Integrand size = 172, antiderivative size = 30 \[ \int \frac {e^{\frac {x^2-2 e^{-48+4 e} x^3+e^{-96+8 e} \left (x+x^4\right )}{1-2 e^{-48+4 e} x+e^{-96+8 e} x^2}} \left (-2 x-2 x^3+e^{-48+4 e} \left (6 x^2+6 x^4\right )+e^{-96+8 e} \left (-x^2-6 x^3-6 x^5\right )+e^{-144+12 e} \left (-x^3+2 x^4+2 x^6\right )\right )}{-16+48 e^{-48+4 e} x-48 e^{-96+8 e} x^2+16 e^{-144+12 e} x^3} \, dx=\frac {1}{16} e^{x^2+\frac {x}{\left (-e^{4 (12-e)}+x\right )^2}} x^2 \]
Time = 1.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.80 \[ \int \frac {e^{\frac {x^2-2 e^{-48+4 e} x^3+e^{-96+8 e} \left (x+x^4\right )}{1-2 e^{-48+4 e} x+e^{-96+8 e} x^2}} \left (-2 x-2 x^3+e^{-48+4 e} \left (6 x^2+6 x^4\right )+e^{-96+8 e} \left (-x^2-6 x^3-6 x^5\right )+e^{-144+12 e} \left (-x^3+2 x^4+2 x^6\right )\right )}{-16+48 e^{-48+4 e} x-48 e^{-96+8 e} x^2+16 e^{-144+12 e} x^3} \, dx=\frac {1}{16} e^{\frac {x \left (e^{96} x-2 e^{48+4 e} x^2+e^{8 e} \left (1+x^3\right )\right )}{\left (e^{48}-e^{4 e} x\right )^2}} x^2 \]
Integrate[(E^((x^2 - 2*E^(-48 + 4*E)*x^3 + E^(-96 + 8*E)*(x + x^4))/(1 - 2 *E^(-48 + 4*E)*x + E^(-96 + 8*E)*x^2))*(-2*x - 2*x^3 + E^(-48 + 4*E)*(6*x^ 2 + 6*x^4) + E^(-96 + 8*E)*(-x^2 - 6*x^3 - 6*x^5) + E^(-144 + 12*E)*(-x^3 + 2*x^4 + 2*x^6)))/(-16 + 48*E^(-48 + 4*E)*x - 48*E^(-96 + 8*E)*x^2 + 16*E ^(-144 + 12*E)*x^3),x]
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.
| \(\displaystyle \int \frac {\left (-2 x^3+e^{4 e-48} \left (6 x^4+6 x^2\right )+e^{12 e-144} \left (2 x^6+2 x^4-x^3\right )+e^{8 e-96} \left (-6 x^5-6 x^3-x^2\right )-2 x\right ) \exp \left (\frac {e^{8 e-96} \left (x^4+x\right )-2 e^{4 e-48} x^3+x^2}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}\right )}{16 e^{12 e-144} x^3-48 e^{8 e-96} x^2+48 e^{4 e-48} x-16} \, dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (-2 x^3+e^{4 e-48} \left (6 x^4+6 x^2\right )+e^{12 e-144} \left (2 x^6+2 x^4-x^3\right )+e^{8 e-96} \left (-6 x^5-6 x^3-x^2\right )-2 x\right ) \exp \left (\frac {e^{8 e-96} \left (x^4+x\right )-2 e^{4 e-48} x^3+x^2}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}\right )}{\left (2 \sqrt [3]{2} e^{\frac {1}{3} (12 e-144)} x-2 \sqrt [3]{2}\right )^3}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (2 x^3-e^{4 e-48} \left (6 x^4+6 x^2\right )-e^{12 e-144} \left (2 x^6+2 x^4-x^3\right )-e^{8 e-96} \left (-6 x^5-6 x^3-x^2\right )+2 x\right ) \exp \left (\frac {x \left (e^{8 e-96} x^3-2 e^{4 e-48} x^2+x+e^{8 e-96}\right )}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}\right )}{\left (2 \sqrt [3]{2}-2 \sqrt [3]{2} e^{\frac {1}{3} (12 e-144)} x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {1}{8} x^3 \exp \left (\frac {x \left (e^{8 e-96} x^3-2 e^{4 e-48} x^2+x+e^{8 e-96}\right )}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}\right )+\frac {1}{8} x \exp \left (\frac {x \left (e^{8 e-96} x^3-2 e^{4 e-48} x^2+x+e^{8 e-96}\right )}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}\right )-\frac {1}{16} \exp \left (\frac {x \left (e^{8 e-96} x^3-2 e^{4 e-48} x^2+x+e^{8 e-96}\right )}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}\right )+\frac {\exp \left (\frac {x \left (e^{8 e-96} x^3-2 e^{4 e-48} x^2+x+e^{8 e-96}\right )}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}+48\right )}{4 \left (e^{48}-e^{4 e} x\right )}-\frac {5 \exp \left (\frac {x \left (e^{8 e-96} x^3-2 e^{4 e-48} x^2+x+e^{8 e-96}\right )}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}+96\right )}{16 \left (e^{48}-e^{4 e} x\right )^2}+\frac {\exp \left (\frac {x \left (e^{8 e-96} x^3-2 e^{4 e-48} x^2+x+e^{8 e-96}\right )}{e^{8 e-96} x^2-2 e^{4 e-48} x+1}+144\right )}{8 \left (e^{48}-e^{4 e} x\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{16} \int \exp \left (\frac {x \left (e^{-96+8 e} x^3-2 e^{-48+4 e} x^2+x+e^{-96+8 e}\right )}{e^{-96+8 e} x^2-2 e^{-48+4 e} x+1}\right )dx+\frac {1}{8} \int \exp \left (\frac {x \left (e^{-96+8 e} x^3-2 e^{-48+4 e} x^2+x+e^{-96+8 e}\right )}{e^{-96+8 e} x^2-2 e^{-48+4 e} x+1}\right ) xdx+\frac {1}{8} \int \exp \left (\frac {x \left (e^{-96+8 e} x^3-2 e^{-48+4 e} x^2+x+e^{-96+8 e}\right )}{e^{-96+8 e} x^2-2 e^{-48+4 e} x+1}\right ) x^3dx+\frac {1}{8} \int \frac {\exp \left (\frac {x \left (e^{-96+8 e} x^3-2 e^{-48+4 e} x^2+x+e^{-96+8 e}\right )}{e^{-96+8 e} x^2-2 e^{-48+4 e} x+1}+144\right )}{\left (e^{48}-e^{4 e} x\right )^3}dx-\frac {5}{16} \int \frac {\exp \left (\frac {x \left (e^{-96+8 e} x^3-2 e^{-48+4 e} x^2+x+e^{-96+8 e}\right )}{e^{-96+8 e} x^2-2 e^{-48+4 e} x+1}+96\right )}{\left (e^{48}-e^{4 e} x\right )^2}dx+\frac {1}{4} \int \frac {\exp \left (\frac {x \left (e^{-96+8 e} x^3-2 e^{-48+4 e} x^2+x+e^{-96+8 e}\right )}{e^{-96+8 e} x^2-2 e^{-48+4 e} x+1}+48\right )}{e^{48}-e^{4 e} x}dx\) |
Int[(E^((x^2 - 2*E^(-48 + 4*E)*x^3 + E^(-96 + 8*E)*(x + x^4))/(1 - 2*E^(-4 8 + 4*E)*x + E^(-96 + 8*E)*x^2))*(-2*x - 2*x^3 + E^(-48 + 4*E)*(6*x^2 + 6* x^4) + E^(-96 + 8*E)*(-x^2 - 6*x^3 - 6*x^5) + E^(-144 + 12*E)*(-x^3 + 2*x^ 4 + 2*x^6)))/(-16 + 48*E^(-48 + 4*E)*x - 48*E^(-96 + 8*E)*x^2 + 16*E^(-144 + 12*E)*x^3),x]
3.23.1.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(25)=50\).
