\(\int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} (230400 e^x-103680 x)}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx\) [6036]

Optimal result

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

[Out]

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

Rubi [F]

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

[In]

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]

[Out]

5760*Defer[Int][E^((-5760 - 40*E^x + 9*x^2)/(40*E^x - 9*x^2))/(40*E^x - 9*x^2), x] - 103680*Defer[Int][(E^((-5
760 - 40*E^x + 9*x^2)/(40*E^x - 9*x^2))*x)/(-40*E^x + 9*x^2)^2, x] + 51840*Defer[Int][(E^((-5760 - 40*E^x + 9*
x^2)/(40*E^x - 9*x^2))*x^2)/(-40*E^x + 9*x^2)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {11520 e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (20 e^x-9 x\right )}{\left (40 e^x-9 x^2\right )^2} \, dx \\ & = 11520 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (20 e^x-9 x\right )}{\left (40 e^x-9 x^2\right )^2} \, dx \\ & = 11520 \int \left (\frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{2 \left (40 e^x-9 x^2\right )}+\frac {9 e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} (-2+x) x}{2 \left (-40 e^x+9 x^2\right )^2}\right ) \, dx \\ & = 5760 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{40 e^x-9 x^2} \, dx+51840 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} (-2+x) x}{\left (-40 e^x+9 x^2\right )^2} \, dx \\ & = 5760 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{40 e^x-9 x^2} \, dx+51840 \int \left (-\frac {2 e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x}{\left (-40 e^x+9 x^2\right )^2}+\frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x^2}{\left (-40 e^x+9 x^2\right )^2}\right ) \, dx \\ & = 5760 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{40 e^x-9 x^2} \, dx+51840 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x^2}{\left (-40 e^x+9 x^2\right )^2} \, dx-103680 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x}{\left (-40 e^x+9 x^2\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[(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]

[Out]

E^(-1 - 5760/(40*E^x - 9*x^2))

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	\(\frac {9 x^{2} {\mathrm e}^{\frac {-40 \,{\mathrm e}^{x}+9 x^{2}-5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}-40 \,{\mathrm e}^{x} {\mathrm e}^{\frac {-40 \,{\mathrm e}^{x}+9 x^{2}-5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}}{9 x^{2}-40 \,{\mathrm e}^{x}}\)	\(74\)

[In]

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)

[Out]

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

Fricas [A] (verification not implemented)

none

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

[In]

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="fricas")

[Out]

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

Sympy [A] (verification not implemented)

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

[In]

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)

[Out]

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

Maxima [A] (verification not implemented)

none

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

[In]

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="maxima")

[Out]

e^(5760/(9*x^2 - 40*e^x) - 1)

Giac [F]

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

[In]

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="giac")

[Out]

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)

Mupad [B] (verification not implemented)

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

[In]

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

[Out]

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))