Integrand size = 16, antiderivative size = 78 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=-\frac {1}{2 a d \left (a+b e^{-c-d x}\right )^2}-\frac {1}{a^2 d \left (a+b e^{-c-d x}\right )}+\frac {x}{a^3}+\frac {\log \left (a+b e^{-c-d x}\right )}{a^3 d} \] Output:
-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
Time = 0.96 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=\frac {\frac {b \left (3 b+4 a e^{c+d x}\right )}{\left (b+a e^{c+d x}\right )^2}+2 \log \left (b+a e^{c+d x}\right )}{2 a^3 d} \] Input:
Integrate[(a + b*E^(-c - d*x))^(-3),x]
Output:
((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)
Time = 0.36 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2720, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\int \frac {e^{c+d x}}{\left (a+b e^{-c-d x}\right )^3}de^{-c-d x}}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -\frac {\int \left (-\frac {b}{a^3 \left (a+b e^{-c-d x}\right )}-\frac {b}{a^2 \left (a+b e^{-c-d x}\right )^2}-\frac {b}{a \left (a+b e^{-c-d x}\right )^3}+\frac {e^{c+d x}}{a^3}\right )de^{-c-d x}}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {\log \left (a+b e^{-c-d x}\right )}{a^3}+\frac {\log \left (e^{-c-d x}\right )}{a^3}+\frac {1}{a^2 \left (a+b e^{-c-d x}\right )}+\frac {1}{2 a \left (a+b e^{-c-d x}\right )^2}}{d}\) |
Input:
Int[(a + b*E^(-c - d*x))^(-3),x]
Output:
-((1/(2*a*(a + b*E^(-c - d*x))^2) + 1/(a^2*(a + b*E^(-c - d*x))) + Log[E^( -c - d*x)]/a^3 - Log[a + b*E^(-c - d*x)]/a^3)/d)
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Time = 0.04 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {x}{a^{3}}+\frac {c}{a^{3} d}-\frac {2 b \,{\mathrm e}^{-d x -c}+3 a}{2 d \,a^{2} \left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{-d x -c}+\frac {a}{b}\right )}{a^{3} d}\) | \(77\) |
derivativedivides | \(-\frac {\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{a^{3}}-\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{-d x -c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}}{d}\) | \(79\) |
default | \(-\frac {\frac {\ln \left ({\mathrm e}^{-d x -c}\right )}{a^{3}}-\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{-d x -c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}}{d}\) | \(79\) |
norman | \(\frac {\frac {x}{a}+\frac {b^{2} x \,{\mathrm e}^{-2 d x -2 c}}{a^{3}}+\frac {2 b x \,{\mathrm e}^{-d x -c}}{a^{2}}+\frac {2 b \,{\mathrm e}^{-d x -c}}{a^{2} d}+\frac {3 b^{2} {\mathrm e}^{-2 d x -2 c}}{2 a^{3} d}}{\left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}+\frac {\ln \left (a +b \,{\mathrm e}^{-d x -c}\right )}{a^{3} d}\) | \(120\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{-2 d x -2 c} b^{2} d x +2 \ln \left (a +b \,{\mathrm e}^{-d x -c}\right ) {\mathrm e}^{-2 d x -2 c} b^{2}+4 \,{\mathrm e}^{-d x -c} a b d x +4 \ln \left (a +b \,{\mathrm e}^{-d x -c}\right ) {\mathrm e}^{-d x -c} a b +2 a^{2} d x +3 b^{2} {\mathrm e}^{-2 d x -2 c}+2 \ln \left (a +b \,{\mathrm e}^{-d x -c}\right ) a^{2}+4 \,{\mathrm e}^{-d x -c} a b}{2 a^{3} d \left (a +b \,{\mathrm e}^{-d x -c}\right )^{2}}\) | \(170\) |
Input:
int(1/(a+b*exp(-d*x-c))^3,x,method=_RETURNVERBOSE)
Output:
x/a^3+1/a^3/d*c-1/2*(2*b*exp(-d*x-c)+3*a)/d/a^2/(a+b*exp(-d*x-c))^2+1/a^3/ d*ln(exp(-d*x-c)+1/b*a)
Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.79 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=\frac {2 \, b^{2} d x e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, a^{2} d x - 3 \, a^{2} + 2 \, {\left (2 \, a b d x - a b\right )} e^{\left (-d x - c\right )} + 2 \, {\left (2 \, a b e^{\left (-d x - c\right )} + b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{2}\right )} \log \left (b e^{\left (-d x - c\right )} + a\right )}{2 \, {\left (2 \, a^{4} b d e^{\left (-d x - c\right )} + a^{3} b^{2} d e^{\left (-2 \, d x - 2 \, c\right )} + a^{5} d\right )}} \] Input:
integrate(1/(a+b*exp(-d*x-c))^3,x, algorithm="fricas")
Output:
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)
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=\frac {- 3 a - 2 b e^{- c - d x}}{2 a^{4} d + 4 a^{3} b d e^{- c - d x} + 2 a^{2} b^{2} d e^{- 2 c - 2 d x}} + \frac {x}{a^{3}} + \frac {\log {\left (\frac {a}{b} + e^{- c - d x} \right )}}{a^{3} d} \] Input:
integrate(1/(a+b*exp(-d*x-c))**3,x)
Output:
(-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)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=-\frac {2 \, b e^{\left (-d x - c\right )} + 3 \, a}{2 \, {\left (2 \, a^{3} b e^{\left (-d x - c\right )} + a^{2} b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{4}\right )} d} + \frac {d x + c}{a^{3} d} + \frac {\log \left (b e^{\left (-d x - c\right )} + a\right )}{a^{3} d} \] Input:
integrate(1/(a+b*exp(-d*x-c))^3,x, algorithm="maxima")
Output:
-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)
Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=\frac {\frac {2 \, {\left (d x + c\right )}}{a^{3}} + \frac {2 \, \log \left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a^{3}} - \frac {2 \, a b e^{\left (-d x - c\right )} + 3 \, a^{2}}{{\left (b e^{\left (-d x - c\right )} + a\right )}^{2} a^{3}}}{2 \, d} \] Input:
integrate(1/(a+b*exp(-d*x-c))^3,x, algorithm="giac")
Output:
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
Time = 23.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.69 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=\frac {\frac {x}{a}+\frac {2\,b\,{\mathrm {e}}^{-c-d\,x}}{a^2\,d}+\frac {b^2\,x\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{a^3}+\frac {3\,b^2\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{2\,a^3\,d}+\frac {2\,b\,x\,{\mathrm {e}}^{-c-d\,x}}{a^2}}{a^2+2\,{\mathrm {e}}^{-c-d\,x}\,a\,b+{\mathrm {e}}^{-2\,c-2\,d\,x}\,b^2}+\frac {\ln \left (a+b\,{\mathrm {e}}^{-c}\,{\mathrm {e}}^{-d\,x}\right )}{a^3\,d} \] Input:
int(1/(a + b*exp(- c - d*x))^3,x)
Output:
(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)
Time = 0.18 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\left (a+b e^{-c-d x}\right )^3} \, dx=\frac {2 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c} a +b \right ) a^{2}-2 e^{2 d x +2 c} a^{2}+4 e^{d x +c} \mathrm {log}\left (e^{d x +c} a +b \right ) a b +2 \,\mathrm {log}\left (e^{d x +c} a +b \right ) b^{2}+b^{2}}{2 a^{3} d \left (e^{2 d x +2 c} a^{2}+2 e^{d x +c} a b +b^{2}\right )} \] Input:
int(1/(a+b*exp(-d*x-c))^3,x)
Output:
(2*e**(2*c + 2*d*x)*log(e**(c + d*x)*a + b)*a**2 - 2*e**(2*c + 2*d*x)*a**2 + 4*e**(c + d*x)*log(e**(c + d*x)*a + b)*a*b + 2*log(e**(c + d*x)*a + b)* b**2 + b**2)/(2*a**3*d*(e**(2*c + 2*d*x)*a**2 + 2*e**(c + d*x)*a*b + b**2) )