3.20.91 \(\int \frac {10 e^{-7+x}}{-3+e^{-7+x}} \, dx\)

Optimal. Leaf size=30 \[ 5 \log \left (\left (3-e^{-7+x}\right )^2 (i \pi +\log (3-\log (5)))^2\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 14, normalized size of antiderivative = 0.47, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 2246, 31} \begin {gather*} 10 \log \left (3 e^7-e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10*E^(-7 + x))/(-3 + E^(-7 + x)),x]

[Out]

10*Log[3*E^7 - E^x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=10 \int \frac {e^{-7+x}}{-3+e^{-7+x}} \, dx\\ &=10 \operatorname {Subst}\left (\int \frac {1}{-3+x} \, dx,x,e^{-7+x}\right )\\ &=10 \log \left (3 e^7-e^x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.47 \begin {gather*} 10 \log \left (3 e^7-e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*E^(-7 + x))/(-3 + E^(-7 + x)),x]

[Out]

10*Log[3*E^7 - E^x]

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fricas [A]  time = 0.53, size = 9, normalized size = 0.30 \begin {gather*} 10 \, \log \left (e^{\left (x - 7\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(x-7)/(exp(x-7)-3),x, algorithm="fricas")

[Out]

10*log(e^(x - 7) - 3)

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giac [A]  time = 0.23, size = 10, normalized size = 0.33 \begin {gather*} 10 \, \log \left ({\left | e^{\left (x - 7\right )} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(x-7)/(exp(x-7)-3),x, algorithm="giac")

[Out]

10*log(abs(e^(x - 7) - 3))

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maple [A]  time = 0.02, size = 10, normalized size = 0.33

method	result	size
derivativedivides	\(10 \ln \left ({\mathrm e}^{x -7}-3\right )\)	\(10\)
default	\(10 \ln \left ({\mathrm e}^{x -7}-3\right )\)	\(10\)
norman	\(10 \ln \left ({\mathrm e}^{x -7}-3\right )\)	\(10\)
risch	\(70+10 \ln \left ({\mathrm e}^{x -7}-3\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(10*exp(x-7)/(exp(x-7)-3),x,method=_RETURNVERBOSE)

[Out]

10*ln(exp(x-7)-3)

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maxima [A]  time = 0.48, size = 9, normalized size = 0.30 \begin {gather*} 10 \, \log \left (e^{\left (x - 7\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(x-7)/(exp(x-7)-3),x, algorithm="maxima")

[Out]

10*log(e^(x - 7) - 3)

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mupad [B]  time = 0.09, size = 9, normalized size = 0.30 \begin {gather*} 10\,\ln \left ({\mathrm {e}}^{x-7}-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*exp(x - 7))/(exp(x - 7) - 3),x)

[Out]

10*log(exp(x - 7) - 3)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.27 \begin {gather*} 10 \log {\left (e^{x - 7} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(10*exp(x-7)/(exp(x-7)-3),x)

[Out]

10*log(exp(x - 7) - 3)

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