Integrand size = 95, antiderivative size = 22 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=4-20736 e^{2 e^{-3+x}} \left (x+\frac {x}{e^3}\right )^4 \] Output:
4-144*exp(exp(x)/exp(3))^2*(x+x/exp(3))^2*(12*x+12*x/exp(3))^2
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=-20736 e^{-12+2 e^{-3+x}} \left (1+e^3\right )^4 x^4 \] Input:
Integrate[E^(-15 + 2*E^(-3 + x))*(-82944*E^3*x^3 - 331776*E^6*x^3 - 497664 *E^9*x^3 - 331776*E^12*x^3 - 82944*E^15*x^3 + E^x*(-41472*x^4 - 165888*E^3 *x^4 - 248832*E^6*x^4 - 165888*E^9*x^4 - 41472*E^12*x^4)),x]
Output:
-20736*E^(-12 + 2*E^(-3 + x))*(1 + E^3)^4*x^4
Leaf count is larger than twice the leaf count of optimal. \(48\) vs. \(2(22)=44\).
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6, 6, 6, 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.
\(\displaystyle \int e^{2 e^{x-3}-15} \left (e^x \left (-41472 e^{12} x^4-165888 e^9 x^4-248832 e^6 x^4-165888 e^3 x^4-41472 x^4\right )-82944 e^{15} x^3-331776 e^{12} x^3-497664 e^9 x^3-331776 e^6 x^3-82944 e^3 x^3\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int e^{2 e^{x-3}-15} \left (e^x \left (-41472 e^{12} x^4-165888 e^9 x^4-248832 e^6 x^4-165888 e^3 x^4-41472 x^4\right )+\left (-82944 e^3-331776 e^6\right ) x^3-82944 e^{15} x^3-331776 e^{12} x^3-497664 e^9 x^3\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int e^{2 e^{x-3}-15} \left (e^x \left (-41472 e^{12} x^4-165888 e^9 x^4-248832 e^6 x^4-165888 e^3 x^4-41472 x^4\right )+\left (-497664 e^9-331776 e^{12}\right ) x^3+\left (-82944 e^3-331776 e^6\right ) x^3-82944 e^{15} x^3\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int e^{2 e^{x-3}-15} \left (e^x \left (-41472 e^{12} x^4-165888 e^9 x^4-248832 e^6 x^4-165888 e^3 x^4-41472 x^4\right )+\left (-82944 e^3-331776 e^6-82944 e^{15}\right ) x^3+\left (-497664 e^9-331776 e^{12}\right ) x^3\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int e^{2 e^{x-3}-15} \left (e^x \left (-41472 e^{12} x^4-165888 e^9 x^4-248832 e^6 x^4-165888 e^3 x^4-41472 x^4\right )+\left (-82944 e^3-331776 e^6-497664 e^9-331776 e^{12}-82944 e^{15}\right ) x^3\right )dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle -20736 e^{2 e^{x-3}-12} \left (e^{12} x^4+4 e^9 x^4+6 e^6 x^4+4 e^3 x^4+x^4\right )\) |
Input:
Int[E^(-15 + 2*E^(-3 + x))*(-82944*E^3*x^3 - 331776*E^6*x^3 - 497664*E^9*x ^3 - 331776*E^12*x^3 - 82944*E^15*x^3 + E^x*(-41472*x^4 - 165888*E^3*x^4 - 248832*E^6*x^4 - 165888*E^9*x^4 - 41472*E^12*x^4)),x]
Output:
-20736*E^(-12 + 2*E^(-3 + x))*(x^4 + 4*E^3*x^4 + 6*E^6*x^4 + 4*E^9*x^4 + E ^12*x^4)
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[(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 = 0.45 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41
method | result | size |
risch | \(-20736 \left ({\mathrm e}^{12}+4 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{6}+4 \,{\mathrm e}^{3}+1\right ) x^{4} {\mathrm e}^{-12+2 \,{\mathrm e}^{-3+x}}\) | \(31\) |
norman | \(\left (-20736 \,{\mathrm e}^{12}-82944 \,{\mathrm e}^{9}-124416 \,{\mathrm e}^{6}-82944 \,{\mathrm e}^{3}-20736\right ) x^{4} {\mathrm e}^{2 \,{\mathrm e}^{-3+x}} {\mathrm e}^{-12}\) | \(43\) |
parallelrisch | \({\mathrm e}^{-15} \left (-20736 \,{\mathrm e}^{15} x^{4} {\mathrm e}^{2 \,{\mathrm e}^{-3+x}}-82944 \,{\mathrm e}^{12} x^{4} {\mathrm e}^{2 \,{\mathrm e}^{-3+x}}-124416 \,{\mathrm e}^{9} x^{4} {\mathrm e}^{2 \,{\mathrm e}^{-3+x}}-82944 \,{\mathrm e}^{6} x^{4} {\mathrm e}^{2 \,{\mathrm e}^{-3+x}}-20736 \,{\mathrm e}^{3} x^{4} {\mathrm e}^{2 \,{\mathrm e}^{-3+x}}\right )\) | \(100\) |
Input:
int(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-165888*x ^4*exp(3)-41472*x^4)*exp(x)-82944*x^3*exp(3)^5-331776*x^3*exp(3)^4-497664* x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(3))^2/ex p(3)^5,x,method=_RETURNVERBOSE)
Output:
-20736*(exp(12)+4*exp(9)+6*exp(6)+4*exp(3)+1)*x^4*exp(-12+2*exp(-3+x))
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (19) = 38\).