Time = 0.70 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20
| method | result | size |
| risch | \(\frac {x^{2} {\mathrm e}^{\frac {x \left ({\mathrm e}^{8 \,{\mathrm e}-96} x^{3}-2 x^{2} {\mathrm e}^{4 \,{\mathrm e}-48}+{\mathrm e}^{8 \,{\mathrm e}-96}+x \right )}{x^{2} {\mathrm e}^{8 \,{\mathrm e}-96}-2 x \,{\mathrm e}^{4 \,{\mathrm e}-48}+1}}}{16}\) | \(66\) |
| parallelrisch | \(\frac {x^{2} {\mathrm e}^{\frac {\left (x^{4}+x \right ) {\mathrm e}^{8 \,{\mathrm e}-96}-2 x^{3} {\mathrm e}^{4 \,{\mathrm e}-48}+x^{2}}{x^{2} {\mathrm e}^{8 \,{\mathrm e}-96}-2 x \,{\mathrm e}^{4 \,{\mathrm e}-48}+1}}}{16}\) | \(66\) |
| gosper | \(\frac {x^{2} {\mathrm e}^{\frac {x \left ({\mathrm e}^{8 \,{\mathrm e}-96} x^{3}-2 x^{2} {\mathrm e}^{4 \,{\mathrm e}-48}+{\mathrm e}^{8 \,{\mathrm e}-96}+x \right )}{x^{2} {\mathrm e}^{8 \,{\mathrm e}-96}-2 x \,{\mathrm e}^{4 \,{\mathrm e}-48}+1}}}{16}\) | \(72\) |
| norman | \(\frac {\frac {x^{2} {\mathrm e}^{\frac {\left (x^{4}+x \right ) {\mathrm e}^{8 \,{\mathrm e}-96}-2 x^{3} {\mathrm e}^{4 \,{\mathrm e}-48}+x^{2}}{x^{2} {\mathrm e}^{8 \,{\mathrm e}-96}-2 x \,{\mathrm e}^{4 \,{\mathrm e}-48}+1}}}{16}-\frac {x^{3} {\mathrm e}^{4 \,{\mathrm e}} {\mathrm e}^{-48} {\mathrm e}^{\frac {\left (x^{4}+x \right ) {\mathrm e}^{8 \,{\mathrm e}-96}-2 x^{3} {\mathrm e}^{4 \,{\mathrm e}-48}+x^{2}}{x^{2} {\mathrm e}^{8 \,{\mathrm e}-96}-2 x \,{\mathrm e}^{4 \,{\mathrm e}-48}+1}}}{8}+\frac {{\mathrm e}^{8 \,{\mathrm e}} {\mathrm e}^{-96} x^{4} {\mathrm e}^{\frac {\left (x^{4}+x \right ) {\mathrm e}^{8 \,{\mathrm e}-96}-2 x^{3} {\mathrm e}^{4 \,{\mathrm e}-48}+x^{2}}{x^{2} {\mathrm e}^{8 \,{\mathrm e}-96}-2 x \,{\mathrm e}^{4 \,{\mathrm e}-48}+1}}}{16}}{\left (x \,{\mathrm e}^{4 \,{\mathrm e}-48}-1\right )^{2}}\) | \(227\) |
int(((2*x^6+2*x^4-x^3)*exp(4*exp(1)-48)^3+(-6*x^5-6*x^3-x^2)*exp(4*exp(1)- 48)^2+(6*x^4+6*x^2)*exp(4*exp(1)-48)-2*x^3-2*x)*exp(((x^4+x)*exp(4*exp(1)- 48)^2-2*x^3*exp(4*exp(1)-48)+x^2)/(x^2*exp(4*exp(1)-48)^2-2*x*exp(4*exp(1) -48)+1))/(16*x^3*exp(4*exp(1)-48)^3-48*x^2*exp(4*exp(1)-48)^2+48*x*exp(4*e xp(1)-48)-16),x,method=_RETURNVERBOSE)
1/16*x^2*exp(x*(exp(8*exp(1)-96)*x^3-2*x^2*exp(4*exp(1)-48)+exp(8*exp(1)-9 6)+x)/(x^2*exp(8*exp(1)-96)-2*x*exp(4*exp(1)-48)+1))
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.17 \[ \int \frac {e^{\frac {x^2-2 e^{-48+4 e} x^3+e^{-96+8 e} \left (x+x^4\right )}{1-2 e^{-48+4 e} x+e^{-96+8 e} x^2}} \left (-2 x-2 x^3+e^{-48+4 e} \left (6 x^2+6 x^4\right )+e^{-96+8 e} \left (-x^2-6 x^3-6 x^5\right )+e^{-144+12 e} \left (-x^3+2 x^4+2 x^6\right )\right )}{-16+48 e^{-48+4 e} x-48 e^{-96+8 e} x^2+16 e^{-144+12 e} x^3} \, dx=\frac {1}{16} \, x^{2} e^{\left (-\frac {2 \, x^{3} e^{\left (4 \, e - 48\right )} - x^{2} - {\left (x^{4} + x\right )} e^{\left (8 \, e - 96\right )}}{x^{2} e^{\left (8 \, e - 96\right )} - 2 \, x e^{\left (4 \, e - 48\right )} + 1}\right )} \]
