Integrand size = 115, antiderivative size = 31 \[ \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (3 x^2+9 x^4+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^2+60 x^4-18 x^6+\left (54-162 x^2\right ) \log (x)+\left (-54+162 x^2\right ) \log ^2(x)\right )\right )}{x^2-6 x^4+9 x^6} \, dx=\frac {e^{e^{\left (-x+\frac {3 \log (x)}{x}\right )^2}}}{\frac {1}{3 x}-x} \]
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (3 x^2+9 x^4+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^2+60 x^4-18 x^6+\left (54-162 x^2\right ) \log (x)+\left (-54+162 x^2\right ) \log ^2(x)\right )\right )}{x^2-6 x^4+9 x^6} \, dx=-\frac {3 e^{\frac {e^{x^2+\frac {9 \log ^2(x)}{x^2}}}{x^6}} x}{-1+3 x^2} \]
Integrate[(E^E^((x^4 - 6*x^2*Log[x] + 9*Log[x]^2)/x^2)*(3*x^2 + 9*x^4 + E^ ((x^4 - 6*x^2*Log[x] + 9*Log[x]^2)/x^2)*(-18*x^2 + 60*x^4 - 18*x^6 + (54 - 162*x^2)*Log[x] + (-54 + 162*x^2)*Log[x]^2)))/(x^2 - 6*x^4 + 9*x^6),x]
Leaf count is larger than twice the leaf count of optimal. \(169\) vs. \(2(31)=62\).
Time = 1.20 (sec) , antiderivative size = 169, normalized size of antiderivative = 5.45, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {2026, 1380, 27, 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.
\(\displaystyle \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (9 x^4+3 x^2+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^6+60 x^4-18 x^2+\left (162 x^2-54\right ) \log ^2(x)+\left (54-162 x^2\right ) \log (x)\right )\right )}{9 x^6-6 x^4+x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (9 x^4+3 x^2+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^6+60 x^4-18 x^2+\left (162 x^2-54\right ) \log ^2(x)+\left (54-162 x^2\right ) \log (x)\right )\right )}{x^2 \left (9 x^4-6 x^2+1\right )}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle 9 \int \frac {e^{\frac {e^{\frac {x^4+9 \log ^2(x)}{x^2}}}{x^6}} \left (3 x^4+x^2-\frac {2 e^{\frac {x^4+9 \log ^2(x)}{x^2}} \left (3 x^6-10 x^4+3 x^2+9 \left (1-3 x^2\right ) \log ^2(x)-9 \left (1-3 x^2\right ) \log (x)\right )}{x^6}\right )}{3 x^2 \left (1-3 x^2\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int \frac {e^{\frac {e^{\frac {x^4+9 \log ^2(x)}{x^2}}}{x^6}} \left (3 x^4+x^2-\frac {2 e^{\frac {x^4+9 \log ^2(x)}{x^2}} \left (3 x^6-10 x^4+3 x^2+9 \left (1-3 x^2\right ) \log ^2(x)-9 \left (1-3 x^2\right ) \log (x)\right )}{x^6}\right )}{x^2 \left (1-3 x^2\right )^2}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {3 \left (3 x^6-10 x^4+3 x^2+9 \left (1-3 x^2\right ) \log ^2(x)-9 \left (1-3 x^2\right ) \log (x)\right ) \exp \left (\frac {x^4+9 \log ^2(x)}{x^2}+\frac {e^{\frac {x^4+9 \log ^2(x)}{x^2}}}{x^6}\right )}{x^8 \left (1-3 x^2\right )^2 \left (\frac {3 e^{\frac {x^4+9 \log ^2(x)}{x^2}}}{x^7}-\frac {e^{\frac {x^4+9 \log ^2(x)}{x^2}} \left (\frac {2 x^3+\frac {9 \log (x)}{x}}{x^2}-\frac {x^4+9 \log ^2(x)}{x^3}\right )}{x^6}\right )}\) |
Int[(E^E^((x^4 - 6*x^2*Log[x] + 9*Log[x]^2)/x^2)*(3*x^2 + 9*x^4 + E^((x^4 - 6*x^2*Log[x] + 9*Log[x]^2)/x^2)*(-18*x^2 + 60*x^4 - 18*x^6 + (54 - 162*x ^2)*Log[x] + (-54 + 162*x^2)*Log[x]^2)))/(x^2 - 6*x^4 + 9*x^6),x]
(3*E^(E^((x^4 + 9*Log[x]^2)/x^2)/x^6 + (x^4 + 9*Log[x]^2)/x^2)*(3*x^2 - 10 *x^4 + 3*x^6 - 9*(1 - 3*x^2)*Log[x] + 9*(1 - 3*x^2)*Log[x]^2))/(x^8*(1 - 3 *x^2)^2*((3*E^((x^4 + 9*Log[x]^2)/x^2))/x^7 - (E^((x^4 + 9*Log[x]^2)/x^2)* ((2*x^3 + (9*Log[x])/x)/x^2 - (x^4 + 9*Log[x]^2)/x^3))/x^6))
3.15.89.3.1 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_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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[(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]
Time = 9.92 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {3 x \,{\mathrm e}^{{\mathrm e}^{\frac {\left (-x^{2}+3 \ln \left (x \right )\right )^{2}}{x^{2}}}}}{3 x^{2}-1}\) | \(31\) |
