Integrand size = 407, antiderivative size = 24 \[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx=\left (-x+e^{2-\frac {1}{x}} (3-\log (4+x))\right )^4 \]
Time = 5.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx=e^{-4/x} \left (-3 e^2+e^{\frac {1}{x}} x+e^2 \log (4+x)\right )^4 \]
Integrate[(16*x^5 + 4*x^6 + E^((3*(-1 + 2*x))/x)*(-1296*x - 756*x^2) + E^( (4*(-1 + 2*x))/x)*(1296 + 324*x - 108*x^2) + E^((2*(-1 + 2*x))/x)*(432*x^2 + 540*x^3 + 72*x^4) + E^((-1 + 2*x)/x)*(-48*x^3 - 156*x^4 - 32*x^5) + (E^ ((4*(-1 + 2*x))/x)*(-1728 - 432*x + 108*x^2) + E^((3*(-1 + 2*x))/x)*(1296* x + 756*x^2 + 36*x^3) + E^((2*(-1 + 2*x))/x)*(-288*x^2 - 360*x^3 - 60*x^4) + E^((-1 + 2*x)/x)*(16*x^3 + 52*x^4 + 12*x^5))*Log[4 + x] + (E^((4*(-1 + 2*x))/x)*(864 + 216*x - 36*x^2) + E^((3*(-1 + 2*x))/x)*(-432*x - 252*x^2 - 24*x^3) + E^((2*(-1 + 2*x))/x)*(48*x^2 + 60*x^3 + 12*x^4))*Log[4 + x]^2 + (E^((4*(-1 + 2*x))/x)*(-192 - 48*x + 4*x^2) + E^((3*(-1 + 2*x))/x)*(48*x + 28*x^2 + 4*x^3))*Log[4 + x]^3 + E^((4*(-1 + 2*x))/x)*(16 + 4*x)*Log[4 + x]^4)/(4*x^2 + x^3),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.
| \(\displaystyle \int \frac {4 x^6+16 x^5+e^{\frac {3 (2 x-1)}{x}} \left (-756 x^2-1296 x\right )+e^{\frac {4 (2 x-1)}{x}} \left (-108 x^2+324 x+1296\right )+\left (e^{\frac {4 (2 x-1)}{x}} \left (4 x^2-48 x-192\right )+e^{\frac {3 (2 x-1)}{x}} \left (4 x^3+28 x^2+48 x\right )\right ) \log ^3(x+4)+e^{\frac {2 x-1}{x}} \left (-32 x^5-156 x^4-48 x^3\right )+e^{\frac {2 (2 x-1)}{x}} \left (72 x^4+540 x^3+432 x^2\right )+\left (e^{\frac {4 (2 x-1)}{x}} \left (-36 x^2+216 x+864\right )+e^{\frac {3 (2 x-1)}{x}} \left (-24 x^3-252 x^2-432 x\right )+e^{\frac {2 (2 x-1)}{x}} \left (12 x^4+60 x^3+48 x^2\right )\right ) \log ^2(x+4)+\left (e^{\frac {4 (2 x-1)}{x}} \left (108 x^2-432 x-1728\right )+e^{\frac {3 (2 x-1)}{x}} \left (36 x^3+756 x^2+1296 x\right )+e^{\frac {2 x-1}{x}} \left (12 x^5+52 x^4+16 x^3\right )+e^{\frac {2 (2 x-1)}{x}} \left (-60 x^4-360 x^3-288 x^2\right )\right ) \log (x+4)+e^{\frac {4 (2 x-1)}{x}} (4 x+16) \log ^4(x+4)}{x^3+4 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {4 x^6+16 x^5+e^{\frac {3 (2 x-1)}{x}} \left (-756 x^2-1296 x\right )+e^{\frac {4 (2 x-1)}{x}} \left (-108 x^2+324 x+1296\right )+\left (e^{\frac {4 (2 x-1)}{x}} \left (4 x^2-48 