Optimal. Leaf size=22 \[ -\frac {1}{2 a \left (b+a e^{2 p x}\right ) p} \]
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Rubi [A]
time = 0.01, 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 = {2320, 267}
\begin {gather*} -\frac {1}{2 a p \left (a e^{2 p x}+b\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{\left (b e^{-p x}+a e^{p x}\right )^2} \, dx &=\frac {\text {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 = 0.03, size = 22, normalized size = 1.00 \begin {gather*} -\frac {1}{2 a \left (b+a e^{2 p x}\right ) p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 0.95
method | result | size |
risch | \(-\frac {1}{2 a \left (b +a \,{\mathrm e}^{2 p x}\right ) p}\) | \(20\) |
derivativedivides | \(-\frac {1}{2 a \left (b +a \,{\mathrm e}^{2 p x}\right ) p}\) | \(21\) |
default | \(-\frac {1}{2 a \left (b +a \,{\mathrm e}^{2 p x}\right ) p}\) | \(21\) |
norman | \(-\frac {1}{2 a \left (b +a \,{\mathrm e}^{2 p x}\right ) p}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{2 \, {\left (b^{2} e^{\left (-2 \, p x\right )} + a b\right )} p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 19, normalized size = 0.86 \begin {gather*} -\frac {1}{2 \, {\left (a^{2} p e^{\left (2 \, p x\right )} + a b p\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 1.00 \begin {gather*} - \frac {1}{2 a^{2} p e^{2 p x} + 2 a b p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 19, normalized size = 0.86 \begin {gather*} -\frac {1}{2 \, {\left (a e^{\left (2 \, p x\right )} + b\right )} a p} \end {gather*}
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 \begin {gather*} \frac {{\mathrm {e}}^{2\,p\,x}}{2\,b\,p\,\left (b+a\,{\mathrm {e}}^{2\,p\,x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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