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 = 0.0386493, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 5546
Rule 5548
Rule 261
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*}
Mathematica [B] time = 0.116666, 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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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm csch} \left (a+2\,\ln \left ( 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 = 1.02855, size = 103, normalized size = 3.96 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.54766, size = 104, normalized size = 4. \begin{align*} -\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}} \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{csch}^{3}{\left (a + 2 \log{\left (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 = 1.18587, size = 51, normalized size = 1.96 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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