Optimal. Leaf size=7 \[ -\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]
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Rubi [A]
time = 0.01, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4372, 3855}
\begin {gather*} -\frac {1}{2} \tanh ^{-1}(\cosh (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 4372
Rubi steps
\begin {align*} \int \cosh (x) \text {csch}(2 x) \, dx &=\frac {1}{2} \int \text {csch}(x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.57 \begin {gather*} \frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.77, size = 6, normalized size = 0.86
method | result | size |
default | \(-\arctanh \left ({\mathrm e}^{x}\right )\) | \(6\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{x}+1\right )}{2}+\frac {\ln \left ({\mathrm e}^{x}-1\right )}{2}\) | \(16\) |
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. 19 vs.
\(2 (5) = 10\).
time = 0.26, size = 19, normalized size = 2.71 \begin {gather*} -\frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \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. 19 vs.
\(2 (5) = 10\).
time = 0.38, size = 19, normalized size = 2.71 \begin {gather*} -\frac {1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac {1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \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 \cosh {\left (x \right )} \operatorname {csch}{\left (2 x \right )}\, 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. 16 vs.
\(2 (5) = 10\).
time = 0.39, size = 16, normalized size = 2.29 \begin {gather*} -\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \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 = 0.06, size = 19, normalized size = 2.71 \begin {gather*} \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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