3.60.89 \(\int \frac {e^x}{-7+e^x} \, dx\)

Optimal. Leaf size=21 \[ \log \left (e \left (-5+e^x+2 x-\frac {2 \left (x+x^2\right )}{x}\right )\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 8, normalized size of antiderivative = 0.38, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2246, 31} \begin {gather*} \log \left (7-e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x/(-7 + E^x),x]

[Out]

Log[7 - E^x]

Rule 31

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

Rule 2246

Int[((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)*((a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.))^(p_.),
x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int[(a + b*x)^p, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b,
c, d, e, n, p}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {1}{-7+x} \, dx,x,e^x\right )\\ &=\log \left (7-e^x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.38 \begin {gather*} \log \left (7-e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x/(-7 + E^x),x]

[Out]

Log[7 - E^x]

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fricas [A]  time = 0.49, size = 5, normalized size = 0.24 \begin {gather*} \log \left (e^{x} - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-7+exp(x)),x, algorithm="fricas")

[Out]

log(e^x - 7)

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giac [A]  time = 0.21, size = 6, normalized size = 0.29 \begin {gather*} \log \left ({\left | e^{x} - 7 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-7+exp(x)),x, algorithm="giac")

[Out]

log(abs(e^x - 7))

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maple [A]  time = 0.03, size = 6, normalized size = 0.29

method	result	size
derivativedivides	\(\ln \left (-7+{\mathrm e}^{x}\right )\)	\(6\)
default	\(\ln \left (-7+{\mathrm e}^{x}\right )\)	\(6\)
norman	\(\ln \left (-7+{\mathrm e}^{x}\right )\)	\(6\)
risch	\(\ln \left (-7+{\mathrm e}^{x}\right )\)	\(6\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(-7+exp(x)),x,method=_RETURNVERBOSE)

[Out]

ln(-7+exp(x))

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maxima [A]  time = 0.35, size = 5, normalized size = 0.24 \begin {gather*} \log \left (e^{x} - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-7+exp(x)),x, algorithm="maxima")

[Out]

log(e^x - 7)

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mupad [B]  time = 0.05, size = 5, normalized size = 0.24 \begin {gather*} \ln \left ({\mathrm {e}}^x-7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(exp(x) - 7),x)

[Out]

log(exp(x) - 7)

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sympy [A]  time = 0.07, size = 5, normalized size = 0.24 \begin {gather*} \log {\left (e^{x} - 7 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(-7+exp(x)),x)

[Out]

log(exp(x) - 7)

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