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

Optimal. Leaf size=19 \[ \frac {x}{-3+e^{-7+2 \left (x-x^3\right )}} \]

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Rubi [F]  time = 1.71, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.58, size = 30, normalized size = 1.58 \begin {gather*} \frac {e^{7+2 x^3} x}{e^{2 x}-3 e^{7+2 x^3}} \end {gather*}

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.55, size = 17, normalized size = 0.89 \begin {gather*} \frac {x}{e^{\left (-2 \, x^{3} + 2 \, x - 7\right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.17, size = 24, normalized size = 1.26 \begin {gather*} -\frac {x e^{7}}{3 \, e^{7} - e^{\left (-2 \, x^{3} + 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.05, size = 18, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.47, size = 30, normalized size = 1.58 \begin {gather*} -\frac {x e^{\left (2 \, x^{3} + 7\right )}}{3 \, e^{\left (2 \, x^{3} + 7\right )} - e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 4.04, size = 19, normalized size = 1.00 \begin {gather*} \frac {x}{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-7}\,{\mathrm {e}}^{-2\,x^3}-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.13, size = 14, normalized size = 0.74 \begin {gather*} \frac {x}{e^{- 2 x^{3} + 2 x - 7} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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