Integrand size = 60, antiderivative size = 28 \[ \int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{195 x-79 x^2+8 x^3} \, dx=\log ^2\left (\frac {4}{3 e^3 \left (5-\frac {5}{8 (5-x)}\right )^2 x^2}\right ) \]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.04 \[ \int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{195 x-79 x^2+8 x^3} \, dx=-4 \left (\log ^2(5-x)-2 \log (5) \log (-5+x)-2 \log (5) \log (x)+\log ^2(x)-\log (5-x) \left (-3+\log \left (\frac {256}{75}\right )+\log \left (\frac {(-5+x)^2}{x^2 (-39+8 x)^2}\right )\right )+\log (x) \left (-3+\log \left (\frac {256}{75}\right )+\log \left (\frac {(-5+x)^2}{x^2 (-39+8 x)^2}\right )\right )+2 \log \left (\frac {39}{8}\right ) \log (-39+8 x)+\log (64) \log (-39+8 x)+2 \log \left (\frac {8 x}{39}\right ) \log (-39+8 x)+\left (-3+\log \left (\frac {256}{75}\right )+\log \left (\frac {(-5+x)^2}{x^2 (-39+8 x)^2}\right )\right ) \log (-39+8 x)+\log ^2(-39+8 x)+2 \operatorname {PolyLog}(2,40-8 x)+2 \operatorname {PolyLog}\left (2,1-\frac {x}{5}\right )+2 \operatorname {PolyLog}\left (2,\frac {x}{5}\right )+2 \operatorname {PolyLog}(2,-39+8 x)\right ) \]
Integrate[((-780 + 320*x - 32*x^2)*Log[(6400 - 2560*x + 256*x^2)/(E^3*(114 075*x^2 - 46800*x^3 + 4800*x^4))])/(195*x - 79*x^2 + 8*x^3),x]
-4*(Log[5 - x]^2 - 2*Log[5]*Log[-5 + x] - 2*Log[5]*Log[x] + Log[x]^2 - Log [5 - x]*(-3 + Log[256/75] + Log[(-5 + x)^2/(x^2*(-39 + 8*x)^2)]) + Log[x]* (-3 + Log[256/75] + Log[(-5 + x)^2/(x^2*(-39 + 8*x)^2)]) + 2*Log[39/8]*Log [-39 + 8*x] + Log[64]*Log[-39 + 8*x] + 2*Log[(8*x)/39]*Log[-39 + 8*x] + (- 3 + Log[256/75] + Log[(-5 + x)^2/(x^2*(-39 + 8*x)^2)])*Log[-39 + 8*x] + Lo g[-39 + 8*x]^2 + 2*PolyLog[2, 40 - 8*x] + 2*PolyLog[2, 1 - x/5] + 2*PolyLo g[2, x/5] + 2*PolyLog[2, -39 + 8*x])
Leaf count is larger than twice the leaf count of optimal. \(203\) vs. \(2(28)=56\).
Time = 1.32 (sec) , antiderivative size = 203, normalized size of antiderivative = 7.25, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2026, 3008, 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.
