Integrand size = 122, antiderivative size = 31 \[ \int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx=\frac {2 e^{-\frac {1}{x^2}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right ) (-4+x) x} \]
Time = 2.55 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx=\frac {2 e^{-\frac {1}{x^2}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right ) (-4+x) x} \]
Integrate[(E^((-1 + x^2*Log[2])/x^2)*(8*x - 2*x^2 - 8*x^3 + 3*x^4 + E^(4/L og[3]^2)*(-8 + 2*x + 4*x^2 - 2*x^3)))/(16*x^6 - 8*x^7 + x^8 + E^(8/Log[3]^ 2)*(16*x^4 - 8*x^5 + x^6) + E^(4/Log[3]^2)*(-32*x^5 + 16*x^6 - 2*x^7)),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.
| \(\displaystyle \int \frac {e^{\frac {x^2 \log (2)-1}{x^2}} \left (3 x^4-8 x^3-2 x^2+\left (-2 x^3+4 x^2+2 x-8\right ) e^{\frac {4}{\log ^2(3)}}+8 x\right )}{x^8-8 x^7+16 x^6+\left (-2 x^7+16 x^6-32 x^5\right ) e^{\frac {4}{\log ^2(3)}}+\left (x^6-8 x^5+16 x^4\right ) e^{\frac {8}{\log ^2(3)}}} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{\frac {x^2 \log (2)-1}{x^2}} \left (3 x^4-8 x^3-2 x^2+\left (-2 x^3+4 x^2+2 x-8\right ) e^{\frac {4}{\log ^2(3)}}+8 x\right )}{x^4 \left (x^4-2 x^3 \left (4+e^{\frac {4}{\log ^2(3)}}\right )+x^2 \left (16+16 e^{\frac {4}{\log ^2(3)}}+e^{\frac {8}{\log ^2(3)}}\right )-8 x e^{\frac {4}{\log ^2(3)}} \left (4+e^{\frac {4}{\log ^2(3)}}\right )+16 e^{\frac {8}{\log ^2(3)}}\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {2 e^{\frac {x^2 \log (2)-1}{x^2}} \left (3 x^4-8 x^3-2 x^2+\left (-2 x^3+4 x^2+2 x-8\right ) e^{\frac {4}{\log ^2(3)}}+8 x\right )}{x^4 \left (e^{\frac {4}{\log ^2(3)}}-4\right )^3 \left (e^{\frac {4}{\log ^2(3)}}-x\right )}+\frac {2 e^{\frac {x^2 \log (2)-1}{x^2}} \left (3 x^4-8 x^3-2 x^2+\left (-2 x^3+4 x^2+2 x-8\right ) e^{\frac {4}{\log ^2(3)}}+8 x\right )}{(x-4) x^4 \left (e^{\frac {4}{\log ^2(3)}}-4\right )^3}+\frac {e^{\frac {x^2 \log (2)-1}{x^2}} \left (3 x^4-8 x^3-2 x^2+\left (-2 x^3+4 x^2+2 x-8\right ) e^{\frac {4}{\log ^2(3)}}+8 x\right )}{x^4 \left (e^{\frac {4}{\log ^2(3)}}-4\right )^2 \left (e^{\frac {4}{\log ^2(3)}}-x\right )^2}+\frac {e^{\frac {x^2 \log (2)-1}{x^2}} \left (3 x^4-8 x^3-2 x^2+\left (-2 x^3+4 x^2+2 x-8\right ) e^{\frac {4}{\log ^2(3)}}+8 x\right )}{(x-4)^2 x^4 \left (e^{\frac {4}{\log ^2(3)}}-4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \int \frac {e^{-\frac {4}{\log ^2(3)}-\frac {1}{x^2}}}{\left (e^{\frac {4}{\log ^2(3)}}-x\right )^2}dx}{4-e^{\frac {4}{\log ^2(3)}}}-\frac {4 \left (4-e^{\frac {4}{\log ^2(3)}}+e^{\frac {12}{\log ^2(3)}}\right ) \int \frac {e^{-\frac {16}{\log ^2(3)}-\frac {1}{x^2}}}{e^{\frac {4}{\log ^2(3)}}-x}dx}{\left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {4 \int \frac {e^{-\frac {4}{\log ^2(3)}-\frac {1}{x^2}}}{e^{\frac {4}{\log ^2(3)}}-x}dx}{\left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {\int \frac {e^{-\frac {1}{x^2}}}{(x-4)^2}dx}{2 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}+\frac {\left (60+e^{\frac {4}{\log ^2(3)}}\right ) \int \frac {e^{-\frac {1}{x^2}}}{x-4}dx}{64 \left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}-\frac {\int \frac {e^{-\frac {1}{x^2}}}{x-4}dx}{\left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {2 \sqrt {\pi } e^{-\frac {12}{\log ^2(3)}} \left (4-e^{\frac {4}{\log ^2(3)}}-2 e^{\frac {8}{\log ^2(3)}}\right ) \text {erf}\left (\frac {1}{x}\right )}{\left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}-\frac {\sqrt {\pi } \left (4+7 e^{\frac {4}{\log ^2(3)}}\right ) \text {erf}\left (\frac {1}{x}\right )}{32 \left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {\sqrt {\pi } e^{\frac {4}{\log ^2(3)}} \text {erf}\left (\frac {1}{x}\right )}{4 \left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {4 \sqrt {\pi } e^{-\frac {4}{\log ^2(3)}} \text {erf}\left (\frac {1}{x}\right )}{\left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {2 e^{-\frac {16}{\log ^2(3)}} \left (4-e^{\frac {4}{\log ^2(3)}}+e^{\frac {12}{\log ^2(3)}}\right ) \operatorname {ExpIntegralEi}\left (-\frac {1}{x^2}\right )}{\left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {\left (60+e^{\frac {4}{\log ^2(3)}}\right ) \operatorname {ExpIntegralEi}\left (-\frac {1}{x^2}\right )}{128 \left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {\left (8+e^{\frac {4}{\log ^2(3)}}\right ) \operatorname {ExpIntegralEi}\left (-\frac {1}{x^2}\right )}{2 \left (4-e^{\frac {4}{\log ^2(3)}}\right )^3}-\frac {4 \left (1+2 e^{-\frac {4}{\log ^2(3)}}\right ) \operatorname {ExpIntegralEi}\left (-\frac {1}{x^2}\right )}{\left (4-e^{\frac {4}{\log ^2(3)}}\right )^3}-\frac {8 e^{-\frac {1}{x^2}-\frac {4}{\log ^2(3)}}}{x \left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}-\frac {e^{\frac {4}{\log ^2(3)}-\frac {1}{x^2}}}{2 x \left (4-e^{\frac {4}{\log ^2(3)}}\right )^2}+\frac {16 e^{-\frac {1}{x^2}}}{x \left (4-e^{\frac {4}{\log ^2(3)}}\right )^3}-\frac {4 e^{\frac {4}{\log ^2(3)}-\frac {1}{x^2}}}{x \left (4-e^{\frac {4}{\log ^2(3)}}\right )^3}+\frac {e^{-\frac {1}{x^2}}}{8 \left (4-e^{\frac {4}{\log ^2(3)}}\right )}-\frac {2 e^{-\frac {1}{x^2}-\frac {8}{\log ^2(3)}}}{4-e^{\frac {4}{\log ^2(3)}}}\) |
Int[(E^((-1 + x^2*Log[2])/x^2)*(8*x - 2*x^2 - 8*x^3 + 3*x^4 + E^(4/Log[3]^ 2)*(-8 + 2*x + 4*x^2 - 2*x^3)))/(16*x^6 - 8*x^7 + x^8 + E^(8/Log[3]^2)*(16 *x^4 - 8*x^5 + x^6) + E^(4/Log[3]^2)*(-32*x^5 + 16*x^6 - 2*x^7)),x]
3.7.28.3.1 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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16
| method | result | size |
| norman | \(\frac {{\mathrm e}^{\frac {x^{2} \ln \left (2\right )-1}{x^{2}}}}{x \left (x -4\right ) \left ({\mathrm e}^{\frac {4}{\ln \left (3\right )^{2}}}-x \right )}\) | \(36\) |
| parallelrisch | \(\frac {{\mathrm e}^{\frac {x^{2} \ln \left (2\right )-1}{x^{2}}}}{x \left ({\mathrm e}^{\frac {4}{\ln \left (3\right )^{2}}} x -x^{2}-4 \,{\mathrm e}^{\frac {4}{\ln \left (3\right )^{2}}}+4 x \right )}\) | \(47\) |
