3.30 \(\int \frac{e^{-n x}}{a+b e^{n x}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \log \left (a+b e^{n x}\right )}{a^2 n}-\frac{b x}{a^2}-\frac{e^{-n x}}{a n} \]

[Out]

-(1/(a*E^(n*x)*n)) - (b*x)/a^2 + (b*Log[a + b*E^(n*x)])/(a^2*n)

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Rubi [A]  time = 0.0438791, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2248, 44} \[ \frac{b \log \left (a+b e^{n x}\right )}{a^2 n}-\frac{b x}{a^2}-\frac{e^{-n x}}{a n} \]

Antiderivative was successfully verified.

[In]

Int[1/(E^(n*x)*(a + b*E^(n*x))),x]

[Out]

-(1/(a*E^(n*x)*n)) - (b*x)/a^2 + (b*Log[a + b*E^(n*x)])/(a^2*n)

Rule 2248

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(g*h*Log[G])/(d*e*Log[F])]}, Dist[(Denominator[m]*G^(f*h - (c*g*h)/d))/(d*e*Log[F]), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^((e*(c + d*x))/Denominator[m])], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{-n x}}{a+b e^{n x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)} \, dx,x,e^{n x}\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx,x,e^{n x}\right )}{n}\\ &=-\frac{e^{-n x}}{a n}-\frac{b x}{a^2}+\frac{b \log \left (a+b e^{n x}\right )}{a^2 n}\\ \end{align*}

Mathematica [A]  time = 0.0382528, size = 34, normalized size = 0.85 \[ -\frac{-b \log \left (a+b e^{n x}\right )+a e^{-n x}+b n x}{a^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(E^(n*x)*(a + b*E^(n*x))),x]

[Out]

-((a/E^(n*x) + b*n*x - b*Log[a + b*E^(n*x)])/(a^2*n))

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Maple [A]  time = 0.01, size = 47, normalized size = 1.2 \begin{align*} -{\frac{1}{a{{\rm e}^{nx}}n}}-{\frac{b\ln \left ({{\rm e}^{nx}} \right ) }{{a}^{2}n}}+{\frac{b\ln \left ( a+b{{\rm e}^{nx}} \right ) }{{a}^{2}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/exp(n*x)/(a+b*exp(n*x)),x)

[Out]

-1/a/exp(n*x)/n-1/n*b/a^2*ln(exp(n*x))+b*ln(a+b*exp(n*x))/a^2/n

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Maxima [A]  time = 1.04324, size = 43, normalized size = 1.08 \begin{align*} -\frac{e^{\left (-n x\right )}}{a n} + \frac{b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(n*x)/(a+b*exp(n*x)),x, algorithm="maxima")

[Out]

-e^(-n*x)/(a*n) + b*log(a*e^(-n*x) + b)/(a^2*n)

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Fricas [A]  time = 1.53373, size = 95, normalized size = 2.38 \begin{align*} -\frac{{\left (b n x e^{\left (n x\right )} - b e^{\left (n x\right )} \log \left (b e^{\left (n x\right )} + a\right ) + a\right )} e^{\left (-n x\right )}}{a^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(n*x)/(a+b*exp(n*x)),x, algorithm="fricas")

[Out]

-(b*n*x*e^(n*x) - b*e^(n*x)*log(b*e^(n*x) + a) + a)*e^(-n*x)/(a^2*n)

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Sympy [A]  time = 0.361883, size = 49, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{e^{- n x}}{a n} & \text{for}\: a n \neq 0 \\x \left (\frac{b}{a^{2}} + \frac{a - b}{a^{2}}\right ) & \text{otherwise} \end{cases} - \frac{b x}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + e^{n x} \right )}}{a^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(n*x)/(a+b*exp(n*x)),x)

[Out]

Piecewise((-exp(-n*x)/(a*n), Ne(a*n, 0)), (x*(b/a**2 + (a - b)/a**2), True)) - b*x/a**2 + b*log(a/b + exp(n*x)
)/(a**2*n)

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Giac [A]  time = 1.26291, size = 51, normalized size = 1.27 \begin{align*} -\frac{\frac{b n x}{a^{2}} + \frac{e^{\left (-n x\right )}}{a} - \frac{b \log \left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{2}}}{n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(n*x)/(a+b*exp(n*x)),x, algorithm="giac")

[Out]

-(b*n*x/a^2 + e^(-n*x)/a - b*log(abs(b*e^(n*x) + a))/a^2)/n