Optimal. Leaf size=42 \[ -\frac{\log \left (a+b e^{p x}\right )}{a^2 p}+\frac{x}{a^2}+\frac{1}{a p \left (a+b e^{p x}\right )} \]
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Rubi [A] time = 0.0334166, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2282, 44} \[ -\frac{\log \left (a+b e^{p x}\right )}{a^2 p}+\frac{x}{a^2}+\frac{1}{a p \left (a+b e^{p x}\right )} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b e^{p x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,e^{p x}\right )}{p}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,e^{p x}\right )}{p}\\ &=\frac{1}{a \left (a+b e^{p x}\right ) p}+\frac{x}{a^2}-\frac{\log \left (a+b e^{p x}\right )}{a^2 p}\\ \end{align*}
Mathematica [A] time = 0.0415607, size = 36, normalized size = 0.86 \[ \frac{\frac{a}{a+b e^{p x}}-\log \left (a+b e^{p x}\right )+p x}{a^2 p} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({{\rm e}^{px}} \right ) }{p{a}^{2}}}-{\frac{\ln \left ( a+b{{\rm e}^{px}} \right ) }{p{a}^{2}}}+{\frac{1}{a \left ( a+b{{\rm e}^{px}} \right ) p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947001, size = 54, normalized size = 1.29 \begin{align*} \frac{x}{a^{2}} + \frac{1}{{\left (a b e^{\left (p x\right )} + a^{2}\right )} p} - \frac{\log \left (b e^{\left (p x\right )} + a\right )}{a^{2} p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0758, size = 124, normalized size = 2.95 \begin{align*} \frac{b p x e^{\left (p x\right )} + a p x -{\left (b e^{\left (p x\right )} + a\right )} \log \left (b e^{\left (p x\right )} + a\right ) + a}{a^{2} b p e^{\left (p x\right )} + a^{3} p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.131613, size = 36, normalized size = 0.86 \begin{align*} \frac{1}{a^{2} p + a b p e^{p x}} + \frac{x}{a^{2}} - \frac{\log{\left (\frac{a}{b} + e^{p x} \right )}}{a^{2} p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07974, size = 55, normalized size = 1.31 \begin{align*} \frac{x}{a^{2}} - \frac{\log \left ({\left | b e^{\left (p x\right )} + a \right |}\right )}{a^{2} p} + \frac{1}{{\left (b e^{\left (p x\right )} + a\right )} a p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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