Integrand size = 279, antiderivative size = 25 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=\left (-\left (-2+e-e^{e^8}+x+x^2\right )^2+\log (1+x)\right )^2 \] Output:
(ln(1+x)-(exp(1)+x^2-exp(exp(4)^2)-2+x)^2)^2
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=\left (\left (-2+e-e^{e^8}+x+x^2\right )^2-\log (1+x)\right )^2 \] Input:
Integrate[(-40 - 40*x + 110*x^2 + 120*x^3 - 98*x^4 - 112*x^5 + 16*x^6 + 36 *x^7 + 8*x^8 + E^(3*E^8)*(-4 - 12*x - 8*x^2) + E^3*(4 + 12*x + 8*x^2) + E^ 2*(-26 - 60*x + 60*x^3 + 24*x^4) + E*(56 + 92*x - 88*x^2 - 180*x^3 + 12*x^ 4 + 84*x^5 + 24*x^6) + E^(2*E^8)*(-26 - 60*x + 60*x^3 + 24*x^4 + E*(12 + 3 6*x + 24*x^2)) + E^E^8*(-56 - 92*x + 88*x^2 + 180*x^3 - 12*x^4 - 84*x^5 - 24*x^6 + E^2*(-12 - 36*x - 24*x^2) + E*(52 + 120*x - 120*x^3 - 48*x^4)) + (10 + 20*x - 20*x^3 - 8*x^4 + E*(-4 - 12*x - 8*x^2) + E^E^8*(4 + 12*x + 8* x^2))*Log[1 + x])/(1 + x),x]
Output:
((-2 + E - E^E^8 + x + x^2)^2 - Log[1 + x])^2
Time = 1.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7239, 27, 25, 7237}
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.
| \(\displaystyle \int \frac {8 x^8+36 x^7+16 x^6-112 x^5-98 x^4+120 x^3+110 x^2+e^{3 e^8} \left (-8 x^2-12 x-4\right )+e^3 \left (8 x^2+12 x+4\right )+e^2 \left (24 x^4+60 x^3-60 x-26\right )+e^{2 e^8} \left (24 x^4+60 x^3+e \left (24 x^2+36 x+12\right )-60 x-26\right )+\left (-8 x^4-20 x^3+e \left (-8 x^2-12 x-4\right )+e^{e^8} \left (8 x^2+12 x+4\right )+20 x+10\right ) \log (x+1)+e \left (24 x^6+84 x^5+12 x^4-180 x^3-88 x^2+92 x+56\right )+e^{e^8} \left (-24 x^6-84 x^5-12 x^4+180 x^3+88 x^2+e^2 \left (-24 x^2-36 x-12\right )+e \left (-48 x^4-120 x^3+120 x+52\right )-92 x-56\right )-40 x-40}{x+1} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-4 x^4-10 x^3-4 \left (e-e^{e^8}\right ) x^2+2 \left (5-3 e+3 e^{e^8}\right ) x+2 e^{e^8}-2 e+5\right ) \left (\log (x+1)-\left (x^2+x-e^{e^8}+e-2\right )^2\right )}{x+1}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\left (-4 x^4-10 x^3-4 \left (e-e^{e^8}\right ) x^2+2 \left (5-3 e+3 e^{e^8}\right ) x+2 e^{e^8}-2 e+5\right ) \left (\left (-x^2-x+e^{e^8}-e+2\right )^2-\log (x+1)\right )}{x+1}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\left (-4 x^4-10 x^3-4 \left (e-e^{e^8}\right ) x^2+2 \left (5-3 e+3 e^{e^8}\right ) x+2 e^{e^8}-2 e+5\right ) \left (\left (-x^2-x+e^{e^8}-e+2\right )^2-\log (x+1)\right )}{x+1}dx\) |
| \(\Big \downarrow \) 7237 |
| \(\displaystyle \left (\left (-x^2-x+e^{e^8}-e+2\right )^2-\log (x+1)\right )^2\) |
Input:
Int[(-40 - 40*x + 110*x^2 + 120*x^3 - 98*x^4 - 112*x^5 + 16*x^6 + 36*x^7 + 8*x^8 + E^(3*E^8)*(-4 - 12*x - 8*x^2) + E^3*(4 + 12*x + 8*x^2) + E^2*(-26 - 60*x + 60*x^3 + 24*x^4) + E*(56 + 92*x - 88*x^2 - 180*x^3 + 12*x^4 + 84 *x^5 + 24*x^6) + E^(2*E^8)*(-26 - 60*x + 60*x^3 + 24*x^4 + E*(12 + 36*x + 24*x^2)) + E^E^8*(-56 - 92*x + 88*x^2 + 180*x^3 - 12*x^4 - 84*x^5 - 24*x^6 + E^2*(-12 - 36*x - 24*x^2) + E*(52 + 120*x - 120*x^3 - 48*x^4)) + (10 + 20*x - 20*x^3 - 8*x^4 + E*(-4 - 12*x - 8*x^2) + E^E^8*(4 + 12*x + 8*x^2))* Log[1 + x])/(1 + x),x]
Output:
((2 - E + E^E^8 - x - x^2)^2 - Log[1 + x])^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 40.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.08
| method | result | size |
| risch | \(\ln \left (1+x \right )^{2}-2 \left (-x^{2}+{\mathrm e}^{{\mathrm e}^{8}}-{\mathrm e}-x +2\right )^{2} \ln \left (1+x \right )+\left (-x^{2}+{\mathrm e}^{{\mathrm e}^{8}}-{\mathrm e}-x +2\right )^{4}\) | \(52\) |
| norman | \(x^{8}+\ln \left (1+x \right )^{2}+\left (-20+12 \,{\mathrm e}-12 \,{\mathrm e}^{{\mathrm e}^{8}}\right ) x^{5}+\left (-2+4 \,{\mathrm e}-4 \,{\mathrm e}^{{\mathrm e}^{8}}\right ) x^{6}+\left (1-12 \,{\mathrm e}+12 \,{\mathrm e}^{{\mathrm e}^{8}}+6 \,{\mathrm e}^{2}+6 \,{\mathrm e}^{2 \,{\mathrm e}^{8}}-12 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}}\right ) x^{4}+\left (40-44 \,{\mathrm e}+44 \,{\mathrm e}^{{\mathrm e}^{8}}+12 \,{\mathrm e}^{2}+12 \,{\mathrm e}^{2 \,{\mathrm e}^{8}}-24 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}}\right ) x^{3}+\left (-2 \,{\mathrm e}^{2}+4 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}}-2 \,{\mathrm e}^{2 \,{\mathrm e}^{8}}+8 \,{\mathrm e}-8 \,{\mathrm e}^{{\mathrm e}^{8}}-8\right ) \ln \left (1+x \right )+\left (-32+48 \,{\mathrm e}-48 \,{\mathrm e}^{{\mathrm e}^{8}}-24 \,{\mathrm e}^{2}-24 \,{\mathrm e}^{2 \,{\mathrm e}^{8}}+48 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}}+4 \,{\mathrm e}^{3}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}}-12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}}+12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}}\right ) x +\left (-8+24 \,{\mathrm e}-24 \,{\mathrm e}^{{\mathrm e}^{8}}-18 \,{\mathrm e}^{2}-18 \,{\mathrm e}^{2 \,{\mathrm e}^{8}}+36 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}}+4 \,{\mathrm e}^{3}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}}-12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}}+12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}}\right ) x^{2}+\left (6-4 \,{\mathrm e}+4 \,{\mathrm e}^{{\mathrm e}^{8}}\right ) x^{2} \ln \left (1+x \right )+\left (8-4 \,{\mathrm e}+4 \,{\mathrm e}^{{\mathrm e}^{8}}\right ) x \ln \left (1+x \right )+4 x^{7}-4 \ln \left (1+x \right ) x^{3}-2 \ln \left (1+x \right ) x^{4}\) | \(386\) |
