Integrand size = 242, antiderivative size = 31 \begin {dmath*} \int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx=\left (2+e^{\frac {x}{\log (\log (4))}}+\frac {x^2}{2 \left (5+x^2\right )-\log (x)}\right )^2 \end {dmath*}
Leaf count is larger than twice the leaf count of optimal. \(68\) vs. \(2(31)=62\).
Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \begin {dmath*} \int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx=4 e^{\frac {x}{\log (\log (4))}}+e^{\frac {2 x}{\log (\log (4))}}+\frac {2 \left (2+e^{\frac {x}{\log (\log (4))}}\right ) x^2}{2 \left (5+x^2\right )-\log (x)}+\frac {x^4}{\left (-2 \left (5+x^2\right )+\log (x)\right )^2} \end {dmath*}
Integrate[(E^((2*x)/Log[Log[4]])*(-2000 - 1200*x^2 - 240*x^4 - 16*x^6 + (6 00 + 240*x^2 + 24*x^4)*Log[x] + (-60 - 12*x^2)*Log[x]^2 + 2*Log[x]^3) + (- 840*x - 210*x^3 + (164*x + 20*x^3)*Log[x] - 8*x*Log[x]^2)*Log[Log[4]] + E^ (x/Log[Log[4]])*(-4000 - 2600*x^2 - 560*x^4 - 40*x^6 + (1200 + 520*x^2 + 5 6*x^4)*Log[x] + (-120 - 26*x^2)*Log[x]^2 + 4*Log[x]^3 + (-420*x - 84*x^3 + (82*x + 8*x^3)*Log[x] - 4*x*Log[x]^2)*Log[Log[4]]))/((-1000 - 600*x^2 - 1 20*x^4 - 8*x^6 + (300 + 120*x^2 + 12*x^4)*Log[x] + (-30 - 6*x^2)*Log[x]^2 + Log[x]^3)*Log[Log[4]]),x]
4*E^(x/Log[Log[4]]) + E^((2*x)/Log[Log[4]]) + (2*(2 + E^(x/Log[Log[4]]))*x ^2)/(2*(5 + x^2) - Log[x]) + x^4/(-2*(5 + x^2) + Log[x])^2
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.
| \(\displaystyle \int \frac {\log (\log (4)) \left (-210 x^3+\left (20 x^3+164 x\right ) \log (x)-840 x-8 x \log ^2(x)\right )+e^{\frac {2 x}{\log (\log (4))}} \left (-16 x^6-240 x^4-1200 x^2+\left (-12 x^2-60\right ) \log ^2(x)+\left (24 x^4+240 x^2+600\right ) \log (x)+2 \log ^3(x)-2000\right )+e^{\frac {x}{\log (\log (4))}} \left (-40 x^6-560 x^4+\log (\log (4)) \left (-84 x^3+\left (8 x^3+82 x\right ) \log (x)-420 x-4 x \log ^2(x)\right )-2600 x^2+\left (-26 x^2-120\right ) \log ^2(x)+\left (56 x^4+520 x^2+1200\right ) \log (x)+4 \log ^3(x)-4000\right )}{\log (\log (4)) \left (-8 x^6-120 x^4-600 x^2+\left (-6 x^2-30\right ) \log ^2(x)+\left (12 x^4+120 x^2+300\right ) \log (x)+\log ^3(x)-1000\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 \left (\log (\log (4)) \left (105 x^3+4 \log ^2(x) x+420 x-2 \left (5 x^3+41 x\right ) \log (x)\right )+e^{\frac {2 x}{\log (\log (4))}} \left (8 x^6+120 x^4+600 x^2-\log ^3(x)+6 \left (x^2+5\right ) \log ^2(x)-12 \left (x^4+10 x^2+25\right ) \log (x)+1000\right )+e^{\frac {x}{\log (\log (4))}} \left (20 x^6+280 x^4+1300 x^2-2 \log ^3(x)+\left (13 x^2+60\right ) \log ^2(x)-4 \left (7 x^4+65 x^2+150\right ) \log (x)+\left (42 x^3+2 \log ^2(x) x+210 x-\left (4 x^3+41 x\right ) \log (x)\right ) \log (\log (4))+2000\right )\right )}{8 x^6+120 x^4+600 x^2-\log ^3(x)+6 \left (x^2+5\right ) \log ^2(x)-12 \left (x^4+10 x^2+25\right ) \log (x)+1000}dx}{\log (\log (4))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {\log (\log (4)) \left (105 x^3+4 \log ^2(x) x+420 x-2 \left (5 x^3+41 x\right ) \log (x)\right )+e^{\frac {2 x}{\log (\log (4))}} \left (8 x^6+120 x^4+600 x^2-\log ^3(x)+6 \left (x^2+5\right ) \log ^2(x)-12 \left (x^4+10 x^2+25\right ) \log (x)+1000\right )+e^{\frac {x}{\log (\log (4))}} \left (20 x^6+280 x^4+1300 x^2-2 \log ^3(x)+\left (13 x^2+60\right ) \log ^2(x)-4 \left (7 x^4+65 x^2+150\right ) \log (x)+\left (42 x^3+2 \log ^2(x) x+210 x-\left (4 x^3+41 x\right ) \log (x)\right ) \log (\log (4))+2000\right )}{8 x^6+120 x^4+600 x^2-\log ^3(x)+6 \left (x^2+5\right ) \log ^2(x)-12 \left (x^4+10 x^2+25\right ) \log (x)+1000}dx}{\log (\log (4))}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {\left (5 \left (x^2+4\right )+2 e^{\frac {x}{\log (\log (4))}} \left (x^2+5\right )-\left (2+e^{\frac {x}{\log (\log (4))}}\right ) \log (x)\right ) \left (4 e^{\frac {x}{\log (\log (4))}} \left (x^2+5\right )^2+e^{\frac {x}{\log (\log (4))}} \log ^2(x)-2 \log (x) \left (\log (\log (4)) x+2 e^{\frac {x}{\log (\log (4))}} \left (x^2+5\right )\right )+21 x \log (\log (4))\right )}{\left (2 \left (x^2+5\right )-\log (x)\right )^3}dx}{\log (\log (4))}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {4 x \log (\log (4)) \log ^2(x)}{\left (2 x^2-\log (x)+10\right )^3}-\frac {42 x \log (\log (4)) \log (x)}{\left (2 x^2-\log (x)+10\right )^3}-\frac {10 x \left (x^2+4\right ) \log (\log (4)) \log (x)}{\left (2 x^2-\log (x)+10\right )^3}+e^{\frac {2 x}{\log (\log (4))}}+\frac {e^{\frac {x}{\log (\log (4))}} \left (10 x^4-9 \log (x) x^2+90 x^2-2 \log (x) \log (\log (4)) x+21 \log (\log (4)) x+2 \log ^2(x)-40 \log (x)+200\right )}{\left (2 x^2-\log (x)+10\right )^2}+\frac {105 x \left (x^2+4\right ) \log (\log (4))}{\left (2 x^2-\log (x)+10\right )^3}\right )dx}{\log (\log (4))}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (2 \log (\log (4)) \int \frac {x}{\left (2 x^2-\log (x)+10\right )^2}dx+4 \log (\log (4)) \int \frac {x}{2 x^2-\log (x)+10}dx-4 \log (\log (4)) \int \frac {x^5}{\left (2 x^2-\log (x)+10\right )^3}dx+\log (\log (4)) \int \frac {x^3}{\left (2 x^2-\log (x)+10\right )^3}dx-6 \log (\log (4)) \int \frac {x^3}{\left (2 x^2-\log (x)+10\right )^2}dx+\frac {\log (\log (4)) e^{\frac {x}{\log (\log (4))}} \left (10 x^4+90 x^2-9 x^2 \log (x)+2 \log ^2(x)-40 \log (x)+200\right )}{\left (2 x^2-\log (x)+10\right )^2}+\frac {1}{2} \log (\log (4)) e^{\frac {2 x}{\log (\log (4))}}\right )}{\log (\log (4))}\) |
