Integrand size = 185, antiderivative size = 25 \[ \int \frac {-4+x+e^5 \left (-48-12 x+6 x^2\right )+e^{10} \left (-144-108 x+9 x^3-8 x^4+2 x^5\right )+\left (e^5 (-24+6 x)+e^{10} \left (-144-36 x+18 x^2\right )\right ) \log (-4+x)+e^{10} (-36+9 x) \log ^2(-4+x)+\left (12-3 x+e^5 \left (144+18 x-12 x^2\right )+e^{10} \left (432+216 x-18 x^2-9 x^3\right )+\left (e^5 (72-18 x)+e^{10} \left (432+54 x-36 x^2\right )\right ) \log (-4+x)+e^{10} (108-27 x) \log ^2(-4+x)\right ) \log (x)}{e^{10} \left (-4 x^4+x^5\right )} \, dx=2 x+\frac {\left (\frac {1}{e^5}+3 (2+x+\log (-4+x))\right )^2 \log (x)}{x^3} \]
Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {-4+x+e^5 \left (-48-12 x+6 x^2\right )+e^{10} \left (-144-108 x+9 x^3-8 x^4+2 x^5\right )+\left (e^5 (-24+6 x)+e^{10} \left (-144-36 x+18 x^2\right )\right ) \log (-4+x)+e^{10} (-36+9 x) \log ^2(-4+x)+\left (12-3 x+e^5 \left (144+18 x-12 x^2\right )+e^{10} \left (432+216 x-18 x^2-9 x^3\right )+\left (e^5 (72-18 x)+e^{10} \left (432+54 x-36 x^2\right )\right ) \log (-4+x)+e^{10} (108-27 x) \log ^2(-4+x)\right ) \log (x)}{e^{10} \left (-4 x^4+x^5\right )} \, dx=\frac {2 e^{10} x+\frac {\left (1+3 e^5 (2+x)+3 e^5 \log (-4+x)\right )^2 \log (x)}{x^3}}{e^{10}} \]
Integrate[(-4 + x + E^5*(-48 - 12*x + 6*x^2) + E^10*(-144 - 108*x + 9*x^3 - 8*x^4 + 2*x^5) + (E^5*(-24 + 6*x) + E^10*(-144 - 36*x + 18*x^2))*Log[-4 + x] + E^10*(-36 + 9*x)*Log[-4 + x]^2 + (12 - 3*x + E^5*(144 + 18*x - 12*x ^2) + E^10*(432 + 216*x - 18*x^2 - 9*x^3) + (E^5*(72 - 18*x) + E^10*(432 + 54*x - 36*x^2))*Log[-4 + x] + E^10*(108 - 27*x)*Log[-4 + x]^2)*Log[x])/(E ^10*(-4*x^4 + x^5)),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.
| \(\displaystyle \int \frac {e^5 \left (6 x^2-12 x-48\right )+\left (e^{10} \left (18 x^2-36 x-144\right )+e^5 (6 x-24)\right ) \log (x-4)+\left (e^5 \left (-12 x^2+18 x+144\right )+\left (e^{10} \left (-36 x^2+54 x+432\right )+e^5 (72-18 x)\right ) \log (x-4)+e^{10} \left (-9 x^3-18 x^2+216 x+432\right )-3 x+e^{10} (108-27 x) \log ^2(x-4)+12\right ) \log (x)+e^{10} \left (2 x^5-8 x^4+9 x^3-108 x-144\right )+x+e^{10} (9 x-36) \log ^2(x-4)-4}{e^{10} \left (x^5-4 x^4\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {9 e^{10} (4-x) \log ^2(x-4)+6 \left (e^5 (4-x)+3 e^{10} \left (-x^2+2 x+8\right )\right ) \log (x-4)-x+6 e^5 \left (-x^2+2 x+8\right )+e^{10} \left (-2 x^5+8 x^4-9 x^3+108 x+144\right )-3 \left (9 e^{10} (4-x) \log ^2(x-4)+6 \left (e^5 (4-x)+e^{10} \left (-2 x^2+3 x+24\right )\right ) \log (x-4)-x+2 e^5 \left (-2 x^2+3 x+24\right )+3 e^{10} \left (-x^3-2 x^2+24 x+48\right )+4\right ) \log (x)+4}{4 x^4-x^5}dx}{e^{10}}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \frac {\int \frac {9 e^{10} (4-x) \log ^2(x-4)+6 \left (e^5 (4-x)+3 e^{10} \left (-x^2+2 x+8\right )\right ) \log (x-4)-x+6 e^5 \left (-x^2+2 x+8\right )+e^{10} \left (-2 x^5+8 x^4-9 x^3+108 x+144\right )-3 \left (9 e^{10} (4-x) \log ^2(x-4)+6 \left (e^5 (4-x)+e^{10} \left (-2 x^2+3 x+24\right )\right ) \log (x-4)-x+2 e^5 \left (-2 x^2+3 x+24\right )+3 e^{10} \left (-x^3-2 x^2+24 x+48\right )+4\right ) \log (x)+4}{(4-x) x^4}dx}{e^{10}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {2 e^{10} x^4+9 e^{10} x^2+18 e^{10} \log (x-4) x+6 e^5 \left (1+6 e^5\right ) x+9 e^{10} \log ^2(x-4)+6 e^5 \left (1+6 e^5\right ) \log (x-4)+12 e^5 \left (1+3 e^5\right )+1}{x^4}+\frac {3 \left (3 e^5 x+3 e^5 \log (x-4)+6 e^5+1\right ) \left (e^5 x^2+3 e^5 \log (x-4) x+x-12 e^5 \log (x-4)-4 \left (1+6 e^5\right )\right ) \log (x)}{(4-x) x^4}\right )dx}{e^{10}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {9}{64} e^5 \left (1+6 e^5\right ) \log \left (\frac {x}{4}\right ) \log ^2(x-4)+\frac {9}{64} e^5 \left (1-3 e^5\right ) \log \left (\frac {x}{4}\right ) \log ^2(x-4)+\frac {9}{8} e^{10} \log \left (\frac {x}{4}\right ) \log ^2(x-4)+\frac {9}{64} e^5 \left (1+6 e^5\right ) \log (x) \log ^2(x-4)-\frac {9}{64} e^5 \left (1-3 e^5\right ) \log (x) \log ^2(x-4)-\frac {9}{8} e^{10} \log (x) \log ^2(x-4)-\frac {3 e^{10} \log ^2(x-4)}{x^3}-\frac {9}{64} e^5 \left (1+6 e^5\right ) \log ^2(x) \log (x-4)+\frac {9}{64} e^5 \left (1-3 e^5\right ) \log ^2(x) \log (x-4)+\frac {9}{8} e^{10} \log ^2(x) \log (x-4)-\frac {3}{32} e^{10} \log \left (1-\frac {4}{4-x}\right ) \log (x-4)+\frac {9 e^5 \left (1+6 e^5\right ) \log (x) \log (x-4)}{8 x}-\frac {9 e^5 \left (1-3 e^5\right ) \log (x) \log (x-4)}{8 x}-\frac {9 e^{10} \log (x) \log (x-4)}{x}+\frac {9 e^5 \left (1+6 e^5\right ) \log (x) \log (x-4)}{4 x^2}-\frac {9 e^5 \left (1-3 e^5\right ) \log (x) \log (x-4)}{4 x^2}+\frac {6 e^5 \left (1+6 e^5\right ) \log (x) \log (x-4)}{x^3}-\frac {9}{32} e^5 \left (1+6 e^5\right ) \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right ) \log (x-4)+\frac {9}{32} e^5 \left (1-3 e^5\right ) \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right ) \log (x-4)+\frac {9}{4} e^{10} \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right ) \log (x-4)+\frac {3 e^{10} (4-x) \log (x-4)}{32 x}+\frac {9 e^5 \left (1+6 e^5\right ) \log (x-4)}{8 x}-\frac {9 e^5 \left (1-3 e^5\right ) \log (x-4)}{8 x}-\frac {9 e^{10} \log (x-4)}{x}+\frac {9 e^5 \left (1+6 e^5\right ) \log (x-4)}{8 x^2}-\frac {9 e^5 \left (1-3 e^5\right ) \log (x-4)}{8 x^2}-\frac {33 e^{10} \log (x-4)}{4 x^2}-\frac {3}{64} \left (1-6 e^5-72 e^{10}\right ) \log (4) \log (x-4)+\frac {3}{64} \left (1+6 e^5\right )^2 \log (4) \log (x-4)-\frac {3}{8} e^5 \left (2+3 e^5\right ) \log (4) \log (x-4)-\frac {9}{4} e^{10} \log (4) \log (x-4)+\frac {9}{64} e^5 \left (1+6 e^5\right ) \log \left (1-\frac {x}{4}\right ) \log ^2(x)-\frac {9}{64} e^5 \left (1-3 e^5\right ) \log \left (1-\frac {x}{4}\right ) \log ^2(x)-\frac {9}{8} e^{10} \log \left (1-\frac {x}{4}\right ) \log ^2(x)+\frac {3}{128} \left (1-6 e^5-72 e^{10}\right ) \log ^2(x)-\frac {3}{128} \left (1+6 e^5\right )^2 \log ^2(x)+\frac {33}{128} e^5 \left (1+6 e^5\right ) \log ^2(x)+\frac {3}{16} e^5 \left (2+3 e^5\right ) \log ^2(x)-\frac {27}{128} e^5 \left (1-3 e^5\right ) \log ^2(x)+2 e^{10} x+\frac {1}{32} e^5 \left (1+24 e^5\right ) \log (4-x)-\frac {49}{128} e^5 \left (1+6 e^5\right ) \log (4-x)+\frac {45}{128} e^5 \left (1-3 e^5\right ) \log (4-x)+\frac {141}{64} e^{10} \log (4-x)-\frac {33}{64} e^5 \left (1+6 e^5\right ) \log (4-x) \log (x)+\frac {27}{64} e^5 \left (1-3 e^5\right ) \log (4-x) \log (x)+\frac {9}{4} e^{10} \log (4-x) \log (x)-\frac {3 \left (1-6 e^5-72 e^{10}\right ) \log (x)}{16 x}+\frac {3 \left (1+6 e^5\right )^2 \log (x)}{16 x}-\frac {15 e^5 \left (1+6 e^5\right ) \log (x)}{16 x}-\frac {3 e^5 \left (2+3 e^5\right ) \log (x)}{2 x}+\frac {9 e^5 \left (1-3 e^5\right ) \log (x)}{16 x}-\frac {3 \left (1-6 e^5-72 e^{10}\right ) \log (x)}{8 x^2}+\frac {3 \left (1+6 e^5\right )^2 \log (x)}{8 x^2}-\frac {3 e^5 \left (1+6 e^5\right ) \log (x)}{4 x^2}+\frac {\left (1+6 e^5\right )^2 \log (x)}{x^3}+\frac {33}{64} e^5 \left (1+6 e^5\right ) \log (4) \log (x)-\frac {27}{64} e^5 \left (1-3 e^5\right ) \log (4) \log (x)-\frac {9}{4} e^{10} \log (4) \log (x)-\frac {1}{32} e^5 \left (1+24 e^5\right ) \log (x)+\frac {49}{128} e^5 \left (1+6 e^5\right ) \log (x)-\frac {45}{128} e^5 \left (1-3 e^5\right ) \log (x)-\frac {135}{64} e^{10} \log (x)+\frac {3}{32} e^{10} \operatorname {PolyLog}\left (2,\frac {4}{4-x}\right )+\frac {3}{64} \left (1-6 e^5-72 e^{10}\right ) \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right )-\frac {3}{64} \left (1+6 e^5\right )^2 \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right )+\frac {3}{8} e^5 \left (2+3 e^5\right ) \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right )+\frac {9}{4} e^{10} \operatorname {PolyLog}\left (2,1-\frac {x}{4}\right )+\frac {9}{32} e^5 \left (1+6 e^5\right ) \log (x) \operatorname {PolyLog}\left (2,\frac {x}{4}\right )-\frac {9}{32} e^5 \left (1-3 e^5\right ) \log (x) \operatorname {PolyLog}\left (2,\frac {x}{4}\right )-\frac {9}{4} e^{10} \log (x) \operatorname {PolyLog}\left (2,\frac {x}{4}\right )-\frac {33}{64} e^5 \left (1+6 e^5\right ) \operatorname {PolyLog}\left (2,\frac {x}{4}\right )+\frac {27}{64} e^5 \left (1-3 e^5\right ) \operatorname {PolyLog}\left (2,\frac {x}{4}\right )+\frac {9}{4} e^{10} \operatorname {PolyLog}\left (2,\frac {x}{4}\right )+\frac {9}{32} e^5 \left (1+6 e^5\right ) \operatorname {PolyLog}\left (3,1-\frac {x}{4}\right )-\frac {9}{32} e^5 \left (1-3 e^5\right ) \operatorname {PolyLog}\left (3,1-\frac {x}{4}\right )-\frac {9}{4} e^{10} \operatorname {PolyLog}\left (3,1-\frac {x}{4}\right )-\frac {9}{32} e^5 \left (1+6 e^5\right ) \operatorname {PolyLog}\left (3,\frac {x}{4}\right )+\frac {9}{32} e^5 \left (1-3 e^5\right ) \operatorname {PolyLog}\left (3,\frac {x}{4}\right )+\frac {9}{4} e^{10} \operatorname {PolyLog}\left (3,\frac {x}{4}\right )-27 e^{10} \int \frac {\log ^2(x-4) \log (x)}{x^4}dx-\frac {3 \left (1-6 e^5-72 e^{10}\right )}{16 x}+\frac {e^5 \left (1+24 e^5\right )}{8 x}+\frac {3 \left (1+6 e^5\right )^2}{16 x}-\frac {43 e^5 \left (1+6 e^5\right )}{32 x}-\frac {3 e^5 \left (2+3 e^5\right )}{2 x}+\frac {27 e^5 \left (1-3 e^5\right )}{32 x}-\frac {147 e^{10}}{16 x}-\frac {3 \left (1-6 e^5-72 e^{10}\right )}{16 x^2}+\frac {3 \left (1+6 e^5\right )^2}{16 x^2}-\frac {27 e^5 \left (1+6 e^5\right )}{8 x^2}}{e^{10}}\) |
Int[(-4 + x + E^5*(-48 - 12*x + 6*x^2) + E^10*(-144 - 108*x + 9*x^3 - 8*x^ 4 + 2*x^5) + (E^5*(-24 + 6*x) + E^10*(-144 - 36*x + 18*x^2))*Log[-4 + x] + E^10*(-36 + 9*x)*Log[-4 + x]^2 + (12 - 3*x + E^5*(144 + 18*x - 12*x^2) + E^10*(432 + 216*x - 18*x^2 - 9*x^3) + (E^5*(72 - 18*x) + E^10*(432 + 54*x - 36*x^2))*Log[-4 + x] + E^10*(108 - 27*x)*Log[-4 + x]^2)*Log[x])/(E^10*(- 4*x^4 + x^5)),x]
3.29.67.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(27)=54\).
