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 = 0.05349, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 4401
Rule 4287
Rule 3770
Rule 4288
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*}
Mathematica [B] time = 0.0093158, 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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Maple [A] time = 0.049, size = 20, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( \sec \left ( x \right ) +\tan \left ( x \right ) \right ) }{2}}+{\frac{\ln \left ( \csc \left ( x \right ) -\cot \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.43778, size = 93, normalized size = 6.2 \begin{align*} -\frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \frac{1}{4} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09555, size = 149, normalized size = 9.93 \begin{align*} -\frac{1}{4} \, \log \left (-\frac{1}{2} \,{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + \frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{4} \, \log \left (-\frac{1}{2} \,{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right ) - \frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.05127, size = 32, normalized size = 2.13 \begin{align*} - \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{4} + \frac{\log{\left (\sin{\left (x \right )} + 1 \right )}}{4} + \frac{\log{\left (\cos{\left (x \right )} - 1 \right )}}{4} - \frac{\log{\left (\cos{\left (x \right )} + 1 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10121, size = 39, normalized size = 2.6 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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