3.1.24 \(\int \frac {1}{(a+b e^{-c-d x})^3} \, dx\) [24]

Optimal. Leaf size=78 \[ -\frac {1}{2 a d \left (a+b e^{-c-d x}\right )^2}-\frac {1}{a^2 d \left (a+b e^{-c-d x}\right )}+\frac {x}{a^3}+\frac {\log \left (a+b e^{-c-d x}\right )}{a^3 d} \]

[Out]

-1/2/a/d/(a+b*exp(-d*x-c))^2-1/a^2/d/(a+b*exp(-d*x-c))+x/a^3+ln(a+b*exp(-d*x-c))/a^3/d

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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2320, 46} \begin {gather*} \frac {\log \left (a+b e^{-c-d x}\right )}{a^3 d}+\frac {x}{a^3}-\frac {1}{a^2 d \left (a+b e^{-c-d x}\right )}-\frac {1}{2 a d \left (a+b e^{-c-d x}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*E^(-c - d*x))^(-3),x]

[Out]

-1/2*1/(a*d*(a + b*E^(-c - d*x))^2) - 1/(a^2*d*(a + b*E^(-c - d*x))) + x/a^3 + Log[a + b*E^(-c - d*x)]/(a^3*d)

Rule 46

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] && Lt
Q[m + n + 2, 0])

Rule 2320

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]]]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x (a+b x)^3} \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {1}{a^3 x}-\frac {b}{a (a+b x)^3}-\frac {b}{a^2 (a+b x)^2}-\frac {b}{a^3 (a+b x)}\right ) \, dx,x,e^{-c-d x}\right )}{d}\\ &=-\frac {1}{2 a d \left (a+b e^{-c-d x}\right )^2}-\frac {1}{a^2 d \left (a+b e^{-c-d x}\right )}+\frac {x}{a^3}+\frac {\log \left (a+b e^{-c-d x}\right )}{a^3 d}\\ \end {align*}

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Mathematica [A]
time = 0.80, size = 54, normalized size = 0.69 \begin {gather*} \frac {\frac {b \left (3 b+4 a e^{c+d x}\right )}{\left (b+a e^{c+d x}\right )^2}+2 \log \left (b+a e^{c+d x}\right )}{2 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*E^(-c - d*x))^(-3),x]

[Out]

((b*(3*b + 4*a*E^(c + d*x)))/(b + a*E^(c + d*x))^2 + 2*Log[b + a*E^(c + d*x)])/(2*a^3*d)

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Maple [A]
time = 0.02, size = 79, normalized size = 1.01

method	result	size
risch	\(\frac {x}{a^{3}}+\frac {c}{a^{3} d}-\frac {2 b \,{\mathrm e}^{-d x -c}+3 a}{2 a^{2} d \left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{-d x -c}+\frac {a}{b}\right )}{a^{3} d}\)	\(77\)
derivativedivides	\(-\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{-d x -c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{a^{3}}}{d}\)	\(79\)
default	\(-\frac {-\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{-d x -c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{a^{3}}}{d}\)	\(79\)
norman	\(\frac {\frac {x}{a}+\frac {b^{2} x \,{\mathrm e}^{-2 d x -2 c}}{a^{3}}+\frac {2 b x \,{\mathrm e}^{-d x -c}}{a^{2}}+\frac {2 b \,{\mathrm e}^{-d x -c}}{a^{2} d}+\frac {3 b^{2} {\mathrm e}^{-2 d x -2 c}}{2 a^{3} d}}{\left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}+\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a^{3} d}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*exp(-d*x-c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/d*(-1/a^3*ln(a+b*exp(-d*x-c))+1/a^2/(a+b*exp(-d*x-c))+1/2/a/(a+b*exp(-d*x-c))^2+1/a^3*ln(exp(-d*x-c)))

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Maxima [A]
time = 0.28, size = 92, normalized size = 1.18 \begin {gather*} -\frac {2 \, b e^{\left (-d x - c\right )} + 3 \, a}{2 \, {\left (2 \, a^{3} b e^{\left (-d x - c\right )} + a^{2} b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{4}\right )} d} + \frac {d x + c}{a^{3} d} + \frac {\log \left (b e^{\left (-d x - c\right )} + a\right )}{a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c))^3,x, algorithm="maxima")