Time = 1.48 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.28
| method | result | size |
| risch | \(\frac {2 x^{4}+9 x^{2} \ln \left (x \right )+18 \ln \left (x \right ) \ln \left (x -4\right ) x +36 x \ln \left (x \right )+36 \ln \left (x \right ) \ln \left (x -4\right )+6 x \ln \left (x \right ) {\mathrm e}^{-5}+36 \ln \left (x \right )+9 \ln \left (x \right ) \ln \left (x -4\right )^{2}+6 \,{\mathrm e}^{-5} \ln \left (x \right ) \ln \left (x -4\right )+12 \ln \left (x \right ) {\mathrm e}^{-5}+\ln \left (x \right ) {\mathrm e}^{-10}}{x^{3}}\) | \(82\) |
| parallelrisch | \(\frac {{\mathrm e}^{-10} \left (2 x^{4} {\mathrm e}^{10}+16 x^{3} {\mathrm e}^{10}+9 x^{2} {\mathrm e}^{10} \ln \left (x \right )+18 \,{\mathrm e}^{10} \ln \left (x -4\right ) \ln \left (x \right ) x +9 \,{\mathrm e}^{10} \ln \left (x -4\right )^{2} \ln \left (x \right )+36 \,{\mathrm e}^{10} \ln \left (x \right ) x +36 \,{\mathrm e}^{10} \ln \left (x -4\right ) \ln \left (x \right )+36 \,{\mathrm e}^{10} \ln \left (x \right )+6 x \,{\mathrm e}^{5} \ln \left (x \right )+6 \ln \left (x -4\right ) {\mathrm e}^{5} \ln \left (x \right )+12 \,{\mathrm e}^{5} \ln \left (x \right )+\ln \left (x \right )\right )}{x^{3}}\) | \(120\) |
int((((-27*x+108)*exp(5)^2*ln(x-4)^2+((-36*x^2+54*x+432)*exp(5)^2+(-18*x+7 2)*exp(5))*ln(x-4)+(-9*x^3-18*x^2+216*x+432)*exp(5)^2+(-12*x^2+18*x+144)*e xp(5)-3*x+12)*ln(x)+(9*x-36)*exp(5)^2*ln(x-4)^2+((18*x^2-36*x-144)*exp(5)^ 2+(6*x-24)*exp(5))*ln(x-4)+(2*x^5-8*x^4+9*x^3-108*x-144)*exp(5)^2+(6*x^2-1 2*x-48)*exp(5)+x-4)/(x^5-4*x^4)/exp(5)^2,x,method=_RETURNVERBOSE)
(2*x^4+9*x^2*ln(x)+18*ln(x)*ln(x-4)*x+36*x*ln(x)+36*ln(x)*ln(x-4)+6*x*ln(x )*exp(-5)+36*ln(x)+9*ln(x)*ln(x-4)^2+6*exp(-5)*ln(x)*ln(x-4)+12*ln(x)*exp( -5)+ln(x)*exp(-10))/x^3
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {-4+x+e^5 \left (-48-12 x+6 x^2\right )+e^{10} \left (-144-108 x+9 x^3-8 x^4+2 x^5\right )+\left (e^5 (-24+6 x)+e^{10} \left (-144-36 x+18 x^2\right )\right ) \log (-4+x)+e^{10} (-36+9 x) \log ^2(-4+x)+\left (12-3 x+e^5 \left (144+18 x-12 x^2\right )+e^{10} \left (432+216 x-18 x^2-9 x^3\right )+\left (e^5 (72-18 x)+e^{10} \left (432+54 x-36 x^2\right )\right ) \log (-4+x)+e^{10} (108-27 x) \log ^2(-4+x)\right ) \log (x)}{e^{10} \left (-4 x^4+x^5\right )} \, dx=\frac {{\left (2 \, x^{4} e^{10} + {\left (9 \, e^{10} \log \left (x - 4\right )^{2} + 9 \, {\left (x^{2} + 4 \, x + 4\right )} e^{10} + 6 \, {\left (x + 2\right )} e^{5} + 6 \, {\left (3 \, {\left (x + 2\right )} e^{10} + e^{5}\right )} \log \left (x - 4\right ) + 1\right )} \log \left (x\right )\right )} e^{\left (-10\right )}}{x^{3}} \]
integrate((((-27*x+108)*exp(5)^2*log(x-4)^2+((-36*x^2+54*x+432)*exp(5)^2+( -18*x+72)*exp(5))*log(x-4)+(-9*x^3-18*x^2+216*x+432)*exp(5)^2+(-12*x^2+18* x+144)*exp(5)-3*x+12)*log(x)+(9*x-36)*exp(5)^2*log(x-4)^2+((18*x^2-36*x-14 4)*exp(5)^2+(6*x-24)*exp(5))*log(x-4)+(2*x^5-8*x^4+9*x^3-108*x-144)*exp(5) ^2+(6*x^2-12*x-48)*exp(5)+x-4)/(x^5-4*x^4)/exp(5)^2,x, algorithm=\
(2*x^4*e^10 + (9*e^10*log(x - 4)^2 + 9*(x^2 + 4*x + 4)*e^10 + 6*(x + 2)*e^ 5 + 6*(3*(x + 2)*e^10 + e^5)*log(x - 4) + 1)*log(x))*e^(-10)/x^3
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (27) = 54\).