x-192\right )+e^{\frac {3 (2 x-1)}{x}} \left (4 x^3+28 x^2+48 x\right )\right ) \log ^3(x+4)+e^{\frac {2 x-1}{x}} \left (-32 x^5-156 x^4-48 x^3\right )+e^{\frac {2 (2 x-1)}{x}} \left (72 x^4+540 x^3+432 x^2\right )+\left (e^{\frac {4 (2 x-1)}{x}} \left (-36 x^2+216 x+864\right )+e^{\frac {3 (2 x-1)}{x}} \left (-24 x^3-252 x^2-432 x\right )+e^{\frac {2 (2 x-1)}{x}} \left (12 x^4+60 x^3+48 x^2\right )\right ) \log ^2(x+4)+\left (e^{\frac {4 (2 x-1)}{x}} \left (108 x^2-432 x-1728\right )+e^{\frac {3 (2 x-1)}{x}} \left (36 x^3+756 x^2+1296 x\right )+e^{\frac {2 x-1}{x}} \left (12 x^5+52 x^4+16 x^3\right )+e^{\frac {2 (2 x-1)}{x}} \left (-60 x^4-360 x^3-288 x^2\right )\right ) \log (x+4)+e^{\frac {4 (2 x-1)}{x}} (4 x+16) \log ^4(x+4)}{x^2 (x+4)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 e^{-4/x} \left (-e^{\frac {1}{x}} x-e^2 \log (x+4)+3 e^2\right )^3 \left (-e^{\frac {1}{x}} (x+4) x^2-e^2 \left (x^2-3 x-12\right )-e^2 (x+4) \log (x+4)\right )}{x^2 (x+4)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {e^{-4/x} \left (-e^{\frac {1}{x}} x-e^2 \log (x+4)+3 e^2\right )^3 \left (e^{\frac {1}{x}} (x+4) x^2-e^2 \left (-x^2+3 x+12\right )+e^2 (x+4) \log (x+4)\right )}{x^2 (x+4)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {e^{-4/x} \left (-e^{\frac {1}{x}} x-e^2 \log (x+4)+3 e^2\right )^3 \left (e^{\frac {1}{x}} (x+4) x^2-e^2 \left (-x^2+3 x+12\right )+e^2 (x+4) \log (x+4)\right )}{x^2 (x+4)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-x^3-\frac {e^{2-\frac {1}{x}} \left (3 \log (x+4) x^2-8 x^2+13 \log (x+4) x-39 x+4 \log (x+4)-12\right ) x}{x+4}-\frac {3 e^{4-\frac {2}{x}} (\log (x+4)-3) \left (\log (x+4) x^2-2 x^2+5 \log (x+4) x-15 x+4 \log (x+4)-12\right )}{x+4}-\frac {e^{6-\frac {3}{x}} (\log (x+4)-3)^2 \left (\log (x+4) x^2+7 \log (x+4) x-21 x+12 \log (x+4)-36\right )}{(x+4) x}-\frac {e^{8-\frac {4}{x}} (\log (x+4)-3)^3 \left (x^2+\log (x+4) x-3 x+4 \log (x+4)-12\right )}{(x+4) x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \left (72 e^{27/4} \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{4}-\frac {3}{x}\right )}{x+4}dx+48 e^{9/2} \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {1}{2}-\frac {2}{x}\right )}{x+4}dx-126 e^6 \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{x}\right )}{x+4}dx-78 e^4 \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {2}{x}\right )}{x+4}dx-\int e^{6-\frac {3}{x}} \log ^3(x+4)dx-3 \int \frac {e^{6-\frac {3}{x}} \log ^3(x+4)}{x}dx+6 \int e^{6-\frac {3}{x}} \log ^2(x+4)dx+9 \int e^{4-\frac {2}{x}} \log ^2(x+4)dx+27 \int \frac {e^{6-\frac {3}{x}} \log ^2(x+4)}{x}dx+12 \int \frac {e^{6-\frac {3}{x}} \log ^2(x+4)}{x+4}dx-3 \int e^{4-\frac {2}{x}} (x+4) \log ^2(x+4)dx-126 e^6 \operatorname {ExpIntegralEi}\left (-\frac {3}{x}\right )-129 e^4 \operatorname {ExpIntegralEi}\left (-\frac {2}{x}\right )+72 e^{27/4} \operatorname {ExpIntegralEi}\left (-\frac {3 (x+4)}{4 x}\right )+84 e^{9/2} \operatorname {ExpIntegralEi}\left (-\frac {x+4}{2 x}\right )+126 e^6 \operatorname {ExpIntegralEi}\left (-\frac {3}{x}\right ) \log (x+4)+78 e^4 \operatorname {ExpIntegralEi}\left (-\frac {2}{x}\right ) \log (x+4)-72 e^{27/4} \operatorname {ExpIntegralEi}\left (-\frac {3 (x+4)}{4 x}\right ) \log (x+4)-48 e^{9/2} \operatorname {ExpIntegralEi}\left (-\frac {x+4}{2 x}\right ) \log (x+4)-\frac {x^4}{4}+3 e^{2-\frac {1}{x}} x^3-e^{2-\frac {1}{x}} x^3 \log (x+4)-\frac {51}{4} e^{4-\frac {2}{x}} x^2+\frac {15}{2} e^{4-\frac {2}{x}} x^2 \log (x+4)+9 e^{6-\frac {3}{x}} x-\frac {45}{2} e^{4-\frac {2}{x}} x-9 e^{6-\frac {3}{x}} x \log (x+4)+15 e^{4-\frac {2}{x}} x \log (x+4)-\frac {e^{8-\frac {4}{x}} (3-\log (x+4))^3 (3 x+x (-\log (x+4))-4 \log (x+4)+12)}{4 (x+4)}\right )\) |
Int[(16*x^5 + 4*x^6 + E^((3*(-1 + 2*x))/x)*(-1296*x - 756*x^2) + E^((4*(-1 + 2*x))/x)*(1296 + 324*x - 108*x^2) + E^((2*(-1 + 2*x))/x)*(432*x^2 + 540 *x^3 + 72*x^4) + E^((-1 + 2*x)/x)*(-48*x^3 - 156*x^4 - 32*x^5) + (E^((4*(- 1 + 2*x))/x)*(-1728 - 432*x + 108*x^2) + E^((3*(-1 + 2*x))/x)*(1296*x + 75 6*x^2 + 36*x^3) + E^((2*(-1 + 2*x))/x)*(-288*x^2 - 360*x^3 - 60*x^4) + E^( (-1 + 2*x)/x)*(16*x^3 + 52*x^4 + 12*x^5))*Log[4 + x] + (E^((4*(-1 + 2*x))/ x)*(864 + 216*x - 36*x^2) + E^((3*(-1 + 2*x))/x)*(-432*x - 252*x^2 - 24*x^ 3) + E^((2*(-1 + 2*x))/x)*(48*x^2 + 60*x^3 + 12*x^4))*Log[4 + x]^2 + (E^(( 4*(-1 + 2*x))/x)*(-192 - 48*x + 4*x^2) + E^((3*(-1 + 2*x))/x)*(48*x + 28*x ^2 + 4*x^3))*Log[4 + x]^3 + E^((4*(-1 + 2*x))/x)*(16 + 4*x)*Log[4 + x]^4)/ (4*x^2 + x^3),x]
3.22.96.3.1 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])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(23)=46\).
