Integrand size = 15, antiderivative size = 21 \[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=x+2 \arctan \left (\frac {\cos (x)-\sin (x)}{2+\cos (x)+\sin (x)}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3245, 3203, 632, 210} \[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=2 \arctan \left (\frac {\cos (x)-\sin (x)}{\sin (x)+\cos (x)+2}\right )+x \]
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Rule 210
Rule 632
Rule 3203
Rule 3245
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{3+2 \cos (x)+2 \sin (x)} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{5+4 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right ) \\ & = -\left (4 \text {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,4+2 \tan \left (\frac {x}{2}\right )\right )\right ) \\ & = x+2 \arctan \left (\frac {\cos (x)-\sin (x)}{2+\cos (x)+\sin (x)}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(21)=42\).
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=-\arctan \left (\frac {\cos \left (\frac {x}{2}\right )}{2 \cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )}\right )+\arctan \left (\sec \left (\frac {x}{2}\right ) \left (2 \cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )\right ) \]
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Time = 0.44 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.48
method | result | size |
default | \(2 \arctan \left (2+\tan \left (\frac {x}{2}\right )\right )\) | \(10\) |
risch | \(-i \ln \left ({\mathrm e}^{i x}+\frac {1}{2}+\frac {i}{2}\right )+i \ln \left ({\mathrm e}^{i x}+1+i\right )\) | \(28\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=-\arctan \left (-\frac {3 \, \cos \left (x\right ) + 3 \, \sin \left (x\right ) + 4}{\cos \left (x\right ) - \sin \left (x\right )}\right ) \]
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\[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=\int \frac {\csc {\left (x \right )}}{2 \cot {\left (x \right )} + 3 \csc {\left (x \right )} + 2}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1} + 2\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=2 \, \pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor + 2 \, \arctan \left (\tan \left (\frac {1}{2} \, x\right ) + 2\right ) \]
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Time = 27.33 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.43 \[ \int \frac {\csc (x)}{2+2 \cot (x)+3 \csc (x)} \, dx=2\,\mathrm {atan}\left (\mathrm {tan}\left (\frac {x}{2}\right )+2\right ) \]
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