Optimal. Leaf size=17 \[ \log (\sinh (x))-\frac {1}{2} \log \left (1-\sinh ^2(x)\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3273, 36, 31,
29} \begin {gather*} \log (\sinh (x))-\frac {1}{2} \log \left (1-\sinh ^2(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 3273
Rubi steps
\begin {align*} \int \frac {\coth (x)}{1-\sinh ^2(x)} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(1-x) x} \, dx,x,\sinh ^2(x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sinh ^2(x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\sinh ^2(x)\right )\\ &=\log (\sinh (x))-\frac {1}{2} \log \left (1-\sinh ^2(x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.35 \begin {gather*} -2 \left (-\frac {1}{2} \log (\sinh (x))+\frac {1}{4} \log \left (1-\sinh ^2(x)\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. \(40\) vs.
\(2(15)=30\).
time = 0.61, size = 41, normalized size = 2.41
method | result | size |
risch | \(\ln \left ({\mathrm e}^{2 x}-1\right )-\frac {\ln \left ({\mathrm e}^{4 x}-6 \,{\mathrm e}^{2 x}+1\right )}{2}\) | \(24\) |
default | \(-\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )-2 \tanh \left (\frac {x}{2}\right )-1\right )}{2}-\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+2 \tanh \left (\frac {x}{2}\right )-1\right )}{2}+\ln \left (\tanh \left (\frac {x}{2}\right )\right )\) | \(41\) |
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. 45 vs.
\(2 (15) = 30\).
time = 0.26, size = 45, normalized size = 2.65 \begin {gather*} -\frac {1}{2} \, \log \left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right ) + \log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) - \frac {1}{2} \, \log \left (-2 \, e^{\left (-x\right )} + e^{\left (-2 \, 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. 47 vs.
\(2 (15) = 30\).
time = 0.48, size = 47, normalized size = 2.76 \begin {gather*} -\frac {1}{2} \, \log \left (\frac {2 \, {\left (\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3\right )}}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}\right ) + \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\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 \frac {\coth {\left (x \right )}}{\sinh ^{2}{\left (x \right )} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 25, normalized size = 1.47 \begin {gather*} -\frac {1}{2} \, \log \left ({\left | e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 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 = 27, normalized size = 1.59 \begin {gather*} \ln \left (5184\,{\mathrm {e}}^{2\,x}-5184\right )-\frac {\ln \left (9\,{\mathrm {e}}^{4\,x}-54\,{\mathrm {e}}^{2\,x}+9\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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