Integrand size = 232, antiderivative size = 24 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {e^{8+x^2} \left (e^4+x+\frac {x}{\log (4)}\right )^4}{x^2} \] Output:
exp(x^2)/x^2*(1/2*x/ln(2)+exp(4)+x)^4*exp(2)^4
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {e^{8+x^2} \left (x+e^4 \log (4)+x \log (4)\right )^4}{x^2 \log ^4(4)} \] Input:
Integrate[(E^x^2*(E^8*(2*x^4 + 2*x^6) + (E^12*(4*x^3 + 8*x^5) + E^8*(8*x^4 + 8*x^6))*Log[4] + (12*E^16*x^4 + E^12*(12*x^3 + 24*x^5) + E^8*(12*x^4 + 12*x^6))*Log[4]^2 + (24*E^16*x^4 + E^20*(-4*x + 8*x^3) + E^12*(12*x^3 + 24 *x^5) + E^8*(8*x^4 + 8*x^6))*Log[4]^3 + (12*E^16*x^4 + E^24*(-2 + 2*x^2) + E^20*(-4*x + 8*x^3) + E^12*(4*x^3 + 8*x^5) + E^8*(2*x^4 + 2*x^6))*Log[4]^ 4))/(x^3*Log[4]^4),x]
Output:
(E^(8 + x^2)*(x + E^4*Log[4] + x*Log[4])^4)/(x^2*Log[4]^4)
Time = 1.88 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {27, 27, 7239, 2726}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{x^2} \left (e^8 \left (2 x^6+2 x^4\right )+\left (24 e^{16} x^4+e^{20} \left (8 x^3-4 x\right )+e^8 \left (8 x^6+8 x^4\right )+e^{12} \left (24 x^5+12 x^3\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^8 \left (12 x^6+12 x^4\right )+e^{12} \left (24 x^5+12 x^3\right )\right ) \log ^2(4)+\left (e^8 \left (8 x^6+8 x^4\right )+e^{12} \left (8 x^5+4 x^3\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{20} \left (8 x^3-4 x\right )+e^{24} \left (2 x^2-2\right )+e^8 \left (2 x^6+2 x^4\right )+e^{12} \left (8 x^5+4 x^3\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 e^{x^2} \left (e^8 \left (x^6+x^4\right )+\left (6 e^{16} x^4-e^{24} \left (1-x^2\right )-2 e^{20} \left (x-2 x^3\right )+2 e^{12} \left (2 x^5+x^3\right )+e^8 \left (x^6+x^4\right )\right ) \log ^4(4)+2 \left (6 e^{16} x^4-e^{20} \left (x-2 x^3\right )+3 e^{12} \left (2 x^5+x^3\right )+2 e^8 \left (x^6+x^4\right )\right ) \log ^3(4)+6 \left (e^{16} x^4+e^{12} \left (2 x^5+x^3\right )+e^8 \left (x^6+x^4\right )\right ) \log ^2(4)+2 \left (e^{12} \left (2 x^5+x^3\right )+2 e^8 \left (x^6+x^4\right )\right ) \log (4)\right )}{x^3}dx}{\log ^4(4)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {e^{x^2} \left (e^8 \left (x^6+x^4\right )+\left (6 e^{16} x^4-e^{24} \left (1-x^2\right )-2 e^{20} \left (x-2 x^3\right )+2 e^{12} \left (2 x^5+x^3\right )+e^8 \left (x^6+x^4\right )\right ) \log ^4(4)+2 \left (6 e^{16} x^4-e^{20} \left (x-2 x^3\right )+3 e^{12} \left (2 x^5+x^3\right )+2 e^8 \left (x^6+x^4\right )\right ) \log ^3(4)+6 \left (e^{16} x^4+e^{12} \left (2 x^5+x^3\right )+e^8 \left (x^6+x^4\right )\right ) \log ^2(4)+2 \left (e^{12} \left (2 x^5+x^3\right )+2 e^8 \left (x^6+x^4\right )\right ) \log (4)\right )}{x^3}dx}{\log ^4(4)}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \frac {e^{x^2+8} \left ((1+\log (4)) x+e^4 \log (4)\right )^3 \left ((1+\log (4)) x^3+e^4 \log (4) x^2+(1+\log (4)) x-e^4 \log (4)\right )}{x^3}dx}{\log ^4(4)}\) |
| \(\Big \downarrow \) 2726 |
| \(\displaystyle \frac {e^{x^2+8} \left (x (1+\log (4))+e^4 \log (4)\right )^3 \left (x^3 (1+\log (4))+e^4 x^2 \log (4)\right )}{x^4 \log ^4(4)}\) |
Input:
Int[(E^x^2*(E^8*(2*x^4 + 2*x^6) + (E^12*(4*x^3 + 8*x^5) + E^8*(8*x^4 + 8*x ^6))*Log[4] + (12*E^16*x^4 + E^12*(12*x^3 + 24*x^5) + E^8*(12*x^4 + 12*x^6 ))*Log[4]^2 + (24*E^16*x^4 + E^20*(-4*x + 8*x^3) + E^12*(12*x^3 + 24*x^5) + E^8*(8*x^4 + 8*x^6))*Log[4]^3 + (12*E^16*x^4 + E^24*(-2 + 2*x^2) + E^20* (-4*x + 8*x^3) + E^12*(4*x^3 + 8*x^5) + E^8*(2*x^4 + 2*x^6))*Log[4]^4))/(x ^3*Log[4]^4),x]
Output:
(E^(8 + x^2)*(E^4*Log[4] + x*(1 + Log[4]))^3*(E^4*x^2*Log[4] + x^3*(1 + Lo g[4])))/(x^4*Log[4]^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, 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. \(154\) vs. \(2(25)=50\).
