Integrand size = 18, antiderivative size = 21 \[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=-\cot (x)-\frac {\cot ^2(x)}{2}-\frac {\cot ^3(x)}{3} \]
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Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {14} \[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=-\frac {1}{3} \cot ^3(x)-\frac {\cot ^2(x)}{2}-\cot (x) \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1+x+x^2}{x^4} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{x^4}+\frac {1}{x^3}+\frac {1}{x^2}\right ) \, dx,x,\tan (x)\right ) \\ & = -\cot (x)-\frac {\cot ^2(x)}{2}-\frac {\cot ^3(x)}{3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=-\frac {2 \cot (x)}{3}-\frac {\csc ^2(x)}{2}-\frac {1}{3} \cot (x) \csc ^2(x) \]
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Time = 0.81 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\cot \left (x \right )-\frac {\cot \left (x \right )^{2}}{2}-\frac {\cot \left (x \right )^{3}}{3}\) | \(18\) |
default | \(-\cot \left (x \right )-\frac {\cot \left (x \right )^{2}}{2}-\frac {\cot \left (x \right )^{3}}{3}\) | \(18\) |
risch | \(\frac {2 \,{\mathrm e}^{4 i x}+4 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{2 i x}-\frac {4 i}{3}}{\left ({\mathrm e}^{2 i x}-1\right )^{3}}\) | \(37\) |
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none
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=-\frac {4 \, \cos \left (x\right )^{3} - 6 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )}{6 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
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\[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=- \int \frac {\cot {\left (x \right )}}{\cos ^{2}{\left (x \right )} - 1}\, dx - \int \frac {\csc ^{2}{\left (x \right )}}{\cos ^{2}{\left (x \right )} - 1}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=-\frac {6 \, \tan \left (x\right )^{2} + 3 \, \tan \left (x\right ) + 2}{6 \, \tan \left (x\right )^{3}} \]
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none
Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=-\frac {6 \, \tan \left (x\right )^{2} + 3 \, \tan \left (x\right ) + 2}{6 \, \tan \left (x\right )^{3}} \]
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Time = 26.58 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \frac {\cot (x)+\csc ^2(x)}{1-\cos ^2(x)} \, dx=-\frac {\mathrm {cot}\left (x\right )\,\left (2\,{\mathrm {cot}\left (x\right )}^2+3\,\mathrm {cot}\left (x\right )+6\right )}{6} \]
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