3.61.8 \(\int -\frac {1}{-2+e-x} \, dx\)

Optimal. Leaf size=7 \[ \log (2-e+x) \]

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Rubi [A]  time = 0.00, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {31} \begin {gather*} \log (x-e+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-(-2 + E - x)^(-1),x]

[Out]

Log[2 - E + x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log (2-e+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} \log (2-e+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(-2 + E - x)^(-1),x]

[Out]

Log[2 - E + x]

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fricas [A]  time = 0.58, size = 8, normalized size = 1.14 \begin {gather*} \log \left (x - e + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - e + 2)

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giac [A]  time = 0.18, size = 9, normalized size = 1.29 \begin {gather*} \log \left ({\left | x - e + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(x - e + 2))

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maple [A]  time = 0.30, size = 9, normalized size = 1.29

method	result	size
default	\(\ln \left ({\mathrm e}-x -2\right )\)	\(9\)
norman	\(\ln \left ({\mathrm e}-x -2\right )\)	\(9\)
risch	\(\ln \left (x -{\mathrm e}+2\right )\)	\(9\)
meijerg	\(-\frac {\left (2-{\mathrm e}\right ) \ln \left (1-\frac {x}{{\mathrm e}-2}\right )}{{\mathrm e}-2}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(exp(1)-x-2)

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maxima [A]  time = 0.35, size = 8, normalized size = 1.14 \begin {gather*} \log \left (x - e + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - e + 2)

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mupad [B]  time = 0.03, size = 8, normalized size = 1.14 \begin {gather*} \ln \left (x-\mathrm {e}+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x - exp(1) + 2),x)

[Out]

log(x - exp(1) + 2)

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sympy [A]  time = 0.06, size = 7, normalized size = 1.00 \begin {gather*} \log {\left (x - e + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x - E + 2)

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