| parts | \(-32 x -2 \,{\mathrm e}+12 x^{5} {\mathrm e}+2 \left (-4+4 \,{\mathrm e}-4 \,{\mathrm e}^{{\mathrm e}^{8}}-{\mathrm e}^{2}-{\mathrm e}^{2 \,{\mathrm e}^{8}}+2 \,{\mathrm e}^{1+{\mathrm e}^{8}}\right ) \ln \left (1+x \right )-18 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{8}}+36 x^{2} {\mathrm e}^{1+{\mathrm e}^{8}}-24 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x +6 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x^{4}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} x^{2}+12 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x^{3}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} x -24 x^{3} {\mathrm e}^{1+{\mathrm e}^{8}}-12 x \,{\mathrm e}^{2+{\mathrm e}^{8}}+12 x \,{\mathrm e}^{1+2 \,{\mathrm e}^{8}}+48 x \,{\mathrm e}^{1+{\mathrm e}^{8}}-12 x^{4} {\mathrm e}^{1+{\mathrm e}^{8}}-12 x^{2} {\mathrm e}^{2+{\mathrm e}^{8}}+12 x^{2} {\mathrm e}^{1+2 \,{\mathrm e}^{8}}+12 x^{3} {\mathrm e}^{2}+24 x^{2} {\mathrm e}-44 x^{3} {\mathrm e}+4 x^{2} {\mathrm e}^{3}-12 x^{4} {\mathrm e}-18 x^{2} {\mathrm e}^{2}+6 x^{4} {\mathrm e}^{2}+48 x \,{\mathrm e}+4 x \,{\mathrm e}^{3}+4 x^{6} {\mathrm e}+6 \ln \left (1+x \right ) \left (1+x \right )^{2}-2 \ln \left (1+x \right ) \left (1+x \right )^{4}+4 \ln \left (1+x \right ) \left (1+x \right )^{3}-8 \left (1+x \right ) \ln \left (1+x \right )+\ln \left (1+x \right )^{2}-24 \,{\mathrm e}^{2} x +4 x^{7}+x^{8}+x^{4}+40 x^{3}-8 x^{2}-2 x^{6}-20 x^{5}+4 \ln \left (1+x \right ) {\mathrm e}^{{\mathrm e}^{8}} x^{2}+4 \ln \left (1+x \right ) {\mathrm e}^{{\mathrm e}^{8}} x -4 \,{\mathrm e}^{{\mathrm e}^{8}} x^{6}-12 \,{\mathrm e}^{{\mathrm e}^{8}} x^{5}+12 \,{\mathrm e}^{{\mathrm e}^{8}} x^{4}+44 \,{\mathrm e}^{{\mathrm e}^{8}} x^{3}-24 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}-48 \,{\mathrm e}^{{\mathrm e}^{8}} x +2 \,{\mathrm e}^{{\mathrm e}^{8}}-4 \,{\mathrm e} \ln \left (1+x \right ) x^{2}-4 \,{\mathrm e} \ln \left (1+x \right ) x +\frac {25}{6}\) | \(445\) |
| parallelrisch | \(72-32 x -120 \,{\mathrm e}+12 x^{5} {\mathrm e}-18 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{8}}-24 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x +6 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x^{4}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} x^{2}+12 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x^{3}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} x +12 x^{3} {\mathrm e}^{2}+24 x^{2} {\mathrm e}-44 x^{3} {\mathrm e}+4 x^{2} {\mathrm e}^{3}-12 x^{4} {\mathrm e}-18 x^{2} {\mathrm e}^{2}+6 x^{4} {\mathrm e}^{2}+48 x \,{\mathrm e}+4 x \,{\mathrm e}^{3}+4 x^{6} {\mathrm e}-2 \ln \left (1+x \right ) x^{4}-4 \ln \left (1+x \right ) x^{3}+6 \ln \left (1+x \right ) x^{2}+8 \,{\mathrm e} \ln \left (1+x \right )+8 \ln \left (1+x \right ) x +\ln \left (1+x \right )^{2}-8 \ln \left (1+x \right )-24 \,{\mathrm e}^{2} x +4 x^{7}+x^{8}+66 \,{\mathrm e}^{2}+x^{4}+40 x^{3}-8 x^{2}-2 x^{6}-20 x^{5}-12 \,{\mathrm e}^{3}-12 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} x^{4}-24 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} x^{3}+36 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}+4 \ln \left (1+x \right ) {\mathrm e}^{{\mathrm e}^{8}} x^{2}+48 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} x +4 \ln \left (1+x \right ) {\mathrm e}^{{\mathrm e}^{8}} x -4 \,{\mathrm e}^{{\mathrm e}^{8}} x^{6}-12 \,{\mathrm e}^{{\mathrm e}^{8}} x^{5}+12 \,{\mathrm e}^{{\mathrm e}^{8}} x^{4}+44 \,{\mathrm e}^{{\mathrm e}^{8}} x^{3}-24 \,{\mathrm e}^{{\mathrm e}^{8}} x^{2}-132 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}}-8 \ln \left (1+x \right ) {\mathrm e}^{{\mathrm e}^{8}}-48 \,{\mathrm e}^{{\mathrm e}^{8}} x +120 \,{\mathrm e}^{{\mathrm e}^{8}}-4 \,{\mathrm e} \ln \left (1+x \right ) x^{2}-4 \,{\mathrm e} \ln \left (1+x \right ) x +12 \,{\mathrm e}^{3 \,{\mathrm e}^{8}}+66 \,{\mathrm e}^{2 \,{\mathrm e}^{8}}+36 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}}-36 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}}+4 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \ln \left (1+x \right )-2 \,{\mathrm e}^{2} \ln \left (1+x \right )-2 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \ln \left (1+x \right )-12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}} x^{2}+12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x^{2}-12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}} x +12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}} x\) | \(570\) |