Int[(E^((2*x)/Log[Log[4]])*(-2000 - 1200*x^2 - 240*x^4 - 16*x^6 + (600 + 2 40*x^2 + 24*x^4)*Log[x] + (-60 - 12*x^2)*Log[x]^2 + 2*Log[x]^3) + (-840*x - 210*x^3 + (164*x + 20*x^3)*Log[x] - 8*x*Log[x]^2)*Log[Log[4]] + E^(x/Log [Log[4]])*(-4000 - 2600*x^2 - 560*x^4 - 40*x^6 + (1200 + 520*x^2 + 56*x^4) *Log[x] + (-120 - 26*x^2)*Log[x]^2 + 4*Log[x]^3 + (-420*x - 84*x^3 + (82*x + 8*x^3)*Log[x] - 4*x*Log[x]^2)*Log[Log[4]]))/((-1000 - 600*x^2 - 120*x^4 - 8*x^6 + (300 + 120*x^2 + 12*x^4)*Log[x] + (-30 - 6*x^2)*Log[x]^2 + Log[ x]^3)*Log[Log[4]]),x]
3.7.48.3.1 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[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. \(263\) vs. \(2(31)=62\).
Time = 10.02 (sec) , antiderivative size = 264, normalized size of antiderivative = 8.52
| method | result | size |
| risch | \(\frac {{\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}} \ln \left (2\right )}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}+\frac {{\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}} \ln \left (\ln \left (2\right )\right )}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}+\frac {4 \ln \left (2\right ) {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}}}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}+\frac {4 \ln \left (\ln \left (2\right )\right ) {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}}}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}+\frac {\left (4 \ln \left (2\right ) x^{2} {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}}+4 \ln \left (\ln \left (2\right )\right ) x^{2} {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}}+9 x^{2} \ln \left (2\right )-2 \ln \left (2\right ) {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}} \ln \left (x \right )+9 x^{2} \ln \left (\ln \left (2\right )\right )-2 \ln \left (\ln \left (2\right )\right ) {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}} \ln \left (x \right )+20 \ln \left (2\right ) {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}}-4 \ln \left (2\right ) \ln \left (x \right )+20 \ln \left (\ln \left (2\right )\right ) {\mathrm e}^{\frac {x}{\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )}}-4 \ln \left (x \right ) \ln \left (\ln \left (2\right )\right )+40 \ln \left (2\right )+40 \ln \left (\ln \left (2\right )\right )\right ) x^{2}}{\left (\ln \left (2\right )+\ln \left (\ln \left (2\right )\right )\right ) \left (2 x^{2}+10-\ln \left (x \right )\right )^{2}}\) | \(264\) |
