Integrand size = 77, antiderivative size = 25 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=e^{-14-x^3} \left (2+\frac {e^{3-x}}{x}\right )^2 \] Output:
exp(ln((2+exp(3)/exp(x)/x)^2)-x^3-14)
Time = 0.89 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=\frac {e^{-14-2 x-x^3} \left (e^3+2 e^x x\right )^2}{x^2} \] Input:
Integrate[(E^(-14 - 2*x - x^3)*(E^6 + 4*E^(3 + x)*x + 4*E^(2*x)*x^2)*(-6*E ^x*x^4 + E^3*(-2 - 2*x - 3*x^3)))/(x^2*(E^3*x + 2*E^x*x^2)),x]
Output:
(E^(-14 - 2*x - x^3)*(E^3 + 2*E^x*x)^2)/x^2
Leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(25)=50\).
Time = 2.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {7239, 7293, 2009}
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.
\(\displaystyle \int \frac {e^{-x^3-2 x-14} \left (4 e^{2 x} x^2+4 e^{x+3} x+e^6\right ) \left (e^3 \left (-3 x^3-2 x-2\right )-6 e^x x^4\right )}{x^2 \left (2 e^x x^2+e^3 x\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{-x^3-2 x-14} \left (2 e^x x+e^3\right ) \left (-6 e^x x^4-e^3 \left (3 x^3+2 x+2\right )\right )}{x^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {e^{-x^3-2 x-8} \left (3 x^3+2 x+2\right )}{x^3}-12 e^{-x^3-14} x^2-\frac {4 e^{-x^3-x-11} \left (3 x^3+x+1\right )}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 e^{-x^3-14}+\frac {4 e^{-x^3-x-11} \left (3 x^3+x\right )}{x^2 \left (3 x^2+1\right )}+\frac {e^{-x^3-2 x-8} \left (3 x^3+2 x\right )}{x^3 \left (3 x^2+2\right )}\) |
Input:
Int[(E^(-14 - 2*x - x^3)*(E^6 + 4*E^(3 + x)*x + 4*E^(2*x)*x^2)*(-6*E^x*x^4 + E^3*(-2 - 2*x - 3*x^3)))/(x^2*(E^3*x + 2*E^x*x^2)),x]
Output:
4*E^(-14 - x^3) + (4*E^(-11 - x - x^3)*(x + 3*x^3))/(x^2*(1 + 3*x^2)) + (E ^(-8 - 2*x - x^3)*(2*x + 3*x^3))/(x^3*(2 + 3*x^2))
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 20.59 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56
method | result | size |
parallelrisch | \({\mathrm e}^{\ln \left (\frac {\left (4 \,{\mathrm e}^{2 x} x^{2}+4 x \,{\mathrm e}^{3} {\mathrm e}^{x}+{\mathrm e}^{6}\right ) {\mathrm e}^{-2 x}}{x^{2}}\right )-x^{3}-14}\) | \(39\) |
risch | \(\frac {\left (4 \,{\mathrm e}^{2 x} x^{2}+4 \,{\mathrm e}^{3+x} x +{\mathrm e}^{6}\right ) {\mathrm e}^{-2 x -14-\frac {i \pi \,\operatorname {csgn}\left (i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x}\right )}{2}+\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-2 x}\right )}{2}+\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )}^{2} \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right )}{2}-\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )}^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right ) {\operatorname {csgn}\left (i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )\right )}^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right )^{3}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right ) \operatorname {csgn}\left (\frac {i}{x^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right )}{2}+\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}}{2}-i \pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-2 x} \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right )^{2}}{2}-\frac {i \pi {\operatorname {csgn}\left (i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (\frac {i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2} {\mathrm e}^{-2 x}}{x^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x^{2}}\right )}{2}+i \pi {\operatorname {csgn}\left (i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )^{2}\right )}^{2} \operatorname {csgn}\left (i \left (2 \,{\mathrm e}^{x} x +{\mathrm e}^{3}\right )\right )-x^{3}}}{x^{2}}\) | \(516\) |
