3.42.19 \(\int \frac {1}{-1+e^4+x} \, dx\)

Optimal. Leaf size=7 \[ \log \left (-1+e^4+x\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.57, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {31} \begin {gather*} \log \left (-x-e^4+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + E^4 + x)^(-1),x]

[Out]

Log[1 - E^4 - 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 \left (1-e^4-x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} \log \left (-1+e^4+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^4 + x)^(-1),x]

[Out]

Log[-1 + E^4 + x]

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fricas [A]  time = 0.56, size = 6, normalized size = 0.86 \begin {gather*} \log \left (x + e^{4} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(exp(4)+x-1),x, algorithm="fricas")

[Out]

log(x + e^4 - 1)

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giac [A]  time = 0.11, size = 7, normalized size = 1.00 \begin {gather*} \log \left ({\left | x + e^{4} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(exp(4)+x-1),x, algorithm="giac")

[Out]

log(abs(x + e^4 - 1))

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maple [A]  time = 0.07, size = 7, normalized size = 1.00

method	result	size
default	\(\ln \left ({\mathrm e}^{4}+x -1\right )\)	\(7\)
norman	\(\ln \left ({\mathrm e}^{4}+x -1\right )\)	\(7\)
risch	\(\ln \left ({\mathrm e}^{4}+x -1\right )\)	\(7\)
meijerg	\(\ln \left (1+\frac {x}{{\mathrm e}^{4}-1}\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(exp(4)+x-1),x,method=_RETURNVERBOSE)

[Out]

ln(exp(4)+x-1)

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maxima [A]  time = 0.36, size = 6, normalized size = 0.86 \begin {gather*} \log \left (x + e^{4} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(exp(4)+x-1),x, algorithm="maxima")

[Out]

log(x + e^4 - 1)

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mupad [B]  time = 0.03, size = 6, normalized size = 0.86 \begin {gather*} \ln \left (x+{\mathrm {e}}^4-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + exp(4) - 1),x)

[Out]

log(x + exp(4) - 1)

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sympy [A]  time = 0.06, size = 7, normalized size = 1.00 \begin {gather*} \log {\left (x - 1 + e^{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(exp(4)+x-1),x)

[Out]

log(x - 1 + exp(4))

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