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 = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 194
Rule 2606
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*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.00 \[ -\frac {1}{30} \text {csch}^5(6 x)-\frac {1}{9} \text {csch}^3(6 x)-\frac {1}{6} \text {csch}(6 x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 250, normalized size = 8.62 \[ -\frac {15 \, \cosh \left (6 \, x\right )^{5} + 75 \, \cosh \left (6 \, x\right ) \sinh \left (6 \, x\right )^{4} + 15 \, \sinh \left (6 \, x\right )^{5} + 5 \, {\left (30 \, \cosh \left (6 \, x\right )^{2} - 7\right )} \sinh \left (6 \, x\right )^{3} - 5 \, \cosh \left (6 \, x\right )^{3} + 15 \, {\left (10 \, \cosh \left (6 \, x\right )^{3} - \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right )^{2} + 3 \, {\left (25 \, \cosh \left (6 \, x\right )^{4} - 35 \, \cosh \left (6 \, x\right )^{2} + 26\right )} \sinh \left (6 \, x\right ) + 38 \, \cosh \left (6 \, x\right )}{45 \, {\left (\cosh \left (6 \, x\right )^{6} + 6 \, \cosh \left (6 \, x\right ) \sinh \left (6 \, x\right )^{5} + \sinh \left (6 \, x\right )^{6} + 3 \, {\left (5 \, \cosh \left (6 \, x\right )^{2} - 2\right )} \sinh \left (6 \, x\right )^{4} - 6 \, \cosh \left (6 \, x\right )^{4} + 4 \, {\left (5 \, \cosh \left (6 \, x\right )^{3} - 4 \, \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right )^{3} + 3 \, {\left (5 \, \cosh \left (6 \, x\right )^{4} - 12 \, \cosh \left (6 \, x\right )^{2} + 5\right )} \sinh \left (6 \, x\right )^{2} + 15 \, \cosh \left (6 \, x\right )^{2} + 2 \, {\left (3 \, \cosh \left (6 \, x\right )^{5} - 8 \, \cosh \left (6 \, x\right )^{3} + 5 \, \cosh \left (6 \, x\right )\right )} \sinh \left (6 \, x\right ) - 10\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 47, normalized size = 1.62 \[ -\frac {15 \, {\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{4} + 40 \, {\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{2} + 48}{45 \, {\left (e^{\left (6 \, x\right )} - e^{\left (-6 \, x\right )}\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 38, normalized size = 1.31 \[ -\frac {\cosh ^{4}\left (6 x \right )}{6 \sinh \left (6 x \right )^{5}}+\frac {2 \left (\cosh ^{2}\left (6 x \right )\right )}{9 \sinh \left (6 x \right )^{5}}-\frac {4}{45 \sinh \left (6 x \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 191, normalized size = 6.59 \[ \frac {e^{\left (-6 \, x\right )}}{3 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} - \frac {4 \, e^{\left (-18 \, x\right )}}{9 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} + \frac {58 \, e^{\left (-30 \, x\right )}}{45 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} - \frac {4 \, e^{\left (-42 \, x\right )}}{9 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} + \frac {e^{\left (-54 \, x\right )}}{3 \, {\left (5 \, e^{\left (-12 \, x\right )} - 10 \, e^{\left (-24 \, x\right )} + 10 \, e^{\left (-36 \, x\right )} - 5 \, e^{\left (-48 \, x\right )} + e^{\left (-60 \, x\right )} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 40, normalized size = 1.38 \[ -\frac {{\mathrm {e}}^{6\,x}\,\left (58\,{\mathrm {e}}^{24\,x}-20\,{\mathrm {e}}^{12\,x}-20\,{\mathrm {e}}^{36\,x}+15\,{\mathrm {e}}^{48\,x}+15\right )}{45\,{\left ({\mathrm {e}}^{12\,x}-1\right )}^5} \]
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 \[ \int \coth ^{5}{\left (6 x \right )} \operatorname {csch}{\left (6 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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