Integrand size = 40, antiderivative size = 22 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=\frac {1}{4}-\left (3+x+e^{-2 e^4} x^3\right )^4 \] Output:
1/4-(x^3/exp(exp(4))^2+3+x)^4
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(22)=44\).
Time = 0.03 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.82 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=-e^{-8 e^4} \left (x^{12}+4 e^{2 e^4} x^9 (3+x)+6 e^{4 e^4} x^6 (3+x)^2+4 e^{6 e^4} x^3 (3+x)^3+e^{8 e^4} x \left (108+54 x+12 x^2+x^3\right )\right ) \] Input:
Integrate[((-4*E^(2*E^4) - 12*x^2)*(x^3 + E^(2*E^4)*(3 + x))^3)/E^(8*E^4), x]
Output:
-((x^12 + 4*E^(2*E^4)*x^9*(3 + x) + 6*E^(4*E^4)*x^6*(3 + x)^2 + 4*E^(6*E^4 )*x^3*(3 + x)^3 + E^(8*E^4)*x*(108 + 54*x + 12*x^2 + x^3))/E^(8*E^4))
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {27, 27, 2021}
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.
\(\displaystyle \int e^{-8 e^4} \left (-12 x^2-4 e^{2 e^4}\right ) \left (x^3+e^{2 e^4} (x+3)\right )^3 \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^{-8 e^4} \int -4 \left (3 x^2+e^{2 e^4}\right ) \left (x^3+e^{2 e^4} (x+3)\right )^3dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 e^{-8 e^4} \int \left (3 x^2+e^{2 e^4}\right ) \left (x^3+e^{2 e^4} (x+3)\right )^3dx\) |
\(\Big \downarrow \) 2021 |
\(\displaystyle -e^{-8 e^4} \left (x^3+e^{2 e^4} (x+3)\right )^4\) |
Input:
Int[((-4*E^(2*E^4) - 12*x^2)*(x^3 + E^(2*E^4)*(3 + x))^3)/E^(8*E^4),x]
Output:
-((x^3 + E^(2*E^4)*(3 + x))^4/E^(8*E^4))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x ]}, Simp[Coeff[Pp, x, p]*x^(p - q + 1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x, q]*Pp , Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; Free Q[m, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && NeQ[m, -1]
Time = 0.44 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
default | \(-{\mathrm e}^{-8 \,{\mathrm e}^{4}} \left (\left (3+x \right ) {\mathrm e}^{2 \,{\mathrm e}^{4}}+x^{3}\right )^{4}\) | \(23\) |
norman | \(\left (-{\mathrm e}^{-{\mathrm e}^{4}} x^{12}-12 \,{\mathrm e}^{{\mathrm e}^{4}} x^{9}-4 \,{\mathrm e}^{{\mathrm e}^{4}} x^{10}-36 \,{\mathrm e}^{3 \,{\mathrm e}^{4}} x^{7}-6 \,{\mathrm e}^{3 \,{\mathrm e}^{4}} x^{8}-36 \,{\mathrm e}^{5 \,{\mathrm e}^{4}} x^{5}-108 \,{\mathrm e}^{7 \,{\mathrm e}^{4}} x -54 \,{\mathrm e}^{7 \,{\mathrm e}^{4}} x^{2}-2 \,{\mathrm e}^{3 \,{\mathrm e}^{4}} \left (2 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}+27\right ) x^{6}-12 \,{\mathrm e}^{5 \,{\mathrm e}^{4}} \left ({\mathrm e}^{2 \,{\mathrm e}^{4}}+9\right ) x^{3}-{\mathrm e}^{5 \,{\mathrm e}^{4}} \left ({\mathrm e}^{2 \,{\mathrm e}^{4}}+108\right ) x^{4}\right ) {\mathrm e}^{-7 \,{\mathrm e}^{4}}\) | \(135\) |
gosper | \(-x \left (x^{3} {\mathrm e}^{8 \,{\mathrm e}^{4}}+4 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{5}+6 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{7}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{9}+x^{11}+12 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x^{2}+36 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{4}+36 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{6}+12 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{8}+54 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x +108 x^{3} {\mathrm e}^{6 \,{\mathrm e}^{4}}+54 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{5}+108 \,{\mathrm e}^{8 \,{\mathrm e}^{4}}+108 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{2}\right ) {\mathrm e}^{-8 \,{\mathrm e}^{4}}\) | \(137\) |
parallelrisch | \({\mathrm e}^{-8 \,{\mathrm e}^{4}} \left (-{\mathrm e}^{8 \,{\mathrm e}^{4}} x^{4}-4 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{6}-6 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{8}-4 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{10}-x^{12}-12 x^{3} {\mathrm e}^{8 \,{\mathrm e}^{4}}-36 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{5}-36 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{7}-12 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{9}-54 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x^{2}-108 \,{\mathrm e}^{6 \,{\mathrm e}^{4}} x^{4}-54 \,{\mathrm e}^{4 \,{\mathrm e}^{4}} x^{6}-108 \,{\mathrm e}^{8 \,{\mathrm e}^{4}} x -108 x^{3} {\mathrm e}^{6 \,{\mathrm e}^{4}}\right )\) | \(141\) |
