Integrand size = 8, antiderivative size = 11 \[ \int \csc ^2(a+b x) \, dx=-\frac {\cot (a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852, 8} \[ \int \csc ^2(a+b x) \, dx=-\frac {\cot (a+b x)}{b} \]
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Rule 8
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}(\int 1 \, dx,x,\cot (a+b x))}{b} \\ & = -\frac {\cot (a+b x)}{b} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \csc ^2(a+b x) \, dx=-\frac {\cot (a+b x)}{b} \]
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Time = 0.39 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(-\frac {\cot \left (x b +a \right )}{b}\) | \(12\) |
default | \(-\frac {\cot \left (x b +a \right )}{b}\) | \(12\) |
risch | \(-\frac {2 i}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )}\) | \(20\) |
parallelrisch | \(\frac {-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{2 b}\) | \(27\) |
norman | \(\frac {-\frac {1}{2 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{2 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\) | \(35\) |
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none
Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \csc ^2(a+b x) \, dx=-\frac {\cos \left (b x + a\right )}{b \sin \left (b x + a\right )} \]
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\[ \int \csc ^2(a+b x) \, dx=\int \csc ^{2}{\left (a + b x \right )}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \csc ^2(a+b x) \, dx=-\frac {1}{b \tan \left (b x + a\right )} \]
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none
Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.18 \[ \int \csc ^2(a+b x) \, dx=-\frac {1}{b \tan \left (b x + a\right )} \]
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Time = 21.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \csc ^2(a+b x) \, dx=-\frac {\mathrm {cot}\left (a+b\,x\right )}{b} \]
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