Time = 0.59 (sec) , antiderivative size = 243, normalized size of antiderivative = 10.12
| method | result | size |
| risch | \({\mathrm e}^{\frac {8 x -4}{x}} \ln \left (4+x \right )^{4}+\left (4 \,{\mathrm e}^{\frac {-3+6 x}{x}} x -12 \,{\mathrm e}^{\frac {8 x -4}{x}}\right ) \ln \left (4+x \right )^{3}+\left (6 \,{\mathrm e}^{\frac {4 x -2}{x}} x^{2}-36 \,{\mathrm e}^{\frac {-3+6 x}{x}} x +54 \,{\mathrm e}^{\frac {8 x -4}{x}}\right ) \ln \left (4+x \right )^{2}+\left (4 \,{\mathrm e}^{\frac {-1+2 x}{x}} x^{3}-36 \,{\mathrm e}^{\frac {4 x -2}{x}} x^{2}+108 \,{\mathrm e}^{\frac {-3+6 x}{x}} x -108 \,{\mathrm e}^{\frac {8 x -4}{x}}\right ) \ln \left (4+x \right )+x^{4}-12 \,{\mathrm e}^{\frac {-1+2 x}{x}} x^{3}+54 \,{\mathrm e}^{\frac {4 x -2}{x}} x^{2}-108 \,{\mathrm e}^{\frac {-3+6 x}{x}} x +81 \,{\mathrm e}^{\frac {8 x -4}{x}}\) | \(243\) |
int(((4*x+16)*exp((-1+2*x)/x)^4*ln(4+x)^4+((4*x^2-48*x-192)*exp((-1+2*x)/x )^4+(4*x^3+28*x^2+48*x)*exp((-1+2*x)/x)^3)*ln(4+x)^3+((-36*x^2+216*x+864)* exp((-1+2*x)/x)^4+(-24*x^3-252*x^2-432*x)*exp((-1+2*x)/x)^3+(12*x^4+60*x^3 +48*x^2)*exp((-1+2*x)/x)^2)*ln(4+x)^2+((108*x^2-432*x-1728)*exp((-1+2*x)/x )^4+(36*x^3+756*x^2+1296*x)*exp((-1+2*x)/x)^3+(-60*x^4-360*x^3-288*x^2)*ex p((-1+2*x)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((-1+2*x)/x))*ln(4+x)+(-108*x^2+ 324*x+1296)*exp((-1+2*x)/x)^4+(-756*x^2-1296*x)*exp((-1+2*x)/x)^3+(72*x^4+ 540*x^3+432*x^2)*exp((-1+2*x)/x)^2+(-32*x^5-156*x^4-48*x^3)*exp((-1+2*x)/x )+4*x^6+16*x^5)/(x^3+4*x^2),x,method=_RETURNVERBOSE)
exp((-1+2*x)/x)^4*ln(4+x)^4+(4*x*exp((-1+2*x)/x)^3-12*exp((-1+2*x)/x)^4)*l n(4+x)^3+(6*exp((-1+2*x)/x)^2*x^2-36*x*exp((-1+2*x)/x)^3+54*exp((-1+2*x)/x )^4)*ln(4+x)^2+(4*exp((-1+2*x)/x)*x^3-36*exp((-1+2*x)/x)^2*x^2+108*x*exp(( -1+2*x)/x)^3-108*exp((-1+2*x)/x)^4)*ln(4+x)+x^4-12*exp((-1+2*x)/x)*x^3+54* exp((-1+2*x)/x)^2*x^2-108*x*exp((-1+2*x)/x)^3+81*exp((-1+2*x)/x)^4
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 230, normalized size of antiderivative = 9.58 \[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx=e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )} \log \left (x + 4\right )^{4} + x^{4} - 12 \, x^{3} e^{\left (\frac {2 \, x - 1}{x}\right )} + 4 \, {\left (x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} - 3 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )} \log \left (x + 4\right )^{3} + 54 \, x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} + 6 \, {\left (x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} - 