| parallelrisch | \(\frac {40 x^{2} \ln \left (2 \ln \left (2\right )\right )+4 \ln \left (2 \ln \left (2\right )\right ) x^{4} {\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}}+40 \ln \left (2 \ln \left (2\right )\right ) x^{2} {\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}}+180 \ln \left (2 \ln \left (2\right )\right ) x^{2} {\mathrm e}^{\frac {x}{\ln \left (2 \ln \left (2\right )\right )}}+\ln \left (2 \ln \left (2\right )\right ) {\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}} \ln \left (x \right )^{2}-4 x^{2} \ln \left (x \right ) \ln \left (2 \ln \left (2\right )\right )+4 \ln \left (2 \ln \left (2\right )\right ) \ln \left (x \right )^{2} {\mathrm e}^{\frac {x}{\ln \left (2 \ln \left (2\right )\right )}}-20 \ln \left (2 \ln \left (2\right )\right ) {\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}} \ln \left (x \right )-80 \ln \left (2 \ln \left (2\right )\right ) \ln \left (x \right ) {\mathrm e}^{\frac {x}{\ln \left (2 \ln \left (2\right )\right )}}+20 \ln \left (2 \ln \left (2\right )\right ) x^{4} {\mathrm e}^{\frac {x}{\ln \left (2 \ln \left (2\right )\right )}}+9 x^{4} \ln \left (2 \ln \left (2\right )\right )+100 \ln \left (2 \ln \left (2\right )\right ) {\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}}+400 \ln \left (2 \ln \left (2\right )\right ) {\mathrm e}^{\frac {x}{\ln \left (2 \ln \left (2\right )\right )}}-18 \ln \left (2 \ln \left (2\right )\right ) x^{2} \ln \left (x \right ) {\mathrm e}^{\frac {x}{\ln \left (2 \ln \left (2\right )\right )}}-4 \ln \left (2 \ln \left (2\right )\right ) x^{2} \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{\ln \left (2 \ln \left (2\right )\right )}}}{\ln \left (2 \ln \left (2\right )\right ) \left (4 x^{4}-4 x^{2} \ln \left (x \right )+40 x^{2}+\ln \left (x \right )^{2}-20 \ln \left (x \right )+100\right )}\) | \(320\) |
int(((2*ln(x)^3+(-12*x^2-60)*ln(x)^2+(24*x^4+240*x^2+600)*ln(x)-16*x^6-240 *x^4-1200*x^2-2000)*exp(x/ln(2*ln(2)))^2+((-4*x*ln(x)^2+(8*x^3+82*x)*ln(x) -84*x^3-420*x)*ln(2*ln(2))+4*ln(x)^3+(-26*x^2-120)*ln(x)^2+(56*x^4+520*x^2 +1200)*ln(x)-40*x^6-560*x^4-2600*x^2-4000)*exp(x/ln(2*ln(2)))+(-8*x*ln(x)^ 2+(20*x^3+164*x)*ln(x)-210*x^3-840*x)*ln(2*ln(2)))/(ln(x)^3+(-6*x^2-30)*ln (x)^2+(12*x^4+120*x^2+300)*ln(x)-8*x^6-120*x^4-600*x^2-1000)/ln(2*ln(2)),x ,method=_RETURNVERBOSE)
1/(ln(2)+ln(ln(2)))*exp(x/(ln(2)+ln(ln(2))))^2*ln(2)+1/(ln(2)+ln(ln(2)))*e xp(x/(ln(2)+ln(ln(2))))^2*ln(ln(2))+4/(ln(2)+ln(ln(2)))*ln(2)*exp(x/(ln(2) +ln(ln(2))))+4/(ln(2)+ln(ln(2)))*ln(ln(2))*exp(x/(ln(2)+ln(ln(2))))+1/(ln( 2)+ln(ln(2)))*(4*ln(2)*x^2*exp(x/(ln(2)+ln(ln(2))))+4*ln(ln(2))*x^2*exp(x/ (ln(2)+ln(ln(2))))+9*x^2*ln(2)-2*ln(2)*exp(x/(ln(2)+ln(ln(2))))*ln(x)+9*x^ 2*ln(ln(2))-2*ln(ln(2))*exp(x/(ln(2)+ln(ln(2))))*ln(x)+20*ln(2)*exp(x/(ln( 2)+ln(ln(2))))-4*ln(2)*ln(x)+20*ln(ln(2))*exp(x/(ln(2)+ln(ln(2))))-4*ln(x) *ln(ln(2))+40*ln(2)+40*ln(ln(2)))*x^2/(2*x^2+10-ln(x))^2
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (31) = 62\).
Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.00 \begin {dmath*} \int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx=\frac {9 \, x^{4} - 4 \, x^{2} \log \left (x\right ) + 40 \, x^{2} + {\left (4 \, x^{4} + 40 \, x^{2} - 4 \, {\left (x^{2} + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 100\right )} e^{\left (\frac {2 \, x}{\log \left (2 \, \log \left (2\right )\right )}\right )} + 2 \, {\left (10 \, x^{4} + 90 \, x^{2} - {\left (9 \, x^{2} + 40\right )} \log \left (x\right ) + 2 \, \log \left (x\right )^{2} + 200\right )} e^{\left (\frac {x}{\log \left (2 \, \log \left (2\right )\right )}\right )}}{4 \, x^{4} + 40 \, x^{2} - 4 \, {\left (x^{2} + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 100} \end {dmath*}
integrate(((2*log(x)^3+(-12*x^2-60)*log(x)^2+(24*x^4+240*x^2+600)*log(x)-1 6*x^6-240*x^4-1200*x^2-2000)*exp(x/log(2*log(2)))^2+((-4*x*log(x)^2+(8*x^3 +82*x)*log(x)-84*x^3-420*x)*log(2*log(2))+4*log(x)^3+(-26*x^2-120)*log(x)^ 2+(56*x^4+520*x^2+1200)*log(x)-40*x^6-560*x^4-2600*x^2-4000)*exp(x/log(2*l og(2)))+(-8*x*log(x)^2+(20*x^3+164*x)*log(x)-210*x^3-840*x)*log(2*log(2))) /(log(x)^3+(-6*x^2-30)*log(x)^2+(12*x^4+120*x^2+300)*log(x)-8*x^6-120*x^4- 600*x^2-1000)/log(2*log(2)),x, algorithm=\
(9*x^4 - 4*x^2*log(x) + 40*x^2 + (4*x^4 + 40*x^2 - 4*(x^2 + 5)*log(x) + lo g(x)^2 + 100)*e^(2*x/log(2*log(2))) + 2*(10*x^4 + 90*x^2 - (9*x^2 + 40)*lo g(x) + 2*log(x)^2 + 200)*e^(x/log(2*log(2))))/(4*x^4 + 40*x^2 - 4*(x^2 + 5 )*log(x) + log(x)^2 + 100)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (26) = 52\).
Time = 0.39 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.23 \begin {dmath*} \int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx=\frac {\left (2 x^{2} - \log {\left (x \right )} + 10\right ) e^{\frac {2 x}{\log {\left (2 \log {\left (2 \right )} \right )}}} + \left (10 x^{2} - 4 \log {\left (x \right )} + 40\right ) e^{\frac {x}{\log {\left (2 \log {\left (2 \right )} \right )}}}}{2 x^{2} - \log {\left (x \right )} + 10} + \frac {9 x^{4} - 4 x^{2} \log {\left (x \right )} + 40 x^{2}}{4 x^{4} + 40 x^{2} + \left (- 4 x^{2} - 20\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 100} \end {dmath*}
integrate(((2*ln(x)**3+(-12*x**2-60)*ln(x)**2+(24*x**4+240*x**2+600)*ln(x) -16*x**6-240*x**4-1200*x**2-2000)*exp(x/ln(2*ln(2)))**2+((-4*x*ln(x)**2+(8 *x**3+82*x)*ln(x)-84*x**3-420*x)*ln(2*ln(2))+4*ln(x)**3+(-26*x**2-120)*ln( x)**2+(56*x**4+520*x**2+1200)*ln(x)-40*x**6-560*x**4-2600*x**2-4000)*exp(x /ln(2*ln(2)))+(-8*x*ln(x)**2+(20*x**3+164*x)*ln(x)-210*x**3-840*x)*ln(2*ln (2)))/(ln(x)**3+(-6*x**2-30)*ln(x)**2+(12*x**4+120*x**2+300)*ln(x)-8*x**6- 120*x**4-600*x**2-1000)/ln(2*ln(2)),x)
((2*x**2 - log(x) + 10)*exp(2*x/log(2*log(2))) + (10*x**2 - 4*log(x) + 40) *exp(x/log(2*log(2))))/(2*x**2 - log(x) + 10) + (9*x**4 - 4*x**2*log(x) + 40*x**2)/(4*x**4 + 40*x**2 + (-4*x**2 - 20)*log(x) + log(x)**2 + 100)
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (31) = 62\).
