Optimal. Leaf size=25 \[ \frac{2 e^{-3 a} c^2}{\left (e^{-2 a}+\frac{c^4}{x^2}\right )^2} \]
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Rubi [A] time = 0.0457748, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5545, 5547, 263, 261} \[ \frac{2 e^{-3 a} c^2}{\left (e^{-2 a}+\frac{c^4}{x^2}\right )^2} \]
Antiderivative was successfully verified.
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Rule 5545
Rule 5547
Rule 263
Rule 261
Rubi steps
\begin{align*} \int \text{sech}^3\left (a+2 \log \left (\frac{c}{\sqrt{x}}\right )\right ) \, dx &=-\left (\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{sech}^3(a+2 \log (x))}{x^3} \, dx,x,\frac{c}{\sqrt{x}}\right )\right )\\ &=-\left (\left (16 c^2 e^{-3 a}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{e^{-2 a}}{x^4}\right )^3 x^9} \, dx,x,\frac{c}{\sqrt{x}}\right )\right )\\ &=-\left (\left (16 c^2 e^{-3 a}\right ) \operatorname{Subst}\left (\int \frac{x^3}{\left (e^{-2 a}+x^4\right )^3} \, dx,x,\frac{c}{\sqrt{x}}\right )\right )\\ &=\frac{2 c^2 e^{-3 a}}{\left (e^{-2 a}+\frac{c^4}{x^2}\right )^2}\\ \end{align*}
Mathematica [B] time = 0.0960164, size = 64, normalized size = 2.56 \[ -\frac{2 c^6 (\sinh (2 a)+\cosh (2 a)) \left (\sinh (a) \left (c^4-2 x^2\right )+\cosh (a) \left (c^4+2 x^2\right )\right )}{\left (\sinh (a) \left (c^4-x^2\right )+\cosh (a) \left (c^4+x^2\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (a+2\,\ln \left ({\frac{c}{\sqrt{x}}} \right ) \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.05959, size = 66, normalized size = 2.64 \begin{align*} -\frac{2 \,{\left (c^{10} e^{\left (5 \, a\right )} + 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} + 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.85147, size = 107, normalized size = 4.28 \begin{align*} -\frac{2 \,{\left (c^{10} e^{\left (5 \, a\right )} + 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} + 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{3}{\left (a + 2 \log{\left (\frac{c}{\sqrt{x}} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11218, size = 50, normalized size = 2. \begin{align*} -\frac{2 \,{\left (c^{10} e^{\left (5 \, a\right )} + 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{{\left (c^{4} e^{\left (2 \, a\right )} + x^{2}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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