6 \, x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} + 9 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )} \log \left (x + 4\right )^{2} - 108 \, x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} + 4 \, {\left (x^{3} e^{\left (\frac {2 \, x - 1}{x}\right )} - 9 \, x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} + 27 \, x e^{\left (\frac {3 \, {\left (2 \, x - 1\right )}}{x}\right )} - 27 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )} \log \left (x + 4\right ) + 81 \, e^{\left (\frac {4 \, {\left (2 \, x - 1\right )}}{x}\right )} \]
integrate(((4*x+16)*exp((-1+2*x)/x)^4*log(4+x)^4+((4*x^2-48*x-192)*exp((-1 +2*x)/x)^4+(4*x^3+28*x^2+48*x)*exp((-1+2*x)/x)^3)*log(4+x)^3+((-36*x^2+216 *x+864)*exp((-1+2*x)/x)^4+(-24*x^3-252*x^2-432*x)*exp((-1+2*x)/x)^3+(12*x^ 4+60*x^3+48*x^2)*exp((-1+2*x)/x)^2)*log(4+x)^2+((108*x^2-432*x-1728)*exp(( -1+2*x)/x)^4+(36*x^3+756*x^2+1296*x)*exp((-1+2*x)/x)^3+(-60*x^4-360*x^3-28 8*x^2)*exp((-1+2*x)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((-1+2*x)/x))*log(4+x)+ (-108*x^2+324*x+1296)*exp((-1+2*x)/x)^4+(-756*x^2-1296*x)*exp((-1+2*x)/x)^ 3+(72*x^4+540*x^3+432*x^2)*exp((-1+2*x)/x)^2+(-32*x^5-156*x^4-48*x^3)*exp( (-1+2*x)/x)+4*x^6+16*x^5)/(x^3+4*x^2),x, algorithm=\
e^(4*(2*x - 1)/x)*log(x + 4)^4 + x^4 - 12*x^3*e^((2*x - 1)/x) + 4*(x*e^(3* (2*x - 1)/x) - 3*e^(4*(2*x - 1)/x))*log(x + 4)^3 + 54*x^2*e^(2*(2*x - 1)/x ) + 6*(x^2*e^(2*(2*x - 1)/x) - 6*x*e^(3*(2*x - 1)/x) + 9*e^(4*(2*x - 1)/x) )*log(x + 4)^2 - 108*x*e^(3*(2*x - 1)/x) + 4*(x^3*e^((2*x - 1)/x) - 9*x^2* e^(2*(2*x - 1)/x) + 27*x*e^(3*(2*x - 1)/x) - 27*e^(4*(2*x - 1)/x))*log(x + 4) + 81*e^(4*(2*x - 1)/x)
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (15) = 30\).
Time = 28.73 (sec) , antiderivative size = 143, normalized size of antiderivative = 5.96 \[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx=x^{4} + \left (4 x^{3} \log {\left (x + 4 \right )} - 12 x^{3}\right ) e^{\frac {2 x - 1}{x}} + \left (6 x^{2} \log {\left (x + 4 \right )}^{2} - 36 x^{2} \log {\left (x + 4 \right )} + 54 x^{2}\right ) e^{\frac {2 \cdot \left (2 x - 1\right )}{x}} + \left (4 x \log {\left (x + 4 \right )}^{3} - 36 x \log {\left (x + 4 \right )}^{2} + 108 x \log {\left (x + 4 \right )} - 108 x\right ) e^{\frac {3 \cdot \left (2 x - 1\right )}{x}} + \left (\log {\left (x + 4 \right )}^{4} - 12 \log {\left (x + 4 \right )}^{3} + 54 \log {\left (x + 4 \right )}^{2} - 108 \log {\left (x + 4 \right )} + 81\right ) e^{\frac {4 \cdot \left (2 x - 1\right )}{x}} \]
