Integrand size = 17, antiderivative size = 55 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {\text {arctanh}(\cos (a+b x))}{8 b}+\frac {\cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b} \]
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Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2691, 3853, 3855} \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {\text {arctanh}(\cos (a+b x))}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac {\cot (a+b x) \csc (a+b x)}{8 b} \]
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Rule 2691
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {1}{4} \int \csc ^3(a+b x) \, dx \\ & = \frac {\cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {1}{8} \int \csc (a+b x) \, dx \\ & = \frac {\text {arctanh}(\cos (a+b x))}{8 b}+\frac {\cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(55)=110\).
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.05 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {\csc ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}+\frac {\log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {\log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {\sec ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \]
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Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93
method | result | size |
norman | \(\frac {-\frac {1}{64 b}+\frac {\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}-\frac {\ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}\) | \(51\) |
parallelrisch | \(\frac {\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )-8 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}{64 b \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}\) | \(53\) |
derivativedivides | \(\frac {-\frac {\cos ^{3}\left (b x +a \right )}{4 \sin \left (b x +a \right )^{4}}-\frac {\cos ^{3}\left (b x +a \right )}{8 \sin \left (b x +a \right )^{2}}-\frac {\cos \left (b x +a \right )}{8}-\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(68\) |
default | \(\frac {-\frac {\cos ^{3}\left (b x +a \right )}{4 \sin \left (b x +a \right )^{4}}-\frac {\cos ^{3}\left (b x +a \right )}{8 \sin \left (b x +a \right )^{2}}-\frac {\cos \left (b x +a \right )}{8}-\frac {\ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(68\) |
risch | \(-\frac {{\mathrm e}^{7 i \left (b x +a \right )}+7 \,{\mathrm e}^{5 i \left (b x +a \right )}+7 \,{\mathrm e}^{3 i \left (b x +a \right )}+{\mathrm e}^{i \left (b x +a \right )}}{4 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{8 b}-\frac {\ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{8 b}\) | \(95\) |
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Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (49) = 98\).
Time = 0.51 (sec) , antiderivative size = 111, normalized size of antiderivative = 2.02 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=-\frac {2 \, \cos \left (b x + a\right )^{3} - {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 2 \, \cos \left (b x + a\right )}{16 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]
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Time = 0.93 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\begin {cases} - \frac {\log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{8 b} + \frac {\tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{64 b} - \frac {1}{64 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{2}{\left (a \right )}}{\sin ^{5}{\left (a \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=-\frac {\frac {2 \, {\left (\cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} - \log \left (\cos \left (b x + a\right ) + 1\right ) + \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \]
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Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.78 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {\frac {{\left (\frac {2 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 4 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \]
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Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \cot ^2(a+b x) \csc ^3(a+b x) \, dx=\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}{64\,b}-\frac {1}{64\,b\,{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{8\,b} \]
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