3.20 \(\int \frac{1}{\left (a+b e^{c+d x}\right )^3} \, dx\)

Optimal. Leaf size=69 \[ -\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d}+\frac{x}{a^3}+\frac{1}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{1}{2 a d \left (a+b e^{c+d x}\right )^2} \]

[Out]

1/(2*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)

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Rubi [A]  time = 0.0847248, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d}+\frac{x}{a^3}+\frac{1}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{1}{2 a d \left (a+b e^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^(c + d*x))^(-3),x]

[Out]

1/(2*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)

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Rubi in Sympy [A]  time = 16.6242, size = 122, normalized size = 1.77 \[ \frac{e^{- c - d x} e^{c + d x}}{2 a d \left (a + b e^{c + d x}\right )^{2}} + \frac{e^{- c - d x} e^{c + d x}}{a^{2} d \left (a + b e^{c + d x}\right )} - \frac{e^{- c - d x} e^{c + d x} \log{\left (a + b e^{c + d x} \right )}}{a^{3} d} + \frac{e^{- c - d x} e^{c + d x} \log{\left (e^{c + d x} \right )}}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.117286, size = 69, normalized size = 1. \[ -\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d}+\frac{x}{a^3}+\frac{1}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{1}{2 a d \left (a+b e^{c+d x}\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

1/(2*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)

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Maple [A]  time = 0.001, size = 74, normalized size = 1.1 \[{\frac{\ln \left ({{\rm e}^{dx+c}} \right ) }{d{a}^{3}}}-{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{d{a}^{3}}}+{\frac{1}{{a}^{2}d \left ( a+b{{\rm e}^{dx+c}} \right ) }}+{\frac{1}{2\,ad \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*exp(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 0.799186, size = 113, normalized size = 1.64 \[ \frac{2 \, b e^{\left (d x + c\right )} + 3 \, a}{2 \,{\left (a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b e^{\left (d x + c\right )} + a^{4}\right )} d} + \frac{d x + c}{a^{3} d} - \frac{\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*b*e^(d*x + c) + 3*a)/((a^2*b^2*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + 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 = 0.245999, size = 171, normalized size = 2.48 \[ \frac{2 \, b^{2} d x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} d x + 3 \, a^{2} + 2 \,{\left (2 \, a b d x + a b\right )} e^{\left (d x + c\right )} - 2 \,{\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{2 \,{\left (a^{3} b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d e^{\left (d x + c\right )} + a^{5} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(d*x + c) + a)^(-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*(b^2*e^(2*d*x + 2*c) + 2*a*b*e^(d*x + c) + a^2)*log(b*e^(d*x + c) + a))
/(a^3*b^2*d*e^(2*d*x + 2*c) + 2*a^4*b*d*e^(d*x + c) + a^5*d)

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Sympy [A]  time = 0.39217, size = 76, normalized size = 1.1 \[ \frac{3 a + 2 b e^{c + d x}}{2 a^{4} d + 4 a^{3} b d e^{c + d x} + 2 a^{2} b^{2} d e^{2 c + 2 d x}} + \frac{x}{a^{3}} - \frac{\log{\left (\frac{a}{b} + e^{c + d x} \right )}}{a^{3} d} \]

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/XCAS [A]  time = 0.250399, size = 93, normalized size = 1.35 \[ \frac{d x + c}{a^{3} d} - \frac{{\rm ln}\left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{a^{3} d} + \frac{2 \, a b e^{\left (d x + c\right )} + 3 \, a^{2}}{2 \,{\left (b e^{\left (d x + c\right )} + a\right )}^{2} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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