| derivativedivides | \(32+32 x +4 \,{\mathrm e} \left (1+x \right )^{6}+6 \ln \left (1+x \right ) \left (1+x \right )^{2}-8 \,{\mathrm e} \left (\frac {\ln \left (1+x \right ) \left (1+x \right )^{2}}{2}-\frac {\left (1+x \right )^{2}}{4}\right )+4 \,{\mathrm e} \left (\left (1+x \right ) \ln \left (1+x \right )-1-x \right )-12 \,{\mathrm e} \left (1+x \right )^{5}-12 \,{\mathrm e} \left (1+x \right )^{4}-2 \ln \left (1+x \right ) \left (1+x \right )^{4}+44 \,{\mathrm e} \left (1+x \right )^{3}+4 \ln \left (1+x \right ) \left (1+x \right )^{3}+22 \,{\mathrm e} \left (1+x \right )^{2}+8 \,{\mathrm e} \ln \left (1+x \right )-44 \left (1+x \right ) {\mathrm e}-8 \left (1+x \right ) \ln \left (1+x \right )-8 \left (1+x \right )^{2}+\ln \left (1+x \right )^{2}+24 \left (1+x \right ) {\mathrm e}^{2}-4 \left (1+x \right ) {\mathrm e}^{3}+\left (1+x \right )^{4}-8 \ln \left (1+x \right )-8 \ln \left (1+x \right ) {\mathrm e}^{{\mathrm e}^{8}}+\left (1+x \right )^{8}-4 \left (1+x \right )^{7}-2 \left (1+x \right )^{6}+20 \left (1+x \right )^{5}-40 \left (1+x \right )^{3}+36 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{2}+12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{2}+24 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{3}+12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )-12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )-48 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )-12 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{4}-12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{2}+4 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \ln \left (1+x \right )-22 \left (1+x \right )^{2} {\mathrm e}^{{\mathrm e}^{8}}+24 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )+44 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )-4 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{6}+8 \,{\mathrm e}^{{\mathrm e}^{8}} \left (\frac {\ln \left (1+x \right ) \left (1+x \right )^{2}}{2}-\frac {\left (1+x \right )^{2}}{4}\right )-4 \,{\mathrm e}^{{\mathrm e}^{8}} \left (\left (1+x \right ) \ln \left (1+x \right )-1-x \right )+6 \,{\mathrm e}^{2} \left (1+x \right )^{4}+6 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{4}+12 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{5}+4 \,{\mathrm e}^{3} \left (1+x \right )^{2}-12 \,{\mathrm e}^{2} \left (1+x \right )^{3}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} \left (1+x \right )^{2}-12 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{3}+12 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{4}-18 \,{\mathrm e}^{2} \left (1+x \right )^{2}+4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} \left (1+x \right )-18 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{2}-44 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{3}-2 \,{\mathrm e}^{2} \ln \left (1+x \right )-2 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \ln \left (1+x \right )\) | \(624\) |
| default | \(32+32 x +4 \,{\mathrm e} \left (1+x \right )^{6}+6 \ln \left (1+x \right ) \left (1+x \right )^{2}-8 \,{\mathrm e} \left (\frac {\ln \left (1+x \right ) \left (1+x \right )^{2}}{2}-\frac {\left (1+x \right )^{2}}{4}\right )+4 \,{\mathrm e} \left (\left (1+x \right ) \ln \left (1+x \right )-1-x \right )-12 \,{\mathrm e} \left (1+x \right )^{5}-12 \,{\mathrm e} \left (1+x \right )^{4}-2 \ln \left (1+x \right ) \left (1+x \right )^{4}+44 \,{\mathrm e} \left (1+x \right )^{3}+4 \ln \left (1+x \right ) \left (1+x \right )^{3}+22 \,{\mathrm e} \left (1+x \right )^{2}+8 \,{\mathrm e} \ln \left (1+x \right )-44 \left (1+x \right ) {\mathrm e}-8 \left (1+x \right ) \ln \left (1+x \right )-8 \left (1+x \right )^{2}+\ln \left (1+x \right )^{2}+24 \left (1+x \right ) {\mathrm e}^{2}-4 \left (1+x \right ) {\mathrm e}^{3}+\left (1+x \right )^{4}-8 \ln \left (1+x \right )-8 \ln \left (1+x \right ) {\mathrm e}^{{\mathrm e}^{8}}+\left (1+x \right )^{8}-4 \left (1+x \right )^{7}-2 \left (1+x \right )^{6}+20 \left (1+x \right )^{5}-40 \left (1+x \right )^{3}+36 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{2}+12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{2}+24 