Time = 0.40 (sec) , antiderivative size = 155, normalized size of antiderivative = 6.46
| method | result | size |
| risch | \(\frac {\left (16 \ln \left (2\right )^{4} {\mathrm e}^{16}+64 \ln \left (2\right )^{4} {\mathrm e}^{12} x +96 \ln \left (2\right )^{4} {\mathrm e}^{8} x^{2}+32 \ln \left (2\right )^{3} {\mathrm e}^{12} x +64 \ln \left (2\right )^{4} {\mathrm e}^{4} x^{3}+96 \ln \left (2\right )^{3} {\mathrm e}^{8} x^{2}+16 x^{4} \ln \left (2\right )^{4}+96 \ln \left (2\right )^{3} {\mathrm e}^{4} x^{3}+24 \ln \left (2\right )^{2} {\mathrm e}^{8} x^{2}+32 x^{4} \ln \left (2\right )^{3}+48 \ln \left (2\right )^{2} {\mathrm e}^{4} x^{3}+24 x^{4} \ln \left (2\right )^{2}+8 \ln \left (2\right ) {\mathrm e}^{4} x^{3}+8 x^{4} \ln \left (2\right )+x^{4}\right ) {\mathrm e}^{x^{2}+8}}{16 x^{2} \ln \left (2\right )^{4}}\) | \(155\) |
| norman | \(\frac {\ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{16} {\mathrm e}^{x^{2}}+\frac {{\mathrm e}^{8} {\mathrm e}^{4} \left (8 \ln \left (2\right )^{3}+12 \ln \left (2\right )^{2}+6 \ln \left (2\right )+1\right ) x^{3} {\mathrm e}^{x^{2}}}{2}+\frac {{\mathrm e}^{8} \left (16 \ln \left (2\right )^{4}+32 \ln \left (2\right )^{3}+24 \ln \left (2\right )^{2}+8 \ln \left (2\right )+1\right ) x^{4} {\mathrm e}^{x^{2}}}{16 \ln \left (2\right )}+\frac {3 \ln \left (2\right ) \left ({\mathrm e}^{8}\right )^{2} \left (4 \ln \left (2\right )^{2}+4 \ln \left (2\right )+1\right ) x^{2} {\mathrm e}^{x^{2}}}{2}+2 \ln \left (2\right )^{2} {\mathrm e}^{8} {\mathrm e}^{12} \left (1+2 \ln \left (2\right )\right ) x \,{\mathrm e}^{x^{2}}}{x^{2} \ln \left (2\right )^{3}}\) | \(157\) |
| gosper | \(\frac {{\mathrm e}^{8} \left (16 \ln \left (2\right )^{4} {\mathrm e}^{16}+64 \ln \left (2\right )^{4} {\mathrm e}^{12} x +96 \ln \left (2\right )^{4} {\mathrm e}^{8} x^{2}+32 \ln \left (2\right )^{3} {\mathrm e}^{12} x +64 \ln \left (2\right )^{4} {\mathrm e}^{4} x^{3}+96 \ln \left (2\right )^{3} {\mathrm e}^{8} x^{2}+16 x^{4} \ln \left (2\right )^{4}+96 \ln \left (2\right )^{3} {\mathrm e}^{4} x^{3}+24 \ln \left (2\right )^{2} {\mathrm e}^{8} x^{2}+32 x^{4} \ln \left (2\right )^{3}+48 \ln \left (2\right )^{2} {\mathrm e}^{4} x^{3}+24 x^{4} \ln \left (2\right )^{2}+8 \ln \left (2\right ) {\mathrm e}^{4} x^{3}+8 x^{4} \ln \left (2\right )+x^{4}\right ) {\mathrm e}^{x^{2}}}{16 \ln \left (2\right )^{4} x^{2}}\) | \(169\) |
| parallelrisch | \(\frac {16 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{16} {\mathrm e}^{x^{2}}+64 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{12} x \,{\mathrm e}^{x^{2}}+96 \ln \left (2\right )^{4} \left ({\mathrm e}^{8}\right )^{2} x^{2} {\mathrm e}^{x^{2}}+64 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{4} x^{3} {\mathrm e}^{x^{2}}+16 \ln \left (2\right )^{4} {\mathrm e}^{8} x^{4} {\mathrm e}^{x^{2}}+32 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{12} x \,{\mathrm e}^{x^{2}}+96 \ln \left (2\right )^{3} \left ({\mathrm e}^{8}\right )^{2} x^{2} {\mathrm e}^{x^{2}}+96 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{4} x^{3} {\mathrm e}^{x^{2}}+32 \ln \left (2\right )^{3} {\mathrm e}^{8} x^{4} {\mathrm e}^{x^{2}}+24 \ln \left (2\right )^{2} \left ({\mathrm e}^{8}\right )^{2} x^{2} {\mathrm e}^{x^{2}}+48 \ln \left (2\right )^{2} {\mathrm e}^{8} {\mathrm e}^{4} x^{3} {\mathrm e}^{x^{2}}+24 \ln \left (2\right )^{2} {\mathrm e}^{8} x^{4} {\mathrm e}^{x^{2}}+8 \ln \left (2\right ) {\mathrm e}^{8} {\mathrm e}^{4} x^{3} {\mathrm e}^{x^{2}}+8 \ln \left (2\right ) {\mathrm e}^{8} x^{4} {\mathrm e}^{x^{2}}+{\mathrm e}^{8} x^{4} {\mathrm e}^{x^{2}}}{16 \ln \left (2\right )^{4} x^{2}}\) | \(282\) |
| meijerg | \({\mathrm e}^{24} \left (\frac {1}{x^{2}}+1-2 \ln \left (x \right )-i \pi -\frac {2 x^{2}+2}{2 x^{2}}+\frac {{\mathrm e}^{x^{2}}}{x^{2}}+\ln \left (-x^{2}\right )+\operatorname {expIntegral}_{1}\left (-x^{2}\right )\right )+\frac {\left (2 \ln \left (2\right )^{4} {\mathrm e}^{8}+4 \ln \left (2\right )^{3} {\mathrm e}^{8}+3 \ln \left (2\right )^{2} {\mathrm e}^{8}+\ln \left (2\right ) {\mathrm e}^{8}+\frac {{\mathrm e}^{8}}{8}\right ) \left (1-\frac {\left (-2 x^{2}+2\right ) {\mathrm e}^{x^{2}}}{2}\right )}{2 \ln \left (2\right )^{4}}-\frac {\left (12 \ln \left (2\right )^{4} {\mathrm e}^{16}+12 \ln \left (2\right )^{3} {\mathrm e}^{16}+2 \ln \left (2\right )^{4} {\mathrm e}^{8}+3 \ln \left (2\right )^{2} {\mathrm e}^{16}+4 \ln \left (2\right )^{3} {\mathrm e}^{8}+3 \ln \left (2\right )^{2} {\mathrm e}^{8}+\ln \left (2\right ) {\mathrm e}^{8}+\frac {{\mathrm e}^{8}}{8}\right ) \left (1-{\mathrm e}^{x^{2}}\right )}{2 \ln \left (2\right )^{4}}+\frac {i \left (8 \ln \left (2\right )^{4} {\mathrm e}^{12}+12 \ln \left (2\right )^{3} {\mathrm e}^{12}+6 \ln \left (2\right )^{2} {\mathrm e}^{12}+{\mathrm e}^{12} \ln \left (2\right )\right ) \left (-i {\mathrm e}^{x^{2}} x +\frac {i \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2}\right )}{2 \ln \left (2\right )^{4}}+\frac {\left (8 \ln \left (2\right )^{4} {\mathrm e}^{20}+4 \ln \left (2\right )^{3} {\mathrm e}^{20}+4 \ln \left (2\right )^{4} {\mathrm e}^{12}+6 \ln \left (2\right )^{3} {\mathrm e}^{12}+3 \ln \left (2\right )^{2} {\mathrm e}^{12}+\frac {{\mathrm e}^{12} \ln \left (2\right )}{2}\right ) \sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2 \ln \left (2\right )^{4}}+\frac {i \left (-4 \ln \left (2\right )^{4} {\mathrm e}^{20}-2 \ln \left (2\right )^{3} {\mathrm e}^{20}\right ) \left (\frac {2 i {\mathrm e}^{x^{2}}}{x}-2 i \sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )}{2 \ln \left (2\right )^{4}}+{\mathrm e}^{24} \left (2 \ln \left (x \right )+i \pi -\ln \left (-x^{2}\right )-\operatorname {expIntegral}_{1}\left (-x^{2}\right )\right )\) | \(362\) |
