Optimal. Leaf size=36 \[ \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x)) \]
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Rubi [A] time = 0.12, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2606, 30, 5208, 12, 453, 206} \[ \frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \text {csch}^3(x) \cot ^{-1}(\cosh (x)) \]
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 206
Rule 453
Rule 2606
Rule 5208
Rubi steps
\begin {align*} \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\int \frac {2 \text {csch}^2(x)}{3 (-3-\cosh (2 x))} \, dx\\ &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\frac {2}{3} \int \frac {\text {csch}^2(x)}{-3-\cosh (2 x)} \, dx\\ &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{2 x^2 \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{x^2 \left (2-x^2\right )} \, dx,x,\tanh (x)\right )\\ &=\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\tanh (x)\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{6 \sqrt {2}}+\frac {\coth (x)}{6}-\frac {1}{3} \cot ^{-1}(\cosh (x)) \text {csch}^3(x)\\ \end {align*}
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Mathematica [A] time = 0.23, size = 40, normalized size = 1.11 \[ \frac {1}{24} \left (2 \sqrt {2} \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {2}}\right )+\text {csch}^3(x) \left (-\cosh (x)+\cosh (3 x)-8 \cot ^{-1}(\cosh (x))\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^{-1}(\cosh (x)) \coth (x) \text {csch}^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.70, size = 423, normalized size = 11.75 \[ \frac {8 \, \cosh \relax (x)^{4} + 32 \, \cosh \relax (x) \sinh \relax (x)^{3} + 8 \, \sinh \relax (x)^{4} + 16 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 64 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3}\right )} \arctan \left (\frac {2 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}\right ) - 16 \, \cosh \relax (x)^{2} + {\left (\sqrt {2} \cosh \relax (x)^{6} + 6 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{5} + \sqrt {2} \sinh \relax (x)^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{4} - 3 \, \sqrt {2} \cosh \relax (x)^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{3} - 3 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{4} - 6 \, \sqrt {2} \cosh \relax (x)^{2} + \sqrt {2}\right )} \sinh \relax (x)^{2} + 3 \, \sqrt {2} \cosh \relax (x)^{2} + 6 \, {\left (\sqrt {2} \cosh \relax (x)^{5} - 2 \, \sqrt {2} \cosh \relax (x)^{3} + \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) - \sqrt {2}\right )} \log \left (-\frac {3 \, {\left (2 \, \sqrt {2} - 3\right )} \cosh \relax (x)^{2} - 4 \, {\left (3 \, \sqrt {2} - 4\right )} \cosh \relax (x) \sinh \relax (x) + 3 \, {\left (2 \, \sqrt {2} - 3\right )} \sinh \relax (x)^{2} + 2 \, \sqrt {2} - 3}{\cosh \relax (x)^{2} + \sinh \relax (x)^{2} + 3}\right ) + 32 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 8}{24 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} - 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 70, normalized size = 1.94 \[ \frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + \frac {1}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} + \frac {8 \, \arctan \left (\frac {2}{e^{\left (-x\right )} + e^{x}}\right )}{3 \, {\left (e^{\left (-x\right )} - e^{x}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.64, size = 854, normalized size = 23.72
method | result | size |
risch | \(\frac {4 i {\mathrm e}^{3 x} \ln \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )}{3 \left (-1+{\mathrm e}^{2 x}\right )^{3}}-\frac {-8+16 \,{\mathrm e}^{2 x}-8 \,{\mathrm e}^{4 x}+16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right ) {\mathrm e}^{3 x}-16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right ) {\mathrm e}^{3 x}-\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (1+\sqrt {2}\right )^{2}\right )+\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (\sqrt {2}-1\right )^{2}\right )-3 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (1+\sqrt {2}\right )^{2}\right ) {\mathrm e}^{4 x}-3 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (\sqrt {2}-1\right )^{2}\right ) {\mathrm e}^{2 x}+3 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (1+\sqrt {2}\right )^{2}\right ) {\mathrm e}^{2 x}-\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (\sqrt {2}-1\right )^{2}\right ) {\mathrm e}^{6 x}+\sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (1+\sqrt {2}\right )^{2}\right ) {\mathrm e}^{6 x}+3 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+\left (\sqrt {2}-1\right )^{2}\right ) {\mathrm e}^{4 x}+16 \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}+16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left ({\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}-16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left ({\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right ) {\mathrm e}^{3 x}+16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left ({\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right ) {\mathrm e}^{3 x}+16 \pi \,\mathrm {csgn}\left (i \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}+16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right ) \mathrm {csgn}\left ({\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}-16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}+16 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}-16 \pi \mathrm {csgn}\left ({\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right )^{3} {\mathrm e}^{3 x}+16 \pi \mathrm {csgn}\left ({\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right )^{3} {\mathrm e}^{3 x}+16 \pi \mathrm {csgn}\left ({\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}-16 \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right )^{3} {\mathrm e}^{3 x}-16 \pi \mathrm {csgn}\left (i {\mathrm e}^{-x} \left (-{\mathrm e}^{2 x}-1+2 i {\mathrm e}^{x}\right )\right )^{3} {\mathrm e}^{3 x}+32 i {\mathrm e}^{3 x} \ln \left ({\mathrm e}^{2 x}+1-2 i {\mathrm e}^{x}\right )+16 \pi \mathrm {csgn}\left ({\mathrm e}^{-x} \left ({\mathrm e}^{2 x}+1+2 i {\mathrm e}^{x}\right )\right )^{2} {\mathrm e}^{3 x}}{24 \left (-1+{\mathrm e}^{2 x}\right )^{3}}\) | \(854\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 54, normalized size = 1.50 \[ -\frac {1}{24} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) - \frac {1}{3 \, {\left (e^{\left (-2 \, x\right )} - 1\right )}} - \frac {\operatorname {arccot}\left (\cosh \relax (x)\right )}{3 \, \sinh \relax (x)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 103, normalized size = 2.86 \[ \frac {\sqrt {2}\,\ln \left (-\frac {2\,{\mathrm {e}}^{2\,x}}{3}-\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}\right )}{24}-\frac {\sqrt {2}\,\ln \left (\frac {\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{24}-\frac {2\,{\mathrm {e}}^{2\,x}}{3}\right )}{24}+\frac {1}{3\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{3\,x}\,\mathrm {acot}\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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