Integrand size = 6, antiderivative size = 12 \[ \int \csc (a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{b} \]
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Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3855} \[ \int \csc (a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{b} \]
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Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arctanh}(\cos (a+b x))}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(12)=24\).
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 3.17 \[ \int \csc (a+b x) \, dx=-\frac {\log \left (\cos \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b}+\frac {\log \left (\sin \left (\frac {a}{2}+\frac {b x}{2}\right )\right )}{b} \]
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Time = 0.14 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25
method | result | size |
norman | \(\frac {\ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{b}\) | \(15\) |
parallelrisch | \(\frac {\ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{b}\) | \(15\) |
derivativedivides | \(-\frac {\ln \left (\csc \left (x b +a \right )+\cot \left (x b +a \right )\right )}{b}\) | \(20\) |
default | \(-\frac {\ln \left (\csc \left (x b +a \right )+\cot \left (x b +a \right )\right )}{b}\) | \(20\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{b}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{b}\) | \(35\) |
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Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).
Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.50 \[ \int \csc (a+b x) \, dx=-\frac {\log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right )}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (10) = 20\).
Time = 0.85 (sec) , antiderivative size = 37, normalized size of antiderivative = 3.08 \[ \int \csc (a+b x) \, dx=\begin {cases} - \frac {\log {\left (\cot {\left (a + b x \right )} + \csc {\left (a + b x \right )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x \left (\cot {\left (a \right )} \csc {\left (a \right )} + \csc ^{2}{\left (a \right )}\right )}{\cot {\left (a \right )} + \csc {\left (a \right )}} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.58 \[ \int \csc (a+b x) \, dx=-\frac {\log \left (\cot \left (b x + a\right ) + \csc \left (b x + a\right )\right )}{b} \]
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none
Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.25 \[ \int \csc (a+b x) \, dx=\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right ) \right |}\right )}{b} \]
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Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \csc (a+b x) \, dx=-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{b} \]
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