Optimal. Leaf size=22 \[ -\frac {1}{2 a p \left (a e^{2 p x}+b\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 261
Rule 2282
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*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 1.00 \[ -\frac {1}{2 a p \left (a e^{2 p x}+b\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 19, normalized size = 0.86 \[ -\frac {1}{2 \, {\left (a^{2} p e^{\left (2 \, p x\right )} + a b p\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 19, normalized size = 0.86 \[ -\frac {1}{2 \, {\left (a e^{\left (2 \, p x\right )} + b\right )} a p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.95 \[ -\frac {1}{2 \left (a \,{\mathrm e}^{2 p x}+b \right ) a p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 20, normalized size = 0.91 \[ \frac {1}{2 \, {\left (b^{2} e^{\left (-2 \, p x\right )} + a b\right )} p} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 24, normalized size = 1.09 \[ \frac {{\mathrm {e}}^{2\,p\,x}}{2\,b\,p\,\left (b+a\,{\mathrm {e}}^{2\,p\,x}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 20, normalized size = 0.91 \[ \frac {1}{2 a b p + 2 b^{2} p e^{- 2 p x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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