Optimal. Leaf size=13 \[ \frac{\csc (x)}{2}-\frac{\cot (x)}{2} \]
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Rubi [A] time = 0.0135493, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {12, 2606, 8, 3767} \[ \frac{\csc (x)}{2}-\frac{\cot (x)}{2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2606
Rule 8
Rule 3767
Rubi steps
\begin{align*} \int \frac{1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx &=\frac{1}{2} \int \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx\\ &=-\left (\frac{1}{2} \int \cot (x) \csc (x) \, dx\right )+\frac{1}{2} \int \csc ^2(x) \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}(\int 1 \, dx,x,\cot (x))\right )+\frac{1}{2} \operatorname{Subst}(\int 1 \, dx,x,\csc (x))\\ &=-\frac{\cot (x)}{2}+\frac{\csc (x)}{2}\\ \end{align*}
Mathematica [A] time = 0.0041819, size = 10, normalized size = 0.77 \[ \frac{1}{2} \tan \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 10, normalized size = 0.8 \begin{align*} -{\frac{\cot \left ( x \right ) }{2}}+{\frac{\csc \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.958026, size = 18, normalized size = 1.38 \begin{align*} \frac{1}{2 \, \sin \left (x\right )} - \frac{1}{2 \, \tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20441, size = 34, normalized size = 2.62 \begin{align*} \frac{\sin \left (x\right )}{2 \,{\left (\cos \left (x\right ) + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070277, size = 14, normalized size = 1.08 \begin{align*} - \frac{\cos{\left (x \right )}}{2 \sin{\left (x \right )}} + \frac{1}{2 \sin{\left (x \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08474, size = 18, normalized size = 1.38 \begin{align*} \frac{1}{2 \, \sin \left (x\right )} - \frac{1}{2 \, \tan \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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