Optimal. Leaf size=26 \[ \frac {2 e^{-3 a} c^2}{\left (e^{-2 a}-\frac {c^4}{x^2}\right )^2} \]
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Rubi [A] time = 0.05, antiderivative size = 26, 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 = {5546, 5548, 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 261
Rule 263
Rule 5546
Rule 5548
Rubi steps
\begin {align*} \int \text {csch}^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 {csch}^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*}
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Mathematica [B] time = 0.11, size = 65, normalized size = 2.50 \[ -\frac {2 c^6 (\sinh (2 a)+\cosh (2 a)) \left (\sinh (a) \left (c^4+2 x^2\right )+\cosh (a) \left (c^4-2 x^2\right )\right )}{\left (\cosh (a) \left (x^2-c^4\right )-\sinh (a) \left (c^4+x^2\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 49, normalized size = 1.88 \[ -\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 39, normalized size = 1.50 \[ -\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{{\left (c^{4} e^{\left (2 \, a\right )} - x^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.73, size = 0, normalized size = 0.00 \[ \int \mathrm {csch}\left (a +2 \ln \left (\frac {c}{\sqrt {x}}\right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 49, normalized size = 1.88 \[ -\frac {2 \, {\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 36, normalized size = 1.38 \[ \frac {2\,c^2\,x^4\,{\mathrm {e}}^a}{{\mathrm {e}}^{4\,a}\,c^8-2\,{\mathrm {e}}^{2\,a}\,c^4\,x^2+x^4} \]
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 \[ \int \operatorname {csch}^{3}{\left (a + 2 \log {\left (\frac {c}{\sqrt {x}} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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