3.32.77 \(\int \frac {e^{\frac {1}{2} (-6-x)} (-2-x)}{e^{\frac {1}{2} (-6-x)} x-x^2} \, dx\)

Optimal. Leaf size=25 \[ \log \left (\frac {9 \left (-e^{\frac {1}{2} (-6+x)-x}+x\right )^2}{x^2}\right ) \]

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Rubi [F]  time = 0.47, 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 {e^{\frac {1}{2} (-6-x)} (-2-x)}{e^{\frac {1}{2} (-6-x)} x-x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-6 - x)/2)*(-2 - x))/(E^((-6 - x)/2)*x - x^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2-x}{x \left (1-e^{3+\frac {x}{2}} x\right )} \, dx\\ &=\int \left (\frac {1}{-1+e^{3+\frac {x}{2}} x}+\frac {2}{x \left (-1+e^{3+\frac {x}{2}} x\right )}\right ) \, dx\\ &=2 \int \frac {1}{x \left (-1+e^{3+\frac {x}{2}} x\right )} \, dx+\int \frac {1}{-1+e^{3+\frac {x}{2}} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 25, normalized size = 1.00 \begin {gather*} -x-2 \log (x)+2 \log \left (1-e^{3+\frac {x}{2}} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-6 - x)/2)*(-2 - x))/(E^((-6 - x)/2)*x - x^2),x]

[Out]

-x - 2*Log[x] + 2*Log[1 - E^(3 + x/2)*x]

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fricas [A]  time = 0.84, size = 18, normalized size = 0.72 \begin {gather*} -2 \, \log \relax (x) + 2 \, \log \left (-x + e^{\left (-\frac {1}{2} \, x - 3\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*log(x) + 2*log(-x + e^(-1/2*x - 3))

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giac [A]  time = 0.23, size = 18, normalized size = 0.72 \begin {gather*} 2 \, \log \left (-x e^{3} + e^{\left (-\frac {1}{2} \, x\right )}\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(-x*e^3 + e^(-1/2*x)) - 2*log(x)

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maple [A]  time = 0.11, size = 19, normalized size = 0.76

method	result	size
norman	\(-2 \ln \relax (x )+2 \ln \left (x -{\mathrm e}^{-\frac {x}{2}-3}\right )\)	\(19\)
risch	\(-2 \ln \relax (x )+6+2 \ln \left ({\mathrm e}^{-\frac {x}{2}-3}-x \right )\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(x)+2*ln(x-exp(-1/2*x-3))

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maxima [A]  time = 0.46, size = 23, normalized size = 0.92 \begin {gather*} -x + 2 \, \log \left (\frac {{\left (x e^{\left (\frac {1}{2} \, x + 3\right )} - 1\right )} e^{\left (-3\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x + 2*log((x*e^(1/2*x + 3) - 1)*e^(-3)/x)

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mupad [B]  time = 0.19, size = 18, normalized size = 0.72 \begin {gather*} 2\,\ln \left (x-\frac {{\mathrm {e}}^{-3}}{\sqrt {{\mathrm {e}}^x}}\right )-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(- x/2 - 3)*(x + 2))/(x*exp(- x/2 - 3) - x^2),x)

[Out]

2*log(x - exp(-3)/exp(x)^(1/2)) - 2*log(x)

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sympy [A]  time = 0.12, size = 17, normalized size = 0.68 \begin {gather*} - 2 \log {\relax (x )} + 2 \log {\left (- x + e^{- \frac {x}{2} - 3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x-2)*exp(-1/2*x-3)/(x*exp(-1/2*x-3)-x**2),x)

[Out]

-2*log(x) + 2*log(-x + exp(-x/2 - 3))

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