Integrand size = 12, antiderivative size = 14 \[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {-\csc ^2(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3738, 4207, 197} \[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {-\csc ^2(x)}} \]
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Rule 197
Rule 3738
Rule 4207
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {-\csc ^2(x)}} \, dx \\ & = \text {Subst}\left (\int \frac {1}{\left (-1-x^2\right )^{3/2}} \, dx,x,\cot (x)\right ) \\ & = -\frac {\cot (x)}{\sqrt {-\csc ^2(x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=-\frac {\cot (x)}{\sqrt {-\csc ^2(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(-\frac {\cot \left (x \right )}{\sqrt {-1-\cot \left (x \right )^{2}}}\) | \(15\) |
default | \(-\frac {\cot \left (x \right )}{\sqrt {-1-\cot \left (x \right )^{2}}}\) | \(15\) |
risch | \(-\frac {i {\mathrm e}^{2 i x}}{2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\) | \(65\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=\frac {1}{2} \, {\left (-i \, e^{\left (2 i \, x\right )} - i\right )} e^{\left (-i \, x\right )} \]
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\[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=\int \frac {1}{\sqrt {- \cot ^{2}{\left (x \right )} - 1}}\, dx \]
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none
Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=-\frac {1}{\sqrt {-\tan \left (x\right )^{2} - 1}} \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=-\frac {2 i}{{\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )} - 2 i \, \mathrm {sgn}\left (\sin \left (x\right )\right ) \]
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Time = 13.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {-1-\cot ^2(x)}} \, dx=\frac {\sin \left (2\,x\right )\,1{}\mathrm {i}}{2\,\sqrt {{\sin \left (x\right )}^2}} \]
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