Optimal. Leaf size=8 \[ \tanh ^{-1}(\cosh (x))-\text {csch}(x) \]
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Rubi [A]
time = 0.03, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3582, 3855}
\begin {gather*} \tanh ^{-1}(\cosh (x))-\text {csch}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 3582
Rule 3855
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(x)}{1+\coth (x)} \, dx &=-\text {csch}(x)-\int \text {csch}(x) \, dx\\ &=\tanh ^{-1}(\cosh (x))-\text {csch}(x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 14, normalized size = 1.75 \begin {gather*} -\text {csch}(x)-\log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(22\) vs.
\(2(8)=16\).
time = 0.56, size = 23, normalized size = 2.88
method | result | size |
default | \(\frac {\tanh \left (\frac {x}{2}\right )}{2}-\frac {1}{2 \tanh \left (\frac {x}{2}\right )}-\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(23\) |
risch | \(-\frac {2 \,{\mathrm e}^{x}}{{\mathrm e}^{2 x}-1}-\ln \left ({\mathrm e}^{x}-1\right )+\ln \left ({\mathrm e}^{x}+1\right )\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (8) = 16\).
time = 0.27, size = 31, normalized size = 3.88 \begin {gather*} \frac {2 \, e^{\left (-x\right )}}{e^{\left (-2 \, x\right )} - 1} + \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs.
\(2 (8) = 16\).
time = 0.39, size = 77, normalized size = 9.62 \begin {gather*} \frac {{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) - 2 \, \cosh \left (x\right ) - 2 \, \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {csch}^{3}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (8) = 16\).
time = 0.41, size = 26, normalized size = 3.25 \begin {gather*} -\frac {2 \, e^{x}}{e^{\left (2 \, x\right )} - 1} + \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 29, normalized size = 3.62 \begin {gather*} \ln \left (2\,{\mathrm {e}}^x+2\right )-\ln \left (2\,{\mathrm {e}}^x-2\right )-\frac {2\,{\mathrm {e}}^x}{{\mathrm {e}}^{2\,x}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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