Optimal. Leaf size=16 \[ \frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3768, 3770} \[ \frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x) \]
Antiderivative was successfully verified.
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Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \text {csch}^3(x) \, dx &=-\frac {1}{2} \coth (x) \text {csch}(x)-\frac {1}{2} \int \text {csch}(x) \, dx\\ &=\frac {1}{2} \tanh ^{-1}(\cosh (x))-\frac {1}{2} \coth (x) \text {csch}(x)\\ \end {align*}
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Mathematica [B] time = 0.00, size = 36, normalized size = 2.25 \[ -\frac {1}{8} \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{8} \text {sech}^2\left (\frac {x}{2}\right )-\frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \text {csch}^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.26, size = 211, normalized size = 13.19 \[ -\frac {2 \, \cosh \relax (x)^{3} + 6 \, \cosh \relax (x) \sinh \relax (x)^{2} + 2 \, \sinh \relax (x)^{3} - {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + 2 \, \cosh \relax (x)}{2 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 45, normalized size = 2.81 \[ -\frac {e^{\left (-x\right )} + e^{x}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} + \frac {1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) - \frac {1}{4} \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 11, normalized size = 0.69
method | result | size |
default | \(-\frac {\coth \relax (x ) \mathrm {csch}\relax (x )}{2}+\arctanh \left ({\mathrm e}^{x}\right )\) | \(11\) |
risch | \(-\frac {{\mathrm e}^{x} \left (1+{\mathrm e}^{2 x}\right )}{\left (-1+{\mathrm e}^{2 x}\right )^{2}}+\frac {\ln \left (1+{\mathrm e}^{x}\right )}{2}-\frac {\ln \left (-1+{\mathrm e}^{x}\right )}{2}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 45, normalized size = 2.81 \[ \frac {e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 16, normalized size = 1.00 \[ -\frac {\ln \left (\mathrm {tanh}\left (\frac {x}{2}\right )\right )}{2}-\frac {\mathrm {cosh}\relax (x)}{2\,{\mathrm {sinh}\relax (x)}^2} \]
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 \[ \int \operatorname {csch}^{3}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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