Integrand size = 15, antiderivative size = 13 \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=-\frac {\cot (x)}{2}+\frac {\csc (x)}{2} \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {12, 2686, 8, 3852} \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=\frac {\csc (x)}{2}-\frac {\cot (x)}{2} \]
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Rule 8
Rule 12
Rule 2686
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \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} \text {Subst}(\int 1 \, dx,x,\cot (x))\right )+\frac {1}{2} \text {Subst}(\int 1 \, dx,x,\csc (x)) \\ & = -\frac {\cot (x)}{2}+\frac {\csc (x)}{2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=\frac {1}{2} \tan \left (\frac {x}{2}\right ) \]
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Time = 1.23 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\csc \left (x \right )}{2}-\frac {\cot \left (x \right )}{2}\) | \(10\) |
parts | \(\frac {\csc \left (x \right )}{2}-\frac {\cot \left (x \right )}{2}\) | \(10\) |
risch | \(\frac {i}{{\mathrm e}^{i x}+1}\) | \(13\) |
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Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=\frac {\sin \left (x\right )}{2 \, {\left (\cos \left (x\right ) + 1\right )}} \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=- \frac {\cos {\left (x \right )}}{2 \sin {\left (x \right )}} + \frac {1}{2 \sin {\left (x \right )}} \]
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Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=\frac {1}{2 \, \sin \left (x\right )} - \frac {1}{2 \, \tan \left (x\right )} \]
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Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=\frac {1}{2 \, \sin \left (x\right )} - \frac {1}{2 \, \tan \left (x\right )} \]
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Time = 25.76 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int \frac {1}{2} \left (-\cot (x) \csc (x)+\csc ^2(x)\right ) \, dx=\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2} \]
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