Integrand size = 156, antiderivative size = 32 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=e^{\frac {9 x \log ^2(3)}{-e^4+\frac {x+x^2}{1+3 x}}} x \] Output:
exp(9*ln(3)^2/((x^2+x)/(1+3*x)-exp(4))*x)*x
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=e^{\frac {9 x (1+3 x) \log ^2(3)}{x (1+x)-e^4 (1+3 x)}} x \] Input:
Integrate[(E^(((-9*x - 27*x^2)*Log[3]^2)/(-x - x^2 + E^4*(1 + 3*x)))*(x^2 + 2*x^3 + x^4 + E^8*(1 + 6*x + 9*x^2) + E^4*(-2*x - 8*x^2 - 6*x^3) + (18*x ^3 + E^4*(-9*x - 54*x^2 - 81*x^3))*Log[3]^2))/(x^2 + 2*x^3 + x^4 + E^8*(1 + 6*x + 9*x^2) + E^4*(-2*x - 8*x^2 - 6*x^3)),x]
Output:
E^((9*x*(1 + 3*x)*Log[3]^2)/(x*(1 + x) - E^4*(1 + 3*x)))*x
Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(32)=64\).
Time = 0.58 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.38, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2463, 6, 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.
\(\displaystyle \int \frac {\left (x^4+2 x^3+x^2+e^8 \left (9 x^2+6 x+1\right )+e^4 \left (-6 x^3-8 x^2-2 x\right )+\left (18 x^3+e^4 \left (-81 x^3-54 x^2-9 x\right )\right ) \log ^2(3)\right ) \exp \left (\frac {\left (-27 x^2-9 x\right ) \log ^2(3)}{-x^2-x+e^4 (3 x+1)}\right )}{x^4+2 x^3+x^2+e^8 \left (9 x^2+6 x+1\right )+e^4 \left (-6 x^3-8 x^2-2 x\right )} \, dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \frac {\left (x^4+2 x^3+x^2+e^8 \left (9 x^2+6 x+1\right )+e^4 \left (-6 x^3-8 x^2-2 x\right )+\left (18 x^3+e^4 \left (-81 x^3-54 x^2-9 x\right )\right ) \log ^2(3)\right ) \exp \left (\frac {\left (-27 x^2-9 x\right ) \log ^2(3)}{-x^2-x+e^4 (3 x+1)}\right )}{\left (x^2-3 e^4 x+x-e^4\right )^2}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (x^4+2 x^3+x^2+e^8 \left (9 x^2+6 x+1\right )+e^4 \left (-6 x^3-8 x^2-2 x\right )+\left (18 x^3+e^4 \left (-81 x^3-54 x^2-9 x\right )\right ) \log ^2(3)\right ) \exp \left (\frac {\left (-27 x^2-9 x\right ) \log ^2(3)}{-x^2-x+e^4 (3 x+1)}\right )}{\left (x^2+\left (1-3 e^4\right ) x-e^4\right )^2}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -\frac {\left (2 x^3-e^4 \left (9 x^3+6 x^2+x\right )\right ) \exp \left (\frac {9 \left (3 x^2+x\right ) \log ^2(3)}{x^2+x-e^4 (3 x+1)}\right )}{\left (-x^2-\left (1-3 e^4\right ) x+e^4\right )^2 \left (\frac {\left (2 x-3 e^4+1\right ) \left (3 x^2+x\right )}{\left (x^2+x-e^4 (3 x+1)\right )^2}-\frac {6 x+1}{x^2+x-e^4 (3 x+1)}\right )}\) |
Input:
Int[(E^(((-9*x - 27*x^2)*Log[3]^2)/(-x - x^2 + E^4*(1 + 3*x)))*(x^2 + 2*x^ 3 + x^4 + E^8*(1 + 6*x + 9*x^2) + E^4*(-2*x - 8*x^2 - 6*x^3) + (18*x^3 + E ^4*(-9*x - 54*x^2 - 81*x^3))*Log[3]^2))/(x^2 + 2*x^3 + x^4 + E^8*(1 + 6*x + 9*x^2) + E^4*(-2*x - 8*x^2 - 6*x^3)),x]
Output:
-((E^((9*(x + 3*x^2)*Log[3]^2)/(x + x^2 - E^4*(1 + 3*x)))*(2*x^3 - E^4*(x + 6*x^2 + 9*x^3)))/((E^4 - (1 - 3*E^4)*x - x^2)^2*(((1 - 3*E^4 + 2*x)*(x + 3*x^2))/(x + x^2 - E^4*(1 + 3*x))^2 - (1 + 6*x)/(x + x^2 - E^4*(1 + 3*x)) )))
