3.32 \(\int \frac{e^{-n x}}{\left (a+b e^{n x}\right )^3} \, dx\)

Optimal. Leaf size=83 \[ \frac{3 b \log \left (a+b e^{n x}\right )}{a^4 n}-\frac{3 b x}{a^4}-\frac{2 b}{a^3 n \left (a+b e^{n x}\right )}-\frac{e^{-n x}}{a^3 n}-\frac{b}{2 a^2 n \left (a+b e^{n x}\right )^2} \]

[Out]

-(1/(a^3*E^(n*x)*n)) - b/(2*a^2*(a + b*E^(n*x))^2*n) - (2*b)/(a^3*(a + b*E^(n*x)
)*n) - (3*b*x)/a^4 + (3*b*Log[a + b*E^(n*x)])/(a^4*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.127716, antiderivative size = 83, 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 \[ \frac{3 b \log \left (a+b e^{n x}\right )}{a^4 n}-\frac{3 b x}{a^4}-\frac{2 b}{a^3 n \left (a+b e^{n x}\right )}-\frac{e^{-n x}}{a^3 n}-\frac{b}{2 a^2 n \left (a+b e^{n x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(E^(n*x)*(a + b*E^(n*x))^3),x]

[Out]

-(1/(a^3*E^(n*x)*n)) - b/(2*a^2*(a + b*E^(n*x))^2*n) - (2*b)/(a^3*(a + b*E^(n*x)
)*n) - (3*b*x)/a^4 + (3*b*Log[a + b*E^(n*x)])/(a^4*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.4443, size = 80, normalized size = 0.96 \[ - \frac{b}{2 a^{2} n \left (a + b e^{n x}\right )^{2}} - \frac{2 b}{a^{3} n \left (a + b e^{n x}\right )} - \frac{e^{- n x}}{a^{3} n} + \frac{3 b \log{\left (a + b e^{n x} \right )}}{a^{4} n} - \frac{3 b \log{\left (e^{n x} \right )}}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/exp(n*x)/(a+b*exp(n*x))**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.136047, size = 66, normalized size = 0.8 \[ \frac{\frac{b^2 e^{n x} \left (6 a+5 b e^{n x}\right )}{\left (a+b e^{n x}\right )^2}+6 b \log \left (a e^{-n x}+b\right )-2 a e^{-n x}}{2 a^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(E^(n*x)*(a + b*E^(n*x))^3),x]

[Out]

((-2*a)/E^(n*x) + (b^2*E^(n*x)*(6*a + 5*b*E^(n*x)))/(a + b*E^(n*x))^2 + 6*b*Log[
b + a/E^(n*x)])/(2*a^4*n)

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 86, normalized size = 1. \[ -{\frac{1}{{a}^{3}{{\rm e}^{nx}}n}}-3\,{\frac{b\ln \left ({{\rm e}^{nx}} \right ) }{n{a}^{4}}}-{\frac{b}{2\,{a}^{2} \left ( a+b{{\rm e}^{nx}} \right ) ^{2}n}}+3\,{\frac{b\ln \left ( a+b{{\rm e}^{nx}} \right ) }{n{a}^{4}}}-2\,{\frac{b}{{a}^{3} \left ( a+b{{\rm e}^{nx}} \right ) n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/exp(n*x)/(a+b*exp(n*x))^3,x)

[Out]

-1/a^3/exp(n*x)/n-3/n/a^4*b*ln(exp(n*x))-1/2*b/a^2/(a+b*exp(n*x))^2/n+3*b*ln(a+b
*exp(n*x))/a^4/n-2*b/a^3/(a+b*exp(n*x))/n

_______________________________________________________________________________________

Maxima [A]  time = 0.806241, size = 115, normalized size = 1.39 \[ \frac{6 \, a b^{2} e^{\left (-n x\right )} + 5 \, b^{3}}{2 \,{\left (2 \, a^{5} b e^{\left (-n x\right )} + a^{6} e^{\left (-2 \, n x\right )} + a^{4} b^{2}\right )} n} - \frac{e^{\left (-n x\right )}}{a^{3} n} + \frac{3 \, b \log \left (a e^{\left (-n x\right )} + b\right )}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(-n*x)/(b*e^(n*x) + a)^3,x, algorithm="maxima")

