Optimal. Leaf size=78 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a^3 d}+\frac{x}{a^3}-\frac{1}{a^2 d \left (a+b e^{-c-d x}\right )}-\frac{1}{2 a d \left (a+b e^{-c-d x}\right )^2} \]
[Out]
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Rubi [A] time = 0.0920927, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\log \left (a+b e^{-c-d x}\right )}{a^3 d}+\frac{x}{a^3}-\frac{1}{a^2 d \left (a+b e^{-c-d x}\right )}-\frac{1}{2 a d \left (a+b e^{-c-d x}\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*E^(-c - d*x))^(-3),x]
[Out]
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Rubi in Sympy [A] time = 17.4879, size = 129, normalized size = 1.65 \[ - \frac{e^{- c - d x} e^{c + d x}}{2 a d \left (a + b e^{- c - d x}\right )^{2}} - \frac{e^{- c - d x} e^{c + d x}}{a^{2} d \left (a + b e^{- c - d x}\right )} + \frac{e^{- c - d x} e^{c + d x} \log{\left (a + b e^{- c - d x} \right )}}{a^{3} d} - \frac{e^{- c - d x} e^{c + d x} \log{\left (e^{- c - d x} \right )}}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*exp(-d*x-c))**3,x)
[Out]
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Mathematica [A] time = 0.0804936, size = 54, normalized size = 0.69 \[ \frac{\frac{b \left (4 a e^{c+d x}+3 b\right )}{\left (a e^{c+d x}+b\right )^2}+2 \log \left (a e^{c+d x}+b\right )}{2 a^3 d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*E^(-c - d*x))^(-3),x]
[Out]
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Maple [A] time = 0.003, size = 87, 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]
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Maxima [A] time = 0.768719, size = 124, normalized size = 1.59 \[ -\frac{2 \, b e^{\left (-d x - c\right )} + 3 \, a}{2 \,{\left (2 \, a^{3} b e^{\left (-d x - c\right )} + a^{2} b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{4}\right )} d} + \frac{d x + c}{a^{3} d} + \frac{\log \left (b e^{\left (-d x - c\right )} + a\right )}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*e^(-d*x - c) + a)^(-3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.262441, size = 189, normalized size = 2.42 \[ \frac{2 \, b^{2} d x e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, a^{2} d x - 3 \, a^{2} + 2 \,{\left (2 \, a b d x - a b\right )} e^{\left (-d x - c\right )} + 2 \,{\left (2 \, a b e^{\left (-d x - c\right )} + b^{2} e^{\left (-2 \, d x - 2 \, c\right )} + a^{2}\right )} \log \left (b e^{\left (-d x - c\right )} + a\right )}{2 \,{\left (2 \, a^{4} b d e^{\left (-d x - c\right )} + a^{3} b^{2} d e^{\left (-2 \, d x - 2 \, c\right )} + a^{5} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*e^(-d*x - c) + a)^(-3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.422483, size = 85, normalized size = 1.09 \[ \frac{- 3 a - 2 b e^{- c - d x}}{2 a^{4} d + 4 a^{3} b d e^{- c - d x} + 2 a^{2} b^{2} d e^{- 2 c - 2 d x}} + \frac{x}{a^{3}} + \frac{\log{\left (\frac{a}{b} + e^{- c - d x} \right )}}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*exp(-d*x-c))**3,x)
[Out]
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GIAC/XCAS [A] time = 0.248816, size = 104, normalized size = 1.33 \[ \frac{d x + c}{a^{3} d} + \frac{{\rm ln}\left ({\left | b e^{\left (-d x - c\right )} + a \right |}\right )}{a^{3} d} - \frac{2 \, a b e^{\left (-d x - c\right )} + 3 \, a^{2}}{2 \,{\left (b e^{\left (-d x - c\right )} + a\right )}^{2} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*e^(-d*x - c) + a)^(-3),x, algorithm="giac")
[Out]