| default | \(\frac {64 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{12} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+32 \ln \left (2\right )^{4} {\mathrm e}^{4} {\mathrm e}^{8} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+32 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{12} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+48 \ln \left (2\right )^{3} {\mathrm e}^{4} {\mathrm e}^{8} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+24 \ln \left (2\right )^{2} {\mathrm e}^{4} {\mathrm e}^{8} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+4 \ln \left (2\right ) {\mathrm e}^{4} {\mathrm e}^{8} \sqrt {\pi }\, \operatorname {erfi}\left (x \right )+192 \ln \left (2\right )^{3} {\mathrm e}^{4} {\mathrm e}^{8} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+128 \ln \left (2\right )^{4} {\mathrm e}^{4} {\mathrm e}^{8} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+16 \ln \left (2\right ) {\mathrm e}^{4} {\mathrm e}^{8} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+96 \ln \left (2\right )^{2} {\mathrm e}^{4} {\mathrm e}^{8} \left (\frac {{\mathrm e}^{x^{2}} x}{2}-\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{4}\right )+96 \ln \left (2\right )^{3} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}+96 \ln \left (2\right )^{4} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}-16 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{16} \operatorname {expIntegral}_{1}\left (-x^{2}\right )+24 \ln \left (2\right )^{2} \left ({\mathrm e}^{8}\right )^{2} {\mathrm e}^{x^{2}}-32 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{12} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )-64 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{12} \left (-\frac {{\mathrm e}^{x^{2}}}{x}+\sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )-32 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{16} \left (-\frac {{\mathrm e}^{x^{2}}}{2 x^{2}}-\frac {\operatorname {expIntegral}_{1}\left (-x^{2}\right )}{2}\right )+32 \ln \left (2\right )^{4} {\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+24 \ln \left (2\right )^{2} {\mathrm e}^{8} {\mathrm e}^{x^{2}}+64 \ln \left (2\right )^{3} {\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+48 \ln \left (2\right )^{2} {\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+16 \ln \left (2\right ) {\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )+16 \ln \left (2\right )^{4} {\mathrm e}^{8} {\mathrm e}^{x^{2}}+32 \ln \left (2\right )^{3} {\mathrm e}^{8} {\mathrm e}^{x^{2}}+8 \ln \left (2\right ) {\mathrm e}^{8} {\mathrm e}^{x^{2}}+{\mathrm e}^{8} {\mathrm e}^{x^{2}}+2 \,{\mathrm e}^{8} \left (\frac {x^{2} {\mathrm e}^{x^{2}}}{2}-\frac {{\mathrm e}^{x^{2}}}{2}\right )}{16 \ln \left (2\right )^{4}}\) | \(573\) |
Input:
int(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4)^3+12 *x^4*exp(2)^4*exp(4)^2+(8*x^5+4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*exp(2)^ 4)*ln(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4)^2+(24*x ^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*ln(2)^3+4*(12*x^4*exp(2 )^4*exp(4)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2)^4)*ln( 2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*ln(2)+(2*x^6 +2*x^4)*exp(2)^4)*exp(x^2)/x^3/ln(2)^4,x,method=_RETURNVERBOSE)