[Out]

1/2*(6*a*b^2*e^(-n*x) + 5*b^3)/((2*a^5*b*e^(-n*x) + a^6*e^(-2*n*x) + a^4*b^2)*n)
- e^(-n*x)/(a^3*n) + 3*b*log(a*e^(-n*x) + b)/(a^4*n)

_______________________________________________________________________________________

Fricas [A]  time = 0.262734, size = 189, normalized size = 2.28 \[ -\frac{6 \, b^{3} n x e^{\left (3 \, n x\right )} + 2 \, a^{3} + 6 \,{\left (2 \, a b^{2} n x + a b^{2}\right )} e^{\left (2 \, n x\right )} + 3 \,{\left (2 \, a^{2} b n x + 3 \, a^{2} b\right )} e^{\left (n x\right )} - 6 \,{\left (b^{3} e^{\left (3 \, n x\right )} + 2 \, a b^{2} e^{\left (2 \, n x\right )} + a^{2} b e^{\left (n x\right )}\right )} \log \left (b e^{\left (n x\right )} + a\right )}{2 \,{\left (a^{4} b^{2} n e^{\left (3 \, n x\right )} + 2 \, a^{5} b n e^{\left (2 \, n x\right )} + a^{6} n e^{\left (n x\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(-n*x)/(b*e^(n*x) + a)^3,x, algorithm="fricas")

[Out]

-1/2*(6*b^3*n*x*e^(3*n*x) + 2*a^3 + 6*(2*a*b^2*n*x + a*b^2)*e^(2*n*x) + 3*(2*a^2
*b*n*x + 3*a^2*b)*e^(n*x) - 6*(b^3*e^(3*n*x) + 2*a*b^2*e^(2*n*x) + a^2*b*e^(n*x)
)*log(b*e^(n*x) + a))/(a^4*b^2*n*e^(3*n*x) + 2*a^5*b*n*e^(2*n*x) + a^6*n*e^(n*x)
)

_______________________________________________________________________________________

Sympy [A]  time = 0.554041, size = 114, normalized size = 1.37 \[ \frac{- 5 a b - 4 b^{2} e^{n x}}{2 a^{5} n + 4 a^{4} b n e^{n x} + 2 a^{3} b^{2} n e^{2 n x}} + \begin{cases} - \frac{e^{- n x}}{a^{3} n} & \text{for}\: a^{3} n \neq 0 \\x \left (\frac{3 b}{a^{4}} + \frac{a - 3 b}{a^{4}}\right ) & \text{otherwise} \end{cases} - \frac{3 b x}{a^{4}} + \frac{3 b \log{\left (\frac{a}{b} + e^{n x} \right )}}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/exp(n*x)/(a+b*exp(n*x))**3,x)

[Out]

(-5*a*b - 4*b**2*exp(n*x))/(2*a**5*n + 4*a**4*b*n*exp(n*x) + 2*a**3*b**2*n*exp(2
*n*x)) + Piecewise((-exp(-n*x)/(a**3*n), Ne(a**3*n, 0)), (x*(3*b/a**4 + (a - 3*b
)/a**4), True)) - 3*b*x/a**4 + 3*b*log(a/b + exp(n*x))/(a**4*n)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269834, size = 103, normalized size = 1.24 \[ -\frac{\frac{6 \, b n x}{a^{4}} - \frac{6 \, b{\rm ln}\left ({\left | b e^{\left (n x\right )} + a \right |}\right )}{a^{4}} + \frac{{\left (6 \, a b^{2} e^{\left (2 \, n x\right )} + 9 \, a^{2} b e^{\left (n x\right )} + 2 \, a^{3}\right )} e^{\left (-n x\right )}}{{\left (b e^{\left (n x\right )} + a\right )}^{2} a^{4}}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(-n*x)/(b*e^(n*x) + a)^3,x, algorithm="giac")

[Out]

-1/2*(6*b*n*x/a^4 - 6*b*ln(abs(b*e^(n*x) + a))/a^4 + (6*a*b^2*e^(2*n*x) + 9*a^2*
b*e^(n*x) + 2*a^3)*e^(-n*x)/((b*e^(n*x) + a)^2*a^4))/n