\(\int \frac {-3+e^{-7+2 x-2 x^3} (1-2 x+6 x^3)}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx\) [6750]

Optimal result

Integrand size = 56, antiderivative size = 19 \[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=\frac {x}{-3+e^{-7+2 \left (x-x^3\right )}} \]

[Out]

x/(exp(-2*x^3+2*x-7)-3)

Rubi [F]

\[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=\int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx \]

[In]

Int[(-3 + E^(-7 + 2*x - 2*x^3)*(1 - 2*x + 6*x^3))/(9 + E^(-14 + 4*x - 4*x^3) - 6*E^(-7 + 2*x - 2*x^3)),x]

[Out]

-1/3*x + Defer[Int][E^(2*x)/(E^(2*x) - 3*E^(7 + 2*x^3)), x]/3 - (2*Defer[Int][(E^(4*x)*x)/(E^(2*x) - 3*E^(7 +
2*x^3))^2, x])/3 + (2*Defer[Int][(E^(2*x)*x)/(E^(2*x) - 3*E^(7 + 2*x^3)), x])/3 + 2*Defer[Int][(E^(4*x)*x^3)/(
E^(2*x) - 3*E^(7 + 2*x^3))^2, x] - 2*Defer[Int][(E^(2*x)*x^3)/(E^(2*x) - 3*E^(7 + 2*x^3)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{3}+\frac {2 e^{4 x} x \left (-1+3 x^2\right )}{3 \left (e^{2 x}-3 e^{7+2 x^3}\right )^2}-\frac {e^{2 x} \left (-1-2 x+6 x^3\right )}{3 \left (e^{2 x}-3 e^{7+2 x^3}\right )}\right ) \, dx \\ & = -\frac {x}{3}-\frac {1}{3} \int \frac {e^{2 x} \left (-1-2 x+6 x^3\right )}{e^{2 x}-3 e^{7+2 x^3}} \, dx+\frac {2}{3} \int \frac {e^{4 x} x \left (-1+3 x^2\right )}{\left (e^{2 x}-3 e^{7+2 x^3}\right )^2} \, dx \\ & = -\frac {x}{3}-\frac {1}{3} \int \left (-\frac {e^{2 x}}{e^{2 x}-3 e^{7+2 x^3}}-\frac {2 e^{2 x} x}{e^{2 x}-3 e^{7+2 x^3}}+\frac {6 e^{2 x} x^3}{e^{2 x}-3 e^{7+2 x^3}}\right ) \, dx+\frac {2}{3} \int \left (-\frac {e^{4 x} x}{\left (e^{2 x}-3 e^{7+2 x^3}\right )^2}+\frac {3 e^{4 x} x^3}{\left (e^{2 x}-3 e^{7+2 x^3}\right )^2}\right ) \, dx \\ & = -\frac {x}{3}+\frac {1}{3} \int \frac {e^{2 x}}{e^{2 x}-3 e^{7+2 x^3}} \, dx-\frac {2}{3} \int \frac {e^{4 x} x}{\left (e^{2 x}-3 e^{7+2 x^3}\right )^2} \, dx+\frac {2}{3} \int \frac {e^{2 x} x}{e^{2 x}-3 e^{7+2 x^3}} \, dx+2 \int \frac {e^{4 x} x^3}{\left (e^{2 x}-3 e^{7+2 x^3}\right )^2} \, dx-2 \int \frac {e^{2 x} x^3}{e^{2 x}-3 e^{7+2 x^3}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=\frac {e^{7+2 x^3} x}{e^{2 x}-3 e^{7+2 x^3}} \]

[In]

Integrate[(-3 + E^(-7 + 2*x - 2*x^3)*(1 - 2*x + 6*x^3))/(9 + E^(-14 + 4*x - 4*x^3) - 6*E^(-7 + 2*x - 2*x^3)),x
]

[Out]

(E^(7 + 2*x^3)*x)/(E^(2*x) - 3*E^(7 + 2*x^3))

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

method	result	size
norman	\(\frac {x}{{\mathrm e}^{-2 x^{3}+2 x -7}-3}\)	\(18\)
risch	\(\frac {x}{{\mathrm e}^{-2 x^{3}+2 x -7}-3}\)	\(18\)
parallelrisch	\(\frac {x}{{\mathrm e}^{-2 x^{3}+2 x -7}-3}\)	\(18\)

[In]

int(((6*x^3-2*x+1)*exp(-2*x^3+2*x-7)-3)/(exp(-2*x^3+2*x-7)^2-6*exp(-2*x^3+2*x-7)+9),x,method=_RETURNVERBOSE)

[Out]

x/(exp(-2*x^3+2*x-7)-3)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=\frac {x}{e^{\left (-2 \, x^{3} + 2 \, x - 7\right )} - 3} \]

[In]

integrate(((6*x^3-2*x+1)*exp(-2*x^3+2*x-7)-3)/(exp(-2*x^3+2*x-7)^2-6*exp(-2*x^3+2*x-7)+9),x, algorithm="fricas
")

[Out]

x/(e^(-2*x^3 + 2*x - 7) - 3)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=\frac {x}{e^{- 2 x^{3} + 2 x - 7} - 3} \]

[In]

integrate(((6*x**3-2*x+1)*exp(-2*x**3+2*x-7)-3)/(exp(-2*x**3+2*x-7)**2-6*exp(-2*x**3+2*x-7)+9),x)

[Out]

x/(exp(-2*x**3 + 2*x - 7) - 3)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=-\frac {x e^{\left (2 \, x^{3} + 7\right )}}{3 \, e^{\left (2 \, x^{3} + 7\right )} - e^{\left (2 \, x\right )}} \]

[In]

integrate(((6*x^3-2*x+1)*exp(-2*x^3+2*x-7)-3)/(exp(-2*x^3+2*x-7)^2-6*exp(-2*x^3+2*x-7)+9),x, algorithm="maxima
")

[Out]

-x*e^(2*x^3 + 7)/(3*e^(2*x^3 + 7) - e^(2*x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=-\frac {x e^{7}}{3 \, e^{7} - e^{\left (-2 \, x^{3} + 2 \, x\right )}} \]

[In]

integrate(((6*x^3-2*x+1)*exp(-2*x^3+2*x-7)-3)/(exp(-2*x^3+2*x-7)^2-6*exp(-2*x^3+2*x-7)+9),x, algorithm="giac")

[Out]

-x*e^7/(3*e^7 - e^(-2*x^3 + 2*x))

Mupad [B] (verification not implemented)

Time = 11.90 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-3+e^{-7+2 x-2 x^3} \left (1-2 x+6 x^3\right )}{9+e^{-14+4 x-4 x^3}-6 e^{-7+2 x-2 x^3}} \, dx=\frac {x}{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-7}\,{\mathrm {e}}^{-2\,x^3}-3} \]

[In]

int((exp(2*x - 2*x^3 - 7)*(6*x^3 - 2*x + 1) - 3)/(exp(4*x - 4*x^3 - 14) - 6*exp(2*x - 2*x^3 - 7) + 9),x)

[Out]

x/(exp(2*x)*exp(-7)*exp(-2*x^3) - 3)