Integrand size = 168, antiderivative size = 21 \[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=5 e^{\left (-1-x+\log \left ((8+i \pi )^2\right )\right )^4} \] Output:
5*exp((2*ln(ln(-exp(4)^2))-1-x)^4)
\[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=\int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx \] Input:
Integrate[E^(1 + 4*x + 6*x^2 + 4*x^3 + x^4 + (-4 - 12*x - 12*x^2 - 4*x^3)* Log[(8 + I*Pi)^2] + (6 + 12*x + 6*x^2)*Log[(8 + I*Pi)^2]^2 + (-4 - 4*x)*Lo g[(8 + I*Pi)^2]^3 + Log[(8 + I*Pi)^2]^4)*(20 + 60*x + 60*x^2 + 20*x^3 + (- 60 - 120*x - 60*x^2)*Log[(8 + I*Pi)^2] + (60 + 60*x)*Log[(8 + I*Pi)^2]^2 - 20*Log[(8 + I*Pi)^2]^3),x]
Output:
Integrate[E^(1 + 4*x + 6*x^2 + 4*x^3 + x^4 + (-4 - 12*x - 12*x^2 - 4*x^3)* Log[(8 + I*Pi)^2] + (6 + 12*x + 6*x^2)*Log[(8 + I*Pi)^2]^2 + (-4 - 4*x)*Lo g[(8 + I*Pi)^2]^3 + Log[(8 + I*Pi)^2]^4)*(20 + 60*x + 60*x^2 + 20*x^3 + (- 60 - 120*x - 60*x^2)*Log[(8 + I*Pi)^2] + (60 + 60*x)*Log[(8 + I*Pi)^2]^2 - 20*Log[(8 + I*Pi)^2]^3), x]
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(21)=42\).
Time = 1.51 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.67, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2006, 7257}
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.
| \(\displaystyle \int \left (20 x^3+60 x^2+\left (-60 x^2-120 x-60\right ) \log \left ((8+i \pi )^2\right )+60 x+(60 x+60) \log ^2\left ((8+i \pi )^2\right )+20-20 \log ^3\left ((8+i \pi )^2\right )\right ) \exp \left (x^4+4 x^3+6 x^2+\left (6 x^2+12 x+6\right ) \log ^2\left ((8+i \pi )^2\right )+\left (-4 x^3-12 x^2-12 x-4\right ) \log \left ((8+i \pi )^2\right )+4 x+(-4 x-4) \log ^3\left ((8+i \pi )^2\right )+1+\log ^4\left ((8+i \pi )^2\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \left (2^{2/3} \sqrt [3]{5} x+2^{2/3} \sqrt [3]{5} \left (1-\log \left ((8+i \pi )^2\right )\right )\right )^3 \exp \left (x^4+4 x^3+6 x^2+\left (6 x^2+12 x+6\right ) \log ^2\left ((8+i \pi )^2\right )+\left (-4 x^3-12 x^2-12 x-4\right ) \log \left ((8+i \pi )^2\right )+4 x+(-4 x-4) \log ^3\left ((8+i \pi )^2\right )+1+\log ^4\left ((8+i \pi )^2\right )\right )dx\) |
| \(\Big \downarrow \) 7257 |
| \(\displaystyle 5 \left ((8+i \pi )^2\right )^{-4 x^3-12 x^2-12 x-4} \exp \left (x^4+4 x^3+6 x^2+6 \left (x^2+2 x+1\right ) \log ^2\left ((8+i \pi )^2\right )+4 x-4 (x+1) \log ^3\left ((8+i \pi )^2\right )+1+\log ^4\left ((8+i \pi )^2\right )\right )\) |
Input:
Int[E^(1 + 4*x + 6*x^2 + 4*x^3 + x^4 + (-4 - 12*x - 12*x^2 - 4*x^3)*Log[(8 + I*Pi)^2] + (6 + 12*x + 6*x^2)*Log[(8 + I*Pi)^2]^2 + (-4 - 4*x)*Log[(8 + I*Pi)^2]^3 + Log[(8 + I*Pi)^2]^4)*(20 + 60*x + 60*x^2 + 20*x^3 + (-60 - 1 20*x - 60*x^2)*Log[(8 + I*Pi)^2] + (60 + 60*x)*Log[(8 + I*Pi)^2]^2 - 20*Lo g[(8 + I*Pi)^2]^3),x]
Output:
5*E^(1 + 4*x + 6*x^2 + 4*x^3 + x^4 + 6*(1 + 2*x + x^2)*Log[(8 + I*Pi)^2]^2 - 4*(1 + x)*Log[(8 + I*Pi)^2]^3 + Log[(8 + I*Pi)^2]^4)*((8 + I*Pi)^2)^(-4 - 12*x - 12*x^2 - 4*x^3)
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Leaf count of result is larger than twice the leaf count of optimal. \(97\) vs. \(2(20)=40\).