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{3}+12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )-12 \,{\mathrm e} \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )-48 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )-12 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{4}-12 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{2}+4 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{8}} \ln \left (1+x \right )-22 \left (1+x \right )^{2} {\mathrm e}^{{\mathrm e}^{8}}+24 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )+44 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )-4 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{6}+8 \,{\mathrm e}^{{\mathrm e}^{8}} \left (\frac {\ln \left (1+x \right ) \left (1+x \right )^{2}}{2}-\frac {\left (1+x \right )^{2}}{4}\right )-4 \,{\mathrm e}^{{\mathrm e}^{8}} \left (\left (1+x \right ) \ln \left (1+x \right )-1-x \right )+6 \,{\mathrm e}^{2} \left (1+x \right )^{4}+6 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{4}+12 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{5}+4 \,{\mathrm e}^{3} \left (1+x \right )^{2}-12 \,{\mathrm e}^{2} \left (1+x \right )^{3}-4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} \left (1+x \right )^{2}-12 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{3}+12 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{4}-18 \,{\mathrm e}^{2} \left (1+x \right )^{2}+4 \,{\mathrm e}^{3 \,{\mathrm e}^{8}} \left (1+x \right )-18 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \left (1+x \right )^{2}-44 \,{\mathrm e}^{{\mathrm e}^{8}} \left (1+x \right )^{3}-2 \,{\mathrm e}^{2} \ln \left (1+x \right )-2 \,{\mathrm e}^{2 \,{\mathrm e}^{8}} \ln \left (1+x \right )\) | \(624\) |
Input:
int((((8*x^2+12*x+4)*exp(exp(4)^2)+(-8*x^2-12*x-4)*exp(1)-8*x^4-20*x^3+20* x+10)*ln(1+x)+(-8*x^2-12*x-4)*exp(exp(4)^2)^3+((24*x^2+36*x+12)*exp(1)+24* x^4+60*x^3-60*x-26)*exp(exp(4)^2)^2+((-24*x^2-36*x-12)*exp(1)^2+(-48*x^4-1 20*x^3+120*x+52)*exp(1)-24*x^6-84*x^5-12*x^4+180*x^3+88*x^2-92*x-56)*exp(e xp(4)^2)+(8*x^2+12*x+4)*exp(1)^3+(24*x^4+60*x^3-60*x-26)*exp(1)^2+(24*x^6+ 84*x^5+12*x^4-180*x^3-88*x^2+92*x+56)*exp(1)+8*x^8+36*x^7+16*x^6-112*x^5-9 8*x^4+120*x^3+110*x^2-40*x-40)/(1+x),x,method=_RETURNVERBOSE)
Output:
ln(1+x)^2-2*(-x^2+exp(exp(8))-exp(1)-x+2)^2*ln(1+x)+(-x^2+exp(exp(8))-exp( 1)-x+2)^4
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 263, normalized size of antiderivative = 10.52 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=x^{8} + 4 \, x^{7} - 2 \, x^{6} - 20 \, x^{5} + x^{4} + 40 \, x^{3} - 8 \, x^{2} + 4 \, {\left (x^{2} + x\right )} e^{3} + 6 \, {\left (x^{4} + 2 \, x^{3} - 3 \, x^{2} - 4 \, x\right )} e^{2} + 4 \, {\left (x^{6} + 3 \, x^{5} - 3 \, x^{4} - 11 \, x^{3} + 6 \, x^{2} + 12 \, x\right )} e - 4 \, {\left (x^{2} + x\right )} e^{\left (3 \, e^{8}\right )} + 6 \, {\left (x^{4} + 2 \, x^{3} - 3 \, x^{2} + 2 \, {\left (x^{2} + x\right )} e - 4 \, x\right )} e^{\left (2 \, e^{8}\right )} - 4 \, {\left (x^{6} + 3 \, x^{5} - 3 \, x^{4} - 11 \, x^{3} + 6 \, x^{2} + 3 \, {\left (x^{2} + x\right )} e^{2} + 3 \, {\left (x^{4} + 2 \, x^{3} - 3 \, x^{2} - 4 \, x\right )} e + 12 \, x\right )} e^{\left (e^{8}\right )} - 2 \, {\left (x^{4} + 2 \, x^{3} - 3 \, x^{2} + 2 \, {\left (x^{2} + x - 2\right )} e - 2 \, {\left (x^{2} + x + e - 2\right )} e^{\left (e^{8}\right )} - 4 \, x + e^{2} + e^{\left (2 \, e^{8}\right )} + 4\right )} \log \left (x + 1\right ) + \log \left (x + 1\right )^{2} - 32 \, x \] Input:
integrate((((8*x^2+12*x+4)*exp(exp(4)^2)+(-8*x^2-12*x-4)*exp(1)-8*x^4-20*x ^3+20*x+10)*log(1+x)+(-8*x^2-12*x-4)*exp(exp(4)^2)^3+((24*x^2+36*x+12)*exp (1)+24*x^4+60*x^3-60*x-26)*exp(exp(4)^2)^2+((-24*x^2-36*x-12)*exp(1)^2+(-4 8*x^4-120*x^3+120*x+52)*exp(1)-24*x^6-84*x^5-12*x^4+180*x^3+88*x^2-92*x-56 )*exp(exp(4)^2)+(8*x^2+12*x+4)*exp(1)^3+(24*x^4+60*x^3-60*x-26)*exp(1)^2+( 24*x^6+84*x^5+12*x^4-180*x^3-88*x^2+92*x+56)*exp(1)+8*x^8+36*x^7+16*x^6-11 2*x^5-98*x^4+120*x^3+110*x^2-40*x-40)/(1+x),x, algorithm="fricas")
Output:
x^8 + 4*x^7 - 2*x^6 - 20*x^5 + x^4 + 40*x^3 - 8*x^2 + 4*(x^2 + x)*e^3 + 6* (x^4 + 2*x^3 - 3*x^2 - 4*x)*e^2 + 4*(x^6 + 3*x^5 - 3*x^4 - 11*x^3 + 6*x^2 + 12*x)*e - 4*(x^2 + x)*e^(3*e^8) + 6*(x^4 + 2*x^3 - 3*x^2 + 2*(x^2 + x)*e - 4*x)*e^(2*e^8) - 4*(x^6 + 3*x^5 - 3*x^4 - 11*x^3 + 6*x^2 + 3*(x^2 + x)* e^2 + 3*(x^4 + 2*x^3 - 3*x^2 - 4*x)*e + 12*x)*e^(e^8) - 2*(x^4 + 2*x^3 - 3 *x^2 + 2*(x^2 + x - 2)*e - 2*(x^2 + x + e - 2)*e^(e^8) - 4*x + e^2 + e^(2* e^8) + 4)*log(x + 1) + log(x + 1)^2 - 32*x
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (22) = 44\).