Input:
int((-6*exp(x)*x^4+(-3*x^3-2*x-2)*exp(3))*exp(ln((4*exp(x)^2*x^2+4*x*exp(3 )*exp(x)+exp(3)^2)/exp(x)^2/x^2)-x^3-14)/(2*exp(x)*x^2+x*exp(3)),x,method= _RETURNVERBOSE)
Output:
exp(ln((4*exp(x)^2*x^2+4*x*exp(3)*exp(x)+exp(3)^2)/exp(x)^2/x^2)-x^3-14)
Time = 0.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=e^{\left (-x^{3} + \log \left (\frac {{\left (4 \, x^{2} e^{\left (2 \, x + 6\right )} + 4 \, x e^{\left (x + 9\right )} + e^{12}\right )} e^{\left (-2 \, x - 6\right )}}{x^{2}}\right ) - 14\right )} \] Input:
integrate((-6*exp(x)*x^4+(-3*x^3-2*x-2)*exp(3))*exp(log((4*exp(x)^2*x^2+4* x*exp(3)*exp(x)+exp(3)^2)/exp(x)^2/x^2)-x^3-14)/(2*exp(x)*x^2+x*exp(3)),x, algorithm="fricas")
Output:
e^(-x^3 + log((4*x^2*e^(2*x + 6) + 4*x*e^(x + 9) + e^12)*e^(-2*x - 6)/x^2) - 14)
Time = 0.14 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=\frac {\left (4 x^{2} + 4 x e^{3} e^{- x} + e^{6} e^{- 2 x}\right ) e^{- x^{3} - 14}}{x^{2}} \] Input:
integrate((-6*exp(x)*x**4+(-3*x**3-2*x-2)*exp(3))*exp(ln((4*exp(x)**2*x**2 +4*x*exp(3)*exp(x)+exp(3)**2)/exp(x)**2/x**2)-x**3-14)/(2*exp(x)*x**2+x*ex p(3)),x)
Output:
(4*x**2 + 4*x*exp(3)*exp(-x) + exp(6)*exp(-2*x))*exp(-x**3 - 14)/x**2
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=\frac {{\left (4 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{\left (x + 3\right )} + e^{6}\right )} e^{\left (-x^{3} - 2 \, x - 14\right )}}{x^{2}} \] Input:
integrate((-6*exp(x)*x^4+(-3*x^3-2*x-2)*exp(3))*exp(log((4*exp(x)^2*x^2+4* x*exp(3)*exp(x)+exp(3)^2)/exp(x)^2/x^2)-x^3-14)/(2*exp(x)*x^2+x*exp(3)),x, algorithm="maxima")
Output:
(4*x^2*e^(2*x) + 4*x*e^(x + 3) + e^6)*e^(-x^3 - 2*x - 14)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (23) = 46\).
Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=\frac {{\left (4 \, x^{2} e^{\left (2 \, x^{3} + 2 \, x\right )} + 4 \, x e^{\left (2 \, x^{3} + x + 3\right )} + e^{\left (2 \, x^{3} + 6\right )}\right )} e^{\left (-3 \, x^{3} - 2 \, x - 14\right )}}{x^{2}} \] Input:
integrate((-6*exp(x)*x^4+(-3*x^3-2*x-2)*exp(3))*exp(log((4*exp(x)^2*x^2+4* x*exp(3)*exp(x)+exp(3)^2)/exp(x)^2/x^2)-x^3-14)/(2*exp(x)*x^2+x*exp(3)),x, algorithm="giac")
Output:
(4*x^2*e^(2*x^3 + 2*x) + 4*x*e^(2*x^3 + x + 3) + e^(2*x^3 + 6))*e^(-3*x^3 - 2*x - 14)/x^2
Time = 3.67 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=4\,{\mathrm {e}}^{-14}\,{\mathrm {e}}^{-x^3}+\frac {{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-8}\,{\mathrm {e}}^{-x^3}}{x^2}+\frac {4\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-11}\,{\mathrm {e}}^{-x^3}}{x} \] Input:
int(-(exp(log((exp(-2*x)*(exp(6) + 4*x^2*exp(2*x) + 4*x*exp(3)*exp(x)))/x^ 2) - x^3 - 14)*(6*x^4*exp(x) + exp(3)*(2*x + 3*x^3 + 2)))/(2*x^2*exp(x) + x*exp(3)),x)
Output:
4*exp(-14)*exp(-x^3) + (exp(-2*x)*exp(-8)*exp(-x^3))/x^2 + (4*exp(-x)*exp( -11)*exp(-x^3))/x
Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {e^{-14-2 x-x^3} \left (e^6+4 e^{3+x} x+4 e^{2 x} x^2\right ) \left (-6 e^x x^4+e^3 \left (-2-2 x-3 x^3\right )\right )}{x^2 \left (e^3 x+2 e^x x^2\right )} \, dx=\frac {4 e^{2 x} x^{2}+4 e^{x} e^{3} x +e^{6}}{e^{x^{3}+2 x} e^{14} x^{2}} \] Input:
int((-6*exp(x)*x^4+(-3*x^3-2*x-2)*exp(3))*exp(log((4*exp(x)^2*x^2+4*x*exp( 3)*exp(x)+exp(3)^2)/exp(x)^2/x^2)-x^3-14)/(2*exp(x)*x^2+x*exp(3)),x)
Output:
(4*e**(2*x)*x**2 + 4*e**x*e**3*x + e**6)/(e**(x**3 + 2*x)*e**14*x**2)