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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]