[Out]

-1/2*(2*b*e^(-d*x - c) + 3*a)/((2*a^3*b*e^(-d*x - c) + a^2*b^2*e^(-2*d*x - 2*c) + a^4)*d) + (d*x + c)/(a^3*d)
+ log(b*e^(-d*x - c) + a)/(a^3*d)

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Fricas [A]
time = 0.38, size = 140, normalized size = 1.79 \begin {gather*} \frac {2 \, b^{2} d x e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, a^{2} d x - 3 \, a^{2} + 2 \, {\left (2 \, a b d x - a b\right )} e^{\left (-d x - c\right )} + 2 \, {\left (2 \, a b e^{\left (-d x - c\right )} + b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{2}\right )} \log \left (b e^{\left (-d x - c\right )} + a\right )}{2 \, {\left (2 \, a^{4} b d e^{\left (-d x - c\right )} + a^{3} b^{2} d e^{\left (-2 \, d x - 2 \, c\right )} + a^{5} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c))^3,x, algorithm="fricas")

[Out]

1/2*(2*b^2*d*x*e^(-2*d*x - 2*c) + 2*a^2*d*x - 3*a^2 + 2*(2*a*b*d*x - a*b)*e^(-d*x - c) + 2*(2*a*b*e^(-d*x - c)
+ b^2*e^(-2*d*x - 2*c) + a^2)*log(b*e^(-d*x - c) + a))/(2*a^4*b*d*e^(-d*x - c) + a^3*b^2*d*e^(-2*d*x - 2*c) +
a^5*d)

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Sympy [A]
time = 0.08, size = 85, normalized size = 1.09 \begin {gather*} \frac {- 3 a - 2 b e^{- c - d x}}{2 a^{4} d + 4 a^{3} b d e^{- c - d x} + 2 a^{2} b^{2} d e^{- 2 c - 2 d x}} + \frac {x}{a^{3}} + \frac {\log {\left (\frac {a}{b} + e^{- c - d x} \right )}}{a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c))**3,x)

[Out]

(-3*a - 2*b*exp(-c - d*x))/(2*a**4*d + 4*a**3*b*d*exp(-c - d*x) + 2*a**2*b**2*d*exp(-2*c - 2*d*x)) + x/a**3 +
log(a/b + exp(-c - d*x))/(a**3*d)

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Giac [A]
time = 1.76, size = 75, normalized size = 0.96 \begin {gather*} \frac {\frac {2 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, \log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a^{3}} - \frac {2 \, a b e^{\left (-d x - c\right )} + 3 \, a^{2}}{{\left (b e^{\left (-d x - c\right )} + a\right )}^{2} a^{3}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(-d*x-c))^3,x, algorithm="giac")

[Out]

1/2*(2*(d*x + c)/a^3 + 2*log(abs(b*e^(-d*x - c) + a))/a^3 - (2*a*b*e^(-d*x - c) + 3*a^2)/((b*e^(-d*x - c) + a)
^2*a^3))/d

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Mupad [B]
time = 3.70, size = 132, normalized size = 1.69 \begin {gather*} \frac {\frac {x}{a}+\frac {2\,b\,{\mathrm {e}}^{-c-d\,x}}{a^2\,d}+\frac {b^2\,x\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{a^3}+\frac {3\,b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{2\,a^3\,d}+\frac {2\,b\,x\,{\mathrm {e}}^{-c-d\,x}}{a^2}}{a^2+2\,{\mathrm {e}}^{-c-d\,x}\,a\,b+{\mathrm {e}}^{-2\,c-2\,d\,x}\,b^2}+\frac {\ln \left (a+b\,{\mathrm {e}}^{-c}\,{\mathrm {e}}^{-d\,x}\right )}{a^3\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*exp(- c - d*x))^3,x)

[Out]

(x/a + (2*b*exp(- c - d*x))/(a^2*d) + (b^2*x*exp(- 2*c - 2*d*x))/a^3 + (3*b^2*exp(- 2*c - 2*d*x))/(2*a^3*d) +
(2*b*x*exp(- c - d*x))/a^2)/(a^2 + b^2*exp(- 2*c - 2*d*x) + 2*a*b*exp(- c - d*x)) + log(a + b*exp(-c)*exp(-d*x
))/(a^3*d)

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