Time = 0.77 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.88 \[ \int \frac {-4+x+e^5 \left (-48-12 x+6 x^2\right )+e^{10} \left (-144-108 x+9 x^3-8 x^4+2 x^5\right )+\left (e^5 (-24+6 x)+e^{10} \left (-144-36 x+18 x^2\right )\right ) \log (-4+x)+e^{10} (-36+9 x) \log ^2(-4+x)+\left (12-3 x+e^5 \left (144+18 x-12 x^2\right )+e^{10} \left (432+216 x-18 x^2-9 x^3\right )+\left (e^5 (72-18 x)+e^{10} \left (432+54 x-36 x^2\right )\right ) \log (-4+x)+e^{10} (108-27 x) \log ^2(-4+x)\right ) \log (x)}{e^{10} \left (-4 x^4+x^5\right )} \, dx=2 x + \frac {\left (18 x e^{5} \log {\left (x \right )} + 6 \log {\left (x \right )} + 36 e^{5} \log {\left (x \right )}\right ) \log {\left (x - 4 \right )}}{x^{3} e^{5}} + \frac {\left (9 x^{2} e^{10} + 6 x e^{5} + 36 x e^{10} + 1 + 12 e^{5} + 36 e^{10}\right ) \log {\left (x \right )}}{x^{3} e^{10}} + \frac {9 \log {\left (x \right )} \log {\left (x - 4 \right )}^{2}}{x^{3}} \]
integrate((((-27*x+108)*exp(5)**2*ln(x-4)**2+((-36*x**2+54*x+432)*exp(5)** 2+(-18*x+72)*exp(5))*ln(x-4)+(-9*x**3-18*x**2+216*x+432)*exp(5)**2+(-12*x* *2+18*x+144)*exp(5)-3*x+12)*ln(x)+(9*x-36)*exp(5)**2*ln(x-4)**2+((18*x**2- 36*x-144)*exp(5)**2+(6*x-24)*exp(5))*ln(x-4)+(2*x**5-8*x**4+9*x**3-108*x-1 44)*exp(5)**2+(6*x**2-12*x-48)*exp(5)+x-4)/(x**5-4*x**4)/exp(5)**2,x)
2*x + (18*x*exp(5)*log(x) + 6*log(x) + 36*exp(5)*log(x))*exp(-5)*log(x - 4 )/x**3 + (9*x**2*exp(10) + 6*x*exp(5) + 36*x*exp(10) + 1 + 12*exp(5) + 36* exp(10))*exp(-10)*log(x)/x**3 + 9*log(x)*log(x - 4)**2/x**3
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 234, normalized size of antiderivative = 9.36 \[ \int \frac {-4+x+e^5 \left (-48-12 x+6 x^2\right )+e^{10} \left (-144-108 x+9 x^3-8 x^4+2 x^5\right )+\left (e^5 (-24+6 x)+e^{10} \left (-144-36 x+18 x^2\right )\right ) \log (-4+x)+e^{10} (-36+9 x) \log ^2(-4+x)+\left (12-3 x+e^5 \left (144+18 x-12 x^2\right )+e^{10} \left (432+216 x-18 x^2-9 x^3\right )+\left (e^5 (72-18 x)+e^{10} \left (432+54 x-36 x^2\right )\right ) \log (-4+x)+e^{10} (108-27 x) \log ^2(-4+x)\right ) \log (x)}{e^{10} \left (-4 x^4+x^5\right )} \, dx=-\frac {1}{192} \, {\left (36 \, {\left (\frac {4 \, {\left (3 \, x^{2} + 6 \, x + 16\right )}}{x^{3}} + 3 \, \log \left (x - 4\right ) - 3 \, \log \left (x\right )\right )} e^{10} + 12 \, {\left (\frac {4 \, {\left (3 \, x^{2} + 6 \, x + 16\right )}}{x^{3}} + 3 \, \log \left (x - 4\right ) - 3 \, \log \left (x\right )\right )} e^{5} - \frac {384 \, x^{4} e^{10} + 1728 \, e^{10} \log \left (x - 4\right )^{2} \log \left (x\right ) + 12 \, x^{2} {\left (36 \, e^{10} + 12 \, e^{5} + 1\right )} + 24 \, x {\left (36 \, e^{10} + 12 \, e^{5} + 1\right )} + 3 \, {\left (x^{3} {\left (36 \, e^{10} + 12 \, e^{5} + 1\right )} + 384 \, {\left (3 \, x e^{10} + 6 \, e^{10} + e^{5}\right )} \log \left (x\right )\right )} \log \left (x - 4\right ) - 3 \, {\left (x^{3} {\left (36 \, e^{10} + 12 \, e^{5} + 1\right )} - 576 \, x^{2} e^{10} - 384 \, x {\left (6 \, e^{10} + e^{5}\right )} - 2304 \, e^{10} - 768 \, e^{5} - 64\right )} \log \left (x\right ) + 2304 \, e^{10} + 768 \, e^{5} + 64}{x^{3}} + \frac {4 \, {\left (3 \, x^{2} + 6 \, x + 16\right )}}{x^{3}} + 3 \, \log \left (x - 4\right ) - 3 \, \log \left (x\right )\right )} e^{\left (-10\right )} \]
integrate((((-27*x+108)*exp(5)^2*log(x-4)^2+((-36*x^2+54*x+432)*exp(5)^2+( -18*x+72)*exp(5))*log(x-4)+(-9*x^3-18*x^2+216*x+432)*exp(5)^2+(-12*x^2+18* x+144)*exp(5)-3*x+12)*log(x)+(9*x-36)*exp(5)^2*log(x-4)^2+((18*x^2-36*x-14 4)*exp(5)^2+(6*x-24)*exp(5))*log(x-4)+(2*x^5-8*x^4+9*x^3-108*x-144)*exp(5) ^2+(6*x^2-12*x-48)*exp(5)+x-4)/(x^5-4*x^4)/exp(5)^2,x, algorithm=\
-1/192*(36*(4*(3*x^2 + 6*x + 16)/x^3 + 3*log(x - 4) - 3*log(x))*e^10 + 12* (4*(3*x^2 + 6*x + 16)/x^3 + 3*log(x - 4) - 3*log(x))*e^5 - (384*x^4*e^10 + 1728*e^10*log(x - 4)^2*log(x) + 12*x^2*(36*e^10 + 12*e^5 + 1) + 24*x*(36* e^10 + 12*e^5 + 1) + 3*(x^3*(36*e^10 + 12*e^5 + 1) + 384*(3*x*e^10 + 6*e^1 0 + e^5)*log(x))*log(x - 4) - 3*(x^3*(36*e^10 + 12*e^5 + 1) - 576*x^2*e^10 - 384*x*(6*e^10 + e^5) - 2304*e^10 - 768*e^5 - 64)*log(x) + 2304*e^10 + 7 68*e^5 + 64)/x^3 + 4*(3*x^2 + 6*x + 16)/x^3 + 3*log(x - 4) - 3*log(x))*e^( -10)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (25) = 50\).
