3.12 \(\int \frac{1}{\left (a+b e^{c+d x}\right )^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{\log \left (a+b e^{c+d x}\right )}{a^2 d}+\frac{x}{a^2}+\frac{1}{a d \left (a+b e^{c+d x}\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0647441, antiderivative size = 46, 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 \[ -\frac{\log \left (a+b e^{c+d x}\right )}{a^2 d}+\frac{x}{a^2}+\frac{1}{a d \left (a+b e^{c+d x}\right )} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*E^(c + d*x))^(-2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.9474, size = 88, normalized size = 1.91 \[ \frac{e^{- c - d x} e^{c + d x}}{a d \left (a + b e^{c + d x}\right )} - \frac{e^{- c - d x} e^{c + d x} \log{\left (a + b e^{c + d x} \right )}}{a^{2} d} + \frac{e^{- c - d x} e^{c + d x} \log{\left (e^{c + d x} \right )}}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*exp(d*x+c))**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0577621, size = 41, normalized size = 0.89 \[ \frac{\frac{a}{a d+b d e^{c+d x}}-\frac{\log \left (a+b e^{c+d x}\right )}{d}+x}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*E^(c + d*x))^(-2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 54, normalized size = 1.2 \[{\frac{\ln \left ({{\rm e}^{dx+c}} \right ) }{d{a}^{2}}}-{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{d{a}^{2}}}+{\frac{1}{ad \left ( a+b{{\rm e}^{dx+c}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*exp(d*x+c))^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.862244, size = 69, normalized size = 1.5 \[ \frac{1}{{\left (a b e^{\left (d x + c\right )} + a^{2}\right )} d} + \frac{d x + c}{a^{2} d} - \frac{\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(d*x + c) + a)^(-2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.254929, size = 81, normalized size = 1.76 \[ \frac{b d x e^{\left (d x + c\right )} + a d x -{\left (b e^{\left (d x + c\right )} + a\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) + a}{a^{2} b d e^{\left (d x + c\right )} + a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(d*x + c) + a)^(-2),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.310755, size = 39, normalized size = 0.85 \[ \frac{1}{a^{2} d + a b d e^{c + d x}} + \frac{x}{a^{2}} - \frac{\log{\left (\frac{a}{b} + e^{c + d x} \right )}}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*exp(d*x+c))**2,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269408, size = 70, normalized size = 1.52 \[ \frac{d x + c}{a^{2} d} - \frac{{\rm ln}\left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{a^{2} d} + \frac{1}{{\left (b e^{\left (d x + c\right )} + a\right )} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*e^(d*x + c) + a)^(-2),x, algorithm="giac")

[Out]

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