3.42.79 \(\int \frac {1}{-1-x} \, dx\)

Optimal. Leaf size=12 \[ e^{3/2}-\log (1+x) \]

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Rubi [A]  time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.50, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {31} \begin {gather*} -\log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - x)^(-1),x]

[Out]

-Log[1 + 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 (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 6, normalized size = 0.50 \begin {gather*} -\log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - x)^(-1),x]

[Out]

-Log[1 + x]

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fricas [A]  time = 1.41, size = 6, normalized size = 0.50 \begin {gather*} -\log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1)

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giac [A]  time = 0.23, size = 7, normalized size = 0.58 \begin {gather*} -\log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(x + 1))

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maple [A]  time = 0.01, size = 7, normalized size = 0.58

method	result	size
norman	\(-\ln \left (x +1\right )\)	\(7\)
risch	\(-\ln \left (x +1\right )\)	\(7\)
default	\(-\ln \left (-x -1\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln(x+1)

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maxima [A]  time = 0.36, size = 6, normalized size = 0.50 \begin {gather*} -\log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(x + 1)

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mupad [B]  time = 0.10, size = 6, normalized size = 0.50 \begin {gather*} -\ln \left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x - log(x*exp(- log(x) - 1))),x)

[Out]

-log(x + 1)

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sympy [A]  time = 0.05, size = 5, normalized size = 0.42 \begin {gather*} - \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(ln(x*exp(-ln(x)-1))-x),x)

[Out]

-log(x + 1)

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