Time = 7.43 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.67
| method | result | size |
| norman | \(5 \,{\mathrm e}^{16 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{4}+8 \left (-4-4 x \right ) {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{3}+4 \left (6 x^{2}+12 x +6\right ) {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{2}+2 \left (-4 x^{3}-12 x^{2}-12 x -4\right ) \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )+x^{4}+4 x^{3}+6 x^{2}+4 x +1}\) | \(98\) |
| parallelrisch | \(5 \,{\mathrm e}^{16 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{4}+8 \left (-4-4 x \right ) {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{3}+4 \left (6 x^{2}+12 x +6\right ) {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{2}+2 \left (-4 x^{3}-12 x^{2}-12 x -4\right ) \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )+x^{4}+4 x^{3}+6 x^{2}+4 x +1}\) | \(98\) |
| risch | \(5 \left (i \pi +8\right )^{-8 \left (1+x \right )^{3}} {\mathrm e}^{16 \ln \left (i \pi +8\right )^{4}-32 \ln \left (i \pi +8\right )^{3} x +24 \ln \left (i \pi +8\right )^{2} x^{2}+x^{4}-32 \ln \left (i \pi +8\right )^{3}+48 \ln \left (i \pi +8\right )^{2} x +4 x^{3}+24 \ln \left (i \pi +8\right )^{2}+6 x^{2}+4 x +1}\) | \(107\) |
| gosper | \(5 \,{\mathrm e}^{16 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{4}-32 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{3} x +24 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{2} x^{2}-8 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right ) x^{3}+x^{4}-32 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{3}+48 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{2} x -24 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right ) x^{2}+4 x^{3}+24 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{2}-24 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right ) x +6 x^{2}-8 \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )+4 x +1}\) | \(146\) |
| orering | \(-\frac {\left (-160 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{3}+4 \left (60 x +60\right ) {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{2}+2 \left (-60 x^{2}-120 x -60\right ) \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )+20 x^{3}+60 x^{2}+60 x +20\right ) {\mathrm e}^{16 {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{4}+8 \left (-4-4 x \right ) {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{3}+4 \left (6 x^{2}+12 x +6\right ) {\ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )}^{2}+2 \left (-4 x^{3}-12 x^{2}-12 x -4\right ) \ln \left (\ln \left (-{\mathrm e}^{8}\right )\right )+x^{4}+4 x^{3}+6 x^{2}+4 x +1}}{4 \left (2 \ln \left (i \left (\pi -8 i\right )\right )-x -1\right )^{3}}\) | \(179\) |
