Integrand size = 127, antiderivative size = 30 \[ \int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx=5+e^{-2 x \left (-e^{\left (3+e^{\frac {x^4}{81}} x\right )^2}+x\right )} x \]
Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx=e^{2 \left (e^{\left (3+e^{\frac {x^4}{81}} x\right )^2}-x\right ) x} x \]
Integrate[(E^(2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*x - 2*x^2)*(81 - 324*x^2 + E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*(162*x + E^(x^4/8 1)*(972*x^2 + 48*x^6) + E^((2*x^4)/81)*(324*x^3 + 16*x^7))))/81,x]
Leaf count is larger than twice the leaf count of optimal. \(248\) vs. \(2(30)=60\).
Time = 1.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 8.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {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.
\(\displaystyle \int \frac {1}{81} \left (-324 x^2+e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} \left (e^{\frac {2 x^4}{81}} \left (16 x^7+324 x^3\right )+e^{\frac {x^4}{81}} \left (48 x^6+972 x^2\right )+162 x\right )+81\right ) \exp \left (2 e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} x-2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{81} \int \exp \left (2 e^{e^{\frac {2 x^4}{81}} x^2+6 e^{\frac {x^4}{81}} x+9} x-2 x^2\right ) \left (-324 x^2+2 e^{e^{\frac {2 x^4}{81}} x^2+6 e^{\frac {x^4}{81}} x+9} \left (81 x+6 e^{\frac {x^4}{81}} \left (4 x^6+81 x^2\right )+2 e^{\frac {2 x^4}{81}} \left (4 x^7+81 x^3\right )\right )+81\right )dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -\frac {\left (162 x^2-e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} \left (2 e^{\frac {2 x^4}{81}} \left (4 x^7+81 x^3\right )+6 e^{\frac {x^4}{81}} \left (4 x^6+81 x^2\right )+81 x\right )\right ) \exp \left (2 e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} x-2 x^2\right )}{81 e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9}+2 e^{6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2+9} \left (12 e^{\frac {x^4}{81}} x^4+81 e^{\frac {2 x^4}{81}} x+243 e^{\frac {x^4}{81}}+4 e^{\frac {2 x^4}{81}} x^5\right ) x-162 x}\) |
Int[(E^(2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*x - 2*x^2)*(81 - 324 *x^2 + E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*(162*x + E^(x^4/81)*(97 2*x^2 + 48*x^6) + E^((2*x^4)/81)*(324*x^3 + 16*x^7))))/81,x]
-((E^(2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*x - 2*x^2)*(162*x^2 - E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2)*(81*x + 6*E^(x^4/81)*(81*x^2 + 4*x^6) + 2*E^((2*x^4)/81)*(81*x^3 + 4*x^7))))/(81*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^2) - 162*x + 2*E^(9 + 6*E^(x^4/81)*x + E^((2*x^4)/81)*x^ 2)*x*(243*E^(x^4/81) + 81*E^((2*x^4)/81)*x + 12*E^(x^4/81)*x^4 + 4*E^((2*x ^4)/81)*x^5)))
3.23.27.3.1 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[(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 = 1.17 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
method | result | size |
risch | \(x \,{\mathrm e}^{-2 x \left (-{\mathrm e}^{x^{2} {\mathrm e}^{\frac {2 x^{4}}{81}}+6 x \,{\mathrm e}^{\frac {x^{4}}{81}}+9}+x \right )}\) | \(33\) |
parallelrisch | \(x \,{\mathrm e}^{2 x \left ({\mathrm e}^{x^{2} {\mathrm e}^{\frac {2 x^{4}}{81}}+6 x \,{\mathrm e}^{\frac {x^{4}}{81}}+9}-x \right )}\) | \(37\) |
int(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/81*x^4) +162*x)*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)-324*x^2+81)/exp(-2*x* exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x,method=_RETURNVERBOS E)
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx=x e^{\left (-2 \, x^{2} + 2 \, x e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )}\right )} \]
integrate(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/8 1*x^4)+162*x)*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)-324*x^2+81)/exp (-2*x*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x, algorithm=\
Time = 4.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx=x e^{- 2 x^{2} + 2 x e^{x^{2} e^{\frac {2 x^{4}}{81}} + 6 x e^{\frac {x^{4}}{81}} + 9}} \]
integrate(1/81*(((16*x**7+324*x**3)*exp(1/81*x**4)**2+(48*x**6+972*x**2)*e xp(1/81*x**4)+162*x)*exp(x**2*exp(1/81*x**4)**2+6*x*exp(1/81*x**4)+9)-324* x**2+81)/exp(-2*x*exp(x**2*exp(1/81*x**4)**2+6*x*exp(1/81*x**4)+9)+2*x**2) ,x)
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx=x e^{\left (-2 \, x^{2} + 2 \, x e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )}\right )} \]
integrate(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/8 1*x^4)+162*x)*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)-324*x^2+81)/exp (-2*x*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x, algorithm=\
\[ \int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx=\int { -\frac {1}{81} \, {\left (324 \, x^{2} - 2 \, {\left (2 \, {\left (4 \, x^{7} + 81 \, x^{3}\right )} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, {\left (4 \, x^{6} + 81 \, x^{2}\right )} e^{\left (\frac {1}{81} \, x^{4}\right )} + 81 \, x\right )} e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )} - 81\right )} e^{\left (-2 \, x^{2} + 2 \, x e^{\left (x^{2} e^{\left (\frac {2}{81} \, x^{4}\right )} + 6 \, x e^{\left (\frac {1}{81} \, x^{4}\right )} + 9\right )}\right )} \,d x } \]
integrate(1/81*(((16*x^7+324*x^3)*exp(1/81*x^4)^2+(48*x^6+972*x^2)*exp(1/8 1*x^4)+162*x)*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)-324*x^2+81)/exp (-2*x*exp(x^2*exp(1/81*x^4)^2+6*x*exp(1/81*x^4)+9)+2*x^2),x, algorithm=\
integrate(-1/81*(324*x^2 - 2*(2*(4*x^7 + 81*x^3)*e^(2/81*x^4) + 6*(4*x^6 + 81*x^2)*e^(1/81*x^4) + 81*x)*e^(x^2*e^(2/81*x^4) + 6*x*e^(1/81*x^4) + 9) - 81)*e^(-2*x^2 + 2*x*e^(x^2*e^(2/81*x^4) + 6*x*e^(1/81*x^4) + 9)), x)
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {1}{81} e^{2 e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} x-2 x^2} \left (81-324 x^2+e^{9+6 e^{\frac {x^4}{81}} x+e^{\frac {2 x^4}{81}} x^2} \left (162 x+e^{\frac {x^4}{81}} \left (972 x^2+48 x^6\right )+e^{\frac {2 x^4}{81}} \left (324 x^3+16 x^7\right )\right )\right ) \, dx=x\,{\mathrm {e}}^{-2\,x^2}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^{\frac {2\,x^4}{81}}}\,{\mathrm {e}}^9\,{\mathrm {e}}^{6\,x\,{\mathrm {e}}^{\frac {x^4}{81}}}} \]