| \(\displaystyle \int \frac {\left (-32 x^2+320 x-780\right ) \log \left (\frac {256 x^2-2560 x+6400}{e^3 \left (4800 x^4-46800 x^3+114075 x^2\right )}\right )}{8 x^3-79 x^2+195 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-32 x^2+320 x-780\right ) \log \left (\frac {256 x^2-2560 x+6400}{e^3 \left (4800 x^4-46800 x^3+114075 x^2\right )}\right )}{x \left (8 x^2-79 x+195\right )}dx\) |
| \(\Big \downarrow \) 3008 |
| \(\displaystyle \int \left (\frac {4 \log \left (\frac {256 x^2-2560 x+6400}{e^3 \left (4800 x^4-46800 x^3+114075 x^2\right )}\right )}{x-5}-\frac {4 \log \left (\frac {256 x^2-2560 x+6400}{e^3 \left (4800 x^4-46800 x^3+114075 x^2\right )}\right )}{x}-\frac {32 \log \left (\frac {256 x^2-2560 x+6400}{e^3 \left (4800 x^4-46800 x^3+114075 x^2\right )}\right )}{8 x-39}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \log \left (\frac {256 \left (x^2-10 x+25\right )}{75 e^3 \left (64 x^4-624 x^3+1521 x^2\right )}\right ) \log (x-5)-4 \log (x) \log \left (\frac {256 \left (x^2-10 x+25\right )}{75 e^3 \left (64 x^4-624 x^3+1521 x^2\right )}\right )-4 \log (8 x-39) \log \left (\frac {256 \left (x^2-10 x+25\right )}{75 e^3 \left (64 x^4-624 x^3+1521 x^2\right )}\right )-4 \log ^2(x-5)-4 \log ^2(x)-4 \log ^2(8 x-39)+8 \log \left (\frac {x}{5}\right ) \log (x-5)+8 \log (8 x-39) \log (x-5)+8 \log (5) \log (x-5)-8 \log \left (\frac {8 x}{39}\right ) \log (8 x-39)-8 \log \left (\frac {39}{8}\right ) \log (8 x-39)\) |
Int[((-780 + 320*x - 32*x^2)*Log[(6400 - 2560*x + 256*x^2)/(E^3*(114075*x^ 2 - 46800*x^3 + 4800*x^4))])/(195*x - 79*x^2 + 8*x^3),x]
8*Log[5]*Log[-5 + x] - 4*Log[-5 + x]^2 + 8*Log[-5 + x]*Log[x/5] - 4*Log[x] ^2 - 8*Log[39/8]*Log[-39 + 8*x] + 8*Log[-5 + x]*Log[-39 + 8*x] - 8*Log[(8* x)/39]*Log[-39 + 8*x] - 4*Log[-39 + 8*x]^2 + 4*Log[-5 + x]*Log[(256*(25 - 10*x + x^2))/(75*E^3*(1521*x^2 - 624*x^3 + 64*x^4))] - 4*Log[x]*Log[(256*( 25 - 10*x + x^2))/(75*E^3*(1521*x^2 - 624*x^3 + 64*x^4))] - 4*Log[-39 + 8* x]*Log[(256*(25 - 10*x + x^2))/(75*E^3*(1521*x^2 - 624*x^3 + 64*x^4))]
3.9.16.3.1 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[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With [{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u ]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFuncti onQ[RGx, x] && IGtQ[n, 0]
Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32
| method | result | size |
| norman | \({\ln \left (\frac {\left (256 x^{2}-2560 x +6400\right ) {\mathrm e}^{-3}}{4800 x^{4}-46800 x^{3}+114075 x^{2}}\right )}^{2}\) | \(37\) |
| default | \(4 \ln \left (75\right ) \left (\ln \left (x \right )-\ln \left (-5+x \right )+\ln \left (8 x -39\right )\right )+12 \ln \left (x \right )-12 \ln \left (-5+x \right )+12 \ln \left (8 x -39\right )-32 \ln \left (2\right ) \left (\ln \left (x \right )-\ln \left (-5+x \right )+\ln \left (8 x -39\right )\right )-4 \ln \left (x \right ) \ln \left (\frac {x^{2}-10 x +25}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-4 \ln \left (x \right )^{2}+8 \left (\ln \left (x \right )-\ln \left (\frac {x}{5}\right )\right ) \ln \left (1-\frac {x}{5}\right )-8 \left (\ln \left (x \right )-\ln \left (\frac {8 x}{39}\right )\right ) \ln \left (-\frac {8 x}{39}+1\right )+4 \ln \left (-5+x \right ) \ln \left (\frac {x^{2}-10 x +25}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-4 \ln \left (-5+x \right )^{2}+8 \ln \left (-5+x \right ) \ln \left (8 x -39\right )+8 \ln \left (-5+x \right ) \ln \left (\frac {x}{5}\right )-4 \ln \left (8 x -39\right ) \ln \left (\frac {x^{2}-10 x +25}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-4 \ln \left (8 x -39\right )^{2}-8 \ln \left (8 x -39\right ) \ln \left (\frac {8 x}{39}\right )\) | \(243\) |
| risch | \(-4 \ln \left (x \right )^{2}-32 \ln \left (2\right ) \ln \left (x \right )+4 \ln \left (75\right ) \ln \left (x \right )-4 \ln \left (x \right ) \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{-3}}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-8 \ln \left (-\frac {8 x}{39}+1\right ) \ln \left (x \right )+8 \ln \left (1-\frac {x}{5}\right ) \ln \left (x \right )+32 \ln \left (2\right ) \ln \left (-5+x \right )-32 \ln \left (2\right ) \ln \left (8 x -39\right )-4 \ln \left (75\right ) \ln \left (-5+x \right )+4 \ln \left (75\right ) \ln \left (8 x -39\right )+4 \ln \left (-5+x \right ) \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{-3}}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )-4 \ln \left (8 x -39\right ) \ln \left (\frac {\left (x^{2}-10 x +25\right ) {\mathrm e}^{-3}}{x^{2} \left (64 x^{2}-624 x +1521\right )}\right )+8 \ln \left (-\frac {8 x}{39}+1\right ) \ln \left (\frac {8 x}{39}\right )-8 \ln \left (1-\frac {x}{5}\right ) \ln \left (\frac {x}{5}\right )-8 \ln \left (8 x -39\right ) \ln \left (\frac {8 x}{39}\right )+8 \ln \left (-5+x \right ) \ln \left (\frac {x}{5}\right )-4 \ln \left (-5+x \right )^{2}+8 \ln \left (-5+x \right ) \ln \left (8 x -39\right )-4 \ln \left (8 x -39\right )^{2}\) | \(251\) |
| parts | \(-4 \ln \left (\frac {\left (256 x^{2}-2560 x +6400\right ) {\mathrm e}^{-3}}{4800 x^{4}-46800 x^{3}+114075 x^{2}}\right ) \ln \left (x \right )+4 \ln \left (\frac {\left (256 x^{2}-2560 x +6400\right ) {\mathrm e}^{-3}}{4800 x^{4}-46800 x^{3}+114075 x^{2}}\right ) \ln \left (-5+x \right )-4 \ln \left (\frac {\left (256 x^{2}-2560 x +6400\right ) {\mathrm e}^{-3}}{4800 x^{4}-46800 x^{3}+114075 x^{2}}\right ) \ln \left (8 x -39\right )-\frac {75 \,{\mathrm e}^{3} \left (\frac {512 \,{\mathrm e}^{-3} \left (\frac {\ln \left (x \right )^{2}}{2}-\left (\ln \left (x \right )-\ln \left (\frac {x}{5}\right )\right ) \ln \left (1-\frac {x}{5}\right )+\operatorname {dilog}\left (\frac {x}{5}\right )+\left (\ln \left (x \right )-\ln \left (\frac {8 x}{39}\right )\right ) \ln \left (-\frac {8 x}{39}+1\right )-\operatorname {dilog}\left (\frac {8 x}{39}\right )\right )}{75}-\frac {512 \,{\mathrm e}^{-3} \left (-\frac {\ln \left (-5+x \right )^{2}}{2}+\operatorname {dilog}\left (8 x -39\right )+\ln \left (-5+x \right ) \ln \left (8 x -39\right )+\operatorname {dilog}\left (\frac {x}{5}\right )+\ln \left (-5+x \right ) \ln \left (\frac {x}{5}\right )\right )}{75}+\frac {512 \,{\mathrm e}^{-3} \left (\frac {\ln \left (8 x -39\right )^{2}}{2}+\operatorname {dilog}\left (8 x -39\right )+\operatorname {dilog}\left (\frac {8 x}{39}\right )+\ln \left (8 x -39\right ) \ln \left (\frac {8 x}{39}\right )\right )}{75}\right )}{64}\) | \(266\) |
