3.21 \(\int \frac{1}{(a+b e^{p x})^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac{\log \left (a+b e^{p x}\right )}{a^2 p}+\frac{x}{a^2}+\frac{1}{a p \left (a+b e^{p x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.0334166, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 44} \[ -\frac{\log \left (a+b e^{p x}\right )}{a^2 p}+\frac{x}{a^2}+\frac{1}{a p \left (a+b e^{p x}\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*E^(p*x))^(-2),x]

[Out]

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[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{1}{\left (a+b e^{p x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,e^{p x}\right )}{p}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,e^{p x}\right )}{p}\\ &=\frac{1}{a \left (a+b e^{p x}\right ) p}+\frac{x}{a^2}-\frac{\log \left (a+b e^{p x}\right )}{a^2 p}\\ \end{align*}

Mathematica [A]  time = 0.0415607, size = 36, normalized size = 0.86 \[ \frac{\frac{a}{a+b e^{p x}}-\log \left (a+b e^{p x}\right )+p x}{a^2 p} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^(p*x))^(-2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 48, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({{\rm e}^{px}} \right ) }{p{a}^{2}}}-{\frac{\ln \left ( a+b{{\rm e}^{px}} \right ) }{p{a}^{2}}}+{\frac{1}{a \left ( a+b{{\rm e}^{px}} \right ) p}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*exp(p*x))^2,x)

[Out]

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

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Maxima [A]  time = 0.947001, size = 54, normalized size = 1.29 \begin{align*} \frac{x}{a^{2}} + \frac{1}{{\left (a b e^{\left (p x\right )} + a^{2}\right )} p} - \frac{\log \left (b e^{\left (p x\right )} + a\right )}{a^{2} p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(p*x))^2,x, algorithm="maxima")

[Out]

x/a^2 + 1/((a*b*e^(p*x) + a^2)*p) - log(b*e^(p*x) + a)/(a^2*p)

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Fricas [A]  time = 2.0758, size = 124, normalized size = 2.95 \begin{align*} \frac{b p x e^{\left (p x\right )} + a p x -{\left (b e^{\left (p x\right )} + a\right )} \log \left (b e^{\left (p x\right )} + a\right ) + a}{a^{2} b p e^{\left (p x\right )} + a^{3} p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(p*x))^2,x, algorithm="fricas")

[Out]

(b*p*x*e^(p*x) + a*p*x - (b*e^(p*x) + a)*log(b*e^(p*x) + a) + a)/(a^2*b*p*e^(p*x) + a^3*p)

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Sympy [A]  time = 0.131613, size = 36, normalized size = 0.86 \begin{align*} \frac{1}{a^{2} p + a b p e^{p x}} + \frac{x}{a^{2}} - \frac{\log{\left (\frac{a}{b} + e^{p x} \right )}}{a^{2} p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(p*x))**2,x)

[Out]

1/(a**2*p + a*b*p*exp(p*x)) + x/a**2 - log(a/b + exp(p*x))/(a**2*p)

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Giac [A]  time = 1.07974, size = 55, normalized size = 1.31 \begin{align*} \frac{x}{a^{2}} - \frac{\log \left ({\left | b e^{\left (p x\right )} + a \right |}\right )}{a^{2} p} + \frac{1}{{\left (b e^{\left (p x\right )} + a\right )} a p} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(p*x))^2,x, algorithm="giac")

[Out]

x/a^2 - log(abs(b*e^(p*x) + a))/(a^2*p) + 1/((b*e^(p*x) + a)*a*p)