Optimal. Leaf size=7 \[ -\frac{1}{2} \tanh ^{-1}(\cos (x)) \]
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Rubi [A] time = 0.0112712, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4287, 3770} \[ -\frac{1}{2} \tanh ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
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Rule 4287
Rule 3770
Rubi steps
\begin{align*} \int \cos (x) \csc (2 x) \, dx &=\frac{1}{2} \int \csc (x) \, dx\\ &=-\frac{1}{2} \tanh ^{-1}(\cos (x))\\ \end{align*}
Mathematica [B] time = 0.0031234, size = 21, normalized size = 3. \[ \frac{1}{2} \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 11, normalized size = 1.6 \begin{align*}{\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 = 0.988998, size = 47, normalized size = 6.71 \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.48182, size = 77, normalized size = 11. \begin{align*} -\frac{1}{4} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{4} \, \log \left (-\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 = 7.94328, size = 15, normalized size = 2.14 \begin{align*} \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 [A] time = 1.11464, size = 11, normalized size = 1.57 \begin{align*} \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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