Optimal. Leaf size=22 \[ -\frac{1}{2 a p \left (a e^{2 p x}+b\right )} \]
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Rubi [A] time = 0.0235268, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2282, 261} \[ -\frac{1}{2 a p \left (a e^{2 p x}+b\right )} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 261
Rubi steps
\begin{align*} \int \frac{1}{\left (b e^{-p x}+a e^{p x}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\left (b+a x^2\right )^2} \, dx,x,e^{p x}\right )}{p}\\ &=-\frac{1}{2 a \left (b+a e^{2 p x}\right ) p}\\ \end{align*}
Mathematica [A] time = 0.0202344, size = 22, normalized size = 1. \[ -\frac{1}{2 a p \left (a e^{2 p x}+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 21, normalized size = 1. \begin{align*} -{\frac{1}{2\,pa \left ( a \left ({{\rm e}^{px}} \right ) ^{2}+b \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.935151, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{2 \,{\left (b^{2} e^{\left (-2 \, p x\right )} + a b\right )} p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95579, size = 43, normalized size = 1.95 \begin{align*} -\frac{1}{2 \,{\left (a^{2} p e^{\left (2 \, p x\right )} + a b p\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.102672, size = 20, normalized size = 0.91 \begin{align*} \frac{1}{2 a b p + 2 b^{2} p e^{- 2 p x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07501, size = 26, normalized size = 1.18 \begin{align*} -\frac{1}{2 \,{\left (a e^{\left (2 \, p x\right )} + b\right )} a p} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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