Time = 0.37 (sec) , antiderivative size = 233, normalized size of antiderivative = 7.52 \begin {dmath*} \int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx=\frac {9 \, x^{4} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} - 4 \, x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (x\right ) + 40 \, x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + {\left (4 \, x^{4} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + 40 \, x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (x\right )^{2} - 4 \, {\left (x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + 5 \, \log \left (2\right ) + 5 \, \log \left (\log \left (2\right )\right )\right )} \log \left (x\right ) + 100 \, \log \left (2\right ) + 100 \, \log \left (\log \left (2\right )\right )\right )} e^{\left (\frac {2 \, x}{\log \left (2\right ) + \log \left (\log \left (2\right )\right )}\right )} + 2 \, {\left (10 \, x^{4} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + 90 \, x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + 2 \, {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} \log \left (x\right )^{2} - {\left (9 \, x^{2} {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right )\right )} + 40 \, \log \left (2\right ) + 40 \, \log \left (\log \left (2\right )\right )\right )} \log \left (x\right ) + 200 \, \log \left (2\right ) + 200 \, \log \left (\log \left (2\right )\right )\right )} e^{\left (\frac {x}{\log \left (2\right ) + \log \left (\log \left (2\right )\right )}\right )}}{{\left (4 \, x^{4} + 40 \, x^{2} - 4 \, {\left (x^{2} + 5\right )} \log \left (x\right ) + \log \left (x\right )^{2} + 100\right )} \log \left (2 \, \log \left (2\right )\right )} \end {dmath*}
integrate(((2*log(x)^3+(-12*x^2-60)*log(x)^2+(24*x^4+240*x^2+600)*log(x)-1 6*x^6-240*x^4-1200*x^2-2000)*exp(x/log(2*log(2)))^2+((-4*x*log(x)^2+(8*x^3 +82*x)*log(x)-84*x^3-420*x)*log(2*log(2))+4*log(x)^3+(-26*x^2-120)*log(x)^ 2+(56*x^4+520*x^2+1200)*log(x)-40*x^6-560*x^4-2600*x^2-4000)*exp(x/log(2*l og(2)))+(-8*x*log(x)^2+(20*x^3+164*x)*log(x)-210*x^3-840*x)*log(2*log(2))) /(log(x)^3+(-6*x^2-30)*log(x)^2+(12*x^4+120*x^2+300)*log(x)-8*x^6-120*x^4- 600*x^2-1000)/log(2*log(2)),x, algorithm=\
(9*x^4*(log(2) + log(log(2))) - 4*x^2*(log(2) + log(log(2)))*log(x) + 40*x ^2*(log(2) + log(log(2))) + (4*x^4*(log(2) + log(log(2))) + 40*x^2*(log(2) + log(log(2))) + (log(2) + log(log(2)))*log(x)^2 - 4*(x^2*(log(2) + log(l og(2))) + 5*log(2) + 5*log(log(2)))*log(x) + 100*log(2) + 100*log(log(2))) *e^(2*x/(log(2) + log(log(2)))) + 2*(10*x^4*(log(2) + log(log(2))) + 90*x^ 2*(log(2) + log(log(2))) + 2*(log(2) + log(log(2)))*log(x)^2 - (9*x^2*(log (2) + log(log(2))) + 40*log(2) + 40*log(log(2)))*log(x) + 200*log(2) + 200 *log(log(2)))*e^(x/(log(2) + log(log(2)))))/((4*x^4 + 40*x^2 - 4*(x^2 + 5) *log(x) + log(x)^2 + 100)*log(2*log(2)))
