Integrand size = 94, antiderivative size = 29 \[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=-2+e^{\frac {4 \left (1-e^3-\frac {9}{x}\right )^2}{x^2}} (2-x) \] Output:
(2-x)*exp(4/x^2*(1-9/x-exp(3))^2)-2
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=-e^{\frac {4 \left (9+\left (-1+e^3\right ) x\right )^2}{x^4}} (-2+x) \] Input:
Integrate[(E^((324 - 72*x + 4*x^2 + 4*E^6*x^2 + E^3*(72*x - 8*x^2))/x^4)*( -2592 + 1728*x - 232*x^2 + 8*x^3 - x^5 + E^3*(-432*x + 248*x^2 - 16*x^3) + E^6*(-16*x^2 + 8*x^3)))/x^5,x]
Output:
-(E^((4*(9 + (-1 + E^3)*x)^2)/x^4)*(-2 + x))
Leaf count is larger than twice the leaf count of optimal. \(150\) vs. \(2(29)=58\).
Time = 0.76 (sec) , antiderivative size = 150, normalized size of antiderivative = 5.17, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {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.
\(\displaystyle \int \frac {\left (-x^5+8 x^3-232 x^2+e^3 \left (-16 x^3+248 x^2-432 x\right )+e^6 \left (8 x^3-16 x^2\right )+1728 x-2592\right ) \exp \left (\frac {4 e^6 x^2+4 x^2+e^3 \left (72 x-8 x^2\right )-72 x+324}{x^4}\right )}{x^5} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {\left (-x^3+29 x^2+e^6 \left (2 x^2-x^3\right )+e^3 \left (2 x^3-31 x^2+54 x\right )-216 x+324\right ) \exp \left (\frac {4 \left (e^6 x^2+x^2+2 e^3 \left (9 x-x^2\right )-18 x+81\right )}{x^4}\right )}{x^5 \left (\frac {-e^3 (9-2 x)-e^6 x-x+9}{x^4}+\frac {2 \left (e^6 x^2+x^2+2 e^3 \left (9 x-x^2\right )-18 x+81\right )}{x^5}\right )}\) |
Input:
Int[(E^((324 - 72*x + 4*x^2 + 4*E^6*x^2 + E^3*(72*x - 8*x^2))/x^4)*(-2592 + 1728*x - 232*x^2 + 8*x^3 - x^5 + E^3*(-432*x + 248*x^2 - 16*x^3) + E^6*( -16*x^2 + 8*x^3)))/x^5,x]
Output:
(E^((4*(81 - 18*x + x^2 + E^6*x^2 + 2*E^3*(9*x - x^2)))/x^4)*(324 - 216*x + 29*x^2 - x^3 + E^6*(2*x^2 - x^3) + E^3*(54*x - 31*x^2 + 2*x^3)))/(x^5*(( 9 - E^3*(9 - 2*x) - x - E^6*x)/x^4 + (2*(81 - 18*x + x^2 + E^6*x^2 + 2*E^3 *(9*x - x^2)))/x^5))
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 = 0.59 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38
method | result | size |
gosper | \(-{\mathrm e}^{\frac {4 x^{2} {\mathrm e}^{6}-8 x^{2} {\mathrm e}^{3}+72 x \,{\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}} \left (-2+x \right )\) | \(40\) |
risch | \(\left (2-x \right ) {\mathrm e}^{-\frac {4 \left (-x^{2} {\mathrm e}^{6}+2 x^{2} {\mathrm e}^{3}-18 x \,{\mathrm e}^{3}-x^{2}+18 x -81\right )}{x^{4}}}\) | \(42\) |
parallelrisch | \(-x \,{\mathrm e}^{\frac {4 x^{2} {\mathrm e}^{6}+\left (-8 x^{2}+72 x \right ) {\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}}+2 \,{\mathrm e}^{\frac {4 x^{2} {\mathrm e}^{6}+\left (-8 x^{2}+72 x \right ) {\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}}\) | \(79\) |
norman | \(\frac {2 x^{4} {\mathrm e}^{\frac {4 x^{2} {\mathrm e}^{6}+\left (-8 x^{2}+72 x \right ) {\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}}-x^{5} {\mathrm e}^{\frac {4 x^{2} {\mathrm e}^{6}+\left (-8 x^{2}+72 x \right ) {\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}}}{x^{4}}\) | \(88\) |
Input:
int(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^3-232* x^2+1728*x-2592)*exp((4*x^2*exp(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x+324)/ x^4)/x^5,x,method=_RETURNVERBOSE)
Output:
-exp(4*(x^2*exp(3)^2-2*x^2*exp(3)+18*x*exp(3)+x^2-18*x+81)/x^4)*(-2+x)
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=-{\left (x - 2\right )} e^{\left (\frac {4 \, {\left (x^{2} e^{6} + x^{2} - 2 \, {\left (x^{2} - 9 \, x\right )} e^{3} - 18 \, x + 81\right )}}{x^{4}}\right )} \] Input:
integrate(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^ 3-232*x^2+1728*x-2592)*exp((4*x^2*exp(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x +324)/x^4)/x^5,x, algorithm="fricas")
Output:
-(x - 2)*e^(4*(x^2*e^6 + x^2 - 2*(x^2 - 9*x)*e^3 - 18*x + 81)/x^4)
Time = 0.55 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=\left (2 - x\right ) e^{\frac {4 x^{2} + 4 x^{2} e^{6} - 72 x + \left (- 8 x^{2} + 72 x\right ) e^{3} + 324}{x^{4}}} \] Input:
integrate(((8*x**3-16*x**2)*exp(3)**2+(-16*x**3+248*x**2-432*x)*exp(3)-x** 5+8*x**3-232*x**2+1728*x-2592)*exp((4*x**2*exp(3)**2+(-8*x**2+72*x)*exp(3) +4*x**2-72*x+324)/x**4)/x**5,x)
Output:
(2 - x)*exp((4*x**2 + 4*x**2*exp(6) - 72*x + (-8*x**2 + 72*x)*exp(3) + 324 )/x**4)
\[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=\int { -\frac {{\left (x^{5} - 8 \, x^{3} + 232 \, x^{2} - 8 \, {\left (x^{3} - 2 \, x^{2}\right )} e^{6} + 8 \, {\left (2 \, x^{3} - 31 \, x^{2} + 54 \, x\right )} e^{3} - 1728 \, x + 2592\right )} e^{\left (\frac {4 \, {\left (x^{2} e^{6} + x^{2} - 2 \, {\left (x^{2} - 9 \, x\right )} e^{3} - 18 \, x + 81\right )}}{x^{4}}\right )}}{x^{5}} \,d x } \] Input:
integrate(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^ 3-232*x^2+1728*x-2592)*exp((4*x^2*exp(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x +324)/x^4)/x^5,x, algorithm="maxima")
Output:
-integrate((x^5 - 8*x^3 + 232*x^2 - 8*(x^3 - 2*x^2)*e^6 + 8*(2*x^3 - 31*x^ 2 + 54*x)*e^3 - 1728*x + 2592)*e^(4*(x^2*e^6 + x^2 - 2*(x^2 - 9*x)*e^3 - 1 8*x + 81)/x^4)/x^5, x)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (24) = 48\).
Time = 0.18 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.41 \[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=-x e^{\left (\frac {4 \, {\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2} + 18 \, x e^{3} - 18 \, x + 81\right )}}{x^{4}}\right )} + 2 \, e^{\left (\frac {4 \, {\left (x^{2} e^{6} - 2 \, x^{2} e^{3} + x^{2} + 18 \, x e^{3} - 18 \, x + 81\right )}}{x^{4}}\right )} \] Input:
integrate(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^ 3-232*x^2+1728*x-2592)*exp((4*x^2*exp(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x +324)/x^4)/x^5,x, algorithm="giac")
Output:
-x*e^(4*(x^2*e^6 - 2*x^2*e^3 + x^2 + 18*x*e^3 - 18*x + 81)/x^4) + 2*e^(4*( x^2*e^6 - 2*x^2*e^3 + x^2 + 18*x*e^3 - 18*x + 81)/x^4)
Time = 4.20 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=-{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^6}{x^2}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^3}{x^2}}\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^3}{x^3}}\,{\mathrm {e}}^{\frac {4}{x^2}}\,{\mathrm {e}}^{-\frac {72}{x^3}}\,{\mathrm {e}}^{\frac {324}{x^4}}\,\left (x-2\right ) \] Input:
int(-(exp((exp(3)*(72*x - 8*x^2) - 72*x + 4*x^2*exp(6) + 4*x^2 + 324)/x^4) *(exp(3)*(432*x - 248*x^2 + 16*x^3) - 1728*x + exp(6)*(16*x^2 - 8*x^3) + 2 32*x^2 - 8*x^3 + x^5 + 2592))/x^5,x)
Output:
-exp((4*exp(6))/x^2)*exp(-(8*exp(3))/x^2)*exp((72*exp(3))/x^3)*exp(4/x^2)* exp(-72/x^3)*exp(324/x^4)*(x - 2)
Time = 0.97 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 \left (72 x-8 x^2\right )}{x^4}} \left (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 \left (-432 x+248 x^2-16 x^3\right )+e^6 \left (-16 x^2+8 x^3\right )\right )}{x^5} \, dx=\frac {e^{\frac {4 e^{6} x^{2}+72 e^{3} x +4 x^{2}+324}{x^{4}}} \left (-x +2\right )}{e^{\frac {8 e^{3} x +72}{x^{3}}}} \] Input:
int(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^3-232* x^2+1728*x-2592)*exp((4*x^2*exp(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x+324)/ x^4)/x^5,x)
Output:
(e**((4*e**6*x**2 + 72*e**3*x + 4*x**2 + 324)/x**4)*( - x + 2))/e**((8*e** 3*x + 72)/x**3)