3.90.53 \(\int \frac {e^x}{-1080-135 e^5+e^x} \, dx\)

Optimal. Leaf size=13 \[ \log \left (8+e^5-\frac {e^x}{135}\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2246, 31} \begin {gather*} \log \left (e^x-135 \left (8+e^5\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x/(-1080 - 135*E^5 + E^x),x]

[Out]

Log[E^x - 135*(8 + E^5)]

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}{-1080-135 e^5+x} \, dx,x,e^x\right )\\ &=\log \left (e^x-135 \left (8+e^5\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \log \left (1080+135 e^5-e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x/(-1080 - 135*E^5 + E^x),x]

[Out]

Log[1080 + 135*E^5 - E^x]

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fricas [A]  time = 0.57, size = 9, normalized size = 0.69 \begin {gather*} \log \left (-135 \, e^{5} + e^{x} - 1080\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(exp(x)-135*exp(5)-1080),x, algorithm="fricas")

[Out]

log(-135*e^5 + e^x - 1080)

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giac [A]  time = 0.23, size = 10, normalized size = 0.77 \begin {gather*} \log \left ({\left | -135 \, e^{5} + e^{x} - 1080 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(exp(x)-135*exp(5)-1080),x, algorithm="giac")

[Out]

log(abs(-135*e^5 + e^x - 1080))

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maple [A]  time = 0.08, size = 10, normalized size = 0.77

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}^{x}-135 \,{\mathrm e}^{5}-1080\right )\)	\(10\)
default	\(\ln \left ({\mathrm e}^{x}-135 \,{\mathrm e}^{5}-1080\right )\)	\(10\)
risch	\(\ln \left ({\mathrm e}^{x}-135 \,{\mathrm e}^{5}-1080\right )\)	\(10\)
norman	\(\ln \left (-{\mathrm e}^{x}+135 \,{\mathrm e}^{5}+1080\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)/(exp(x)-135*exp(5)-1080),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x)-135*exp(5)-1080)

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maxima [A]  time = 0.34, size = 11, normalized size = 0.85 \begin {gather*} \log \left (135 \, e^{5} - e^{x} + 1080\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(exp(x)-135*exp(5)-1080),x, algorithm="maxima")

[Out]

log(135*e^5 - e^x + 1080)

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mupad [B]  time = 0.11, size = 9, normalized size = 0.69 \begin {gather*} \ln \left ({\mathrm {e}}^x-135\,{\mathrm {e}}^5-1080\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x)/(135*exp(5) - exp(x) + 1080),x)

[Out]

log(exp(x) - 135*exp(5) - 1080)

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sympy [A]  time = 0.09, size = 10, normalized size = 0.77 \begin {gather*} \log {\left (e^{x} - 135 e^{5} - 1080 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x)/(exp(x)-135*exp(5)-1080),x)

[Out]

log(exp(x) - 135*exp(5) - 1080)

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