Input:
int((-160*ln(ln(-exp(4)^2))^3+4*(60*x+60)*ln(ln(-exp(4)^2))^2+2*(-60*x^2-1 20*x-60)*ln(ln(-exp(4)^2))+20*x^3+60*x^2+60*x+20)*exp(16*ln(ln(-exp(4)^2)) ^4+8*(-4-4*x)*ln(ln(-exp(4)^2))^3+4*(6*x^2+12*x+6)*ln(ln(-exp(4)^2))^2+2*( -4*x^3-12*x^2-12*x-4)*ln(ln(-exp(4)^2))+x^4+4*x^3+6*x^2+4*x+1),x,method=_R ETURNVERBOSE)
Output:
5*exp(16*ln(ln(-exp(4)^2))^4+8*(-4-4*x)*ln(ln(-exp(4)^2))^3+4*(6*x^2+12*x+ 6)*ln(ln(-exp(4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*ln(ln(-exp(4)^2))+x^4+4*x^ 3+6*x^2+4*x+1)
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 255, normalized size of antiderivative = 12.14 \[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=5 \, \cosh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x + 8 \, \log \left (i \, \pi + 8\right ) - 1\right ) - 5 \, \sinh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x + 8 \, \log \left (i \, \pi + 8\right ) - 1\right ) \] Input:
integrate((-160*log(log(-exp(4)^2))^3+4*(60*x+60)*log(log(-exp(4)^2))^2+2* (-60*x^2-120*x-60)*log(log(-exp(4)^2))+20*x^3+60*x^2+60*x+20)*exp(16*log(l og(-exp(4)^2))^4+8*(-4-4*x)*log(log(-exp(4)^2))^3+4*(6*x^2+12*x+6)*log(log (-exp(4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*log(log(-exp(4)^2))+x^4+4*x^3+6*x^ 2+4*x+1),x, algorithm="fricas")
Output:
5*cosh(-x^4 + 8*x^3*log(I*pi + 8) - 24*x^2*log(I*pi + 8)^2 + 32*x*log(I*pi + 8)^3 - 16*log(I*pi + 8)^4 - 4*x^3 + 24*x^2*log(I*pi + 8) - 48*x*log(I*p i + 8)^2 + 32*log(I*pi + 8)^3 - 6*x^2 + 24*x*log(I*pi + 8) - 24*log(I*pi + 8)^2 - 4*x + 8*log(I*pi + 8) - 1) - 5*sinh(-x^4 + 8*x^3*log(I*pi + 8) - 2 4*x^2*log(I*pi + 8)^2 + 32*x*log(I*pi + 8)^3 - 16*log(I*pi + 8)^4 - 4*x^3 + 24*x^2*log(I*pi + 8) - 48*x*log(I*pi + 8)^2 + 32*log(I*pi + 8)^3 - 6*x^2 + 24*x*log(I*pi + 8) - 24*log(I*pi + 8)^2 - 4*x + 8*log(I*pi + 8) - 1)
Timed out. \[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=\text {Timed out} \] Input:
integrate((-160*ln(ln(-exp(4)**2))**3+4*(60*x+60)*ln(ln(-exp(4)**2))**2+2* (-60*x**2-120*x-60)*ln(ln(-exp(4)**2))+20*x**3+60*x**2+60*x+20)*exp(16*ln( ln(-exp(4)**2))**4+8*(-4-4*x)*ln(ln(-exp(4)**2))**3+4*(6*x**2+12*x+6)*ln(l n(-exp(4)**2))**2+2*(-4*x**3-12*x**2-12*x-4)*ln(ln(-exp(4)**2))+x**4+4*x** 3+6*x**2+4*x+1),x)
Output:
