Optimal. Leaf size=7 \[ -\log (\coth (x)+1) \]
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Rubi [A] time = 0.04, antiderivative size = 7, 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 = {3487, 31} \[ -\log (\coth (x)+1) \]
Antiderivative was successfully verified.
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Rule 31
Rule 3487
Rubi steps
\begin {align*} \int \frac {\text {csch}^2(x)}{1+\coth (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\coth (x)\right )\\ &=-\log (1+\coth (x))\\ \end {align*}
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Mathematica [A] time = 0.00, size = 7, normalized size = 1.00 \[ \log (\sinh (x))-x \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 18, normalized size = 2.57 \[ -2 \, x + \log \left (\frac {2 \, \sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 12, normalized size = 1.71 \[ -2 \, x + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 8, normalized size = 1.14 \[ -\ln \left (1+\coth \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 7, normalized size = 1.00 \[ -\log \left (\coth \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 11, normalized size = 1.57 \[ \ln \left ({\mathrm {e}}^{2\,x}-1\right )-2\,x \]
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 \[ \int \frac {\operatorname {csch}^{2}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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