Integrand size = 204, antiderivative size = 32 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^2 \left (-4-\frac {x}{e^3}\right )^2}{(4-2 x)^2 \left (e^{3+x}+x^2\right )^2} \]
Time = 4.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^2 \left (4 e^3+x\right )^2}{4 e^6 (-2+x)^2 \left (e^{3+x}+x^2\right )^2} \]
Integrate[(-x^6 + E^6*(32*x^3 - 32*x^4) + E^3*(8*x^4 - 12*x^5) + E^(3 + x) *(-4*x^3 + 3*x^4 - x^5 + E^6*(-32*x + 32*x^2 - 16*x^3) + E^3*(-24*x^2 + 20 *x^3 - 8*x^4)))/(E^(15 + 3*x)*(-16 + 24*x - 12*x^2 + 2*x^3) + E^(12 + 2*x) *(-48*x^2 + 72*x^3 - 36*x^4 + 6*x^5) + E^(9 + x)*(-48*x^4 + 72*x^5 - 36*x^ 6 + 6*x^7) + E^6*(-16*x^6 + 24*x^7 - 12*x^8 + 2*x^9)),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.
| \(\displaystyle \int \frac {-x^6+e^3 \left (8 x^4-12 x^5\right )+e^6 \left (32 x^3-32 x^4\right )+e^{x+3} \left (-x^5+3 x^4-4 x^3+e^6 \left (-16 x^3+32 x^2-32 x\right )+e^3 \left (-8 x^4+20 x^3-24 x^2\right )\right )}{e^{3 x+15} \left (2 x^3-12 x^2+24 x-16\right )+e^6 \left (2 x^9-12 x^8+24 x^7-16 x^6\right )+e^{x+9} \left (6 x^7-36 x^6+72 x^5-48 x^4\right )+e^{2 x+12} \left (6 x^5-36 x^4+72 x^3-48 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x+4 e^3\right ) \left (x^4+8 e^3 (x-1) x^2+e^{x+3} \left (x^2-3 x+4\right ) x+4 e^{x+6} \left (x^2-2 x+2\right )\right )}{2 e^6 (2-x)^3 \left (x^2+e^{x+3}\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {x \left (x+4 e^3\right ) \left (-x^4+8 e^3 (1-x) x^2-e^{x+3} \left (x^2-3 x+4\right ) x-4 e^{x+6} \left (x^2-2 x+2\right )\right )}{(2-x)^3 \left (x^2+e^{x+3}\right )^3}dx}{2 e^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {x \left (x+4 e^3\right ) \left (-x^4+8 e^3 (1-x) x^2-e^{x+3} \left (x^2-3 x+4\right ) x-4 e^{x+6} \left (x^2-2 x+2\right )\right )}{(2-x)^3 \left (x^2+e^{x+3}\right )^3}dx}{2 e^6}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (\frac {x \left (-x^4+\left (3-8 e^3\right ) x^3-4 \left (1-5 e^3+4 e^6\right ) x^2-8 e^3 \left (3-4 e^3\right ) x-32 e^6\right )}{(2-x)^3 \left (x^2+e^{x+3}\right )^2}-\frac {x^3 \left (x+4 e^3\right )^2}{(x-2) \left (x^2+e^{x+3}\right )^3}\right )dx}{2 e^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-16 \left (1+2 e^3\right )^2 \int \frac {1}{\left (x^2+e^{x+3}\right )^3}dx-32 \left (1+2 e^3\right )^2 \int \frac {1}{(x-2) \left (x^2+e^{x+3}\right )^3}dx-8 \left (1+2 e^3\right )^2 \int \frac {x}{\left (x^2+e^{x+3}\right )^3}dx-4 \left (1+2 e^3\right )^2 \int \frac {x^2}{\left (x^2+e^{x+3}\right )^3}dx+2 \left (5+14 e^3+8 e^6\right ) \int \frac {1}{\left (x^2+e^{x+3}\right )^2}dx+16 \left (1+2 e^3\right )^2 \int \frac {1}{(x-2)^3 \left (x^2+e^{x+3}\right )^2}dx+16 \left (2+7 e^3+6 e^6\right ) \int \frac {1}{(x-2)^2 \left (x^2+e^{x+3}\right )^2}dx+32 \left (1+3 e^3+2 e^6\right ) \int \frac {1}{(x-2) \left (x^2+e^{x+3}\right )^2}dx+\left (3+8 e^3\right ) \int \frac {x}{\left (x^2+e^{x+3}\right )^2}dx+\int \frac {x^2}{\left (x^2+e^{x+3}\right )^2}dx-\int \frac {x^4}{\left (x^2+e^{x+3}\right )^3}dx-2 \left (1+4 e^3\right ) \int \frac {x^3}{\left (x^2+e^{x+3}\right )^3}dx}{2 e^6}\) |
Int[(-x^6 + E^6*(32*x^3 - 32*x^4) + E^3*(8*x^4 - 12*x^5) + E^(3 + x)*(-4*x ^3 + 3*x^4 - x^5 + E^6*(-32*x + 32*x^2 - 16*x^3) + E^3*(-24*x^2 + 20*x^3 - 8*x^4)))/(E^(15 + 3*x)*(-16 + 24*x - 12*x^2 + 2*x^3) + E^(12 + 2*x)*(-48* x^2 + 72*x^3 - 36*x^4 + 6*x^5) + E^(9 + x)*(-48*x^4 + 72*x^5 - 36*x^6 + 6* x^7) + E^6*(-16*x^6 + 24*x^7 - 12*x^8 + 2*x^9)),x]
3.15.46.3.1 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[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(\frac {x^{2} \left (16 \,{\mathrm e}^{6}+8 x \,{\mathrm e}^{3}+x^{2}\right ) {\mathrm e}^{-6}}{4 \left (x^{2}-4 x +4\right ) \left (x^{2}+{\mathrm e}^{3+x}\right )^{2}}\) | \(41\) |
| parallelrisch | \(\frac {\left (16 x^{2} {\mathrm e}^{6}+8 x^{3} {\mathrm e}^{3}+x^{4}\right ) {\mathrm e}^{-6}}{4 x^{6}-16 x^{5}+8 x^{4} {\mathrm e}^{3+x}+16 x^{4}-32 x^{3} {\mathrm e}^{3+x}+4 x^{2} {\mathrm e}^{2 x +6}+32 x^{2} {\mathrm e}^{3+x}-16 x \,{\mathrm e}^{2 x +6}+16 \,{\mathrm e}^{2 x +6}}\) | \(97\) |
int((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x^5+3*x ^4-4*x^3)*exp(3+x)+(-32*x^4+32*x^3)*exp(3)^2+(-12*x^5+8*x^4)*exp(3)-x^6)/( (2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48*x^2)*ex p(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2*x^9-12 *x^8+24*x^7-16*x^6)*exp(3)^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.44 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^{4} e^{6} + 8 \, x^{3} e^{9} + 16 \, x^{2} e^{12}}{4 \, {\left ({\left (x^{6} - 4 \, x^{5} + 4 \, x^{4}\right )} e^{12} + {\left (x^{2} - 4 \, x + 4\right )} e^{\left (2 \, x + 18\right )} + 2 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} e^{\left (x + 15\right )}\right )}} \]
integrate((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x ^5+3*x^4-4*x^3)*exp(3+x)+(-32*x^4+32*x^3)*exp(3)^2+(-12*x^5+8*x^4)*exp(3)- x^6)/((2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48*x ^2)*exp(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2* x^9-12*x^8+24*x^7-16*x^6)*exp(3)^2),x, algorithm=\
1/4*(x^4*e^6 + 8*x^3*e^9 + 16*x^2*e^12)/((x^6 - 4*x^5 + 4*x^4)*e^12 + (x^2 - 4*x + 4)*e^(2*x + 18) + 2*(x^4 - 4*x^3 + 4*x^2)*e^(x + 15))
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (29) = 58\).