Timed out
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 281, normalized size of antiderivative = 13.38 \[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=\frac {20 \, {\left (\cosh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x - 1\right ) - \sinh \left (-x^{4} + 8 \, x^{3} \log \left (i \, \pi + 8\right ) - 24 \, x^{2} \log \left (i \, \pi + 8\right )^{2} + 32 \, x \log \left (i \, \pi + 8\right )^{3} - 16 \, \log \left (i \, \pi + 8\right )^{4} - 4 \, x^{3} + 24 \, x^{2} \log \left (i \, \pi + 8\right ) - 48 \, x \log \left (i \, \pi + 8\right )^{2} + 32 \, \log \left (i \, \pi + 8\right )^{3} - 6 \, x^{2} + 24 \, x \log \left (i \, \pi + 8\right ) - 24 \, \log \left (i \, \pi + 8\right )^{2} - 4 \, x - 1\right )\right )}}{67108864 i \, \pi + 4 \, \pi ^{8} - 256 i \, \pi ^{7} - 7168 \, \pi ^{6} + 114688 i \, \pi ^{5} + 1146880 \, \pi ^{4} - 7340032 i \, \pi ^{3} - 29360128 \, \pi ^{2} + 67108864} \] Input:
integrate((-160*log(log(-exp(4)^2))^3+4*(60*x+60)*log(log(-exp(4)^2))^2+2* (-60*x^2-120*x-60)*log(log(-exp(4)^2))+20*x^3+60*x^2+60*x+20)*exp(16*log(l og(-exp(4)^2))^4+8*(-4-4*x)*log(log(-exp(4)^2))^3+4*(6*x^2+12*x+6)*log(log (-exp(4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*log(log(-exp(4)^2))+x^4+4*x^3+6*x^ 2+4*x+1),x, algorithm="maxima")
Output:
20*(cosh(-x^4 + 8*x^3*log(I*pi + 8) - 24*x^2*log(I*pi + 8)^2 + 32*x*log(I* pi + 8)^3 - 16*log(I*pi + 8)^4 - 4*x^3 + 24*x^2*log(I*pi + 8) - 48*x*log(I *pi + 8)^2 + 32*log(I*pi + 8)^3 - 6*x^2 + 24*x*log(I*pi + 8) - 24*log(I*pi + 8)^2 - 4*x - 1) - sinh(-x^4 + 8*x^3*log(I*pi + 8) - 24*x^2*log(I*pi + 8 )^2 + 32*x*log(I*pi + 8)^3 - 16*log(I*pi + 8)^4 - 4*x^3 + 24*x^2*log(I*pi + 8) - 48*x*log(I*pi + 8)^2 + 32*log(I*pi + 8)^3 - 6*x^2 + 24*x*log(I*pi + 8) - 24*log(I*pi + 8)^2 - 4*x - 1))/(67108864*I*pi + 4*pi^8 - 256*I*pi^7 - 7168*pi^6 + 114688*I*pi^5 + 1146880*pi^4 - 7340032*I*pi^3 - 29360128*pi^ 2 + 67108864)
Leaf count of result is larger than twice the leaf count of optimal. 14865 vs. \(2 (16) = 32\).
Time = 0.78 (sec) , antiderivative size = 14865, normalized size of antiderivative = 707.86 \[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=\text {Too large to display} \] Input:
integrate((-160*log(log(-exp(4)^2))^3+4*(60*x+60)*log(log(-exp(4)^2))^2+2* (-60*x^2-120*x-60)*log(log(-exp(4)^2))+20*x^3+60*x^2+60*x+20)*exp(16*log(l og(-exp(4)^2))^4+8*(-4-4*x)*log(log(-exp(4)^2))^3+4*(6*x^2+12*x+6)*log(log (-exp(4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*log(log(-exp(4)^2))+x^4+4*x^3+6*x^ 2+4*x+1),x, algorithm="giac")
