Integrand size = 119, antiderivative size = 27 \[ \int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx=\frac {16}{\left (e^{2 x-x (5+x) (4+x (6+x))}-x\right )^2} \]
Time = 5.96 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx=\frac {16 e^{2 x \left (18+34 x+11 x^2+x^3\right )}}{\left (-1+e^{x \left (18+34 x+11 x^2+x^3\right )} x\right )^2} \]
Integrate[(32 + E^(-18*x - 34*x^2 - 11*x^3 - x^4)*(576 + 2176*x + 1056*x^2 + 128*x^3))/(E^(-54*x - 102*x^2 - 33*x^3 - 3*x^4) - 3*E^(-36*x - 68*x^2 - 22*x^3 - 2*x^4)*x + 3*E^(-18*x - 34*x^2 - 11*x^3 - x^4)*x^2 - 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.
\(\displaystyle \int \frac {e^{-x^4-11 x^3-34 x^2-18 x} \left (128 x^3+1056 x^2+2176 x+576\right )+32}{-x^3+3 e^{-x^4-11 x^3-34 x^2-18 x} x^2-3 e^{-2 x^4-22 x^3-68 x^2-36 x} x+e^{-3 x^4-33 x^3-102 x^2-54 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{3 x \left (x^3+11 x^2+34 x+18\right )} \left (e^{-x^4-11 x^3-34 x^2-18 x} \left (128 x^3+1056 x^2+2176 x+576\right )+32\right )}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {128 x^3 \exp \left (-x^4-11 x^3-34 x^2+3 \left (x^3+11 x^2+34 x+18\right ) x-18 x\right )}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}+\frac {1056 x^2 \exp \left (-x^4-11 x^3-34 x^2+3 \left (x^3+11 x^2+34 x+18\right ) x-18 x\right )}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}+\frac {2176 x \exp \left (-x^4-11 x^3-34 x^2+3 \left (x^3+11 x^2+34 x+18\right ) x-18 x\right )}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}+\frac {576 \exp \left (-x^4-11 x^3-34 x^2+3 \left (x^3+11 x^2+34 x+18\right ) x-18 x\right )}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}-\frac {32 e^{3 x \left (x^3+11 x^2+34 x+18\right )}}{\left (e^{x \left (x^3+11 x^2+34 x+18\right )} x-1\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 576 \int \frac {e^{2 x \left (x^3+11 x^2+34 x+18\right )}}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}dx+2176 \int \frac {e^{2 x \left (x^3+11 x^2+34 x+18\right )} x}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}dx+1056 \int \frac {e^{2 x \left (x^3+11 x^2+34 x+18\right )} x^2}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}dx+128 \int \frac {e^{2 x \left (x^3+11 x^2+34 x+18\right )} x^3}{\left (1-e^{x \left (x^3+11 x^2+34 x+18\right )} x\right )^3}dx-32 \int \frac {e^{3 x \left (x^3+11 x^2+34 x+18\right )}}{\left (e^{x \left (x^3+11 x^2+34 x+18\right )} x-1\right )^3}dx\) |
Int[(32 + E^(-18*x - 34*x^2 - 11*x^3 - x^4)*(576 + 2176*x + 1056*x^2 + 128 *x^3))/(E^(-54*x - 102*x^2 - 33*x^3 - 3*x^4) - 3*E^(-36*x - 68*x^2 - 22*x^ 3 - 2*x^4)*x + 3*E^(-18*x - 34*x^2 - 11*x^3 - x^4)*x^2 - x^3),x]
3.6.75.3.1 Defintions of rubi rules used
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96
method | result | size |
risch | \(\frac {16}{\left (x -{\mathrm e}^{-x \left (x^{3}+11 x^{2}+34 x +18\right )}\right )^{2}}\) | \(26\) |
norman | \(\frac {16}{\left (x -{\mathrm e}^{-x^{4}-11 x^{3}-34 x^{2}-18 x}\right )^{2}}\) | \(29\) |