Output:
1/16/x^2*(16*ln(2)^4*exp(16)+64*ln(2)^4*exp(12)*x+96*ln(2)^4*exp(8)*x^2+32 *ln(2)^3*exp(12)*x+64*ln(2)^4*exp(4)*x^3+96*ln(2)^3*exp(8)*x^2+16*x^4*ln(2 )^4+96*ln(2)^3*exp(4)*x^3+24*ln(2)^2*exp(8)*x^2+32*x^4*ln(2)^3+48*ln(2)^2* exp(4)*x^3+24*x^4*ln(2)^2+8*ln(2)*exp(4)*x^3+8*x^4*ln(2)+x^4)/ln(2)^4*exp( x^2+8)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 5.33 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {{\left (x^{4} e^{8} + 16 \, {\left (x^{4} e^{8} + 4 \, x^{3} e^{12} + 6 \, x^{2} e^{16} + 4 \, x e^{20} + e^{24}\right )} \log \left (2\right )^{4} + 32 \, {\left (x^{4} e^{8} + 3 \, x^{3} e^{12} + 3 \, x^{2} e^{16} + x e^{20}\right )} \log \left (2\right )^{3} + 24 \, {\left (x^{4} e^{8} + 2 \, x^{3} e^{12} + x^{2} e^{16}\right )} \log \left (2\right )^{2} + 8 \, {\left (x^{4} e^{8} + x^{3} e^{12}\right )} \log \left (2\right )\right )} e^{\left (x^{2}\right )}}{16 \, x^{2} \log \left (2\right )^{4}} \] Input:
integrate(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4 )^3+12*x^4*exp(2)^4*exp(4)^2+(8*x^5+4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*e xp(2)^4)*log(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4)^ 2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)^3+4*(12*x ^4*exp(2)^4*exp(4)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2 )^4)*log(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log (2)+(2*x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/log(2)^4,x, algorithm="fricas")
Output:
1/16*(x^4*e^8 + 16*(x^4*e^8 + 4*x^3*e^12 + 6*x^2*e^16 + 4*x*e^20 + e^24)*l og(2)^4 + 32*(x^4*e^8 + 3*x^3*e^12 + 3*x^2*e^16 + x*e^20)*log(2)^3 + 24*(x ^4*e^8 + 2*x^3*e^12 + x^2*e^16)*log(2)^2 + 8*(x^4*e^8 + x^3*e^12)*log(2))* e^(x^2)/(x^2*log(2)^4)
Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (24) = 48\).
Time = 0.18 (sec) , antiderivative size = 201, normalized size of antiderivative = 8.38 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {\left (x^{4} e^{8} + 16 x^{4} e^{8} \log {\left (2 \right )}^{4} + 8 x^{4} e^{8} \log {\left (2 \right )} + 32 x^{4} e^{8} \log {\left (2 \right )}^{3} + 24 x^{4} e^{8} \log {\left (2 \right )}^{2} + 8 x^{3} e^{12} \log {\left (2 \right )} + 64 x^{3} e^{12} \log {\left (2 \right )}^{4} + 48 x^{3} e^{12} \log {\left (2 \right )}^{2} + 96 x^{3} e^{12} \log {\left (2 \right )}^{3} + 24 x^{2} e^{16} \log {\left (2 \right )}^{2} + 96 x^{2} e^{16} \log {\left (2 \right )}^{4} + 96 x^{2} e^{16} \log {\left (2 \right )}^{3} + 32 x e^{20} \log {\left (2 \right )}^{3} + 64 x e^{20} \log {\left (2 \right )}^{4} + 16 e^{24} \log {\left (2 \right )}^{4}\right ) e^{x^{2}}}{16 x^{2} \log {\left (2 \right )}^{4}} \] Input:
