Optimal. Leaf size=29 \[ -\frac{1}{30} \text{csch}^5(6 x)-\frac{1}{9} \text{csch}^3(6 x)-\frac{1}{6} \text{csch}(6 x) \]
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Rubi [A] time = 0.0191935, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2606, 194} \[ -\frac{1}{30} \text{csch}^5(6 x)-\frac{1}{9} \text{csch}^3(6 x)-\frac{1}{6} \text{csch}(6 x) \]
Antiderivative was successfully verified.
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Rule 2606
Rule 194
Rubi steps
\begin{align*} \int \coth ^5(6 x) \text{csch}(6 x) \, dx &=-\left (\frac{1}{6} i \operatorname{Subst}\left (\int \left (-1+x^2\right )^2 \, dx,x,-i \text{csch}(6 x)\right )\right )\\ &=-\left (\frac{1}{6} i \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-i \text{csch}(6 x)\right )\right )\\ &=-\frac{1}{6} \text{csch}(6 x)-\frac{1}{9} \text{csch}^3(6 x)-\frac{1}{30} \text{csch}^5(6 x)\\ \end{align*}
Mathematica [A] time = 0.0171361, size = 29, normalized size = 1. \[ -\frac{1}{30} \text{csch}^5(6 x)-\frac{1}{9} \text{csch}^3(6 x)-\frac{1}{6} \text{csch}(6 x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.017, size = 64, normalized size = 2.2 \begin{align*} -{\frac{ \left ( \cosh \left ( 6\,x \right ) \right ) ^{4}}{6\, \left ( \sinh \left ( 6\,x \right ) \right ) ^{5}}}+{\frac{2\, \left ( \cosh \left ( 6\,x \right ) \right ) ^{2}}{15\, \left ( \sinh \left ( 6\,x \right ) \right ) ^{5}}}+{\frac{4\, \left ( \cosh \left ( 6\,x \right ) \right ) ^{2}}{45\, \left ( \sinh \left ( 6\,x \right ) \right ) ^{3}}}-{\frac{4\, \left ( \cosh \left ( 6\,x \right ) \right ) ^{2}}{45\,\sinh \left ( 6\,x \right ) }}+{\frac{4\,\sinh \left ( 6\,x \right ) }{45}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09573, size = 258, normalized size = 8.9 \begin{align*} \frac{e^{\left (-6 \, x\right )}}{3 \,{\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} - \frac{4 \, e^{\left (-18 \, x\right )}}{9 \,{\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} + \frac{58 \, e^{\left (-30 \, x\right )}}{45 \,{\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} - \frac{4 \, e^{\left (-42 \, x\right )}}{9 \,{\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} + \frac{e^{\left (-54 \, x\right )}}{3 \,{\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68757, size = 705, normalized size = 24.31 \begin{align*} -\frac{15 \, \cosh \left (6 \, x\right )^{5} + 75 \, \cosh \left (6 \, x\right ) \sinh \left (6 \, x\right )^{4} + 15 \, \sinh \left (6 \, x\right )^{5} + 5 \,{\left (30 \, \cosh \left (6 \, x\right )^{2} - 7\right )} \sinh \left (6 \, x\right )^{3} - 5 \, \cosh \left (6 \, x\right )^{3} + 15 \,{\left (10 \, \cosh \left (6 \, x\right )^{3} - \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right )^{2} + 3 \,{\left (25 \, \cosh \left (6 \, x\right )^{4} - 35 \, \cosh \left (6 \, x\right )^{2} + 26\right )} \sinh \left (6 \, x\right ) + 38 \, \cosh \left (6 \, x\right )}{45 \,{\left (\cosh \left (6 \, x\right )^{6} + 6 \, \cosh \left (6 \, x\right ) \sinh \left (6 \, x\right )^{5} + \sinh \left (6 \, x\right )^{6} + 3 \,{\left (5 \, \cosh \left (6 \, x\right )^{2} - 2\right )} \sinh \left (6 \, x\right )^{4} - 6 \, \cosh \left (6 \, x\right )^{4} + 4 \,{\left (5 \, \cosh \left (6 \, x\right )^{3} - 4 \, \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right )^{3} + 3 \,{\left (5 \, \cosh \left (6 \, x\right )^{4} - 12 \, \cosh \left (6 \, x\right )^{2} + 5\right )} \sinh \left (6 \, x\right )^{2} + 15 \, \cosh \left (6 \, x\right )^{2} + 2 \,{\left (3 \, \cosh \left (6 \, x\right )^{5} - 8 \, \cosh \left (6 \, x\right )^{3} + 5 \, \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right ) - 10\right )}} \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 \coth ^{5}{\left (6 x \right )} \operatorname{csch}{\left (6 x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14253, size = 63, normalized size = 2.17 \begin{align*} -\frac{15 \,{\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{4} + 40 \,{\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{2} + 48}{45 \,{\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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