2+x \right )\right ) \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )^{2}+i \ln \left (x \right ) \pi \operatorname {csgn}\left (i \left (2+x \right )^{2}\right )^{3}-i \ln \left (x \right ) \pi \operatorname {csgn}\left (i \left (\ln \left (3\right )-x \right )\right )^{2} \operatorname {csgn}\left (i \left (\ln \left (3\right )-x \right )^{2}\right )+2 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )-x \right )\right ) \operatorname {csgn}\left (i \left (\ln \left (3\right )-x \right )^{2}\right )^{2}-i \ln \left (x \right ) \pi \operatorname {csgn}\left (i \left (\ln \left (3\right )-x \right )^{2}\right )^{3}+i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )-x \right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right )-x \right )^{2}}{\left (2+x \right )^{2}}\right )^{2}-i \ln \left (x \right ) \pi \operatorname {csgn}\left (\frac {i \left (\ln \left (3\right )-x \right )^{2}}{\left (2+x \right )^{2}}\right )^{3}-4 \ln \left (x \right ) \ln \left (2+x \right )+4 \ln \left (x \right )\) | \(310\) |
| default | \(4 \ln \left (x \right )-2 \ln \left (\frac {1}{2+x}\right ) \ln \left (\frac {\ln \left (3\right )^{2}}{\left (2+x \right )^{2}}+\frac {4 \ln \left (3\right )}{\left (2+x \right )^{2}}-\frac {2 \ln \left (3\right )}{2+x}+\frac {4}{\left (2+x \right )^{2}}-\frac {4}{2+x}+1\right )+4 \left (2+\ln \left (3\right )\right ) \left (\frac {\left (\ln \left (\frac {1}{2+x}\right )-\ln \left (\frac {2+\ln \left (3\right )}{2+x}\right )\right ) \ln \left (-\frac {2+\ln \left (3\right )}{2+x}+1\right )}{2+\ln \left (3\right )}-\frac {\operatorname {dilog}\left (\frac {2+\ln \left (3\right )}{2+x}\right )}{2+\ln \left (3\right )}\right )+2 \ln \left (-1+\frac {2}{2+x}\right ) \ln \left (\frac {\ln \left (3\right )^{2}}{\left (2+x \right )^{2}}+\frac {4 \ln \left (3\right )}{\left (2+x \right )^{2}}-\frac {2 \ln \left (3\right )}{2+x}+\frac {4}{\left (2+x \right )^{2}}-\frac {4}{2+x}+1\right )-2 \left (2 \ln \left (3\right )+4\right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (2+\ln \left (3\right )\right ) \left (-1+\frac {2}{2+x}\right )+\ln \left (3\right )}{\ln \left (3\right )}\right )}{2+\ln \left (3\right )}+\frac {\ln \left (-1+\frac {2}{2+x}\right ) \ln \left (\frac {\left (2+\ln \left (3\right )\right ) \left (-1+\frac {2}{2+x}\right )+\ln \left (3\right )}{\ln \left (3\right )}\right )}{2+\ln \left (3\right )}\right )+\left (-4 \ln \left (3\right )-8\right ) \left (-\frac {\left (\ln \left (x \right )-\ln \left (\frac {x}{\ln \left (3\right )}\right )\right ) \ln \left (\frac {\ln \left (3\right )-x}{\ln \left (3\right )}\right )-\operatorname {dilog}\left (\frac {x}{\ln \left (3\right )}\right )}{2+\ln \left (3\right )}+\frac {\operatorname {dilog}\left (1+\frac {x}{2}\right )+\ln \left (x \right ) \ln \left (1+\frac {x}{2}\right )}{2+\ln \left (3\right )}\right )\) | \(336\) |