integrate(1/16*(16*((2*x**2-2)*exp(2)**4*exp(4)**4+(8*x**3-4*x)*exp(2)**4* exp(4)**3+12*x**4*exp(2)**4*exp(4)**2+(8*x**5+4*x**3)*exp(2)**4*exp(4)+(2* x**6+2*x**4)*exp(2)**4)*ln(2)**4+8*((8*x**3-4*x)*exp(2)**4*exp(4)**3+24*x* *4*exp(2)**4*exp(4)**2+(24*x**5+12*x**3)*exp(2)**4*exp(4)+(8*x**6+8*x**4)* exp(2)**4)*ln(2)**3+4*(12*x**4*exp(2)**4*exp(4)**2+(24*x**5+12*x**3)*exp(2 )**4*exp(4)+(12*x**6+12*x**4)*exp(2)**4)*ln(2)**2+2*((8*x**5+4*x**3)*exp(2 )**4*exp(4)+(8*x**6+8*x**4)*exp(2)**4)*ln(2)+(2*x**6+2*x**4)*exp(2)**4)*ex p(x**2)/x**3/ln(2)**4,x)
Output:
(x**4*exp(8) + 16*x**4*exp(8)*log(2)**4 + 8*x**4*exp(8)*log(2) + 32*x**4*e xp(8)*log(2)**3 + 24*x**4*exp(8)*log(2)**2 + 8*x**3*exp(12)*log(2) + 64*x* *3*exp(12)*log(2)**4 + 48*x**3*exp(12)*log(2)**2 + 96*x**3*exp(12)*log(2)* *3 + 24*x**2*exp(16)*log(2)**2 + 96*x**2*exp(16)*log(2)**4 + 96*x**2*exp(1 6)*log(2)**3 + 32*x*exp(20)*log(2)**3 + 64*x*exp(20)*log(2)**4 + 16*exp(24 )*log(2)**4)*exp(x**2)/(16*x**2*log(2)**4)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.09 (sec) , antiderivative size = 467, normalized size of antiderivative = 19.46 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx =\text {Too large to display} \] Input:
integrate(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4 )^3+12*x^4*exp(2)^4*exp(4)^2+(8*x^5+4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*e xp(2)^4)*log(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4)^ 2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)^3+4*(12*x ^4*exp(2)^4*exp(4)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2 )^4)*log(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log (2)+(2*x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/log(2)^4,x, algorithm="maxima")
Output:
-1/16*(64*I*sqrt(pi)*erf(I*x)*e^20*log(2)^4 + 32*I*sqrt(pi)*erf(I*x)*e^12* log(2)^4 - 16*Ei(x^2)*e^24*log(2)^4 - 16*(x^2*e^8 - e^8)*e^(x^2)*log(2)^4 + 16*e^24*gamma(-1, -x^2)*log(2)^4 + 32*I*sqrt(pi)*erf(I*x)*e^20*log(2)^3 + 48*I*sqrt(pi)*erf(I*x)*e^12*log(2)^3 - 32*sqrt(-x^2)*e^20*gamma(-1/2, -x ^2)*log(2)^4/x - 32*(x^2*e^8 - e^8)*e^(x^2)*log(2)^3 + 32*(-I*sqrt(pi)*erf (I*x)*e^12 - 2*x*e^(x^2 + 12))*log(2)^4 - 96*e^(x^2 + 16)*log(2)^4 - 16*e^ (x^2 + 8)*log(2)^4 + 24*I*sqrt(pi)*erf(I*x)*e^12*log(2)^2 - 16*sqrt(-x^2)* e^20*gamma(-1/2, -x^2)*log(2)^3/x - 24*(x^2*e^8 - e^8)*e^(x^2)*log(2)^2 + 48*(-I*sqrt(pi)*erf(I*x)*e^12 - 2*x*e^(x^2 + 12))*log(2)^3 - 96*e^(x^2 + 1 6)*log(2)^3 - 32*e^(x^2 + 8)*log(2)^3 + 4*I*sqrt(pi)*erf(I*x)*e^12*log(2) - 8*(x^2*e^8 - e^8)*e^(x^2)*log(2) + 24*(-I*sqrt(pi)*erf(I*x)*e^12 - 2*x*e ^(x^2 + 12))*log(2)^2 - 24*e^(x^2 + 16)*log(2)^2 - 24*e^(x^2 + 8)*log(2)^2 - (x^2*e^8 - e^8)*e^(x^2) + 4*(-I*sqrt(pi)*erf(I*x)*e^12 - 2*x*e^(x^2 + 1 2))*log(2) - 8*e^(x^2 + 8)*log(2) - e^(x^2 + 8))/log(2)^4