int((-32*x^2+320*x-780)*ln((256*x^2-2560*x+6400)/(4800*x^4-46800*x^3+11407 5*x^2)/exp(3))/(8*x^3-79*x^2+195*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{195 x-79 x^2+8 x^3} \, dx=\log \left (\frac {256 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (-3\right )}}{75 \, {\left (64 \, x^{4} - 624 \, x^{3} + 1521 \, x^{2}\right )}}\right )^{2} \]
integrate((-32*x^2+320*x-780)*log((256*x^2-2560*x+6400)/(4800*x^4-46800*x^ 3+114075*x^2)/exp(3))/(8*x^3-79*x^2+195*x),x, algorithm=\
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{195 x-79 x^2+8 x^3} \, dx=\log {\left (\frac {256 x^{2} - 2560 x + 6400}{\left (4800 x^{4} - 46800 x^{3} + 114075 x^{2}\right ) e^{3}} \right )}^{2} \]
integrate((-32*x**2+320*x-780)*ln((256*x**2-2560*x+6400)/(4800*x**4-46800* x**3+114075*x**2)/exp(3))/(8*x**3-79*x**2+195*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (19) = 38\).
Time = 0.21 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.50 \[ \int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{195 x-79 x^2+8 x^3} \, dx=8 \, {\left (\log \left (x - 5\right ) - \log \left (x\right )\right )} \log \left (8 \, x - 39\right ) - 4 \, \log \left (8 \, x - 39\right )^{2} - 4 \, \log \left (x - 5\right )^{2} + 8 \, \log \left (x - 5\right ) \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - 4 \, {\left (\log \left (8 \, x - 39\right ) - \log \left (x - 5\right ) + \log \left (x\right )\right )} \log \left (\frac {256 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (-3\right )}}{75 \, {\left (64 \, x^{4} - 624 \, x^{3} + 1521 \, x^{2}\right )}}\right ) \]
integrate((-32*x^2+320*x-780)*log((256*x^2-2560*x+6400)/(4800*x^4-46800*x^ 3+114075*x^2)/exp(3))/(8*x^3-79*x^2+195*x),x, algorithm=\
8*(log(x - 5) - log(x))*log(8*x - 39) - 4*log(8*x - 39)^2 - 4*log(x - 5)^2 + 8*log(x - 5)*log(x) - 4*log(x)^2 - 4*(log(8*x - 39) - log(x - 5) + log( x))*log(256/75*(x^2 - 10*x + 25)*e^(-3)/(64*x^4 - 624*x^3 + 1521*x^2))
\[ \int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{195 x-79 x^2+8 x^3} \, dx=\int { -\frac {4 \, {\left (8 \, x^{2} - 80 \, x + 195\right )} \log \left (\frac {256 \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (-3\right )}}{75 \, {\left (64 \, x^{4} - 624 \, x^{3} + 1521 \, x^{2}\right )}}\right )}{8 \, x^{3} - 79 \, x^{2} + 195 \, x} \,d x } \]
integrate((-32*x^2+320*x-780)*log((256*x^2-2560*x+6400)/(4800*x^4-46800*x^ 3+114075*x^2)/exp(3))/(8*x^3-79*x^2+195*x),x, algorithm=\
integrate(-4*(8*x^2 - 80*x + 195)*log(256/75*(x^2 - 10*x + 25)*e^(-3)/(64* x^4 - 624*x^3 + 1521*x^2))/(8*x^3 - 79*x^2 + 195*x), x)
Time = 9.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-780+320 x-32 x^2\right ) \log \left (\frac {6400-2560 x+256 x^2}{e^3 \left (114075 x^2-46800 x^3+4800 x^4\right )}\right )}{195 x-79 x^2+8 x^3} \, dx={\left (\ln \left (\frac {256\,x^2-2560\,x+6400}{4800\,x^4-46800\,x^3+114075\,x^2}\right )-3\right )}^2 \]