| parts | \(4 \ln \left (x \right )-2 \ln \left (\frac {1}{2+x}\right ) \ln \left (\frac {\ln \left (3\right )^{2}}{\left (2+x \right )^{2}}+\frac {4 \ln \left (3\right )}{\left (2+x \right )^{2}}-\frac {2 \ln \left (3\right )}{2+x}+\frac {4}{\left (2+x \right )^{2}}-\frac {4}{2+x}+1\right )+4 \left (2+\ln \left (3\right )\right ) \left (\frac {\left (\ln \left (\frac {1}{2+x}\right )-\ln \left (\frac {2+\ln \left (3\right )}{2+x}\right )\right ) \ln \left (-\frac {2+\ln \left (3\right )}{2+x}+1\right )}{2+\ln \left (3\right )}-\frac {\operatorname {dilog}\left (\frac {2+\ln \left (3\right )}{2+x}\right )}{2+\ln \left (3\right )}\right )+2 \ln \left (-1+\frac {2}{2+x}\right ) \ln \left (\frac {\ln \left (3\right )^{2}}{\left (2+x \right )^{2}}+\frac {4 \ln \left (3\right )}{\left (2+x \right )^{2}}-\frac {2 \ln \left (3\right )}{2+x}+\frac {4}{\left (2+x \right )^{2}}-\frac {4}{2+x}+1\right )-2 \left (2 \ln \left (3\right )+4\right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (2+\ln \left (3\right )\right ) \left (-1+\frac {2}{2+x}\right )+\ln \left (3\right )}{\ln \left (3\right )}\right )}{2+\ln \left (3\right )}+\frac {\ln \left (-1+\frac {2}{2+x}\right ) \ln \left (\frac {\left (2+\ln \left (3\right )\right ) \left (-1+\frac {2}{2+x}\right )+\ln \left (3\right )}{\ln \left (3\right )}\right )}{2+\ln \left (3\right )}\right )+\left (-4 \ln \left (3\right )-8\right ) \left (-\frac {\left (\ln \left (x \right )-\ln \left (\frac {x}{\ln \left (3\right )}\right )\right ) \ln \left (\frac {\ln \left (3\right )-x}{\ln \left (3\right )}\right )-\operatorname {dilog}\left (\frac {x}{\ln \left (3\right )}\right )}{2+\ln \left (3\right )}+\frac {\operatorname {dilog}\left (1+\frac {x}{2}\right )+\ln \left (x \right ) \ln \left (1+\frac {x}{2}\right )}{2+\ln \left (3\right )}\right )\) | \(336\) |
int((((4+2*x)*ln(3)-2*x^2-4*x)*ln((ln(3)^2-2*x*ln(3)+x^2)/(x^2+4*x+4))+(-4 *x*ln(3)-8*x)*ln(x)+(4*x+8)*ln(3)-4*x^2-8*x)/((x^2+2*x)*ln(3)-x^3-2*x^2),x ,method=_RETURNVERBOSE)
4*ln(x)*ln(ln(3)-x)-I*ln(x)*Pi*csgn(I/(2+x)^2)*csgn(I*(ln(3)-x)^2)*csgn(I/ (2+x)^2*(ln(3)-x)^2)+I*ln(x)*Pi*csgn(I/(2+x)^2)*csgn(I/(2+x)^2*(ln(3)-x)^2 )^2+I*ln(x)*Pi*csgn(I*(2+x))^2*csgn(I*(2+x)^2)-2*I*ln(x)*Pi*csgn(I*(2+x))* csgn(I*(2+x)^2)^2+I*ln(x)*Pi*csgn(I*(2+x)^2)^3-I*ln(x)*Pi*csgn(I*(ln(3)-x) )^2*csgn(I*(ln(3)-x)^2)+2*I*ln(x)*Pi*csgn(I*(ln(3)-x))*csgn(I*(ln(3)-x)^2) ^2-I*ln(x)*Pi*csgn(I*(ln(3)-x)^2)^3+I*ln(x)*Pi*csgn(I*(ln(3)-x)^2)*csgn(I/ (2+x)^2*(ln(3)-x)^2)^2-I*ln(x)*Pi*csgn(I/(2+x)^2*(ln(3)-x)^2)^3-4*ln(x)*ln (2+x)+4*ln(x)
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+\left (-4 x-2 x^2+(4+2 x) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2}\right )}{-2 x^2-x^3+\left (2 x+x^2\right ) \log (3)} \, dx=2 \, \log \left (x\right ) \log \left (\frac {x^{2} - 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}}{x^{2} + 4 \, x + 4}\right ) + 4 \, \log \left (x\right ) \]
integrate((((4+2*x)*log(3)-2*x^2-4*x)*log((log(3)^2-2*x*log(3)+x^2)/(x^2+4 *x+4))+(-4*x*log(3)-8*x)*log(x)+(4*x+8)*log(3)-4*x^2-8*x)/((x^2+2*x)*log(3 )-x^3-2*x^2),x, algorithm=\
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+\left (-4 x-2 x^2+(4+2 x) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2}\right )}{-2 x^2-x^3+\left (2 x+x^2\right ) \log (3)} \, dx=2 \log {\left (x \right )} \log {\left (\frac {x^{2} - 2 x \log {\left (3 \right )} + \log {\left (3 \right )}^{2}}{x^{2} + 4 x + 4} \right )} + 4 \log {\left (x \right )} \]