integrate(((4*x+16)*exp((-1+2*x)/x)**4*ln(4+x)**4+((4*x**2-48*x-192)*exp(( -1+2*x)/x)**4+(4*x**3+28*x**2+48*x)*exp((-1+2*x)/x)**3)*ln(4+x)**3+((-36*x **2+216*x+864)*exp((-1+2*x)/x)**4+(-24*x**3-252*x**2-432*x)*exp((-1+2*x)/x )**3+(12*x**4+60*x**3+48*x**2)*exp((-1+2*x)/x)**2)*ln(4+x)**2+((108*x**2-4 32*x-1728)*exp((-1+2*x)/x)**4+(36*x**3+756*x**2+1296*x)*exp((-1+2*x)/x)**3 +(-60*x**4-360*x**3-288*x**2)*exp((-1+2*x)/x)**2+(12*x**5+52*x**4+16*x**3) *exp((-1+2*x)/x))*ln(4+x)+(-108*x**2+324*x+1296)*exp((-1+2*x)/x)**4+(-756* x**2-1296*x)*exp((-1+2*x)/x)**3+(72*x**4+540*x**3+432*x**2)*exp((-1+2*x)/x )**2+(-32*x**5-156*x**4-48*x**3)*exp((-1+2*x)/x)+4*x**6+16*x**5)/(x**3+4*x **2),x)
x**4 + (4*x**3*log(x + 4) - 12*x**3)*exp((2*x - 1)/x) + (6*x**2*log(x + 4) **2 - 36*x**2*log(x + 4) + 54*x**2)*exp(2*(2*x - 1)/x) + (4*x*log(x + 4)** 3 - 36*x*log(x + 4)**2 + 108*x*log(x + 4) - 108*x)*exp(3*(2*x - 1)/x) + (l og(x + 4)**4 - 12*log(x + 4)**3 + 54*log(x + 4)**2 - 108*log(x + 4) + 81)* exp(4*(2*x - 1)/x)
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 6.75 \[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx=x^{4} + 4 \, {\left (x^{3} e^{2} \log \left (x + 4\right ) - 3 \, x^{3} e^{2}\right )} e^{\left (-\frac {1}{x}\right )} + 6 \, {\left (x^{2} e^{4} \log \left (x + 4\right )^{2} - 6 \, x^{2} e^{4} \log \left (x + 4\right ) + 9 \, x^{2} e^{4}\right )} e^{\left (-\frac {2}{x}\right )} + 4 \, {\left (x e^{6} \log \left (x + 4\right )^{3} - 9 \, x e^{6} \log \left (x + 4\right )^{2} + 27 \, x e^{6} \log \left (x + 4\right ) - 27 \, x e^{6}\right )} e^{\left (-\frac {3}{x}\right )} + {\left (e^{8} \log \left (x + 4\right )^{4} - 12 \, e^{8} \log \left (x + 4\right )^{3} + 54 \, e^{8} \log \left (x + 4\right )^{2} - 108 \, e^{8} \log \left (x + 4\right ) + 81 \, e^{8}\right )} e^{\left (-\frac {4}{x}\right )} \]
integrate(((4*x+16)*exp((-1+2*x)/x)^4*log(4+x)^4+((4*x^2-48*x-192)*exp((-1 +2*x)/x)^4+(4*x^3+28*x^2+48*x)*exp((-1+2*x)/x)^3)*log(4+x)^3+((-36*x^2+216 *x+864)*exp((-1+2*x)/x)^4+(-24*x^3-252*x^2-432*x)*exp((-1+2*x)/x)^3+(12*x^ 4+60*x^3+48*x^2)*exp((-1+2*x)/x)^2)*log(4+x)^2+((108*x^2-432*x-1728)*exp(( -1+2*x)/x)^4+(36*x^3+756*x^2+1296*x)*exp((-1+2*x)/x)^3+(-60*x^4-360*x^3-28 8*x^2)*exp((-1+2*x)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((-1+2*x)/x))*log(4+x)+ (-108*x^2+324*x+1296)*exp((-1+2*x)/x)^4+(-756*x^2-1296*x)*exp((-1+2*x)/x)^ 3+(72*x^4+540*x^3+432*x^2)*exp((-1+2*x)/x)^2+(-32*x^5-156*x^4-48*x^3)*exp( (-1+2*x)/x)+4*x^6+16*x^5)/(x^3+4*x^2),x, algorithm=\