parallelrisch | \(-\frac {3 x \,{\mathrm e}^{{\mathrm e}^{\frac {9 \ln \left (x \right )^{2}-6 x^{2} \ln \left (x \right )+x^{4}}{x^{2}}}}}{3 x^{2}-1}\) | \(36\) |
int((((162*x^2-54)*ln(x)^2+(-162*x^2+54)*ln(x)-18*x^6+60*x^4-18*x^2)*exp(( 9*ln(x)^2-6*x^2*ln(x)+x^4)/x^2)+9*x^4+3*x^2)*exp(exp((9*ln(x)^2-6*x^2*ln(x )+x^4)/x^2))/(9*x^6-6*x^4+x^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (3 x^2+9 x^4+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^2+60 x^4-18 x^6+\left (54-162 x^2\right ) \log (x)+\left (-54+162 x^2\right ) \log ^2(x)\right )\right )}{x^2-6 x^4+9 x^6} \, dx=-\frac {3 \, x e^{\left (e^{\left (\frac {x^{4} - 6 \, x^{2} \log \left (x\right ) + 9 \, \log \left (x\right )^{2}}{x^{2}}\right )}\right )}}{3 \, x^{2} - 1} \]
integrate((((162*x^2-54)*log(x)^2+(-162*x^2+54)*log(x)-18*x^6+60*x^4-18*x^ 2)*exp((9*log(x)^2-6*x^2*log(x)+x^4)/x^2)+9*x^4+3*x^2)*exp(exp((9*log(x)^2 -6*x^2*log(x)+x^4)/x^2))/(9*x^6-6*x^4+x^2),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (3 x^2+9 x^4+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^2+60 x^4-18 x^6+\left (54-162 x^2\right ) \log (x)+\left (-54+162 x^2\right ) \log ^2(x)\right )\right )}{x^2-6 x^4+9 x^6} \, dx=- \frac {3 x e^{e^{\frac {x^{4} - 6 x^{2} \log {\left (x \right )} + 9 \log {\left (x \right )}^{2}}{x^{2}}}}}{3 x^{2} - 1} \]
integrate((((162*x**2-54)*ln(x)**2+(-162*x**2+54)*ln(x)-18*x**6+60*x**4-18 *x**2)*exp((9*ln(x)**2-6*x**2*ln(x)+x**4)/x**2)+9*x**4+3*x**2)*exp(exp((9* ln(x)**2-6*x**2*ln(x)+x**4)/x**2))/(9*x**6-6*x**4+x**2),x)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (3 x^2+9 x^4+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^2+60 x^4-18 x^6+\left (54-162 x^2\right ) \log (x)+\left (-54+162 x^2\right ) \log ^2(x)\right )\right )}{x^2-6 x^4+9 x^6} \, dx=-\frac {3 \, x e^{\left (\frac {e^{\left (x^{2} + \frac {9 \, \log \left (x\right )^{2}}{x^{2}}\right )}}{x^{6}}\right )}}{3 \, x^{2} - 1} \]
integrate((((162*x^2-54)*log(x)^2+(-162*x^2+54)*log(x)-18*x^6+60*x^4-18*x^ 2)*exp((9*log(x)^2-6*x^2*log(x)+x^4)/x^2)+9*x^4+3*x^2)*exp(exp((9*log(x)^2 -6*x^2*log(x)+x^4)/x^2))/(9*x^6-6*x^4+x^2),x, algorithm=\
\[ \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (3 x^2+9 x^4+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^2+60 x^4-18 x^6+\left (54-162 x^2\right ) \log (x)+\left (-54+162 x^2\right ) \log ^2(x)\right )\right )}{x^2-6 x^4+9 x^6} \, dx=\int { \frac {3 \, {\left (3 \, x^{4} + x^{2} - 2 \, {\left (3 \, x^{6} - 10 \, x^{4} - 9 \, {\left (3 \, x^{2} - 1\right )} \log \left (x\right )^{2} + 3 \, x^{2} + 9 \, {\left (3 \, x^{2} - 1\right )} \log \left (x\right )\right )} e^{\left (\frac {x^{4} - 6 \, x^{2} \log \left (x\right ) + 9 \, \log \left (x\right )^{2}}{x^{2}}\right )}\right )} e^{\left (e^{\left (\frac {x^{4} - 6 \, x^{2} \log \left (x\right ) + 9 \, \log \left (x\right )^{2}}{x^{2}}\right )}\right )}}{9 \, x^{6} - 6 \, x^{4} + x^{2}} \,d x } \]
integrate((((162*x^2-54)*log(x)^2+(-162*x^2+54)*log(x)-18*x^6+60*x^4-18*x^ 2)*exp((9*log(x)^2-6*x^2*log(x)+x^4)/x^2)+9*x^4+3*x^2)*exp(exp((9*log(x)^2 -6*x^2*log(x)+x^4)/x^2))/(9*x^6-6*x^4+x^2),x, algorithm=\
integrate(3*(3*x^4 + x^2 - 2*(3*x^6 - 10*x^4 - 9*(3*x^2 - 1)*log(x)^2 + 3* x^2 + 9*(3*x^2 - 1)*log(x))*e^((x^4 - 6*x^2*log(x) + 9*log(x)^2)/x^2))*e^( e^((x^4 - 6*x^2*log(x) + 9*log(x)^2)/x^2))/(9*x^6 - 6*x^4 + x^2), x)
Time = 14.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {e^{e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}}} \left (3 x^2+9 x^4+e^{\frac {x^4-6 x^2 \log (x)+9 \log ^2(x)}{x^2}} \left (-18 x^2+60 x^4-18 x^6+\left (54-162 x^2\right ) \log (x)+\left (-54+162 x^2\right ) \log ^2(x)\right )\right )}{x^2-6 x^4+9 x^6} \, dx=-\frac {x\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {9\,{\ln \left (x\right )}^2}{x^2}}}{x^6}}}{x^2-\frac {1}{3}} \]