risch | \(-{\mathrm e}^{-8 \,{\mathrm e}^{4}} x^{12}-4 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{10}-6 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{4 \,{\mathrm e}^{4}} x^{8}-12 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{9}-4 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{6 \,{\mathrm e}^{4}} x^{6}-36 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{4 \,{\mathrm e}^{4}} x^{7}-{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{8 \,{\mathrm e}^{4}} x^{4}-36 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{6 \,{\mathrm e}^{4}} x^{5}-54 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{4 \,{\mathrm e}^{4}} x^{6}-12 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{8 \,{\mathrm e}^{4}} x^{3}-108 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{6 \,{\mathrm e}^{4}} x^{4}-54 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{8 \,{\mathrm e}^{4}} x^{2}-108 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{6 \,{\mathrm e}^{4}} x^{3}-108 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{8 \,{\mathrm e}^{4}} x -81 \,{\mathrm e}^{-8 \,{\mathrm e}^{4}} {\mathrm e}^{8 \,{\mathrm e}^{4}}\) | \(241\) |
Input:
int((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4))^8,x, method=_RETURNVERBOSE)
Output:
-1/exp(exp(4))^8*((3+x)*exp(exp(4))^2+x^3)^4
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (18) = 36\).
Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.41 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=-{\left (x^{12} + {\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x\right )} e^{\left (8 \, e^{4}\right )} + 4 \, {\left (x^{6} + 9 \, x^{5} + 27 \, x^{4} + 27 \, x^{3}\right )} e^{\left (6 \, e^{4}\right )} + 6 \, {\left (x^{8} + 6 \, x^{7} + 9 \, x^{6}\right )} e^{\left (4 \, e^{4}\right )} + 4 \, {\left (x^{10} + 3 \, x^{9}\right )} e^{\left (2 \, e^{4}\right )}\right )} e^{\left (-8 \, e^{4}\right )} \] Input:
integrate((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4) )^8,x, algorithm="fricas")
Output:
-(x^12 + (x^4 + 12*x^3 + 54*x^2 + 108*x)*e^(8*e^4) + 4*(x^6 + 9*x^5 + 27*x ^4 + 27*x^3)*e^(6*e^4) + 6*(x^8 + 6*x^7 + 9*x^6)*e^(4*e^4) + 4*(x^10 + 3*x ^9)*e^(2*e^4))*e^(-8*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (17) = 34\).
Time = 0.04 (sec) , antiderivative size = 141, normalized size of antiderivative = 6.41 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=- \frac {x^{12}}{e^{8 e^{4}}} - \frac {4 x^{10}}{e^{6 e^{4}}} - \frac {12 x^{9}}{e^{6 e^{4}}} - \frac {6 x^{8}}{e^{4 e^{4}}} - \frac {36 x^{7}}{e^{4 e^{4}}} + \frac {x^{6} \left (- 4 e^{2 e^{4}} - 54\right )}{e^{4 e^{4}}} - \frac {36 x^{5}}{e^{2 e^{4}}} + \frac {x^{4} \left (- e^{2 e^{4}} - 108\right )}{e^{2 e^{4}}} + \frac {x^{3} \left (- 12 e^{2 e^{4}} - 108\right )}{e^{2 e^{4}}} - 54 x^{2} - 108 x \] Input:
integrate((-4*exp(exp(4))**2-12*x**2)*((3+x)*exp(exp(4))**2+x**3)**3/exp(e xp(4))**8,x)
Output:
-x**12*exp(-8*exp(4)) - 4*x**10*exp(-6*exp(4)) - 12*x**9*exp(-6*exp(4)) - 6*x**8*exp(-4*exp(4)) - 36*x**7*exp(-4*exp(4)) + x**6*(-4*exp(2*exp(4)) - 54)*exp(-4*exp(4)) - 36*x**5*exp(-2*exp(4)) + x**4*(-exp(2*exp(4)) - 108)* exp(-2*exp(4)) + x**3*(-12*exp(2*exp(4)) - 108)*exp(-2*exp(4)) - 54*x**2 - 108*x
Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=-{\left (x^{3} + {\left (x + 3\right )} e^{\left (2 \, e^{4}\right )}\right )}^{4} e^{\left (-8 \, e^{4}\right )} \] Input:
integrate((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4) )^8,x, algorithm="maxima")
Output:
-(x^3 + (x + 3)*e^(2*e^4))^4*e^(-8*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (18) = 36\).