Output:
5*(pi^8*e^(x^4 - 24*x^2*arctan(1/8*pi)^2 + 16*arctan(1/8*pi)^4 - 4*x^3*log (pi^2 + 64) + 48*x*arctan(1/8*pi)^2*log(pi^2 + 64) + 6*x^2*log(pi^2 + 64)^ 2 - 24*arctan(1/8*pi)^2*log(pi^2 + 64)^2 - 4*x*log(pi^2 + 64)^3 + log(pi^2 + 64)^4 + 4*x^3 - 48*x*arctan(1/8*pi)^2 - 12*x^2*log(pi^2 + 64) + 48*arct an(1/8*pi)^2*log(pi^2 + 64) + 12*x*log(pi^2 + 64)^2 - 4*log(pi^2 + 64)^3 + 6*x^2 - 24*arctan(1/8*pi)^2 - 12*x*log(pi^2 + 64) + 6*log(pi^2 + 64)^2 + 4*x + 1)*tan(4*x^3*arctan(1/8*pi) - 16*x*arctan(1/8*pi)^3 - 12*x^2*arctan( 1/8*pi)*log(pi^2 + 64) + 12*x*arctan(1/8*pi)*log(pi^2 + 64)^2 + 12*x^2*arc tan(1/8*pi) - 24*x*arctan(1/8*pi)*log(pi^2 + 64) + 12*x*arctan(1/8*pi))^2* tan(16*arctan(1/8*pi)^3*log(pi^2 + 64) - 4*arctan(1/8*pi)*log(pi^2 + 64)^3 - 16*arctan(1/8*pi)^3 + 12*arctan(1/8*pi)*log(pi^2 + 64)^2 - 12*arctan(1/ 8*pi)*log(pi^2 + 64))^2 - pi^8*e^(x^4 - 24*x^2*arctan(1/8*pi)^2 + 16*arcta n(1/8*pi)^4 - 4*x^3*log(pi^2 + 64) + 48*x*arctan(1/8*pi)^2*log(pi^2 + 64) + 6*x^2*log(pi^2 + 64)^2 - 24*arctan(1/8*pi)^2*log(pi^2 + 64)^2 - 4*x*log( pi^2 + 64)^3 + log(pi^2 + 64)^4 + 4*x^3 - 48*x*arctan(1/8*pi)^2 - 12*x^2*l og(pi^2 + 64) + 48*arctan(1/8*pi)^2*log(pi^2 + 64) + 12*x*log(pi^2 + 64)^2 - 4*log(pi^2 + 64)^3 + 6*x^2 - 24*arctan(1/8*pi)^2 - 12*x*log(pi^2 + 64) + 6*log(pi^2 + 64)^2 + 4*x + 1)*tan(4*x^3*arctan(1/8*pi) - 16*x*arctan(1/8 *pi)^3 - 12*x^2*arctan(1/8*pi)*log(pi^2 + 64) + 12*x*arctan(1/8*pi)*log(pi ^2 + 64)^2 + 12*x^2*arctan(1/8*pi) - 24*x*arctan(1/8*pi)*log(pi^2 + 64)...
Time = 3.38 (sec) , antiderivative size = 168, normalized size of antiderivative = 8.00 \[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=\frac {5\,{\mathrm {e}}^{-32\,x\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^3}\,{\mathrm {e}}^{48\,x\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^2}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^4}\,\mathrm {e}\,{\mathrm {e}}^{24\,x^2\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^2}\,{\mathrm {e}}^{4\,x^3}\,{\mathrm {e}}^{6\,x^2}\,{\mathrm {e}}^{16\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^4}\,{\mathrm {e}}^{24\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^2}\,{\mathrm {e}}^{-32\,{\ln \left (8+\pi \,1{}\mathrm {i}\right )}^3}}{{\left (8+\pi \,1{}\mathrm {i}\right )}^{8\,x^3+24\,x^2+24\,x}\,\left (286720\,\pi ^4-7340032\,\pi ^2-1792\,\pi ^6+\pi ^8+16777216+\pi \,16777216{}\mathrm {i}-\pi ^3\,1835008{}\mathrm {i}+\pi ^5\,28672{}\mathrm {i}-\pi ^7\,64{}\mathrm {i}\right )} \] Input:
