Optimal. Leaf size=26 \[ -\frac {2 e^{-a} c^6}{\left (c^4-\frac {e^{-2 a}}{x^2}\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5546, 5548, 261} \[ -\frac {2 e^{-a} c^6}{\left (c^4-\frac {e^{-2 a}}{x^2}\right )^2} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5546
Rule 5548
Rubi steps
\begin {align*} \int \text {csch}^3\left (a+2 \log \left (c \sqrt {x}\right )\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int x \text {csch}^3(a+2 \log (x)) \, dx,x,c \sqrt {x}\right )}{c^2}\\ &=\frac {\left (16 e^{-3 a}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {e^{-2 a}}{x^4}\right )^3 x^5} \, dx,x,c \sqrt {x}\right )}{c^2}\\ &=-\frac {2 c^6 e^{-a}}{\left (c^4-\frac {e^{-2 a}}{x^2}\right )^2}\\ \end {align*}
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Mathematica [B] time = 0.13, size = 62, normalized size = 2.38 \[ \frac {2 (\cosh (a)-\sinh (a)) \left (\sinh ^2(a)+\cosh ^2(a)-2 \sinh (a) \cosh (a)-2 c^4 x^2\right )}{c^2 \left (\sinh (a) \left (c^4 x^2+1\right )+\cosh (a) \left (c^4 x^2-1\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 48, normalized size = 1.85 \[ -\frac {2 \, {\left (2 \, c^{4} x^{2} e^{\left (2 \, a\right )} - 1\right )}}{c^{10} x^{4} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )} + c^{2} e^{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 38, normalized size = 1.46 \[ -\frac {2 \, {\left (2 \, c^{4} x^{2} e^{\left (2 \, a\right )} - 1\right )} e^{\left (-a\right )}}{{\left (c^{4} x^{2} e^{\left (2 \, a\right )} - 1\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int \mathrm {csch}\left (a +2 \ln \left (c \sqrt {x}\right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 76, normalized size = 2.92 \[ -\frac {2 \, {\left (\frac {2 \, c^{4} x^{2} e^{\left (2 \, a\right )}}{c^{8} x^{4} e^{\left (5 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (3 \, a\right )} + e^{a}} - \frac {1}{c^{8} x^{4} e^{\left (5 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (3 \, a\right )} + e^{a}}\right )}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.65, size = 48, normalized size = 1.85 \[ \frac {\frac {2\,{\mathrm {e}}^{-a}}{c^2}-4\,c^2\,x^2\,{\mathrm {e}}^a}{{\mathrm {e}}^{4\,a}\,c^8\,x^4-2\,{\mathrm {e}}^{2\,a}\,c^4\,x^2+1} \]
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 \[ \int \operatorname {csch}^{3}{\left (a + 2 \log {\left (c \sqrt {x} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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