parallelrisch | \(\frac {16}{x^{2}-2 x \,{\mathrm e}^{-x^{4}-11 x^{3}-34 x^{2}-18 x}+{\mathrm e}^{-2 x^{4}-22 x^{3}-68 x^{2}-36 x}}\) | \(54\) |
int(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/(exp(- x^4-11*x^3-34*x^2-18*x)^3-3*x*exp(-x^4-11*x^3-34*x^2-18*x)^2+3*x^2*exp(-x^ 4-11*x^3-34*x^2-18*x)-x^3),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx=\frac {16}{x^{2} - 2 \, x e^{\left (-x^{4} - 11 \, x^{3} - 34 \, x^{2} - 18 \, x\right )} + e^{\left (-2 \, x^{4} - 22 \, x^{3} - 68 \, x^{2} - 36 \, x\right )}} \]
integrate(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/ (exp(-x^4-11*x^3-34*x^2-18*x)^3-3*x*exp(-x^4-11*x^3-34*x^2-18*x)^2+3*x^2*e xp(-x^4-11*x^3-34*x^2-18*x)-x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx=\frac {16}{x^{2} - 2 x e^{- x^{4} - 11 x^{3} - 34 x^{2} - 18 x} + e^{- 2 x^{4} - 22 x^{3} - 68 x^{2} - 36 x}} \]
integrate(((128*x**3+1056*x**2+2176*x+576)*exp(-x**4-11*x**3-34*x**2-18*x) +32)/(exp(-x**4-11*x**3-34*x**2-18*x)**3-3*x*exp(-x**4-11*x**3-34*x**2-18* x)**2+3*x**2*exp(-x**4-11*x**3-34*x**2-18*x)-x**3),x)
16/(x**2 - 2*x*exp(-x**4 - 11*x**3 - 34*x**2 - 18*x) + exp(-2*x**4 - 22*x* *3 - 68*x**2 - 36*x))
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.63 \[ \int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx=\frac {16 \, e^{\left (2 \, x^{4} + 22 \, x^{3} + 68 \, x^{2} + 36 \, x\right )}}{x^{2} e^{\left (2 \, x^{4} + 22 \, x^{3} + 68 \, x^{2} + 36 \, x\right )} - 2 \, x e^{\left (x^{4} + 11 \, x^{3} + 34 \, x^{2} + 18 \, x\right )} + 1} \]
integrate(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/ (exp(-x^4-11*x^3-34*x^2-18*x)^3-3*x*exp(-x^4-11*x^3-34*x^2-18*x)^2+3*x^2*e xp(-x^4-11*x^3-34*x^2-18*x)-x^3),x, algorithm=\
16*e^(2*x^4 + 22*x^3 + 68*x^2 + 36*x)/(x^2*e^(2*x^4 + 22*x^3 + 68*x^2 + 36 *x) - 2*x*e^(x^4 + 11*x^3 + 34*x^2 + 18*x) + 1)
Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx=\frac {16}{x^{2} - 2 \, x e^{\left (-x^{4} - 11 \, x^{3} - 34 \, x^{2} - 18 \, x\right )} + e^{\left (-2 \, x^{4} - 22 \, x^{3} - 68 \, x^{2} - 36 \, x\right )}} \]
integrate(((128*x^3+1056*x^2+2176*x+576)*exp(-x^4-11*x^3-34*x^2-18*x)+32)/ (exp(-x^4-11*x^3-34*x^2-18*x)^3-3*x*exp(-x^4-11*x^3-34*x^2-18*x)^2+3*x^2*e xp(-x^4-11*x^3-34*x^2-18*x)-x^3),x, algorithm=\
Time = 8.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {32+e^{-18 x-34 x^2-11 x^3-x^4} \left (576+2176 x+1056 x^2+128 x^3\right )}{e^{-54 x-102 x^2-33 x^3-3 x^4}-3 e^{-36 x-68 x^2-22 x^3-2 x^4} x+3 e^{-18 x-34 x^2-11 x^3-x^4} x^2-x^3} \, dx=\frac {16}{x^2+{\mathrm {e}}^{-36\,x}\,{\mathrm {e}}^{-2\,x^4}\,{\mathrm {e}}^{-22\,x^3}\,{\mathrm {e}}^{-68\,x^2}-2\,x\,{\mathrm {e}}^{-18\,x}\,{\mathrm {e}}^{-x^4}\,{\mathrm {e}}^{-11\,x^3}\,{\mathrm {e}}^{-34\,x^2}} \]
int((exp(- 18*x - 34*x^2 - 11*x^3 - x^4)*(2176*x + 1056*x^2 + 128*x^3 + 57 6) + 32)/(exp(- 54*x - 102*x^2 - 33*x^3 - 3*x^4) - 3*x*exp(- 36*x - 68*x^2 - 22*x^3 - 2*x^4) + 3*x^2*exp(- 18*x - 34*x^2 - 11*x^3 - x^4) - x^3),x)