Integrand size = 73, antiderivative size = 29 \[ \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx=\frac {e^{4-\frac {1}{x}-\frac {16 x^2}{9}+\frac {x}{3+x^2}}}{x} \]
Time = 3.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx=\frac {e^{4-\frac {1}{x}-\frac {16 x^2}{9}+\frac {x}{3+x^2}}}{x} \]
Integrate[(E^((-27 + 108*x - 12*x^3 - 16*x^5)/(27*x + 9*x^3))*(81 - 81*x + 81*x^2 - 342*x^3 - 201*x^5 - 32*x^7))/(81*x^3 + 54*x^5 + 9*x^7),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.
\(\displaystyle \int \frac {e^{\frac {-16 x^5-12 x^3+108 x-27}{9 x^3+27 x}} \left (-32 x^7-201 x^5-342 x^3+81 x^2-81 x+81\right )}{9 x^7+54 x^5+81 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {e^{\frac {-16 x^5-12 x^3+108 x-27}{9 x^3+27 x}} \left (-32 x^7-201 x^5-342 x^3+81 x^2-81 x+81\right )}{x^3 \left (9 x^4+54 x^2+81\right )}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle 9 \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 \left (x^3+3 x\right )}} \left (-32 x^7-201 x^5-342 x^3+81 x^2-81 x+81\right )}{81 x^3 \left (x^2+3\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{9} \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 \left (x^3+3 x\right )}} \left (-32 x^7-201 x^5-342 x^3+81 x^2-81 x+81\right )}{x^3 \left (x^2+3\right )^2}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \frac {1}{9} \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}} \left (-32 x^7-201 x^5-342 x^3+81 x^2-81 x+81\right )}{x^3 \left (x^2+3\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{9} \int \left (-\frac {3 e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}} x}{x^2+3}-\frac {18 e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}} x}{\left (x^2+3\right )^2}-32 e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}+\frac {3 e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{x}-\frac {9 e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{x^2}+\frac {9 e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{9} \left (-32 \int e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}dx+\frac {3}{2} \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{i \sqrt {3}-x}dx+9 \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{x^3}dx-9 \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{x^2}dx+3 \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{x}dx-\frac {3}{2} \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}}}{x+i \sqrt {3}}dx-18 \int \frac {e^{-\frac {16 x^5+12 x^3-108 x+27}{9 x \left (x^2+3\right )}} x}{\left (x^2+3\right )^2}dx\right )\) |
Int[(E^((-27 + 108*x - 12*x^3 - 16*x^5)/(27*x + 9*x^3))*(81 - 81*x + 81*x^ 2 - 342*x^3 - 201*x^5 - 32*x^7))/(81*x^3 + 54*x^5 + 9*x^7),x]
3.25.29.3.1 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_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14
method | result | size |
gosper | \(\frac {{\mathrm e}^{-\frac {16 x^{5}+12 x^{3}-108 x +27}{9 \left (x^{2}+3\right ) x}}}{x}\) | \(33\) |
risch | \(\frac {{\mathrm e}^{-\frac {16 x^{5}+12 x^{3}-108 x +27}{9 \left (x^{2}+3\right ) x}}}{x}\) | \(33\) |
parallelrisch | \(\frac {{\mathrm e}^{-\frac {16 x^{5}+12 x^{3}-108 x +27}{9 \left (x^{2}+3\right ) x}}}{x}\) | \(33\) |
norman | \(\frac {x^{3} {\mathrm e}^{\frac {-16 x^{5}-12 x^{3}+108 x -27}{9 x^{3}+27 x}}+3 x \,{\mathrm e}^{\frac {-16 x^{5}-12 x^{3}+108 x -27}{9 x^{3}+27 x}}}{x^{2} \left (x^{2}+3\right )}\) | \(76\) |
int((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108*x-27) /(9*x^3+27*x))/(9*x^7+54*x^5+81*x^3),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx=\frac {e^{\left (-\frac {16 \, x^{5} + 12 \, x^{3} - 108 \, x + 27}{9 \, {\left (x^{3} + 3 \, x\right )}}\right )}}{x} \]
integrate((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108 *x-27)/(9*x^3+27*x))/(9*x^7+54*x^5+81*x^3),x, algorithm=\
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx=\frac {e^{\frac {- 16 x^{5} - 12 x^{3} + 108 x - 27}{9 x^{3} + 27 x}}}{x} \]
integrate((-32*x**7-201*x**5-342*x**3+81*x**2-81*x+81)*exp((-16*x**5-12*x* *3+108*x-27)/(9*x**3+27*x))/(9*x**7+54*x**5+81*x**3),x)
Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx=\frac {e^{\left (-\frac {16}{9} \, x^{2} + \frac {x}{x^{2} + 3} - \frac {1}{x} + 4\right )}}{x} \]
integrate((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108 *x-27)/(9*x^3+27*x))/(9*x^7+54*x^5+81*x^3),x, algorithm=\
Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx=\frac {e^{\left (-\frac {16 \, x^{5} + 12 \, x^{3} - 108 \, x + 27}{9 \, {\left (x^{3} + 3 \, x\right )}}\right )}}{x} \]
integrate((-32*x^7-201*x^5-342*x^3+81*x^2-81*x+81)*exp((-16*x^5-12*x^3+108 *x-27)/(9*x^3+27*x))/(9*x^7+54*x^5+81*x^3),x, algorithm=\
Time = 14.17 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {e^{\frac {-27+108 x-12 x^3-16 x^5}{27 x+9 x^3}} \left (81-81 x+81 x^2-342 x^3-201 x^5-32 x^7\right )}{81 x^3+54 x^5+9 x^7} \, dx=\frac {{\mathrm {e}}^{-\frac {3}{x^3+3\,x}}\,{\mathrm {e}}^{-\frac {4\,x^2}{3\,\left (x^2+3\right )}}\,{\mathrm {e}}^{-\frac {16\,x^4}{9\,\left (x^2+3\right )}}\,{\mathrm {e}}^{\frac {12}{x^2+3}}}{x} \]
int(-(exp(-(12*x^3 - 108*x + 16*x^5 + 27)/(27*x + 9*x^3))*(81*x - 81*x^2 + 342*x^3 + 201*x^5 + 32*x^7 - 81))/(81*x^3 + 54*x^5 + 9*x^7),x)