Integrand size = 7, antiderivative size = 26 \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=x-\frac {2 \sin (x)}{1+\cos (x)}+\frac {2 \sin ^3(x)}{3 (1+\cos (x))^3} \]
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Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4477, 2759, 8} \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=x+\frac {2 \sin ^3(x)}{3 (\cos (x)+1)^3}-\frac {2 \sin (x)}{\cos (x)+1} \]
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Rule 8
Rule 2759
Rule 4477
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin ^4(x)}{(1+\cos (x))^4} \, dx \\ & = \frac {2 \sin ^3(x)}{3 (1+\cos (x))^3}-\int \frac {\sin ^2(x)}{(1+\cos (x))^2} \, dx \\ & = -\frac {2 \sin (x)}{1+\cos (x)}+\frac {2 \sin ^3(x)}{3 (1+\cos (x))^3}+\int 1 \, dx \\ & = x-\frac {2 \sin (x)}{1+\cos (x)}+\frac {2 \sin ^3(x)}{3 (1+\cos (x))^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=x-\frac {8}{3} \tan \left (\frac {x}{2}\right )+\frac {2}{3} \sec ^2\left (\frac {x}{2}\right ) \tan \left (\frac {x}{2}\right ) \]
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Time = 1.60 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {2 \tan \left (\frac {x}{2}\right )^{3}}{3}-2 \tan \left (\frac {x}{2}\right )+2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )\) | \(23\) |
risch | \(x -\frac {8 i \left (3 \,{\mathrm e}^{2 i x}+3 \,{\mathrm e}^{i x}+2\right )}{3 \left ({\mathrm e}^{i x}+1\right )^{3}}\) | \(31\) |
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Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=\frac {3 \, x \cos \left (x\right )^{2} + 6 \, x \cos \left (x\right ) - 4 \, {\left (2 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 3 \, x}{3 \, {\left (\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}} \]
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Timed out. \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=\text {Timed out} \]
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Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=-\frac {2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, \sin \left (x\right )^{3}}{3 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} + 2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=\frac {2}{3} \, \tan \left (\frac {1}{2} \, x\right )^{3} + x - 2 \, \tan \left (\frac {1}{2} \, x\right ) \]
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Time = 28.95 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(\cot (x)+\csc (x))^4} \, dx=\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{3}-2\,\mathrm {tan}\left (\frac {x}{2}\right )+x \]
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