Time = 0.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.19 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^{4} + 8 x^{3} e^{3} + 16 x^{2} e^{6}}{4 x^{6} e^{6} - 16 x^{5} e^{6} + 16 x^{4} e^{6} + \left (4 x^{2} e^{6} - 16 x e^{6} + 16 e^{6}\right ) e^{2 x + 6} + \left (8 x^{4} e^{6} - 32 x^{3} e^{6} + 32 x^{2} e^{6}\right ) e^{x + 3}} \]
integrate((((-16*x**3+32*x**2-32*x)*exp(3)**2+(-8*x**4+20*x**3-24*x**2)*ex p(3)-x**5+3*x**4-4*x**3)*exp(3+x)+(-32*x**4+32*x**3)*exp(3)**2+(-12*x**5+8 *x**4)*exp(3)-x**6)/((2*x**3-12*x**2+24*x-16)*exp(3)**2*exp(3+x)**3+(6*x** 5-36*x**4+72*x**3-48*x**2)*exp(3)**2*exp(3+x)**2+(6*x**7-36*x**6+72*x**5-4 8*x**4)*exp(3)**2*exp(3+x)+(2*x**9-12*x**8+24*x**7-16*x**6)*exp(3)**2),x)
(x**4 + 8*x**3*exp(3) + 16*x**2*exp(6))/(4*x**6*exp(6) - 16*x**5*exp(6) + 16*x**4*exp(6) + (4*x**2*exp(6) - 16*x*exp(6) + 16*exp(6))*exp(2*x + 6) + (8*x**4*exp(6) - 32*x**3*exp(6) + 32*x**2*exp(6))*exp(x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (28) = 56\).
Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.78 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {x^{4} + 8 \, x^{3} e^{3} + 16 \, x^{2} e^{6}}{4 \, {\left (x^{6} e^{6} - 4 \, x^{5} e^{6} + 4 \, x^{4} e^{6} + {\left (x^{2} e^{12} - 4 \, x e^{12} + 4 \, e^{12}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} e^{9} - 4 \, x^{3} e^{9} + 4 \, x^{2} e^{9}\right )} e^{x}\right )}} \]
integrate((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x ^5+3*x^4-4*x^3)*exp(3+x)+(-32*x^4+32*x^3)*exp(3)^2+(-12*x^5+8*x^4)*exp(3)- x^6)/((2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48*x ^2)*exp(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2* x^9-12*x^8+24*x^7-16*x^6)*exp(3)^2),x, algorithm=\
1/4*(x^4 + 8*x^3*e^3 + 16*x^2*e^6)/(x^6*e^6 - 4*x^5*e^6 + 4*x^4*e^6 + (x^2 *e^12 - 4*x*e^12 + 4*e^12)*e^(2*x) + 2*(x^4*e^9 - 4*x^3*e^9 + 4*x^2*e^9)*e ^x)
Leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (28) = 56\).
Time = 0.36 (sec) , antiderivative size = 226, normalized size of antiderivative = 7.06 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=\frac {{\left (x + 3\right )}^{4} e^{3} + 8 \, {\left (x + 3\right )}^{3} e^{6} - 12 \, {\left (x + 3\right )}^{3} e^{3} + 16 \, {\left (x + 3\right )}^{2} e^{9} - 72 \, {\left (x + 3\right )}^{2} e^{6} + 54 \, {\left (x + 3\right )}^{2} e^{3} - 96 \, {\left (x + 3\right )} e^{9} + 216 \, {\left (x + 3\right )} e^{6} - 108 \, {\left (x + 3\right )} e^{3} + 144 \, e^{9} - 216 \, e^{6} + 81 \, e^{3}}{4 \, {\left ({\left (x + 3\right )}^{6} e^{9} - 22 \, {\left (x + 3\right )}^{5} e^{9} + 199 \, {\left (x + 3\right )}^{4} e^{9} + 2 \, {\left (x + 3\right )}^{4} e^{\left (x + 12\right )} - 948 \, {\left (x + 3\right )}^{3} e^{9} - 32 \, {\left (x + 3\right )}^{3} e^{\left (x + 12\right )} + 2511 \, {\left (x + 3\right )}^{2} e^{9} + {\left (x + 3\right )}^{2} e^{\left (2 \, x + 15\right )} + 188 \, {\left (x + 3\right )}^{2} e^{\left (x + 12\right )} - 3510 \, {\left (x + 3\right )} e^{9} - 10 \, {\left (x + 3\right )} e^{\left (2 \, x + 15\right )} - 480 \, {\left (x + 3\right )} e^{\left (x + 12\right )} + 2025 \, e^{9} + 25 \, e^{\left (2 \, x + 15\right )} + 450 \, e^{\left (x + 12\right )}\right )}} \]
integrate((((-16*x^3+32*x^2-32*x)*exp(3)^2+(-8*x^4+20*x^3-24*x^2)*exp(3)-x ^5+3*x^4-4*x^3)*exp(3+x)+(-32*x^4+32*x^3)*exp(3)^2+(-12*x^5+8*x^4)*exp(3)- x^6)/((2*x^3-12*x^2+24*x-16)*exp(3)^2*exp(3+x)^3+(6*x^5-36*x^4+72*x^3-48*x ^2)*exp(3)^2*exp(3+x)^2+(6*x^7-36*x^6+72*x^5-48*x^4)*exp(3)^2*exp(3+x)+(2* x^9-12*x^8+24*x^7-16*x^6)*exp(3)^2),x, algorithm=\
1/4*((x + 3)^4*e^3 + 8*(x + 3)^3*e^6 - 12*(x + 3)^3*e^3 + 16*(x + 3)^2*e^9 - 72*(x + 3)^2*e^6 + 54*(x + 3)^2*e^3 - 96*(x + 3)*e^9 + 216*(x + 3)*e^6 - 108*(x + 3)*e^3 + 144*e^9 - 216*e^6 + 81*e^3)/((x + 3)^6*e^9 - 22*(x + 3 )^5*e^9 + 199*(x + 3)^4*e^9 + 2*(x + 3)^4*e^(x + 12) - 948*(x + 3)^3*e^9 - 32*(x + 3)^3*e^(x + 12) + 2511*(x + 3)^2*e^9 + (x + 3)^2*e^(2*x + 15) + 1 88*(x + 3)^2*e^(x + 12) - 3510*(x + 3)*e^9 - 10*(x + 3)*e^(2*x + 15) - 480 *(x + 3)*e^(x + 12) + 2025*e^9 + 25*e^(2*x + 15) + 450*e^(x + 12))
Time = 12.67 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.06 \[ \int \frac {-x^6+e^6 \left (32 x^3-32 x^4\right )+e^3 \left (8 x^4-12 x^5\right )+e^{3+x} \left (-4 x^3+3 x^4-x^5+e^6 \left (-32 x+32 x^2-16 x^3\right )+e^3 \left (-24 x^2+20 x^3-8 x^4\right )\right )}{e^{15+3 x} \left (-16+24 x-12 x^2+2 x^3\right )+e^{12+2 x} \left (-48 x^2+72 x^3-36 x^4+6 x^5\right )+e^{9+x} \left (-48 x^4+72 x^5-36 x^6+6 x^7\right )+e^6 \left (-16 x^6+24 x^7-12 x^8+2 x^9\right )} \, dx=-\frac {\frac {{\mathrm {e}}^{-6}\,x^7}{4}+\frac {{\mathrm {e}}^{-6}\,\left (8\,{\mathrm {e}}^3-4\right )\,x^6}{4}+\frac {{\mathrm {e}}^{-6}\,\left (16\,{\mathrm {e}}^6-32\,{\mathrm {e}}^3+4\right )\,x^5}{4}+\frac {{\mathrm {e}}^{-6}\,\left (32\,{\mathrm {e}}^3-64\,{\mathrm {e}}^6\right )\,x^4}{4}+16\,x^3}{\left (2\,x-x^2\right )\,{\left (x-2\right )}^3\,\left ({\mathrm {e}}^{2\,x+6}+2\,x^2\,{\mathrm {e}}^{x+3}+x^4\right )} \]
int((exp(x + 3)*(exp(6)*(32*x - 32*x^2 + 16*x^3) + exp(3)*(24*x^2 - 20*x^3 + 8*x^4) + 4*x^3 - 3*x^4 + x^5) - exp(3)*(8*x^4 - 12*x^5) - exp(6)*(32*x^ 3 - 32*x^4) + x^6)/(exp(6)*(16*x^6 - 24*x^7 + 12*x^8 - 2*x^9) - exp(6)*exp (3*x + 9)*(24*x - 12*x^2 + 2*x^3 - 16) + exp(x + 3)*exp(6)*(48*x^4 - 72*x^ 5 + 36*x^6 - 6*x^7) + exp(6)*exp(2*x + 6)*(48*x^2 - 72*x^3 + 36*x^4 - 6*x^ 5)),x)