Optimal. Leaf size=15 \[ \frac {1}{2} \tanh ^{-1}(\sin (x))-\frac {1}{2} \tanh ^{-1}(\cos (x)) \]
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Rubi [A] time = 0.05, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4401, 4287, 3770, 4288} \[ \frac {1}{2} \tanh ^{-1}(\sin (x))-\frac {1}{2} \tanh ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3770
Rule 4287
Rule 4288
Rule 4401
Rubi steps
\begin {align*} \int \csc (2 x) (\cos (x)+\sin (x)) \, dx &=\int (\cos (x) \csc (2 x)+\csc (2 x) \sin (x)) \, dx\\ &=\int \cos (x) \csc (2 x) \, dx+\int \csc (2 x) \sin (x) \, dx\\ &=\frac {1}{2} \int \csc (x) \, dx+\frac {1}{2} \int \sec (x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\cos (x))+\frac {1}{2} \tanh ^{-1}(\sin (x))\\ \end {align*}
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Mathematica [B] time = 0.01, size = 61, normalized size = 4.07 \[ \frac {1}{2} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {1}{2} \log \left (\cos \left (\frac {x}{2}\right )\right )-\frac {1}{2} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\frac {1}{2} \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 35, normalized size = 2.33 \[ -\frac {1}{4} \, \log \left (-\frac {1}{2} \, {\left (\cos \relax (x) + 1\right )} \sin \relax (x) + \frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{4} \, \log \left (-\frac {1}{2} \, {\left (\cos \relax (x) - 1\right )} \sin \relax (x) - \frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 29, normalized size = 1.93 \[ \frac {1}{2} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 20, normalized size = 1.33 \[ \frac {\ln \left (\sec \relax (x )+\tan \relax (x )\right )}{2}+\frac {\ln \left (\csc \relax (x )-\cot \relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 69, normalized size = 4.60 \[ -\frac {1}{4} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) + \frac {1}{4} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) - \frac {1}{4} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.11, size = 24, normalized size = 1.60 \[ \frac {\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+\mathrm {tan}\left (\frac {x}{2}\right )\right )}{2}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.86, size = 32, normalized size = 2.13 \[ - \frac {\log {\left (\sin {\relax (x )} - 1 \right )}}{4} + \frac {\log {\left (\sin {\relax (x )} + 1 \right )}}{4} + \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{4} - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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