Time = 0.54 (sec) , antiderivative size = 357, normalized size of antiderivative = 14.28 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=x^{8} + 4 x^{7} + x^{6} \left (-2 + 4 e - 4 e^{e^{8}}\right ) + x^{5} \left (-20 + 12 e - 12 e^{e^{8}}\right ) + x^{4} \left (- 12 e + 1 + 6 e^{2} + 6 e^{2 e^{8}} + 12 e^{e^{8}} - 12 e e^{e^{8}}\right ) + x^{3} \left (- 44 e + 40 + 12 e^{2} + 12 e^{2 e^{8}} + 44 e^{e^{8}} - 24 e e^{e^{8}}\right ) + x^{2} \left (- 18 e^{2} - 8 + 24 e + 4 e^{3} - 24 e^{e^{8}} - 18 e^{2 e^{8}} - 4 e^{3 e^{8}} - 12 e^{2} e^{e^{8}} + 12 e e^{2 e^{8}} + 36 e e^{e^{8}}\right ) + x \left (- 24 e^{2} - 32 + 4 e^{3} + 48 e - 48 e^{e^{8}} - 24 e^{2 e^{8}} - 4 e^{3 e^{8}} - 12 e^{2} e^{e^{8}} + 12 e e^{2 e^{8}} + 48 e e^{e^{8}}\right ) + \left (- 2 x^{4} - 4 x^{3} - 4 e x^{2} + 6 x^{2} + 4 x^{2} e^{e^{8}} - 4 e x + 8 x + 4 x e^{e^{8}}\right ) \log {\left (x + 1 \right )} + \log {\left (x + 1 \right )}^{2} - 2 \left (-2 + e - e^{e^{8}}\right )^{2} \log {\left (x + 1 \right )} \] Input:
integrate((((8*x**2+12*x+4)*exp(exp(4)**2)+(-8*x**2-12*x-4)*exp(1)-8*x**4- 20*x**3+20*x+10)*ln(1+x)+(-8*x**2-12*x-4)*exp(exp(4)**2)**3+((24*x**2+36*x +12)*exp(1)+24*x**4+60*x**3-60*x-26)*exp(exp(4)**2)**2+((-24*x**2-36*x-12) *exp(1)**2+(-48*x**4-120*x**3+120*x+52)*exp(1)-24*x**6-84*x**5-12*x**4+180 *x**3+88*x**2-92*x-56)*exp(exp(4)**2)+(8*x**2+12*x+4)*exp(1)**3+(24*x**4+6 0*x**3-60*x-26)*exp(1)**2+(24*x**6+84*x**5+12*x**4-180*x**3-88*x**2+92*x+5 6)*exp(1)+8*x**8+36*x**7+16*x**6-112*x**5-98*x**4+120*x**3+110*x**2-40*x-4 0)/(1+x),x)
Output:
x**8 + 4*x**7 + x**6*(-2 + 4*E - 4*exp(exp(8))) + x**5*(-20 + 12*E - 12*ex p(exp(8))) + x**4*(-12*E + 1 + 6*exp(2) + 6*exp(2*exp(8)) + 12*exp(exp(8)) - 12*E*exp(exp(8))) + x**3*(-44*E + 40 + 12*exp(2) + 12*exp(2*exp(8)) + 4 4*exp(exp(8)) - 24*E*exp(exp(8))) + x**2*(-18*exp(2) - 8 + 24*E + 4*exp(3) - 24*exp(exp(8)) - 18*exp(2*exp(8)) - 4*exp(3*exp(8)) - 12*exp(2)*exp(exp (8)) + 12*E*exp(2*exp(8)) + 36*E*exp(exp(8))) + x*(-24*exp(2) - 32 + 4*exp (3) + 48*E - 48*exp(exp(8)) - 24*exp(2*exp(8)) - 4*exp(3*exp(8)) - 12*exp( 2)*exp(exp(8)) + 12*E*exp(2*exp(8)) + 48*E*exp(exp(8))) + (-2*x**4 - 4*x** 3 - 4*E*x**2 + 6*x**2 + 4*x**2*exp(exp(8)) - 4*E*x + 8*x + 4*x*exp(exp(8)) )*log(x + 1) + log(x + 1)**2 - 2*(-2 + E - exp(exp(8)))**2*log(x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 1065 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 1065, normalized size of antiderivative = 42.60 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=\text {Too large to display} \] Input:
integrate((((8*x^2+12*x+4)*exp(exp(4)^2)+(-8*x^2-12*x-4)*exp(1)-8*x^4-20*x ^3+20*x+10)*log(1+x)+(-8*x^2-12*x-4)*exp(exp(4)^2)^3+((24*x^2+36*x+12)*exp (1)+24*x^4+60*x^3-60*x-26)*exp(exp(4)^2)^2+((-24*x^2-36*x-12)*exp(1)^2+(-4 8*x^4-120*x^3+120*x+52)*exp(1)-24*x^6-84*x^5-12*x^4+180*x^3+88*x^2-92*x-56 )*exp(exp(4)^2)+(8*x^2+12*x+4)*exp(1)^3+(24*x^4+60*x^3-60*x-26)*exp(1)^2+( 24*x^6+84*x^5+12*x^4-180*x^3-88*x^2+92*x+56)*exp(1)+8*x^8+36*x^7+16*x^6-11 2*x^5-98*x^4+120*x^3+110*x^2-40*x-40)/(1+x),x, algorithm="maxima")
Output:
x^8 + 4*x^7 - 2*x^6 - 20*x^5 + x^4 + 40*x^3 - 4*(x^2 - 2*x + 2*log(x + 1)) *e*log(x + 1) - 12*(x - log(x + 1))*e*log(x + 1) + 4*(x^2 - 2*x + 2*log(x + 1))*e^(e^8)*log(x + 1) + 12*(x - log(x + 1))*e^(e^8)*log(x + 1) - 2*e*lo g(x + 1)^2 + 2*e^(e^8)*log(x + 1)^2 - 8*x^2 + 4*(x^2 - 2*x + 2*log(x + 1)) *e^3 + 12*(x - log(x + 1))*e^3 + 2*(3*x^4 - 4*x^3 + 6*x^2 - 12*x + 12*log( x + 1))*e^2 + 10*(2*x^3 - 3*x^2 + 6*x - 6*log(x + 1))*e^2 - 60*(x - log(x + 1))*e^2 + 2/5*(10*x^6 - 12*x^5 + 15*x^4 - 20*x^3 + 30*x^2 - 60*x + 60*lo g(x + 1))*e + 7/5*(12*x^5 - 15*x^4 + 20*x^3 - 30*x^2 + 60*x - 60*log(x + 1 ))*e + (3*x^4 - 4*x^3 + 6*x^2 - 12*x + 12*log(x + 1))*e - 30*(2*x^3 - 3*x^ 2 + 6*x - 6*log(x + 1))*e + 2*(x^2 + 2*log(x + 1)^2 - 6*x + 6*log(x + 1))* e - 44*(x^2 - 2*x + 2*log(x + 1))*e - 6*(log(x + 1)^2 - 2*x + 2*log(x + 1) )*e + 92*(x - log(x + 1))*e - 4*(x^2 - 2*x + 2*log(x + 1))*e^(3*e^8) - 12* (x - log(x + 1))*e^(3*e^8) + 2*(3*x^4 - 4*x^3 + 6*x^2 - 12*x + 12*log(x + 1))*e^(2*e^8) + 10*(2*x^3 - 3*x^2 + 6*x - 6*log(x + 1))*e^(2*e^8) - 60*(x - log(x + 1))*e^(2*e^8) + 12*(x^2 - 2*x + 2*log(x + 1))*e^(2*e^8 + 1) + 36 *(x - log(x + 1))*e^(2*e^8 + 1) - 12*(x^2 - 2*x + 2*log(x + 1))*e^(e^8 + 2 ) - 36*(x - log(x + 1))*e^(e^8 + 2) - 4*(3*x^4 - 4*x^3 + 6*x^2 - 12*x + 12 *log(x + 1))*e^(e^8 + 1) - 20*(2*x^3 - 3*x^2 + 6*x - 6*log(x + 1))*e^(e^8 + 1) + 120*(x - log(x + 1))*e^(e^8 + 1) - 2/5*(10*x^6 - 12*x^5 + 15*x^4 - 20*x^3 + 30*x^2 - 60*x + 60*log(x + 1))*e^(e^8) - 7/5*(12*x^5 - 15*x^4 ...
