Integrand size = 8, antiderivative size = 34 \[ \int \csc ^3(a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]
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Time = 0.02 (sec) , antiderivative size = 34, 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 = {3853, 3855} \[ \int \csc ^3(a+b x) \, dx=-\frac {\text {arctanh}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \]
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Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (a+b x) \csc (a+b x)}{2 b}+\frac {1}{2} \int \csc (a+b x) \, dx \\ & = -\frac {\text {arctanh}(\cos (a+b x))}{2 b}-\frac {\cot (a+b x) \csc (a+b x)}{2 b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(34)=68\).
Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.21 \[ \int \csc ^3(a+b x) \, dx=-\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{8 b} \]
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Time = 0.47 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12
method | result | size |
derivativedivides | \(\frac {-\frac {\csc \left (x b +a \right ) \cot \left (x b +a \right )}{2}+\frac {\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{2}}{b}\) | \(38\) |
default | \(\frac {-\frac {\csc \left (x b +a \right ) \cot \left (x b +a \right )}{2}+\frac {\ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{2}}{b}\) | \(38\) |
parallelrisch | \(\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+4 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{8 b}\) | \(43\) |
norman | \(\frac {-\frac {1}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}{8 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{2 b}\) | \(51\) |
risch | \(\frac {{\mathrm e}^{3 i \left (x b +a \right )}+{\mathrm e}^{i \left (x b +a \right )}}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{2 b}\) | \(72\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (30) = 60\).
Time = 0.24 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.12 \[ \int \csc ^3(a+b x) \, dx=-\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - {\left (\cos \left (b x + a\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 2 \, \cos \left (b x + a\right )}{4 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]
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\[ \int \csc ^3(a+b x) \, dx=\int \csc ^{3}{\left (a + b x \right )}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.35 \[ \int \csc ^3(a+b x) \, dx=\frac {\frac {2 \, \cos \left (b x + a\right )}{\cos \left (b x + a\right )^{2} - 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{4 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (30) = 60\).
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.71 \[ \int \csc ^3(a+b x) \, dx=-\frac {\frac {{\left (\frac {2 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}}{\cos \left (b x + a\right ) - 1} + \frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 2 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{8 \, b} \]
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Time = 21.28 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \csc ^3(a+b x) \, dx=\frac {\cos \left (a+b\,x\right )}{2\,b\,\left ({\cos \left (a+b\,x\right )}^2-1\right )}-\frac {\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{2\,b} \]
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