3.1.36 \(\int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} (230400 e^x-103680 x)}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx\) [36]

3.1.36.1 Optimal result

Integrand size = 61, antiderivative size = 23 \[ \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx=e^{-1+\frac {40}{-\frac {5 e^x}{18}+\frac {x^2}{16}}} \]

output

exp(40/(1/16*x^2-5/18*exp(x))-1)
 

3.1.36.2 Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx=e^{-1-\frac {5760}{40 e^x-9 x^2}} \]

input

Integrate[(E^((-5760 - 40*E^x + 9*x^2)/(40*E^x - 9*x^2))*(230400*E^x - 103 
680*x))/(1600*E^(2*x) - 720*E^x*x^2 + 81*x^4),x]
 

output

E^(-1 - 5760/(40*E^x - 9*x^2))
 

3.1.36.3 Rubi [F]

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.

\(\displaystyle \int \frac {e^{\frac {9 x^2-40 e^x-5760}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{81 x^4-720 e^x x^2+1600 e^{2 x}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {11520 e^{\frac {9 x^2-40 e^x-5760}{40 e^x-9 x^2}} \left (20 e^x-9 x\right )}{\left (40 e^x-9 x^2\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 11520 \int \frac {e^{-\frac {-9 x^2+40 e^x+5760}{40 e^x-9 x^2}} \left (20 e^x-9 x\right )}{\left (40 e^x-9 x^2\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 11520 \int \left (\frac {9 e^{-\frac {-9 x^2+40 e^x+5760}{40 e^x-9 x^2}} (x-2) x}{2 \left (9 x^2-40 e^x\right )^2}+\frac {e^{-\frac {-9 x^2+40 e^x+5760}{40 e^x-9 x^2}}}{2 \left (40 e^x-9 x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 11520 \left (\frac {1}{2} \int \frac {e^{-\frac {-9 x^2+40 e^x+5760}{40 e^x-9 x^2}}}{40 e^x-9 x^2}dx-9 \int \frac {e^{-\frac {-9 x^2+40 e^x+5760}{40 e^x-9 x^2}} x}{\left (9 x^2-40 e^x\right )^2}dx+\frac {9}{2} \int \frac {e^{-\frac {-9 x^2+40 e^x+5760}{40 e^x-9 x^2}} x^2}{\left (9 x^2-40 e^x\right )^2}dx\right )\)

input

Int[(E^((-5760 - 40*E^x + 9*x^2)/(40*E^x - 9*x^2))*(230400*E^x - 103680*x) 
)/(1600*E^(2*x) - 720*E^x*x^2 + 81*x^4),x]
 

output

$Aborted
 

3.1.36.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.1.36.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17

method	result	size
risch	\({\mathrm e}^{-\frac {-9 x^{2}+40 \,{\mathrm e}^{x}+5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}\)	\(27\)
parallelrisch	\({\mathrm e}^{-\frac {-9 x^{2}+40 \,{\mathrm e}^{x}+5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}\)	\(27\)
norman	\(\frac {9 x^{2} {\mathrm e}^{\frac {-40 \,{\mathrm e}^{x}+9 x^{2}-5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}-40 \,{\mathrm e}^{x} {\mathrm e}^{\frac {-40 \,{\mathrm e}^{x}+9 x^{2}-5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}}{9 x^{2}-40 \,{\mathrm e}^{x}}\)	\(74\)

input

int((230400*exp(x)-103680*x)*exp((-40*exp(x)+9*x^2-5760)/(40*exp(x)-9*x^2) 
)/(1600*exp(x)^2-720*exp(x)*x^2+81*x^4),x,method=_RETURNVERBOSE)
 

output

exp(-(-9*x^2+40*exp(x)+5760)/(40*exp(x)-9*x^2))
 

3.1.36.5 Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx=e^{\left (-\frac {9 \, x^{2} - 40 \, e^{x} - 5760}{9 \, x^{2} - 40 \, e^{x}}\right )} \]

input

integrate((230400*exp(x)-103680*x)*exp((-40*exp(x)+9*x^2-5760)/(40*exp(x)- 
9*x^2))/(1600*exp(x)^2-720*exp(x)*x^2+81*x^4),x, algorithm=\
 

output

e^(-(9*x^2 - 40*e^x - 5760)/(9*x^2 - 40*e^x))
 

3.1.36.6 Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx=e^{\frac {9 x^{2} - 40 e^{x} - 5760}{- 9 x^{2} + 40 e^{x}}} \]

input

integrate((230400*exp(x)-103680*x)*exp((-40*exp(x)+9*x**2-5760)/(40*exp(x) 
-9*x**2))/(1600*exp(x)**2-720*exp(x)*x**2+81*x**4),x)
 

output

exp((9*x**2 - 40*exp(x) - 5760)/(-9*x**2 + 40*exp(x)))
 

3.1.36.7 Maxima [A] (verification not implemented)

Time = 1.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx=e^{\left (\frac {5760}{9 \, x^{2} - 40 \, e^{x}} - 1\right )} \]

input

integrate((230400*exp(x)-103680*x)*exp((-40*exp(x)+9*x^2-5760)/(40*exp(x)- 
9*x^2))/(1600*exp(x)^2-720*exp(x)*x^2+81*x^4),x, algorithm=\
 

output

e^(5760/(9*x^2 - 40*e^x) - 1)
 

3.1.36.8 Giac [F]

\[ \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx=\int { -\frac {11520 \, {\left (9 \, x - 20 \, e^{x}\right )} e^{\left (-\frac {9 \, x^{2} - 40 \, e^{x} - 5760}{9 \, x^{2} - 40 \, e^{x}}\right )}}{81 \, x^{4} - 720 \, x^{2} e^{x} + 1600 \, e^{\left (2 \, x\right )}} \,d x } \]

input

integrate((230400*exp(x)-103680*x)*exp((-40*exp(x)+9*x^2-5760)/(40*exp(x)- 
9*x^2))/(1600*exp(x)^2-720*exp(x)*x^2+81*x^4),x, algorithm=\
 

output

integrate(-11520*(9*x - 20*e^x)*e^(-(9*x^2 - 40*e^x - 5760)/(9*x^2 - 40*e^ 
x))/(81*x^4 - 720*x^2*e^x + 1600*e^(2*x)), x)
 

3.1.36.9 Mupad [B] (verification not implemented)

Time = 11.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx={\mathrm {e}}^{\frac {9\,x^2}{40\,{\mathrm {e}}^x-9\,x^2}}\,{\mathrm {e}}^{-\frac {40\,{\mathrm {e}}^x}{40\,{\mathrm {e}}^x-9\,x^2}}\,{\mathrm {e}}^{-\frac {5760}{40\,{\mathrm {e}}^x-9\,x^2}} \]

input

int(-(exp(-(40*exp(x) - 9*x^2 + 5760)/(40*exp(x) - 9*x^2))*(103680*x - 230 
400*exp(x)))/(1600*exp(2*x) - 720*x^2*exp(x) + 81*x^4),x)
 

output

exp((9*x^2)/(40*exp(x) - 9*x^2))*exp(-(40*exp(x))/(40*exp(x) - 9*x^2))*exp 
(-5760/(40*exp(x) - 9*x^2))