integrate(((2*x^6+2*x^4-x^3)*exp(4*exp(1)-48)^3+(-6*x^5-6*x^3-x^2)*exp(4*e xp(1)-48)^2+(6*x^4+6*x^2)*exp(4*exp(1)-48)-2*x^3-2*x)*exp(((x^4+x)*exp(4*e xp(1)-48)^2-2*x^3*exp(4*exp(1)-48)+x^2)/(x^2*exp(4*exp(1)-48)^2-2*x*exp(4* exp(1)-48)+1))/(16*x^3*exp(4*exp(1)-48)^3-48*x^2*exp(4*exp(1)-48)^2+48*x*e xp(4*exp(1)-48)-16),x, algorithm=\
1/16*x^2*e^(-(2*x^3*e^(4*e - 48) - x^2 - (x^4 + x)*e^(8*e - 96))/(x^2*e^(8 *e - 96) - 2*x*e^(4*e - 48) + 1))
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (22) = 44\).
Time = 0.44 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {x^2-2 e^{-48+4 e} x^3+e^{-96+8 e} \left (x+x^4\right )}{1-2 e^{-48+4 e} x+e^{-96+8 e} x^2}} \left (-2 x-2 x^3+e^{-48+4 e} \left (6 x^2+6 x^4\right )+e^{-96+8 e} \left (-x^2-6 x^3-6 x^5\right )+e^{-144+12 e} \left (-x^3+2 x^4+2 x^6\right )\right )}{-16+48 e^{-48+4 e} x-48 e^{-96+8 e} x^2+16 e^{-144+12 e} x^3} \, dx=\frac {x^{2} e^{\frac {- \frac {2 x^{3}}{e^{48 - 4 e}} + x^{2} + \frac {x^{4} + x}{e^{96 - 8 e}}}{\frac {x^{2}}{e^{96 - 8 e}} - \frac {2 x}{e^{48 - 4 e}} + 1}}}{16} \]
integrate(((2*x**6+2*x**4-x**3)*exp(4*exp(1)-48)**3+(-6*x**5-6*x**3-x**2)* exp(4*exp(1)-48)**2+(6*x**4+6*x**2)*exp(4*exp(1)-48)-2*x**3-2*x)*exp(((x** 4+x)*exp(4*exp(1)-48)**2-2*x**3*exp(4*exp(1)-48)+x**2)/(x**2*exp(4*exp(1)- 48)**2-2*x*exp(4*exp(1)-48)+1))/(16*x**3*exp(4*exp(1)-48)**3-48*x**2*exp(4 *exp(1)-48)**2+48*x*exp(4*exp(1)-48)-16),x)
x**2*exp((-2*x**3*exp(-48 + 4*E) + x**2 + (x**4 + x)*exp(-96 + 8*E))/(x**2 *exp(-96 + 8*E) - 2*x*exp(-48 + 4*E) + 1))/16
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).
Time = 0.53 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {e^{\frac {x^2-2 e^{-48+4 e} x^3+e^{-96+8 e} \left (x+x^4\right )}{1-2 e^{-48+4 e} x+e^{-96+8 e} x^2}} \left (-2 x-2 x^3+e^{-48+4 e} \left (6 x^2+6 x^4\right )+e^{-96+8 e} \left (-x^2-6 x^3-6 x^5\right )+e^{-144+12 e} \left (-x^3+2 x^4+2 x^6\right )\right )}{-16+48 e^{-48+4 e} x-48 e^{-96+8 e} x^2+16 e^{-144+12 e} x^3} \, dx=\frac {1}{16} \, x^{2} e^{\left (x^{2} + \frac {e^{\left (4 \, e\right )}}{x e^{\left (4 \, e\right )} - e^{48}} + \frac {e^{\left (4 \, e + 48\right )}}{x^{2} e^{\left (8 \, e\right )} - 2 \, x e^{\left (4 \, e + 48\right )} + e^{96}}\right )} \]
integrate(((2*x^6+2*x^4-x^3)*exp(4*exp(1)-48)^3+(-6*x^5-6*x^3-x^2)*exp(4*e xp(1)-48)^2+(6*x^4+6*x^2)*exp(4*exp(1)-48)-2*x^3-2*x)*exp(((x^4+x)*exp(4*e xp(1)-48)^2-2*x^3*exp(4*exp(1)-48)+x^2)/(x^2*exp(4*exp(1)-48)^2-2*x*exp(4* exp(1)-48)+1))/(16*x^3*exp(4*exp(1)-48)^3-48*x^2*exp(4*exp(1)-48)^2+48*x*e xp(4*exp(1)-48)-16),x, algorithm=\
1/16*x^2*e^(x^2 + e^(4*e)/(x*e^(4*e) - e^48) + e^(4*e + 48)/(x^2*e^(8*e) - 2*x*e^(4*e + 48) + e^96))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (25) = 50\).