Leaf count of result is larger than twice the leaf count of optimal. 428 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 428, normalized size of antiderivative = 17.12 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=x^{8} + 4 \, x^{7} + 4 \, x^{6} e - 4 \, x^{6} e^{\left (e^{8}\right )} - 2 \, x^{6} + 12 \, x^{5} e - 12 \, x^{5} e^{\left (e^{8}\right )} - 20 \, x^{5} + 6 \, x^{4} e^{2} - 12 \, x^{4} e + 6 \, x^{4} e^{\left (2 \, e^{8}\right )} - 12 \, x^{4} e^{\left (e^{8} + 1\right )} + 12 \, x^{4} e^{\left (e^{8}\right )} - 2 \, x^{4} \log \left (x + 1\right ) + x^{4} + 12 \, x^{3} e^{2} - 44 \, x^{3} e + 12 \, x^{3} e^{\left (2 \, e^{8}\right )} - 24 \, x^{3} e^{\left (e^{8} + 1\right )} + 44 \, x^{3} e^{\left (e^{8}\right )} - 4 \, x^{3} \log \left (x + 1\right ) - 4 \, x^{2} e \log \left (x + 1\right ) + 4 \, x^{2} e^{\left (e^{8}\right )} \log \left (x + 1\right ) + 40 \, x^{3} + 4 \, x^{2} e^{3} - 18 \, x^{2} e^{2} + 24 \, x^{2} e - 4 \, x^{2} e^{\left (3 \, e^{8}\right )} - 18 \, x^{2} e^{\left (2 \, e^{8}\right )} + 12 \, x^{2} e^{\left (2 \, e^{8} + 1\right )} - 12 \, x^{2} e^{\left (e^{8} + 2\right )} + 36 \, x^{2} e^{\left (e^{8} + 1\right )} - 24 \, x^{2} e^{\left (e^{8}\right )} + 6 \, x^{2} \log \left (x + 1\right ) - 4 \, x e \log \left (x + 1\right ) + 4 \, x e^{\left (e^{8}\right )} \log \left (x + 1\right ) - 8 \, x^{2} + 4 \, x e^{3} - 24 \, x e^{2} + 48 \, x e - 4 \, x e^{\left (3 \, e^{8}\right )} - 24 \, x e^{\left (2 \, e^{8}\right )} + 12 \, x e^{\left (2 \, e^{8} + 1\right )} - 12 \, x e^{\left (e^{8} + 2\right )} + 48 \, x e^{\left (e^{8} + 1\right )} - 48 \, x e^{\left (e^{8}\right )} + 8 \, x \log \left (x + 1\right ) - 2 \, e^{2} \log \left (x + 1\right ) + 8 \, e \log \left (x + 1\right ) - 2 \, e^{\left (2 \, e^{8}\right )} \log \left (x + 1\right ) + 4 \, e^{\left (e^{8} + 1\right )} \log \left (x + 1\right ) - 8 \, e^{\left (e^{8}\right )} \log \left (x + 1\right ) + \log \left (x + 1\right )^{2} - 32 \, x - 8 \, \log \left (x + 1\right ) \] Input:
integrate((((8*x^2+12*x+4)*exp(exp(4)^2)+(-8*x^2-12*x-4)*exp(1)-8*x^4-20*x ^3+20*x+10)*log(1+x)+(-8*x^2-12*x-4)*exp(exp(4)^2)^3+((24*x^2+36*x+12)*exp (1)+24*x^4+60*x^3-60*x-26)*exp(exp(4)^2)^2+((-24*x^2-36*x-12)*exp(1)^2+(-4 8*x^4-120*x^3+120*x+52)*exp(1)-24*x^6-84*x^5-12*x^4+180*x^3+88*x^2-92*x-56 )*exp(exp(4)^2)+(8*x^2+12*x+4)*exp(1)^3+(24*x^4+60*x^3-60*x-26)*exp(1)^2+( 24*x^6+84*x^5+12*x^4-180*x^3-88*x^2+92*x+56)*exp(1)+8*x^8+36*x^7+16*x^6-11 2*x^5-98*x^4+120*x^3+110*x^2-40*x-40)/(1+x),x, algorithm="giac")
Output:
x^8 + 4*x^7 + 4*x^6*e - 4*x^6*e^(e^8) - 2*x^6 + 12*x^5*e - 12*x^5*e^(e^8) - 20*x^5 + 6*x^4*e^2 - 12*x^4*e + 6*x^4*e^(2*e^8) - 12*x^4*e^(e^8 + 1) + 1 2*x^4*e^(e^8) - 2*x^4*log(x + 1) + x^4 + 12*x^3*e^2 - 44*x^3*e + 12*x^3*e^ (2*e^8) - 24*x^3*e^(e^8 + 1) + 44*x^3*e^(e^8) - 4*x^3*log(x + 1) - 4*x^2*e *log(x + 1) + 4*x^2*e^(e^8)*log(x + 1) + 40*x^3 + 4*x^2*e^3 - 18*x^2*e^2 + 24*x^2*e - 4*x^2*e^(3*e^8) - 18*x^2*e^(2*e^8) + 12*x^2*e^(2*e^8 + 1) - 12 *x^2*e^(e^8 + 2) + 36*x^2*e^(e^8 + 1) - 24*x^2*e^(e^8) + 6*x^2*log(x + 1) - 4*x*e*log(x + 1) + 4*x*e^(e^8)*log(x + 1) - 8*x^2 + 4*x*e^3 - 24*x*e^2 + 48*x*e - 4*x*e^(3*e^8) - 24*x*e^(2*e^8) + 12*x*e^(2*e^8 + 1) - 12*x*e^(e^ 8 + 2) + 48*x*e^(e^8 + 1) - 48*x*e^(e^8) + 8*x*log(x + 1) - 2*e^2*log(x + 1) + 8*e*log(x + 1) - 2*e^(2*e^8)*log(x + 1) + 4*e^(e^8 + 1)*log(x + 1) - 8*e^(e^8)*log(x + 1) + log(x + 1)^2 - 32*x - 8*log(x + 1)
