Optimal. Leaf size=15 \[ \tanh (x)-\log (\tanh (x))-\log (\coth (x)+1) \]
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Rubi [A] time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3516, 44} \[ \tanh (x)-\log (\tanh (x))-\log (\coth (x)+1) \]
Antiderivative was successfully verified.
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Rule 44
Rule 3516
Rubi steps
\begin {align*} \int \frac {\text {sech}^2(x)}{1+\coth (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 (1+x)} \, dx,x,\coth (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {1}{x}+\frac {1}{1+x}\right ) \, dx,x,\coth (x)\right )\\ &=-\log (1+\coth (x))-\log (\tanh (x))+\tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 9, normalized size = 0.60 \[ -x+\tanh (x)+\log (\cosh (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 78, normalized size = 5.20 \[ -\frac {2 \, x \cosh \relax (x)^{2} + 4 \, x \cosh \relax (x) \sinh \relax (x) + 2 \, x \sinh \relax (x)^{2} - {\left (\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1\right )} \log \left (\frac {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, x + 2}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 27, normalized size = 1.80 \[ -2 \, x - \frac {e^{\left (2 \, x\right )} + 3}{e^{\left (2 \, x\right )} + 1} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 36, normalized size = 2.40 \[ -2 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {2 \tanh \left (\frac {x}{2}\right )}{\tanh ^{2}\left (\frac {x}{2}\right )+1}+\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 18, normalized size = 1.20 \[ \frac {2}{e^{\left (-2 \, x\right )} + 1} + \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 21, normalized size = 1.40 \[ \ln \left ({\mathrm {e}}^{2\,x}+1\right )-2\,x-\frac {2}{{\mathrm {e}}^{2\,x}+1} \]
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 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\coth {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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