integrate((((4+2*x)*ln(3)-2*x**2-4*x)*ln((ln(3)**2-2*x*ln(3)+x**2)/(x**2+4 *x+4))+(-4*x*ln(3)-8*x)*ln(x)+(4*x+8)*ln(3)-4*x**2-8*x)/((x**2+2*x)*ln(3)- x**3-2*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.38 \[ \int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+\left (-4 x-2 x^2+(4+2 x) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2}\right )}{-2 x^2-x^3+\left (2 x+x^2\right ) \log (3)} \, dx=-4 \, {\left (\frac {2 \, \log \left (x - \log \left (3\right )\right )}{\log \left (3\right )^{2} + 2 \, \log \left (3\right )} + \frac {\log \left (x + 2\right )}{\log \left (3\right ) + 2} - \frac {\log \left (x\right )}{\log \left (3\right )}\right )} \log \left (3\right ) - 4 \, {\left (\frac {\log \left (x - \log \left (3\right )\right )}{\log \left (3\right ) + 2} - \frac {\log \left (x + 2\right )}{\log \left (3\right ) + 2}\right )} \log \left (3\right ) + 4 \, \log \left (x - \log \left (3\right )\right ) \log \left (x\right ) - 4 \, \log \left (x + 2\right ) \log \left (x\right ) + \frac {4 \, \log \left (3\right ) \log \left (x - \log \left (3\right )\right )}{\log \left (3\right ) + 2} + \frac {8 \, \log \left (x - \log \left (3\right )\right )}{\log \left (3\right ) + 2} \]
integrate((((4+2*x)*log(3)-2*x^2-4*x)*log((log(3)^2-2*x*log(3)+x^2)/(x^2+4 *x+4))+(-4*x*log(3)-8*x)*log(x)+(4*x+8)*log(3)-4*x^2-8*x)/((x^2+2*x)*log(3 )-x^3-2*x^2),x, algorithm=\
-4*(2*log(x - log(3))/(log(3)^2 + 2*log(3)) + log(x + 2)/(log(3) + 2) - lo g(x)/log(3))*log(3) - 4*(log(x - log(3))/(log(3) + 2) - log(x + 2)/(log(3) + 2))*log(3) + 4*log(x - log(3))*log(x) - 4*log(x + 2)*log(x) + 4*log(3)* log(x - log(3))/(log(3) + 2) + 8*log(x - log(3))/(log(3) + 2)
Time = 0.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+\left (-4 x-2 x^2+(4+2 x) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2}\right )}{-2 x^2-x^3+\left (2 x+x^2\right ) \log (3)} \, dx=2 \, \log \left (x^{2} - 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right ) \log \left (x\right ) - 2 \, \log \left (x^{2} + 4 \, x + 4\right ) \log \left (x\right ) + 4 \, \log \left (x\right ) \]
integrate((((4+2*x)*log(3)-2*x^2-4*x)*log((log(3)^2-2*x*log(3)+x^2)/(x^2+4 *x+4))+(-4*x*log(3)-8*x)*log(x)+(4*x+8)*log(3)-4*x^2-8*x)/((x^2+2*x)*log(3 )-x^3-2*x^2),x, algorithm=\
Timed out. \[ \int \frac {-8 x-4 x^2+(8+4 x) \log (3)+(-8 x-4 x \log (3)) \log (x)+\left (-4 x-2 x^2+(4+2 x) \log (3)\right ) \log \left (\frac {x^2-2 x \log (3)+\log ^2(3)}{4+4 x+x^2}\right )}{-2 x^2-x^3+\left (2 x+x^2\right ) \log (3)} \, dx=\int \frac {8\,x-\ln \left (3\right )\,\left (4\,x+8\right )+\ln \left (\frac {x^2-2\,\ln \left (3\right )\,x+{\ln \left (3\right )}^2}{x^2+4\,x+4}\right )\,\left (4\,x-\ln \left (3\right )\,\left (2\,x+4\right )+2\,x^2\right )+\ln \left (x\right )\,\left (8\,x+4\,x\,\ln \left (3\right )\right )+4\,x^2}{2\,x^2+x^3-\ln \left (3\right )\,\left (x^2+2\,x\right )} \,d x \]
int((8*x - log(3)*(4*x + 8) + log((log(3)^2 - 2*x*log(3) + x^2)/(4*x + x^2 + 4))*(4*x - log(3)*(2*x + 4) + 2*x^2) + log(x)*(8*x + 4*x*log(3)) + 4*x^ 2)/(2*x^2 + x^3 - log(3)*(2*x + x^2)),x)