Time = 3.24 (sec) , antiderivative size = 428, normalized size of antiderivative = 17.12 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=48\,x\,{\mathrm {e}}^{{\mathrm {e}}^8+1}-8\,\ln \left (x+1\right )-24\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^8}-4\,x\,{\mathrm {e}}^{3\,{\mathrm {e}}^8}-32\,x-12\,x\,{\mathrm {e}}^{{\mathrm {e}}^8+2}-24\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^8}+44\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^8}+12\,x^4\,{\mathrm {e}}^{{\mathrm {e}}^8}-12\,x^5\,{\mathrm {e}}^{{\mathrm {e}}^8}-4\,x^6\,{\mathrm {e}}^{{\mathrm {e}}^8}+8\,\ln \left (x+1\right )\,\mathrm {e}-2\,\ln \left (x+1\right )\,{\mathrm {e}}^2+8\,x\,\ln \left (x+1\right )+48\,x\,\mathrm {e}-24\,x\,{\mathrm {e}}^2+4\,x\,{\mathrm {e}}^3+12\,x\,{\mathrm {e}}^{2\,{\mathrm {e}}^8+1}-18\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^8}-4\,x^2\,{\mathrm {e}}^{3\,{\mathrm {e}}^8}+12\,x^3\,{\mathrm {e}}^{2\,{\mathrm {e}}^8}+6\,x^4\,{\mathrm {e}}^{2\,{\mathrm {e}}^8}+36\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^8+1}-12\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^8+2}-24\,x^3\,{\mathrm {e}}^{{\mathrm {e}}^8+1}-12\,x^4\,{\mathrm {e}}^{{\mathrm {e}}^8+1}-8\,\ln \left (x+1\right )\,{\mathrm {e}}^{{\mathrm {e}}^8}+6\,x^2\,\ln \left (x+1\right )-4\,x^3\,\ln \left (x+1\right )-2\,x^4\,\ln \left (x+1\right )+24\,x^2\,\mathrm {e}-18\,x^2\,{\mathrm {e}}^2-44\,x^3\,\mathrm {e}+4\,x^2\,{\mathrm {e}}^3+12\,x^3\,{\mathrm {e}}^2-12\,x^4\,\mathrm {e}+6\,x^4\,{\mathrm {e}}^2+12\,x^5\,\mathrm {e}+4\,x^6\,\mathrm {e}-48\,x\,{\mathrm {e}}^{{\mathrm {e}}^8}+{\ln \left (x+1\right )}^2+12\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^8+1}-2\,\ln \left (x+1\right )\,{\mathrm {e}}^{2\,{\mathrm {e}}^8}+4\,\ln \left (x+1\right )\,{\mathrm {e}}^{{\mathrm {e}}^8+1}-8\,x^2+40\,x^3+x^4-20\,x^5-2\,x^6+4\,x^7+x^8+4\,x^2\,\ln \left (x+1\right )\,{\mathrm {e}}^{{\mathrm {e}}^8}-4\,x\,\ln \left (x+1\right )\,\mathrm {e}-4\,x^2\,\ln \left (x+1\right )\,\mathrm {e}+4\,x\,\ln \left (x+1\right )\,{\mathrm {e}}^{{\mathrm {e}}^8} \] Input:
int((exp(1)*(92*x - 88*x^2 - 180*x^3 + 12*x^4 + 84*x^5 + 24*x^6 + 56) - 40 *x - exp(exp(8))*(92*x + exp(2)*(36*x + 24*x^2 + 12) - exp(1)*(120*x - 120 *x^3 - 48*x^4 + 52) - 88*x^2 - 180*x^3 + 12*x^4 + 84*x^5 + 24*x^6 + 56) + log(x + 1)*(20*x - exp(1)*(12*x + 8*x^2 + 4) + exp(exp(8))*(12*x + 8*x^2 + 4) - 20*x^3 - 8*x^4 + 10) + exp(3)*(12*x + 8*x^2 + 4) + exp(2*exp(8))*(ex p(1)*(36*x + 24*x^2 + 12) - 60*x + 60*x^3 + 24*x^4 - 26) - exp(2)*(60*x - 60*x^3 - 24*x^4 + 26) + 110*x^2 + 120*x^3 - 98*x^4 - 112*x^5 + 16*x^6 + 36 *x^7 + 8*x^8 - exp(3*exp(8))*(12*x + 8*x^2 + 4) - 40)/(x + 1),x)
Output:
48*x*exp(exp(8) + 1) - 8*log(x + 1) - 24*x*exp(2*exp(8)) - 4*x*exp(3*exp(8 )) - 32*x - 12*x*exp(exp(8) + 2) - 24*x^2*exp(exp(8)) + 44*x^3*exp(exp(8)) + 12*x^4*exp(exp(8)) - 12*x^5*exp(exp(8)) - 4*x^6*exp(exp(8)) + 8*log(x + 1)*exp(1) - 2*log(x + 1)*exp(2) + 8*x*log(x + 1) + 48*x*exp(1) - 24*x*exp (2) + 4*x*exp(3) + 12*x*exp(2*exp(8) + 1) - 18*x^2*exp(2*exp(8)) - 4*x^2*e xp(3*exp(8)) + 12*x^3*exp(2*exp(8)) + 6*x^4*exp(2*exp(8)) + 36*x^2*exp(exp (8) + 1) - 12*x^2*exp(exp(8) + 2) - 24*x^3*exp(exp(8) + 1) - 12*x^4*exp(ex p(8) + 1) - 8*log(x + 1)*exp(exp(8)) + 6*x^2*log(x + 1) - 4*x^3*log(x + 1) - 2*x^4*log(x + 1) + 24*x^2*exp(1) - 18*x^2*exp(2) - 44*x^3*exp(1) + 4*x^ 2*exp(3) + 12*x^3*exp(2) - 12*x^4*exp(1) + 6*x^4*exp(2) + 12*x^5*exp(1) + 4*x^6*exp(1) - 48*x*exp(exp(8)) + log(x + 1)^2 + 12*x^2*exp(2*exp(8) + 1) - 2*log(x + 1)*exp(2*exp(8)) + 4*log(x + 1)*exp(exp(8) + 1) - 8*x^2 + 40*x ^3 + x^4 - 20*x^5 - 2*x^6 + 4*x^7 + x^8 + 4*x^2*log(x + 1)*exp(exp(8)) - 4 *x*log(x + 1)*exp(1) - 4*x^2*log(x + 1)*exp(1) + 4*x*log(x + 1)*exp(exp(8) )