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.59 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=-{\left ({\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )}^{4} + 12 \, {\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )}^{3} e^{\left (2 \, e^{4}\right )} + 54 \, {\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )}^{2} e^{\left (4 \, e^{4}\right )} + 108 \, {\left (x^{3} + x e^{\left (2 \, e^{4}\right )}\right )} e^{\left (6 \, e^{4}\right )}\right )} e^{\left (-8 \, e^{4}\right )} \] Input:
integrate((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4) )^8,x, algorithm="giac")
Output:
-((x^3 + x*e^(2*e^4))^4 + 12*(x^3 + x*e^(2*e^4))^3*e^(2*e^4) + 54*(x^3 + x *e^(2*e^4))^2*e^(4*e^4) + 108*(x^3 + x*e^(2*e^4))*e^(6*e^4))*e^(-8*e^4)
Time = 3.40 (sec) , antiderivative size = 122, normalized size of antiderivative = 5.55 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=-{\mathrm {e}}^{-8\,{\mathrm {e}}^4}\,x^{12}-4\,{\mathrm {e}}^{-6\,{\mathrm {e}}^4}\,x^{10}-12\,{\mathrm {e}}^{-6\,{\mathrm {e}}^4}\,x^9-6\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,x^8-36\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,x^7-2\,{\mathrm {e}}^{-4\,{\mathrm {e}}^4}\,\left (2\,{\mathrm {e}}^{2\,{\mathrm {e}}^4}+27\right )\,x^6-36\,{\mathrm {e}}^{-2\,{\mathrm {e}}^4}\,x^5-{\mathrm {e}}^{-2\,{\mathrm {e}}^4}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^4}+108\right )\,x^4-12\,{\mathrm {e}}^{-2\,{\mathrm {e}}^4}\,\left ({\mathrm {e}}^{2\,{\mathrm {e}}^4}+9\right )\,x^3-54\,x^2-108\,x \] Input:
int(-exp(-8*exp(4))*(4*exp(2*exp(4)) + 12*x^2)*(exp(2*exp(4))*(x + 3) + x^ 3)^3,x)
Output:
- 108*x - 36*x^5*exp(-2*exp(4)) - 36*x^7*exp(-4*exp(4)) - 6*x^8*exp(-4*exp (4)) - 12*x^9*exp(-6*exp(4)) - 4*x^10*exp(-6*exp(4)) - x^12*exp(-8*exp(4)) - 54*x^2 - 2*x^6*exp(-4*exp(4))*(2*exp(2*exp(4)) + 27) - 12*x^3*exp(-2*ex p(4))*(exp(2*exp(4)) + 9) - x^4*exp(-2*exp(4))*(exp(2*exp(4)) + 108)
Time = 0.20 (sec) , antiderivative size = 168, normalized size of antiderivative = 7.64 \[ \int e^{-8 e^4} \left (-4 e^{2 e^4}-12 x^2\right ) \left (x^3+e^{2 e^4} (3+x)\right )^3 \, dx=\frac {x \left (-e^{8 e^{4}} x^{3}-12 e^{8 e^{4}} x^{2}-54 e^{8 e^{4}} x -108 e^{8 e^{4}}-4 e^{6 e^{4}} x^{5}-36 e^{6 e^{4}} x^{4}-108 e^{6 e^{4}} x^{3}-108 e^{6 e^{4}} x^{2}-6 e^{4 e^{4}} x^{7}-36 e^{4 e^{4}} x^{6}-54 e^{4 e^{4}} x^{5}-4 e^{2 e^{4}} x^{9}-12 e^{2 e^{4}} x^{8}-x^{11}\right )}{e^{8 e^{4}}} \] Input:
int((-4*exp(exp(4))^2-12*x^2)*((3+x)*exp(exp(4))^2+x^3)^3/exp(exp(4))^8,x)
Output:
(x*( - e**(8*e**4)*x**3 - 12*e**(8*e**4)*x**2 - 54*e**(8*e**4)*x - 108*e** (8*e**4) - 4*e**(6*e**4)*x**5 - 36*e**(6*e**4)*x**4 - 108*e**(6*e**4)*x**3 - 108*e**(6*e**4)*x**2 - 6*e**(4*e**4)*x**7 - 36*e**(4*e**4)*x**6 - 54*e* *(4*e**4)*x**5 - 4*e**(2*e**4)*x**9 - 12*e**(2*e**4)*x**8 - x**11))/e**(8* e**4)