Time = 0.88 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27 \[ \int \frac {e^{\frac {x^2-2 e^{-48+4 e} x^3+e^{-96+8 e} \left (x+x^4\right )}{1-2 e^{-48+4 e} x+e^{-96+8 e} x^2}} \left (-2 x-2 x^3+e^{-48+4 e} \left (6 x^2+6 x^4\right )+e^{-96+8 e} \left (-x^2-6 x^3-6 x^5\right )+e^{-144+12 e} \left (-x^3+2 x^4+2 x^6\right )\right )}{-16+48 e^{-48+4 e} x-48 e^{-96+8 e} x^2+16 e^{-144+12 e} x^3} \, dx=\frac {1}{16} \, x^{2} e^{\left (\frac {x^{4} e^{\left (8 \, e - 96\right )} - 2 \, x^{3} e^{\left (4 \, e - 48\right )} + x^{2} + x e^{\left (8 \, e - 96\right )}}{x^{2} e^{\left (8 \, e - 96\right )} - 2 \, x e^{\left (4 \, e - 48\right )} + 1}\right )} \]
integrate(((2*x^6+2*x^4-x^3)*exp(4*exp(1)-48)^3+(-6*x^5-6*x^3-x^2)*exp(4*e xp(1)-48)^2+(6*x^4+6*x^2)*exp(4*exp(1)-48)-2*x^3-2*x)*exp(((x^4+x)*exp(4*e xp(1)-48)^2-2*x^3*exp(4*exp(1)-48)+x^2)/(x^2*exp(4*exp(1)-48)^2-2*x*exp(4* exp(1)-48)+1))/(16*x^3*exp(4*exp(1)-48)^3-48*x^2*exp(4*exp(1)-48)^2+48*x*e xp(4*exp(1)-48)-16),x, algorithm=\
1/16*x^2*e^((x^4*e^(8*e - 96) - 2*x^3*e^(4*e - 48) + x^2 + x*e^(8*e - 96)) /(x^2*e^(8*e - 96) - 2*x*e^(4*e - 48) + 1))
Timed out. \[ \int \frac {e^{\frac {x^2-2 e^{-48+4 e} x^3+e^{-96+8 e} \left (x+x^4\right )}{1-2 e^{-48+4 e} x+e^{-96+8 e} x^2}} \left (-2 x-2 x^3+e^{-48+4 e} \left (6 x^2+6 x^4\right )+e^{-96+8 e} \left (-x^2-6 x^3-6 x^5\right )+e^{-144+12 e} \left (-x^3+2 x^4+2 x^6\right )\right )}{-16+48 e^{-48+4 e} x-48 e^{-96+8 e} x^2+16 e^{-144+12 e} x^3} \, dx=\text {Hanged} \]
int(-(exp((exp(8*exp(1) - 96)*(x + x^4) - 2*x^3*exp(4*exp(1) - 48) + x^2)/ (x^2*exp(8*exp(1) - 96) - 2*x*exp(4*exp(1) - 48) + 1))*(2*x + exp(8*exp(1) - 96)*(x^2 + 6*x^3 + 6*x^5) - exp(4*exp(1) - 48)*(6*x^2 + 6*x^4) + 2*x^3 - exp(12*exp(1) - 144)*(2*x^4 - x^3 + 2*x^6)))/(48*x*exp(4*exp(1) - 48) - 48*x^2*exp(8*exp(1) - 96) + 16*x^3*exp(12*exp(1) - 144) - 16),x)