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, 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]
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 = 1.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06
method | result | size |
gosper | \(x \,{\mathrm e}^{-\frac {9 x \left (1+3 x \right ) \ln \left (3\right )^{2}}{3 x \,{\mathrm e}^{4}-x^{2}+{\mathrm e}^{4}-x}}\) | \(34\) |
risch | \(x \,{\mathrm e}^{-\frac {9 x \left (1+3 x \right ) \ln \left (3\right )^{2}}{3 x \,{\mathrm e}^{4}-x^{2}+{\mathrm e}^{4}-x}}\) | \(34\) |
parallelrisch | \(x \,{\mathrm e}^{\frac {\ln \left (3\right )^{2} \left (-27 x^{2}-9 x \right )}{3 x \,{\mathrm e}^{4}-x^{2}+{\mathrm e}^{4}-x}}\) | \(36\) |
norman | \(\frac {x \,{\mathrm e}^{4} {\mathrm e}^{\frac {\left (-27 x^{2}-9 x \right ) \ln \left (3\right )^{2}}{\left (1+3 x \right ) {\mathrm e}^{4}-x^{2}-x}}+\left (3 \,{\mathrm e}^{4}-1\right ) x^{2} {\mathrm e}^{\frac {\left (-27 x^{2}-9 x \right ) \ln \left (3\right )^{2}}{\left (1+3 x \right ) {\mathrm e}^{4}-x^{2}-x}}-x^{3} {\mathrm e}^{\frac {\left (-27 x^{2}-9 x \right ) \ln \left (3\right )^{2}}{\left (1+3 x \right ) {\mathrm e}^{4}-x^{2}-x}}}{3 x \,{\mathrm e}^{4}-x^{2}+{\mathrm e}^{4}-x}\) | \(142\) |
Input:
int((((-81*x^3-54*x^2-9*x)*exp(4)+18*x^3)*ln(3)^2+(9*x^2+6*x+1)*exp(4)^2+( -6*x^3-8*x^2-2*x)*exp(4)+x^4+2*x^3+x^2)*exp((-27*x^2-9*x)*ln(3)^2/((1+3*x) *exp(4)-x^2-x))/((9*x^2+6*x+1)*exp(4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+x^4+2*x^ 3+x^2),x,method=_RETURNVERBOSE)
Output:
x*exp(-9*x*(1+3*x)*ln(3)^2/(3*x*exp(4)-x^2+exp(4)-x))
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=x e^{\left (\frac {9 \, {\left (3 \, x^{2} + x\right )} \log \left (3\right )^{2}}{x^{2} - {\left (3 \, x + 1\right )} e^{4} + x}\right )} \] Input:
integrate((((-81*x^3-54*x^2-9*x)*exp(4)+18*x^3)*log(3)^2+(9*x^2+6*x+1)*exp (4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+x^4+2*x^3+x^2)*exp((-27*x^2-9*x)*log(3)^2/ ((1+3*x)*exp(4)-x^2-x))/((9*x^2+6*x+1)*exp(4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+ x^4+2*x^3+x^2),x, algorithm="fricas")
Output:
x*e^(9*(3*x^2 + x)*log(3)^2/(x^2 - (3*x + 1)*e^4 + x))
Time = 1.92 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=x e^{\frac {\left (- 27 x^{2} - 9 x\right ) \log {\left (3 \right )}^{2}}{- x^{2} - x + \left (3 x + 1\right ) e^{4}}} \] Input:
integrate((((-81*x**3-54*x**2-9*x)*exp(4)+18*x**3)*ln(3)**2+(9*x**2+6*x+1) *exp(4)**2+(-6*x**3-8*x**2-2*x)*exp(4)+x**4+2*x**3+x**2)*exp((-27*x**2-9*x )*ln(3)**2/((1+3*x)*exp(4)-x**2-x))/((9*x**2+6*x+1)*exp(4)**2+(-6*x**3-8*x **2-2*x)*exp(4)+x**4+2*x**3+x**2),x)
Output:
x*exp((-27*x**2 - 9*x)*log(3)**2/(-x**2 - x + (3*x + 1)*exp(4)))
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (30) = 60\).
Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.84 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=x e^{\left (\frac {81 \, x e^{4} \log \left (3\right )^{2}}{x^{2} - x {\left (3 \, e^{4} - 1\right )} - e^{4}} - \frac {18 \, x \log \left (3\right )^{2}}{x^{2} - x {\left (3 \, e^{4} - 1\right )} - e^{4}} + \frac {27 \, e^{4} \log \left (3\right )^{2}}{x^{2} - x {\left (3 \, e^{4} - 1\right )} - e^{4}} + 27 \, \log \left (3\right )^{2}\right )} \] Input:
integrate((((-81*x^3-54*x^2-9*x)*exp(4)+18*x^3)*log(3)^2+(9*x^2+6*x+1)*exp (4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+x^4+2*x^3+x^2)*exp((-27*x^2-9*x)*log(3)^2/ ((1+3*x)*exp(4)-x^2-x))/((9*x^2+6*x+1)*exp(4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+ x^4+2*x^3+x^2),x, algorithm="maxima")
Output:
x*e^(81*x*e^4*log(3)^2/(x^2 - x*(3*e^4 - 1) - e^4) - 18*x*log(3)^2/(x^2 - x*(3*e^4 - 1) - e^4) + 27*e^4*log(3)^2/(x^2 - x*(3*e^4 - 1) - e^4) + 27*lo g(3)^2)
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=x e^{\left (\frac {9 \, {\left (3 \, x^{2} \log \left (3\right )^{2} + x \log \left (3\right )^{2}\right )}}{x^{2} - 3 \, x e^{4} + x - e^{4}}\right )} \] Input:
integrate((((-81*x^3-54*x^2-9*x)*exp(4)+18*x^3)*log(3)^2+(9*x^2+6*x+1)*exp (4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+x^4+2*x^3+x^2)*exp((-27*x^2-9*x)*log(3)^2/ ((1+3*x)*exp(4)-x^2-x))/((9*x^2+6*x+1)*exp(4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+ x^4+2*x^3+x^2),x, algorithm="giac")
Output:
x*e^(9*(3*x^2*log(3)^2 + x*log(3)^2)/(x^2 - 3*x*e^4 + x - e^4))
Time = 6.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=x\,{\mathrm {e}}^{\frac {27\,{\ln \left (3\right )}^2\,x^2+9\,{\ln \left (3\right )}^2\,x}{x-{\mathrm {e}}^4-3\,x\,{\mathrm {e}}^4+x^2}} \] Input:
int((exp((log(3)^2*(9*x + 27*x^2))/(x + x^2 - exp(4)*(3*x + 1)))*(exp(8)*( 6*x + 9*x^2 + 1) - log(3)^2*(exp(4)*(9*x + 54*x^2 + 81*x^3) - 18*x^3) - ex p(4)*(2*x + 8*x^2 + 6*x^3) + x^2 + 2*x^3 + x^4))/(exp(8)*(6*x + 9*x^2 + 1) - exp(4)*(2*x + 8*x^2 + 6*x^3) + x^2 + 2*x^3 + x^4),x)
Output:
x*exp((27*x^2*log(3)^2 + 9*x*log(3)^2)/(x - exp(4) - 3*x*exp(4) + x^2))
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\frac {\left (-9 x-27 x^2\right ) \log ^2(3)}{-x-x^2+e^4 (1+3 x)}} \left (x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )+\left (18 x^3+e^4 \left (-9 x-54 x^2-81 x^3\right )\right ) \log ^2(3)\right )}{x^2+2 x^3+x^4+e^8 \left (1+6 x+9 x^2\right )+e^4 \left (-2 x-8 x^2-6 x^3\right )} \, dx=\frac {x}{e^{\frac {27 \mathrm {log}\left (3\right )^{2} x^{2}+9 \mathrm {log}\left (3\right )^{2} x}{3 e^{4} x +e^{4}-x^{2}-x}}} \] Input:
int((((-81*x^3-54*x^2-9*x)*exp(4)+18*x^3)*log(3)^2+(9*x^2+6*x+1)*exp(4)^2+ (-6*x^3-8*x^2-2*x)*exp(4)+x^4+2*x^3+x^2)*exp((-27*x^2-9*x)*log(3)^2/((1+3* x)*exp(4)-x^2-x))/((9*x^2+6*x+1)*exp(4)^2+(-6*x^3-8*x^2-2*x)*exp(4)+x^4+2* x^3+x^2),x)
Output:
x/e**((27*log(3)**2*x**2 + 9*log(3)**2*x)/(3*e**4*x + e**4 - x**2 - x))