\begin {dmath*} \int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx=\int { \frac {2 \, {\left ({\left (8 \, x^{6} + 120 \, x^{4} + 6 \, {\left (x^{2} + 5\right )} \log \left (x\right )^{2} - \log \left (x\right )^{3} + 600 \, x^{2} - 12 \, {\left (x^{4} + 10 \, x^{2} + 25\right )} \log \left (x\right ) + 1000\right )} e^{\left (\frac {2 \, x}{\log \left (2 \, \log \left (2\right )\right )}\right )} + {\left (20 \, x^{6} + 280 \, x^{4} + {\left (13 \, x^{2} + 60\right )} \log \left (x\right )^{2} - 2 \, \log \left (x\right )^{3} + 1300 \, x^{2} - 4 \, {\left (7 \, x^{4} + 65 \, x^{2} + 150\right )} \log \left (x\right ) + {\left (42 \, x^{3} + 2 \, x \log \left (x\right )^{2} - {\left (4 \, x^{3} + 41 \, x\right )} \log \left (x\right ) + 210 \, x\right )} \log \left (2 \, \log \left (2\right )\right ) + 2000\right )} e^{\left (\frac {x}{\log \left (2 \, \log \left (2\right )\right )}\right )} + {\left (105 \, x^{3} + 4 \, x \log \left (x\right )^{2} - 2 \, {\left (5 \, x^{3} + 41 \, x\right )} \log \left (x\right ) + 420 \, x\right )} \log \left (2 \, \log \left (2\right )\right )\right )}}{{\left (8 \, x^{6} + 120 \, x^{4} + 6 \, {\left (x^{2} + 5\right )} \log \left (x\right )^{2} - \log \left (x\right )^{3} + 600 \, x^{2} - 12 \, {\left (x^{4} + 10 \, x^{2} + 25\right )} \log \left (x\right ) + 1000\right )} \log \left (2 \, \log \left (2\right )\right )} \,d x } \end {dmath*}
integrate(((2*log(x)^3+(-12*x^2-60)*log(x)^2+(24*x^4+240*x^2+600)*log(x)-1 6*x^6-240*x^4-1200*x^2-2000)*exp(x/log(2*log(2)))^2+((-4*x*log(x)^2+(8*x^3 +82*x)*log(x)-84*x^3-420*x)*log(2*log(2))+4*log(x)^3+(-26*x^2-120)*log(x)^ 2+(56*x^4+520*x^2+1200)*log(x)-40*x^6-560*x^4-2600*x^2-4000)*exp(x/log(2*l og(2)))+(-8*x*log(x)^2+(20*x^3+164*x)*log(x)-210*x^3-840*x)*log(2*log(2))) /(log(x)^3+(-6*x^2-30)*log(x)^2+(12*x^4+120*x^2+300)*log(x)-8*x^6-120*x^4- 600*x^2-1000)/log(2*log(2)),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {e^{\frac {2 x}{\log (\log (4))}} \left (-2000-1200 x^2-240 x^4-16 x^6+\left (600+240 x^2+24 x^4\right ) \log (x)+\left (-60-12 x^2\right ) \log ^2(x)+2 \log ^3(x)\right )+\left (-840 x-210 x^3+\left (164 x+20 x^3\right ) \log (x)-8 x \log ^2(x)\right ) \log (\log (4))+e^{\frac {x}{\log (\log (4))}} \left (-4000-2600 x^2-560 