Optimal. Leaf size=10 \[ \sin (x)-\frac {1}{2} \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {388, 206} \[ \sin (x)-\frac {1}{2} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 206
Rule 388
Rubi steps
\begin {align*} \int \cot (2 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1-2 x^2}{2-2 x^2} \, dx,x,\sin (x)\right )\\ &=\sin (x)-\operatorname {Subst}\left (\int \frac {1}{2-2 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{2} \tanh ^{-1}(\sin (x))+\sin (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 10, normalized size = 1.00 \[ \sin (x)-\frac {1}{2} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 19, normalized size = 1.90 \[ -\frac {1}{4} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (-\sin \relax (x) + 1\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 19, normalized size = 1.90 \[ -\frac {1}{4} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (-\sin \relax (x) + 1\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 12, normalized size = 1.20 \[ \sin \relax (x )-\frac {\ln \left (\sec \relax (x )+\tan \relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 37, normalized size = 3.70 \[ -\frac {1}{4} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.32, size = 10, normalized size = 1.00 \[ \sin \relax (x)-\mathrm {atanh}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.82, size = 19, normalized size = 1.90 \[ \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{4} - \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{4} + \sin {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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