int(exp(4*x - 8*log(log(-exp(8)))^3*(4*x + 4) - 2*log(log(-exp(8)))*(12*x + 12*x^2 + 4*x^3 + 4) + 4*log(log(-exp(8)))^2*(12*x + 6*x^2 + 6) + 6*x^2 + 4*x^3 + x^4 + 16*log(log(-exp(8)))^4 + 1)*(60*x - 2*log(log(-exp(8)))*(12 0*x + 60*x^2 + 60) + 4*log(log(-exp(8)))^2*(60*x + 60) + 60*x^2 + 20*x^3 - 160*log(log(-exp(8)))^3 + 20),x)
Output:
(5*exp(-32*x*log(pi*1i + 8)^3)*exp(48*x*log(pi*1i + 8)^2)*exp(4*x)*exp(x^4 )*exp(1)*exp(24*x^2*log(pi*1i + 8)^2)*exp(4*x^3)*exp(6*x^2)*exp(16*log(pi* 1i + 8)^4)*exp(24*log(pi*1i + 8)^2)*exp(-32*log(pi*1i + 8)^3))/((pi*1i + 8 )^(24*x + 24*x^2 + 8*x^3)*(pi*16777216i - 7340032*pi^2 - pi^3*1835008i + 2 86720*pi^4 + pi^5*28672i - 1792*pi^6 - pi^7*64i + pi^8 + 16777216))
Time = 4.30 (sec) , antiderivative size = 129, normalized size of antiderivative = 6.14 \[ \int e^{1+4 x+6 x^2+4 x^3+x^4+\left (-4-12 x-12 x^2-4 x^3\right ) \log \left ((8+i \pi )^2\right )+\left (6+12 x+6 x^2\right ) \log ^2\left ((8+i \pi )^2\right )+(-4-4 x) \log ^3\left ((8+i \pi )^2\right )+\log ^4\left ((8+i \pi )^2\right )} \left (20+60 x+60 x^2+20 x^3+\left (-60-120 x-60 x^2\right ) \log \left ((8+i \pi )^2\right )+(60+60 x) \log ^2\left ((8+i \pi )^2\right )-20 \log ^3\left ((8+i \pi )^2\right )\right ) \, dx=\frac {5 e^{16 {\mathrm {log}\left (\mathrm {log}\left (-e^{8}\right )\right )}^{4}+24 {\mathrm {log}\left (\mathrm {log}\left (-e^{8}\right )\right )}^{2} x^{2}+48 {\mathrm {log}\left (\mathrm {log}\left (-e^{8}\right )\right )}^{2} x +24 {\mathrm {log}\left (\mathrm {log}\left (-e^{8}\right )\right )}^{2}+x^{4}+4 x^{3}+6 x^{2}+4 x} e}{e^{32 {\mathrm {log}\left (\mathrm {log}\left (-e^{8}\right )\right )}^{3} x +32 {\mathrm {log}\left (\mathrm {log}\left (-e^{8}\right )\right )}^{3}} \mathrm {log}\left (-e^{8}\right )^{8 x^{3}+24 x^{2}+24 x} \mathrm {log}\left (-e^{8}\right )^{8}} \] Input:
int((-160*log(log(-exp(4)^2))^3+4*(60*x+60)*log(log(-exp(4)^2))^2+2*(-60*x ^2-120*x-60)*log(log(-exp(4)^2))+20*x^3+60*x^2+60*x+20)*exp(16*log(log(-ex p(4)^2))^4+8*(-4-4*x)*log(log(-exp(4)^2))^3+4*(6*x^2+12*x+6)*log(log(-exp( 4)^2))^2+2*(-4*x^3-12*x^2-12*x-4)*log(log(-exp(4)^2))+x^4+4*x^3+6*x^2+4*x+ 1),x)
Output:
(5*e**(16*log(log( - e**8))**4 + 24*log(log( - e**8))**2*x**2 + 48*log(log ( - e**8))**2*x + 24*log(log( - e**8))**2 + x**4 + 4*x**3 + 6*x**2 + 4*x)* e)/(e**(32*log(log( - e**8))**3*x + 32*log(log( - e**8))**3)*log( - e**8)* *(8*x**3 + 24*x**2 + 24*x)*log( - e**8)**8)