3.82.50 \(\int -\frac {5 e^{390622+\frac {x}{4}}}{4+20 e^{390622+\frac {x}{4}}} \, dx\)

Optimal. Leaf size=20 \[ 5-\log \left (\frac {1}{5 e^{390622}}+e^{x/4}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 0.80, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {12, 2246, 31} \begin {gather*} -\log \left (5 e^{\frac {x}{4}+390622}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5*E^(390622 + x/4))/(4 + 20*E^(390622 + x/4)),x]

[Out]

-Log[1 + 5*E^(390622 + x/4)]

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} &=-\left (5 \int \frac {e^{390622+\frac {x}{4}}}{4+20 e^{390622+\frac {x}{4}}} \, dx\right )\\ &=-\left (20 \operatorname {Subst}\left (\int \frac {1}{4+20 x} \, dx,x,e^{390622+\frac {x}{4}}\right )\right )\\ &=-\log \left (1+5 e^{390622+\frac {x}{4}}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.80 \begin {gather*} -\log \left (1+5 e^{390622+\frac {x}{4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*E^(390622 + x/4))/(4 + 20*E^(390622 + x/4)),x]

[Out]

-Log[1 + 5*E^(390622 + x/4)]

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fricas [A]  time = 1.32, size = 13, normalized size = 0.65 \begin {gather*} -\log \left (5 \, e^{\left (\frac {1}{4} \, x + 390622\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-5*exp(390622)*exp(1/4*x)/(20*exp(390622)*exp(1/4*x)+4),x, algorithm="fricas")

[Out]

-log(5*e^(1/4*x + 390622) + 1)

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giac [A]  time = 0.17, size = 13, normalized size = 0.65 \begin {gather*} -\log \left (5 \, e^{\left (\frac {1}{4} \, x + 390622\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-5*exp(390622)*exp(1/4*x)/(20*exp(390622)*exp(1/4*x)+4),x, algorithm="giac")

[Out]

-log(5*e^(1/4*x + 390622) + 1)

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maple [A]  time = 0.04, size = 13, normalized size = 0.65

method	result	size
risch	\(-\ln \left ({\mathrm e}^{\frac {x}{4}}+\frac {{\mathrm e}^{-390622}}{5}\right )\)	\(13\)
derivativedivides	\(-\ln \left (20 \,{\mathrm e}^{390622} {\mathrm e}^{\frac {x}{4}}+4\right )\)	\(14\)
default	\(-\ln \left (5 \,{\mathrm e}^{390622} {\mathrm e}^{\frac {x}{4}}+1\right )\)	\(14\)
norman	\(-\ln \left (20 \,{\mathrm e}^{390622} {\mathrm e}^{\frac {x}{4}}+4\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-5*exp(390622)*exp(1/4*x)/(20*exp(390622)*exp(1/4*x)+4),x,method=_RETURNVERBOSE)

[Out]

-ln(exp(1/4*x)+1/5*exp(-390622))

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maxima [A]  time = 0.39, size = 13, normalized size = 0.65 \begin {gather*} -\log \left (5 \, e^{\left (\frac {1}{4} \, x + 390622\right )} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-5*exp(390622)*exp(1/4*x)/(20*exp(390622)*exp(1/4*x)+4),x, algorithm="maxima")

[Out]

-log(5*e^(1/4*x + 390622) + 1)

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mupad [B]  time = 0.10, size = 13, normalized size = 0.65 \begin {gather*} -\ln \left (5\,{\mathrm {e}}^{\frac {x}{4}+390622}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*exp(x/4)*exp(390622))/(20*exp(x/4)*exp(390622) + 4),x)

[Out]

-log(5*exp(x/4 + 390622) + 1)

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sympy [A]  time = 0.10, size = 14, normalized size = 0.70 \begin {gather*} - \log {\left (e^{\frac {x}{4}} + \frac {1}{5 e^{390622}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-5*exp(390622)*exp(1/4*x)/(20*exp(390622)*exp(1/4*x)+4),x)

[Out]

-log(exp(x/4) + exp(-390622)/5)

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