Time = 0.08 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.27 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=-20736 \, {\left (x^{4} e^{15} + 4 \, x^{4} e^{12} + 6 \, x^{4} e^{9} + 4 \, x^{4} e^{6} + x^{4} e^{3}\right )} e^{\left (-{\left (15 \, e^{3} - 2 \, e^{x}\right )} e^{\left (-3\right )}\right )} \] Input:
integrate(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-16 5888*x^4*exp(3)-41472*x^4)*exp(x)-82944*x^3*exp(3)^5-331776*x^3*exp(3)^4-4 97664*x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(3) )^2/exp(3)^5,x, algorithm="fricas")
Output:
-20736*(x^4*e^15 + 4*x^4*e^12 + 6*x^4*e^9 + 4*x^4*e^6 + x^4*e^3)*e^(-(15*e ^3 - 2*e^x)*e^(-3))
Time = 0.21 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.41 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=\frac {\left (- 20736 x^{4} e^{12} - 82944 x^{4} e^{9} - 124416 x^{4} e^{6} - 82944 x^{4} e^{3} - 20736 x^{4}\right ) e^{\frac {2 e^{x}}{e^{3}}}}{e^{12}} \] Input:
integrate(((-41472*x**4*exp(3)**4-165888*x**4*exp(3)**3-248832*x**4*exp(3) **2-165888*x**4*exp(3)-41472*x**4)*exp(x)-82944*x**3*exp(3)**5-331776*x**3 *exp(3)**4-497664*x**3*exp(3)**3-331776*x**3*exp(3)**2-82944*x**3*exp(3))* exp(exp(x)/exp(3))**2/exp(3)**5,x)
Output:
(-20736*x**4*exp(12) - 82944*x**4*exp(9) - 124416*x**4*exp(6) - 82944*x**4 *exp(3) - 20736*x**4)*exp(-12)*exp(2*exp(-3)*exp(x))
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=-20736 \, x^{4} {\left (e^{12} + 4 \, e^{9} + 6 \, e^{6} + 4 \, e^{3} + 1\right )} e^{\left (2 \, e^{\left (x - 3\right )} - 12\right )} \] Input:
integrate(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-16 5888*x^4*exp(3)-41472*x^4)*exp(x)-82944*x^3*exp(3)^5-331776*x^3*exp(3)^4-4 97664*x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(3) )^2/exp(3)^5,x, algorithm="maxima")
Output:
-20736*x^4*(e^12 + 4*e^9 + 6*e^6 + 4*e^3 + 1)*e^(2*e^(x - 3) - 12)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.23 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=-20736 \, {\left (x^{4} e^{\left (2 \, e^{\left (x - 3\right )}\right )} + x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 12\right )} + 4 \, x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 9\right )} + 6 \, x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 6\right )} + 4 \, x^{4} e^{\left (2 \, e^{\left (x - 3\right )} + 3\right )}\right )} e^{\left (-12\right )} \] Input:
integrate(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-16 5888*x^4*exp(3)-41472*x^4)*exp(x)-82944*x^3*exp(3)^5-331776*x^3*exp(3)^4-4 97664*x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(3) )^2/exp(3)^5,x, algorithm="giac")
Output:
-20736*(x^4*e^(2*e^(x - 3)) + x^4*e^(2*e^(x - 3) + 12) + 4*x^4*e^(2*e^(x - 3) + 9) + 6*x^4*e^(2*e^(x - 3) + 6) + 4*x^4*e^(2*e^(x - 3) + 3))*e^(-12)
Time = 2.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=-20736\,x^4\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-12}\,\left (4\,{\mathrm {e}}^3+6\,{\mathrm {e}}^6+4\,{\mathrm {e}}^9+{\mathrm {e}}^{12}+1\right ) \] Input:
int(-exp(2*exp(-3)*exp(x))*exp(-15)*(exp(x)*(165888*x^4*exp(3) + 248832*x^ 4*exp(6) + 165888*x^4*exp(9) + 41472*x^4*exp(12) + 41472*x^4) + 82944*x^3* exp(3) + 331776*x^3*exp(6) + 497664*x^3*exp(9) + 331776*x^3*exp(12) + 8294 4*x^3*exp(15)),x)
Output:
-20736*x^4*exp(2*exp(-3)*exp(x))*exp(-12)*(4*exp(3) + 6*exp(6) + 4*exp(9) + exp(12) + 1)
Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82 \[ \int e^{-15+2 e^{-3+x}} \left (-82944 e^3 x^3-331776 e^6 x^3-497664 e^9 x^3-331776 e^{12} x^3-82944 e^{15} x^3+e^x \left (-41472 x^4-165888 e^3 x^4-248832 e^6 x^4-165888 e^9 x^4-41472 e^{12} x^4\right )\right ) \, dx=\frac {20736 e^{\frac {2 e^{x}}{e^{3}}} x^{4} \left (-e^{12}-4 e^{9}-6 e^{6}-4 e^{3}-1\right )}{e^{12}} \] Input:
int(((-41472*x^4*exp(3)^4-165888*x^4*exp(3)^3-248832*x^4*exp(3)^2-165888*x ^4*exp(3)-41472*x^4)*exp(x)-82944*x^3*exp(3)^5-331776*x^3*exp(3)^4-497664* x^3*exp(3)^3-331776*x^3*exp(3)^2-82944*x^3*exp(3))*exp(exp(x)/exp(3))^2/ex p(3)^5,x)
Output:
(20736*e**((2*e**x)/e**3)*x**4*( - e**12 - 4*e**9 - 6*e**6 - 4*e**3 - 1))/ e**12