3.93.76 \(\int \frac {3-e^x-x}{1+e^x-x} \, dx\)

Optimal. Leaf size=16 \[ 1+x-\log \left (\left (-1-e^x+x\right )^2\right ) \]

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Rubi [F]  time = 0.16, 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^x-x}{1+e^x-x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 - E^x - x)/(1 + E^x - x),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {2 (-2+x)}{-1-e^x+x}\right ) \, dx\\ &=-x+2 \int \frac {-2+x}{-1-e^x+x} \, dx\\ &=-x+2 \int \left (\frac {2}{1+e^x-x}+\frac {x}{-1-e^x+x}\right ) \, dx\\ &=-x+2 \int \frac {x}{-1-e^x+x} \, dx+4 \int \frac {1}{1+e^x-x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 13, normalized size = 0.81 \begin {gather*} x-2 \log \left (1+e^x-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 - E^x - x)/(1 + E^x - x),x]

[Out]

x - 2*Log[1 + E^x - x]

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fricas [A]  time = 0.87, size = 12, normalized size = 0.75 \begin {gather*} x - 2 \, \log \left (-x + e^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-x-exp(x))/(1+exp(x)-x),x, algorithm="fricas")

[Out]

x - 2*log(-x + e^x + 1)

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giac [A]  time = 0.18, size = 12, normalized size = 0.75 \begin {gather*} x - 2 \, \log \left (x - e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-x-exp(x))/(1+exp(x)-x),x, algorithm="giac")

[Out]

x - 2*log(x - e^x - 1)

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maple [A]  time = 0.02, size = 13, normalized size = 0.81

method	result	size
norman	\(x -2 \ln \left (x -{\mathrm e}^{x}-1\right )\)	\(13\)
risch	\(x -2 \ln \left (1+{\mathrm e}^{x}-x \right )\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3-x-exp(x))/(1+exp(x)-x),x,method=_RETURNVERBOSE)

[Out]

x-2*ln(x-exp(x)-1)

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maxima [A]  time = 0.38, size = 12, normalized size = 0.75 \begin {gather*} x - 2 \, \log \left (-x + e^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-x-exp(x))/(1+exp(x)-x),x, algorithm="maxima")

[Out]

x - 2*log(-x + e^x + 1)

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mupad [B]  time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} x-2\,\ln \left (x-{\mathrm {e}}^x-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + exp(x) - 3)/(exp(x) - x + 1),x)

[Out]

x - 2*log(x - exp(x) - 1)

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sympy [A]  time = 0.10, size = 10, normalized size = 0.62 \begin {gather*} x - 2 \log {\left (- x + e^{x} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-x-exp(x))/(1+exp(x)-x),x)

[Out]

x - 2*log(-x + exp(x) + 1)

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