Time = 0.23 (sec) , antiderivative size = 471, normalized size of antiderivative = 18.84 \[ \int \frac {-40-40 x+110 x^2+120 x^3-98 x^4-112 x^5+16 x^6+36 x^7+8 x^8+e^{3 e^8} \left (-4-12 x-8 x^2\right )+e^3 \left (4+12 x+8 x^2\right )+e^2 \left (-26-60 x+60 x^3+24 x^4\right )+e \left (56+92 x-88 x^2-180 x^3+12 x^4+84 x^5+24 x^6\right )+e^{2 e^8} \left (-26-60 x+60 x^3+24 x^4+e \left (12+36 x+24 x^2\right )\right )+e^{e^8} \left (-56-92 x+88 x^2+180 x^3-12 x^4-84 x^5-24 x^6+e^2 \left (-12-36 x-24 x^2\right )+e \left (52+120 x-120 x^3-48 x^4\right )\right )+\left (10+20 x-20 x^3-8 x^4+e \left (-4-12 x-8 x^2\right )+e^{e^8} \left (4+12 x+8 x^2\right )\right ) \log (1+x)}{1+x} \, dx=-32 x -18 e^{2} x^{2}+12 e^{2} x^{3}-24 e^{2} x -8 x^{2}+4 e^{3} x^{2}-20 x^{5}+4 e^{3} x +4 x^{7}-4 e^{3 e^{8}} x^{2}-4 e^{3 e^{8}} x -2 e^{2 e^{8}} \mathrm {log}\left (x +1\right )+6 e^{2 e^{8}} x^{4}+12 e^{2 e^{8}} x^{3}-18 e^{2 e^{8}} x^{2}-24 e^{2 e^{8}} x -8 e^{e^{8}} \mathrm {log}\left (x +1\right )-4 e^{e^{8}} x^{6}-12 e^{e^{8}} x^{5}+12 e^{e^{8}} x^{4}+44 e^{e^{8}} x^{3}-24 e^{e^{8}} x^{2}-48 e^{e^{8}} x -2 \,\mathrm {log}\left (x +1\right ) e^{2}+8 \,\mathrm {log}\left (x +1\right ) e -2 \,\mathrm {log}\left (x +1\right ) x^{4}-4 \,\mathrm {log}\left (x +1\right ) x^{3}+6 \,\mathrm {log}\left (x +1\right ) x^{2}+8 \,\mathrm {log}\left (x +1\right ) x +6 e^{2} x^{4}+4 e \,x^{6}+12 e \,x^{5}-12 e \,x^{4}+40 x^{3}+x^{8}-2 x^{6}-8 \,\mathrm {log}\left (x +1\right )+48 e x +x^{4}+\mathrm {log}\left (x +1\right )^{2}-44 e \,x^{3}+12 e^{2 e^{8}} e \,x^{2}+12 e^{2 e^{8}} e x +4 e^{e^{8}} \mathrm {log}\left (x +1\right ) e +4 e^{e^{8}} \mathrm {log}\left (x +1\right ) x^{2}+4 e^{e^{8}} \mathrm {log}\left (x +1\right ) x -12 e^{e^{8}} e^{2} x^{2}-12 e^{e^{8}} e^{2} x -12 e^{e^{8}} e \,x^{4}-24 e^{e^{8}} e \,x^{3}+36 e^{e^{8}} e \,x^{2}+48 e^{e^{8}} e x -4 \,\mathrm {log}\left (x +1\right ) e \,x^{2}-4 \,\mathrm {log}\left (x +1\right ) e x +24 e \,x^{2} \] Input:
int((((8*x^2+12*x+4)*exp(exp(4)^2)+(-8*x^2-12*x-4)*exp(1)-8*x^4-20*x^3+20* x+10)*log(1+x)+(-8*x^2-12*x-4)*exp(exp(4)^2)^3+((24*x^2+36*x+12)*exp(1)+24 *x^4+60*x^3-60*x-26)*exp(exp(4)^2)^2+((-24*x^2-36*x-12)*exp(1)^2+(-48*x^4- 120*x^3+120*x+52)*exp(1)-24*x^6-84*x^5-12*x^4+180*x^3+88*x^2-92*x-56)*exp( exp(4)^2)+(8*x^2+12*x+4)*exp(1)^3+(24*x^4+60*x^3-60*x-26)*exp(1)^2+(24*x^6 +84*x^5+12*x^4-180*x^3-88*x^2+92*x+56)*exp(1)+8*x^8+36*x^7+16*x^6-112*x^5- 98*x^4+120*x^3+110*x^2-40*x-40)/(1+x),x)
Output:
- 4*e**(3*e**8)*x**2 - 4*e**(3*e**8)*x - 2*e**(2*e**8)*log(x + 1) + 12*e* *(2*e**8)*e*x**2 + 12*e**(2*e**8)*e*x + 6*e**(2*e**8)*x**4 + 12*e**(2*e**8 )*x**3 - 18*e**(2*e**8)*x**2 - 24*e**(2*e**8)*x + 4*e**(e**8)*log(x + 1)*e + 4*e**(e**8)*log(x + 1)*x**2 + 4*e**(e**8)*log(x + 1)*x - 8*e**(e**8)*lo g(x + 1) - 12*e**(e**8)*e**2*x**2 - 12*e**(e**8)*e**2*x - 12*e**(e**8)*e*x **4 - 24*e**(e**8)*e*x**3 + 36*e**(e**8)*e*x**2 + 48*e**(e**8)*e*x - 4*e** (e**8)*x**6 - 12*e**(e**8)*x**5 + 12*e**(e**8)*x**4 + 44*e**(e**8)*x**3 - 24*e**(e**8)*x**2 - 48*e**(e**8)*x + log(x + 1)**2 - 2*log(x + 1)*e**2 - 4 *log(x + 1)*e*x**2 - 4*log(x + 1)*e*x + 8*log(x + 1)*e - 2*log(x + 1)*x**4 - 4*log(x + 1)*x**3 + 6*log(x + 1)*x**2 + 8*log(x + 1)*x - 8*log(x + 1) + 4*e**3*x**2 + 4*e**3*x + 6*e**2*x**4 + 12*e**2*x**3 - 18*e**2*x**2 - 24*e **2*x + 4*e*x**6 + 12*e*x**5 - 12*e*x**4 - 44*e*x**3 + 24*e*x**2 + 48*e*x + x**8 + 4*x**7 - 2*x**6 - 20*x**5 + x**4 + 40*x**3 - 8*x**2 - 32*x