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 219, normalized size of antiderivative = 9.12 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {16 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right )^{4} + 32 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right )^{3} + 64 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right )^{4} + 24 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right )^{2} + 96 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right )^{3} + 96 \, x^{2} e^{\left (x^{2} + 16\right )} \log \left (2\right )^{4} + 8 \, x^{4} e^{\left (x^{2} + 8\right )} \log \left (2\right ) + 48 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right )^{2} + 96 \, x^{2} e^{\left (x^{2} + 16\right )} \log \left (2\right )^{3} + 64 \, x e^{\left (x^{2} + 20\right )} \log \left (2\right )^{4} + x^{4} e^{\left (x^{2} + 8\right )} + 8 \, x^{3} e^{\left (x^{2} + 12\right )} \log \left (2\right ) + 24 \, x^{2} e^{\left (x^{2} + 16\right )} \log \left (2\right )^{2} + 32 \, x e^{\left (x^{2} + 20\right )} \log \left (2\right )^{3} + 16 \, e^{\left (x^{2} + 24\right )} \log \left (2\right )^{4}}{16 \, x^{2} \log \left (2\right )^{4}} \] Input:
integrate(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4 )^3+12*x^4*exp(2)^4*exp(4)^2+(8*x^5+4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*e xp(2)^4)*log(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4)^ 2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)^3+4*(12*x ^4*exp(2)^4*exp(4)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2 )^4)*log(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log (2)+(2*x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/log(2)^4,x, algorithm="giac")
Output:
1/16*(16*x^4*e^(x^2 + 8)*log(2)^4 + 32*x^4*e^(x^2 + 8)*log(2)^3 + 64*x^3*e ^(x^2 + 12)*log(2)^4 + 24*x^4*e^(x^2 + 8)*log(2)^2 + 96*x^3*e^(x^2 + 12)*l og(2)^3 + 96*x^2*e^(x^2 + 16)*log(2)^4 + 8*x^4*e^(x^2 + 8)*log(2) + 48*x^3 *e^(x^2 + 12)*log(2)^2 + 96*x^2*e^(x^2 + 16)*log(2)^3 + 64*x*e^(x^2 + 20)* log(2)^4 + x^4*e^(x^2 + 8) + 8*x^3*e^(x^2 + 12)*log(2) + 24*x^2*e^(x^2 + 1 6)*log(2)^2 + 32*x*e^(x^2 + 20)*log(2)^3 + 16*e^(x^2 + 24)*log(2)^4)/(x^2* log(2)^4)
Time = 3.65 (sec) , antiderivative size = 159, normalized size of antiderivative = 6.62 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {{\mathrm {e}}^{x^2}\,\left (3\,{\mathrm {e}}^{16}+12\,{\mathrm {e}}^{16}\,\ln \left (2\right )+12\,{\mathrm {e}}^{16}\,{\ln \left (2\right )}^2\right )}{2\,{\ln \left (2\right )}^2}+\frac {{\mathrm {e}}^{x^2+24}\,{\ln \left (2\right )}^4+2\,x\,{\mathrm {e}}^{x^2+20}\,{\ln \left (2\right )}^3\,\left (2\,\ln \left (2\right )+1\right )}{x^2\,{\ln \left (2\right )}^4}+\frac {x\,{\mathrm {e}}^{x^2}\,\left (4\,{\mathrm {e}}^{12}\,\ln \left (2\right )+24\,{\mathrm {e}}^{12}\,{\ln \left (2\right )}^2+48\,{\mathrm {e}}^{12}\,{\ln \left (2\right )}^3+32\,{\mathrm {e}}^{12}\,{\ln \left (2\right )}^4\right )}{8\,{\ln \left (2\right )}^4}+\frac {x^2\,{\mathrm {e}}^{x^2}\,\left (\frac {{\mathrm {e}}^8}{2}+4\,{\mathrm {e}}^8\,\ln \left (2\right )+12\,{\mathrm {e}}^8\,{\ln \left (2\right )}^2+16\,{\mathrm {e}}^8\,{\ln \left (2\right )}^3+8\,{\mathrm {e}}^8\,{\ln \left (2\right )}^4\right )}{8\,{\ln \left (2\right )}^4} \] Input:
int((exp(x^2)*(4*log(2)^2*(exp(8)*(12*x^4 + 12*x^6) + exp(12)*(12*x^3 + 24 *x^5) + 12*x^4*exp(16)) + 2*log(2)*(exp(12)*(4*x^3 + 8*x^5) + exp(8)*(8*x^ 4 + 8*x^6)) + 8*log(2)^3*(exp(8)*(8*x^4 + 8*x^6) - exp(20)*(4*x - 8*x^3) + exp(12)*(12*x^3 + 24*x^5) + 24*x^4*exp(16)) + exp(8)*(2*x^4 + 2*x^6) + 16 *log(2)^4*(exp(24)*(2*x^2 - 2) - exp(20)*(4*x - 8*x^3) + exp(8)*(2*x^4 + 2 *x^6) + exp(12)*(4*x^3 + 8*x^5) + 12*x^4*exp(16))))/(16*x^3*log(2)^4),x)
Output:
(exp(x^2)*(3*exp(16) + 12*exp(16)*log(2) + 12*exp(16)*log(2)^2))/(2*log(2) ^2) + (exp(x^2 + 24)*log(2)^4 + 2*x*exp(x^2 + 20)*log(2)^3*(2*log(2) + 1)) /(x^2*log(2)^4) + (x*exp(x^2)*(4*exp(12)*log(2) + 24*exp(12)*log(2)^2 + 48 *exp(12)*log(2)^3 + 32*exp(12)*log(2)^4))/(8*log(2)^4) + (x^2*exp(x^2)*(ex p(8)/2 + 4*exp(8)*log(2) + 12*exp(8)*log(2)^2 + 16*exp(8)*log(2)^3 + 8*exp (8)*log(2)^4))/(8*log(2)^4)
Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 6.92 \[ \int \frac {e^{x^2} \left (e^8 \left (2 x^4+2 x^6\right )+\left (e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log (4)+\left (12 e^{16} x^4+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (12 x^4+12 x^6\right )\right ) \log ^2(4)+\left (24 e^{16} x^4+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (12 x^3+24 x^5\right )+e^8 \left (8 x^4+8 x^6\right )\right ) \log ^3(4)+\left (12 e^{16} x^4+e^{24} \left (-2+2 x^2\right )+e^{20} \left (-4 x+8 x^3\right )+e^{12} \left (4 x^3+8 x^5\right )+e^8 \left (2 x^4+2 x^6\right )\right ) \log ^4(4)\right )}{x^3 \log ^4(4)} \, dx=\frac {e^{x^{2}} e^{8} \left (16 \mathrm {log}\left (2\right )^{4} e^{16}+64 \mathrm {log}\left (2\right )^{4} e^{12} x +96 \mathrm {log}\left (2\right )^{4} e^{8} x^{2}+64 \mathrm {log}\left (2\right )^{4} e^{4} x^{3}+16 \mathrm {log}\left (2\right )^{4} x^{4}+32 \mathrm {log}\left (2\right )^{3} e^{12} x +96 \mathrm {log}\left (2\right )^{3} e^{8} x^{2}+96 \mathrm {log}\left (2\right )^{3} e^{4} x^{3}+32 \mathrm {log}\left (2\right )^{3} x^{4}+24 \mathrm {log}\left (2\right )^{2} e^{8} x^{2}+48 \mathrm {log}\left (2\right )^{2} e^{4} x^{3}+24 \mathrm {log}\left (2\right )^{2} x^{4}+8 \,\mathrm {log}\left (2\right ) e^{4} x^{3}+8 \,\mathrm {log}\left (2\right ) x^{4}+x^{4}\right )}{16 \mathrm {log}\left (2\right )^{4} x^{2}} \] Input:
int(1/16*(16*((2*x^2-2)*exp(2)^4*exp(4)^4+(8*x^3-4*x)*exp(2)^4*exp(4)^3+12 *x^4*exp(2)^4*exp(4)^2+(8*x^5+4*x^3)*exp(2)^4*exp(4)+(2*x^6+2*x^4)*exp(2)^ 4)*log(2)^4+8*((8*x^3-4*x)*exp(2)^4*exp(4)^3+24*x^4*exp(2)^4*exp(4)^2+(24* x^5+12*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)^3+4*(12*x^4*exp (2)^4*exp(4)^2+(24*x^5+12*x^3)*exp(2)^4*exp(4)+(12*x^6+12*x^4)*exp(2)^4)*l og(2)^2+2*((8*x^5+4*x^3)*exp(2)^4*exp(4)+(8*x^6+8*x^4)*exp(2)^4)*log(2)+(2 *x^6+2*x^4)*exp(2)^4)*exp(x^2)/x^3/log(2)^4,x)
Output:
(e**(x**2)*e**8*(16*log(2)**4*e**16 + 64*log(2)**4*e**12*x + 96*log(2)**4* e**8*x**2 + 64*log(2)**4*e**4*x**3 + 16*log(2)**4*x**4 + 32*log(2)**3*e**1 2*x + 96*log(2)**3*e**8*x**2 + 96*log(2)**3*e**4*x**3 + 32*log(2)**3*x**4 + 24*log(2)**2*e**8*x**2 + 48*log(2)**2*e**4*x**3 + 24*log(2)**2*x**4 + 8* log(2)*e**4*x**3 + 8*log(2)*x**4 + x**4))/(16*log(2)**4*x**2)