x^4 + 4*(x^3*e^2*log(x + 4) - 3*x^3*e^2)*e^(-1/x) + 6*(x^2*e^4*log(x + 4)^ 2 - 6*x^2*e^4*log(x + 4) + 9*x^2*e^4)*e^(-2/x) + 4*(x*e^6*log(x + 4)^3 - 9 *x*e^6*log(x + 4)^2 + 27*x*e^6*log(x + 4) - 27*x*e^6)*e^(-3/x) + (e^8*log( x + 4)^4 - 12*e^8*log(x + 4)^3 + 54*e^8*log(x + 4)^2 - 108*e^8*log(x + 4) + 81*e^8)*e^(-4/x)
Timed out. \[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx=\text {Timed out} \]
integrate(((4*x+16)*exp((-1+2*x)/x)^4*log(4+x)^4+((4*x^2-48*x-192)*exp((-1 +2*x)/x)^4+(4*x^3+28*x^2+48*x)*exp((-1+2*x)/x)^3)*log(4+x)^3+((-36*x^2+216 *x+864)*exp((-1+2*x)/x)^4+(-24*x^3-252*x^2-432*x)*exp((-1+2*x)/x)^3+(12*x^ 4+60*x^3+48*x^2)*exp((-1+2*x)/x)^2)*log(4+x)^2+((108*x^2-432*x-1728)*exp(( -1+2*x)/x)^4+(36*x^3+756*x^2+1296*x)*exp((-1+2*x)/x)^3+(-60*x^4-360*x^3-28 8*x^2)*exp((-1+2*x)/x)^2+(12*x^5+52*x^4+16*x^3)*exp((-1+2*x)/x))*log(4+x)+ (-108*x^2+324*x+1296)*exp((-1+2*x)/x)^4+(-756*x^2-1296*x)*exp((-1+2*x)/x)^ 3+(72*x^4+540*x^3+432*x^2)*exp((-1+2*x)/x)^2+(-32*x^5-156*x^4-48*x^3)*exp( (-1+2*x)/x)+4*x^6+16*x^5)/(x^3+4*x^2),x, algorithm=\
Timed out. \[ \int \frac {16 x^5+4 x^6+e^{\frac {3 (-1+2 x)}{x}} \left (-1296 x-756 x^2\right )+e^{\frac {4 (-1+2 x)}{x}} \left (1296+324 x-108 x^2\right )+e^{\frac {2 (-1+2 x)}{x}} \left (432 x^2+540 x^3+72 x^4\right )+e^{\frac {-1+2 x}{x}} \left (-48 x^3-156 x^4-32 x^5\right )+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-1728-432 x+108 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (1296 x+756 x^2+36 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (-288 x^2-360 x^3-60 x^4\right )+e^{\frac {-1+2 x}{x}} \left (16 x^3+52 x^4+12 x^5\right )\right ) \log (4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (864+216 x-36 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (-432 x-252 x^2-24 x^3\right )+e^{\frac {2 (-1+2 x)}{x}} \left (48 x^2+60 x^3+12 x^4\right )\right ) \log ^2(4+x)+\left (e^{\frac {4 (-1+2 x)}{x}} \left (-192-48 x+4 x^2\right )+e^{\frac {3 (-1+2 x)}{x}} \left (48 x+28 x^2+4 x^3\right )\right ) \log ^3(4+x)+e^{\frac {4 (-1+2 x)}{x}} (16+4 x) \log ^4(4+x)}{4 x^2+x^3} \, dx=\int \frac {{\ln \left (x+4\right )}^2\,\left ({\mathrm {e}}^{\frac {2\,\left (2\,x-1\right )}{x}}\,\left (12\,x^4+60\,x^3+48\,x^2\right )+{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-36\,x^2+216\,x+864\right )-{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (24\,x^3+252\,x^2+432\,x\right )\right )-{\mathrm {e}}^{\frac {2\,x-1}{x}}\,\left (32\,x^5+156\,x^4+48\,x^3\right )+{\mathrm {e}}^{\frac {2\,\left (2\,x-1\right )}{x}}\,\left (72\,x^4+540\,x^3+432\,x^2\right )+{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-108\,x^2+324\,x+1296\right )-{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (756\,x^2+1296\,x\right )+16\,x^5+4\,x^6-{\ln \left (x+4\right )}^3\,\left ({\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-4\,x^2+48\,x+192\right )-{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (4\,x^3+28\,x^2+48\,x\right )\right )+\ln \left (x+4\right )\,\left ({\mathrm {e}}^{\frac {2\,x-1}{x}}\,\left (12\,x^5+52\,x^4+16\,x^3\right )-{\mathrm {e}}^{\frac {2\,\left (2\,x-1\right )}{x}}\,\left (60\,x^4+360\,x^3+288\,x^2\right )-{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (-108\,x^2+432\,x+1728\right )+{\mathrm {e}}^{\frac {3\,\left (2\,x-1\right )}{x}}\,\left (36\,x^3+756\,x^2+1296\,x\right )\right )+{\ln \left (x+4\right )}^4\,{\mathrm {e}}^{\frac {4\,\left (2\,x-1\right )}{x}}\,\left (4\,x+16\right )}{x^3+4\,x^2} \,d x \]
int((log(x + 4)^2*(exp((2*(2*x - 1))/x)*(48*x^2 + 60*x^3 + 12*x^4) + exp(( 4*(2*x - 1))/x)*(216*x - 36*x^2 + 864) - exp((3*(2*x - 1))/x)*(432*x + 252 *x^2 + 24*x^3)) - exp((2*x - 1)/x)*(48*x^3 + 156*x^4 + 32*x^5) + exp((2*(2 *x - 1))/x)*(432*x^2 + 540*x^3 + 72*x^4) + exp((4*(2*x - 1))/x)*(324*x - 1 08*x^2 + 1296) - exp((3*(2*x - 1))/x)*(1296*x + 756*x^2) + 16*x^5 + 4*x^6 - log(x + 4)^3*(exp((4*(2*x - 1))/x)*(48*x - 4*x^2 + 192) - exp((3*(2*x - 1))/x)*(48*x + 28*x^2 + 4*x^3)) + log(x + 4)*(exp((2*x - 1)/x)*(16*x^3 + 5 2*x^4 + 12*x^5) - exp((2*(2*x - 1))/x)*(288*x^2 + 360*x^3 + 60*x^4) - exp( (4*(2*x - 1))/x)*(432*x - 108*x^2 + 1728) + exp((3*(2*x - 1))/x)*(1296*x + 756*x^2 + 36*x^3)) + log(x + 4)^4*exp((4*(2*x - 1))/x)*(4*x + 16))/(4*x^2 + x^3),x)
int((log(x + 4)^2*(exp((2*(2*x - 1))/x)*(48*x^2 + 60*x^3 + 12*x^4) + exp(( 4*(2*x - 1))/x)*(216*x - 36*x^2 + 864) - exp((3*(2*x - 1))/x)*(432*x + 252 *x^2 + 24*x^3)) - exp((2*x - 1)/x)*(48*x^3 + 156*x^4 + 32*x^5) + exp((2*(2 *x - 1))/x)*(432*x^2 + 540*x^3 + 72*x^4) + exp((4*(2*x - 1))/x)*(324*x - 1 08*x^2 + 1296) - exp((3*(2*x - 1))/x)*(1296*x + 756*x^2) + 16*x^5 + 4*x^6 - log(x + 4)^3*(exp((4*(2*x - 1))/x)*(48*x - 4*x^2 + 192) - exp((3*(2*x - 1))/x)*(48*x + 28*x^2 + 4*x^3)) + log(x + 4)*(exp((2*x - 1)/x)*(16*x^3 + 5 2*x^4 + 12*x^5) - exp((2*(2*x - 1))/x)*(288*x^2 + 360*x^3 + 60*x^4) - exp( (4*(2*x - 1))/x)*(432*x - 108*x^2 + 1728) + exp((3*(2*x - 1))/x)*(1296*x + 756*x^2 + 36*x^3)) + log(x + 4)^4*exp((4*(2*x - 1))/x)*(4*x + 16))/(4*x^2 + x^3), x)