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.76 \[ \int \frac {-4+x+e^5 \left (-48-12 x+6 x^2\right )+e^{10} \left (-144-108 x+9 x^3-8 x^4+2 x^5\right )+\left (e^5 (-24+6 x)+e^{10} \left (-144-36 x+18 x^2\right )\right ) \log (-4+x)+e^{10} (-36+9 x) \log ^2(-4+x)+\left (12-3 x+e^5 \left (144+18 x-12 x^2\right )+e^{10} \left (432+216 x-18 x^2-9 x^3\right )+\left (e^5 (72-18 x)+e^{10} \left (432+54 x-36 x^2\right )\right ) \log (-4+x)+e^{10} (108-27 x) \log ^2(-4+x)\right ) \log (x)}{e^{10} \left (-4 x^4+x^5\right )} \, dx=\frac {{\left (2 \, x^{4} e^{10} + 9 \, x^{2} e^{10} \log \left (x\right ) + 18 \, x e^{10} \log \left (x - 4\right ) \log \left (x\right ) + 9 \, e^{10} \log \left (x - 4\right )^{2} \log \left (x\right ) + 36 \, x e^{10} \log \left (x\right ) + 6 \, x e^{5} \log \left (x\right ) + 36 \, e^{10} \log \left (x - 4\right ) \log \left (x\right ) + 6 \, e^{5} \log \left (x - 4\right ) \log \left (x\right ) + 36 \, e^{10} \log \left (x\right ) + 12 \, e^{5} \log \left (x\right ) + \log \left (x\right )\right )} e^{\left (-10\right )}}{x^{3}} \]
integrate((((-27*x+108)*exp(5)^2*log(x-4)^2+((-36*x^2+54*x+432)*exp(5)^2+( -18*x+72)*exp(5))*log(x-4)+(-9*x^3-18*x^2+216*x+432)*exp(5)^2+(-12*x^2+18* x+144)*exp(5)-3*x+12)*log(x)+(9*x-36)*exp(5)^2*log(x-4)^2+((18*x^2-36*x-14 4)*exp(5)^2+(6*x-24)*exp(5))*log(x-4)+(2*x^5-8*x^4+9*x^3-108*x-144)*exp(5) ^2+(6*x^2-12*x-48)*exp(5)+x-4)/(x^5-4*x^4)/exp(5)^2,x, algorithm=\
(2*x^4*e^10 + 9*x^2*e^10*log(x) + 18*x*e^10*log(x - 4)*log(x) + 9*e^10*log (x - 4)^2*log(x) + 36*x*e^10*log(x) + 6*x*e^5*log(x) + 36*e^10*log(x - 4)* log(x) + 6*e^5*log(x - 4)*log(x) + 36*e^10*log(x) + 12*e^5*log(x) + log(x) )*e^(-10)/x^3
Time = 13.68 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.96 \[ \int \frac {-4+x+e^5 \left (-48-12 x+6 x^2\right )+e^{10} \left (-144-108 x+9 x^3-8 x^4+2 x^5\right )+\left (e^5 (-24+6 x)+e^{10} \left (-144-36 x+18 x^2\right )\right ) \log (-4+x)+e^{10} (-36+9 x) \log ^2(-4+x)+\left (12-3 x+e^5 \left (144+18 x-12 x^2\right )+e^{10} \left (432+216 x-18 x^2-9 x^3\right )+\left (e^5 (72-18 x)+e^{10} \left (432+54 x-36 x^2\right )\right ) \log (-4+x)+e^{10} (108-27 x) \log ^2(-4+x)\right ) \log (x)}{e^{10} \left (-4 x^4+x^5\right )} \, dx=2\,x+\frac {\ln \left (x\right )\,\left (9\,x^2+6\,{\mathrm {e}}^{-5}\,\left (6\,{\mathrm {e}}^5+1\right )\,x+{\mathrm {e}}^{-10}\,{\left (6\,{\mathrm {e}}^5+1\right )}^2\right )}{x^3}+\frac {9\,{\ln \left (x-4\right )}^2\,\ln \left (x\right )}{x^3}+\frac {\ln \left (x-4\right )\,{\mathrm {e}}^{-5}\,\ln \left (x\right )\,\left (36\,{\mathrm {e}}^5+18\,x\,{\mathrm {e}}^5+6\right )}{x^3} \]
int((exp(-10)*(log(x - 4)*(exp(10)*(36*x - 18*x^2 + 144) - exp(5)*(6*x - 2 4)) - x + exp(5)*(12*x - 6*x^2 + 48) + exp(10)*(108*x - 9*x^3 + 8*x^4 - 2* x^5 + 144) - log(x)*(log(x - 4)*(exp(10)*(54*x - 36*x^2 + 432) - exp(5)*(1 8*x - 72)) - 3*x + exp(5)*(18*x - 12*x^2 + 144) + exp(10)*(216*x - 18*x^2 - 9*x^3 + 432) - log(x - 4)^2*exp(10)*(27*x - 108) + 12) - log(x - 4)^2*ex p(10)*(9*x - 36) + 4))/(4*x^4 - x^5),x)