Integrand size = 90, antiderivative size = 24 \[ \int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx=e^{1+e^{x^2} \left (x+\log \left (\frac {x^2}{256+x^2}\right )\right )} \]
Time = 3.77 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx=e^{1+e^{x^2} x} \left (\frac {x^2}{256+x^2}\right )^{e^{x^2}} \]
Integrate[(E^(1 + E^x^2*x + E^x^2*Log[x^2/(256 + x^2)])*(E^x^2*(512 + 256* x + 513*x^3 + 2*x^5) + E^x^2*(512*x^2 + 2*x^4)*Log[x^2/(256 + x^2)]))/(256 *x + x^3),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1} \left (e^{x^2} \left (2 x^4+512 x^2\right ) \log \left (\frac {x^2}{x^2+256}\right )+e^{x^2} \left (2 x^5+513 x^3+256 x+512\right )\right )}{x^3+256 x} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1} \left (e^{x^2} \left (2 x^4+512 x^2\right ) \log \left (\frac {x^2}{x^2+256}\right )+e^{x^2} \left (2 x^5+513 x^3+256 x+512\right )\right )}{x \left (x^2+256\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1} \left (2 x^5+513 x^3+512 x^2 \log \left (\frac {x^2}{x^2+256}\right )+2 x^4 \log \left (\frac {x^2}{x^2+256}\right )+256 x+512\right )}{256 x}-\frac {x e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1} \left (2 x^5+513 x^3+512 x^2 \log \left (\frac {x^2}{x^2+256}\right )+2 x^4 \log \left (\frac {x^2}{x^2+256}\right )+256 x+512\right )}{256 \left (x^2+256\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1}dx+\int \frac {e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1}}{16 i-x}dx+2 \int \frac {e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1}}{x}dx+2 \int e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1} x^2dx-\int \frac {e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1}}{x+16 i}dx+2 \int e^{x^2+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{x^2+256}\right )+1} x \log \left (\frac {x^2}{x^2+256}\right )dx\) |
Int[(E^(1 + E^x^2*x + E^x^2*Log[x^2/(256 + x^2)])*(E^x^2*(512 + 256*x + 51 3*x^3 + 2*x^5) + E^x^2*(512*x^2 + 2*x^4)*Log[x^2/(256 + x^2)]))/(256*x + x ^3),x]
3.19.79.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 1.70 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12
method | result | size |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{x^{2}} \ln \left (\frac {x^{2}}{x^{2}+256}\right )+{\mathrm e}^{x^{2}} x +1}\) | \(27\) |
risch | \(x^{2 \,{\mathrm e}^{x^{2}}} \left (x^{2}+256\right )^{-{\mathrm e}^{x^{2}}} {\mathrm e}^{1-\frac {i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}+i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-\frac {i {\mathrm e}^{x^{2}} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}}{2}+\frac {i {\mathrm e}^{x^{2}} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{2}}{2}-\frac {i {\mathrm e}^{x^{2}} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}+256}\right )}{2}-\frac {i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{3}}{2}+\frac {i {\mathrm e}^{x^{2}} \pi \operatorname {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{2} \operatorname {csgn}\left (\frac {i}{x^{2}+256}\right )}{2}+{\mathrm e}^{x^{2}} x}\) | \(222\) |
int(((2*x^4+512*x^2)*exp(x^2)*ln(x^2/(x^2+256))+(2*x^5+513*x^3+256*x+512)* exp(x^2))*exp(exp(x^2)*ln(x^2/(x^2+256))+exp(x^2)*x+1)/(x^3+256*x),x,metho d=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx=e^{\left (x e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )} \log \left (\frac {x^{2}}{x^{2} + 256}\right ) + 1\right )} \]
integrate(((2*x^4+512*x^2)*exp(x^2)*log(x^2/(x^2+256))+(2*x^5+513*x^3+256* x+512)*exp(x^2))*exp(exp(x^2)*log(x^2/(x^2+256))+exp(x^2)*x+1)/(x^3+256*x) ,x, algorithm=\
Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx=e^{x e^{x^{2}} + e^{x^{2}} \log {\left (\frac {x^{2}}{x^{2} + 256} \right )} + 1} \]
integrate(((2*x**4+512*x**2)*exp(x**2)*ln(x**2/(x**2+256))+(2*x**5+513*x** 3+256*x+512)*exp(x**2))*exp(exp(x**2)*ln(x**2/(x**2+256))+exp(x**2)*x+1)/( x**3+256*x),x)
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx=e^{\left (x e^{\left (x^{2}\right )} - e^{\left (x^{2}\right )} \log \left (x^{2} + 256\right ) + 2 \, e^{\left (x^{2}\right )} \log \left (x\right ) + 1\right )} \]
integrate(((2*x^4+512*x^2)*exp(x^2)*log(x^2/(x^2+256))+(2*x^5+513*x^3+256* x+512)*exp(x^2))*exp(exp(x^2)*log(x^2/(x^2+256))+exp(x^2)*x+1)/(x^3+256*x) ,x, algorithm=\
Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx=e^{\left (x e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )} \log \left (\frac {x^{2}}{x^{2} + 256}\right ) + 1\right )} \]
integrate(((2*x^4+512*x^2)*exp(x^2)*log(x^2/(x^2+256))+(2*x^5+513*x^3+256* x+512)*exp(x^2))*exp(exp(x^2)*log(x^2/(x^2+256))+exp(x^2)*x+1)/(x^3+256*x) ,x, algorithm=\
Time = 11.96 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx=\mathrm {e}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{x^2}}\,{\left (\frac {x^2}{x^2+256}\right )}^{{\mathrm {e}}^{x^2}} \]