int(((-2*x^3+4*x^2+2*x-8)*exp(4/ln(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp((x^2*l n(2)-1)/x^2)/((x^6-8*x^5+16*x^4)*exp(4/ln(3)^2)^2+(-2*x^7+16*x^6-32*x^5)*e xp(4/ln(3)^2)+x^8-8*x^7+16*x^6),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx=-\frac {e^{\left (\frac {x^{2} \log \left (2\right ) - 1}{x^{2}}\right )}}{x^{3} - 4 \, x^{2} - {\left (x^{2} - 4 \, x\right )} e^{\left (\frac {4}{\log \left (3\right )^{2}}\right )}} \]
integrate(((-2*x^3+4*x^2+2*x-8)*exp(4/log(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp ((x^2*log(2)-1)/x^2)/((x^6-8*x^5+16*x^4)*exp(4/log(3)^2)^2+(-2*x^7+16*x^6- 32*x^5)*exp(4/log(3)^2)+x^8-8*x^7+16*x^6),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx=- \frac {e^{\frac {x^{2} \log {\left (2 \right )} - 1}{x^{2}}}}{x^{3} - x^{2} e^{\frac {4}{\log {\left (3 \right )}^{2}}} - 4 x^{2} + 4 x e^{\frac {4}{\log {\left (3 \right )}^{2}}}} \]
integrate(((-2*x**3+4*x**2+2*x-8)*exp(4/ln(3)**2)+3*x**4-8*x**3-2*x**2+8*x )*exp((x**2*ln(2)-1)/x**2)/((x**6-8*x**5+16*x**4)*exp(4/ln(3)**2)**2+(-2*x **7+16*x**6-32*x**5)*exp(4/ln(3)**2)+x**8-8*x**7+16*x**6),x)
Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx=-\frac {2 \, e^{\left (-\frac {1}{x^{2}}\right )}}{x^{3} - x^{2} {\left (e^{\left (\frac {4}{\log \left (3\right )^{2}}\right )} + 4\right )} + 4 \, x e^{\left (\frac {4}{\log \left (3\right )^{2}}\right )}} \]
integrate(((-2*x^3+4*x^2+2*x-8)*exp(4/log(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp ((x^2*log(2)-1)/x^2)/((x^6-8*x^5+16*x^4)*exp(4/log(3)^2)^2+(-2*x^7+16*x^6- 32*x^5)*exp(4/log(3)^2)+x^8-8*x^7+16*x^6),x, algorithm=\
\[ \int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx=\int { \frac {{\left (3 \, x^{4} - 8 \, x^{3} - 2 \, x^{2} - 2 \, {\left (x^{3} - 2 \, x^{2} - x + 4\right )} e^{\left (\frac {4}{\log \left (3\right )^{2}}\right )} + 8 \, x\right )} e^{\left (\frac {x^{2} \log \left (2\right ) - 1}{x^{2}}\right )}}{x^{8} - 8 \, x^{7} + 16 \, x^{6} + {\left (x^{6} - 8 \, x^{5} + 16 \, x^{4}\right )} e^{\left (\frac {8}{\log \left (3\right )^{2}}\right )} - 2 \, {\left (x^{7} - 8 \, x^{6} + 16 \, x^{5}\right )} e^{\left (\frac {4}{\log \left (3\right )^{2}}\right )}} \,d x } \]
integrate(((-2*x^3+4*x^2+2*x-8)*exp(4/log(3)^2)+3*x^4-8*x^3-2*x^2+8*x)*exp ((x^2*log(2)-1)/x^2)/((x^6-8*x^5+16*x^4)*exp(4/log(3)^2)^2+(-2*x^7+16*x^6- 32*x^5)*exp(4/log(3)^2)+x^8-8*x^7+16*x^6),x, algorithm=\
Time = 9.39 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {-1+x^2 \log (2)}{x^2}} \left (8 x-2 x^2-8 x^3+3 x^4+e^{\frac {4}{\log ^2(3)}} \left (-8+2 x+4 x^2-2 x^3\right )\right )}{16 x^6-8 x^7+x^8+e^{\frac {8}{\log ^2(3)}} \left (16 x^4-8 x^5+x^6\right )+e^{\frac {4}{\log ^2(3)}} \left (-32 x^5+16 x^6-2 x^7\right )} \, dx=-\frac {2\,{\mathrm {e}}^{-\frac {1}{x^2}}}{4\,x\,{\mathrm {e}}^{\frac {4}{{\ln \left (3\right )}^2}}-x^2\,{\mathrm {e}}^{\frac {4}{{\ln \left (3\right )}^2}}-4\,x^2+x^3} \]