x^4-40 x^6+\left (1200+520 x^2+56 x^4\right ) \log (x)+\left (-120-26 x^2\right ) \log ^2(x)+4 \log ^3(x)+\left (-420 x-84 x^3+\left (82 x+8 x^3\right ) \log (x)-4 x \log ^2(x)\right ) \log (\log (4))\right )}{\left (-1000-600 x^2-120 x^4-8 x^6+\left (300+120 x^2+12 x^4\right ) \log (x)+\left (-30-6 x^2\right ) \log ^2(x)+\log ^3(x)\right ) \log (\log (4))} \, dx=\int \frac {{\mathrm {e}}^{\frac {2\,x}{\ln \left (2\,\ln \left (2\right )\right )}}\,\left ({\ln \left (x\right )}^2\,\left (12\,x^2+60\right )-\ln \left (x\right )\,\left (24\,x^4+240\,x^2+600\right )-2\,{\ln \left (x\right )}^3+1200\,x^2+240\,x^4+16\,x^6+2000\right )+\ln \left (2\,\ln \left (2\right )\right )\,\left (840\,x+8\,x\,{\ln \left (x\right )}^2-\ln \left (x\right )\,\left (20\,x^3+164\,x\right )+210\,x^3\right )+{\mathrm {e}}^{\frac {x}{\ln \left (2\,\ln \left (2\right )\right )}}\,\left ({\ln \left (x\right )}^2\,\left (26\,x^2+120\right )-\ln \left (x\right )\,\left (56\,x^4+520\,x^2+1200\right )-4\,{\ln \left (x\right )}^3+\ln \left (2\,\ln \left (2\right )\right )\,\left (420\,x+4\,x\,{\ln \left (x\right )}^2-\ln \left (x\right )\,\left (8\,x^3+82\,x\right )+84\,x^3\right )+2600\,x^2+560\,x^4+40\,x^6+4000\right )}{\ln \left (2\,\ln \left (2\right )\right )\,\left ({\ln \left (x\right )}^2\,\left (6\,x^2+30\right )-\ln \left (x\right )\,\left (12\,x^4+120\,x^2+300\right )-{\ln \left (x\right )}^3+600\,x^2+120\,x^4+8\,x^6+1000\right )} \,d x \end {dmath*}
int((exp((2*x)/log(2*log(2)))*(log(x)^2*(12*x^2 + 60) - log(x)*(240*x^2 + 24*x^4 + 600) - 2*log(x)^3 + 1200*x^2 + 240*x^4 + 16*x^6 + 2000) + log(2*l og(2))*(840*x + 8*x*log(x)^2 - log(x)*(164*x + 20*x^3) + 210*x^3) + exp(x/ log(2*log(2)))*(log(x)^2*(26*x^2 + 120) - log(x)*(520*x^2 + 56*x^4 + 1200) - 4*log(x)^3 + log(2*log(2))*(420*x + 4*x*log(x)^2 - log(x)*(82*x + 8*x^3 ) + 84*x^3) + 2600*x^2 + 560*x^4 + 40*x^6 + 4000))/(log(2*log(2))*(log(x)^ 2*(6*x^2 + 30) - log(x)*(120*x^2 + 12*x^4 + 300) - log(x)^3 + 600*x^2 + 12 0*x^4 + 8*x^6 + 1000)),x)
int((exp((2*x)/log(2*log(2)))*(log(x)^2*(12*x^2 + 60) - log(x)*(240*x^2 + 24*x^4 + 600) - 2*log(x)^3 + 1200*x^2 + 240*x^4 + 16*x^6 + 2000) + log(2*l og(2))*(840*x + 8*x*log(x)^2 - log(x)*(164*x + 20*x^3) + 210*x^3) + exp(x/ log(2*log(2)))*(log(x)^2*(26*x^2 + 120) - log(x)*(520*x^2 + 56*x^4 + 1200) - 4*log(x)^3 + log(2*log(2))*(420*x + 4*x*log(x)^2 - log(x)*(82*x + 8*x^3 ) + 84*x^3) + 2600*x^2 + 560*x^4 + 40*x^6 + 4000))/(log(2*log(2))*(log(x)^ 2*(6*x^2 + 30) - log(x)*(120*x^2 + 12*x^4 + 300) - log(x)^3 + 600*x^2 + 12 0*x^4 + 8*x^6 + 1000)), x)