Optimal. Leaf size=25 \[ \frac {2 c^2 e^{-3 a}}{\left (e^{-2 a}+\frac {c^4}{x^2}\right )^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {5664, 5666,
269, 267} \begin {gather*} \frac {2 e^{-3 a} c^2}{\left (e^{-2 a}+\frac {c^4}{x^2}\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 269
Rule 5664
Rule 5666
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 ) \text {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 ) \text {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 ) \text {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*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(25)=50\).
time = 0.07, size = 64, normalized size = 2.56 \begin {gather*} -\frac {2 c^6 \left (\left (c^4+2 x^2\right ) \cosh (a)+\left (c^4-2 x^2\right ) \sinh (a)\right ) (\cosh (2 a)+\sinh (2 a))}{\left (\left (c^4+x^2\right ) \cosh (a)+\left (c^4-x^2\right ) \sinh (a)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 3.65, size = 0, normalized size = 0.00 \[\int \mathrm {sech}\left (a +2 \ln \left (\frac {c}{\sqrt {x}}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (23) = 46\).
time = 0.28, size = 49, normalized size = 1.96 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs.
\(2 (23) = 46\).
time = 0.39, size = 49, normalized size = 1.96 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {sech}^{3}{\left (a + 2 \log {\left (\frac {c}{\sqrt {x}} \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 37, normalized size = 1.48 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.45, size = 36, normalized size = 1.44 \begin {gather*} \frac {2\,c^2\,x^4\,{\mathrm {e}}^a}{{\mathrm {e}}^{4\,a}\,c^8+2\,{\mathrm {e}}^{2\,a}\,c^4\,x^2+x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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