Integrand size = 15, antiderivative size = 55 \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}+\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot ^3(a+b x) \csc (a+b x)}{4 b} \]
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Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2691, 3855} \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}-\frac {\cot ^3(a+b x) \csc (a+b x)}{4 b}+\frac {3 \cot (a+b x) \csc (a+b x)}{8 b} \]
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Rule 2691
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^3(a+b x) \csc (a+b x)}{4 b}-\frac {3}{4} \int \cot ^2(a+b x) \csc (a+b x) \, dx \\ & = \frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot ^3(a+b x) \csc (a+b x)}{4 b}+\frac {3}{8} \int \csc (a+b x) \, dx \\ & = -\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}+\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot ^3(a+b x) \csc (a+b x)}{4 b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(55)=110\).
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.05 \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=\frac {5 \csc ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {3 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}-\frac {5 \sec ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \]
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Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.25
method | result | size |
parallelrisch | \(\frac {\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )-\left (\cot ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-8 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+8 \left (\cot ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{64 b}\) | \(69\) |
derivativedivides | \(\frac {-\frac {\cos ^{5}\left (b x +a \right )}{4 \sin \left (b x +a \right )^{4}}+\frac {\cos ^{5}\left (b x +a \right )}{8 \sin \left (b x +a \right )^{2}}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{8}+\frac {3 \cos \left (b x +a \right )}{8}+\frac {3 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(78\) |
default | \(\frac {-\frac {\cos ^{5}\left (b x +a \right )}{4 \sin \left (b x +a \right )^{4}}+\frac {\cos ^{5}\left (b x +a \right )}{8 \sin \left (b x +a \right )^{2}}+\frac {\left (\cos ^{3}\left (b x +a \right )\right )}{8}+\frac {3 \cos \left (b x +a \right )}{8}+\frac {3 \ln \left (\csc \left (b x +a \right )-\cot \left (b x +a \right )\right )}{8}}{b}\) | \(78\) |
norman | \(\frac {-\frac {1}{64 b}+\frac {\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}-\frac {\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )}{8 b}+\frac {\tan ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )}{64 b}}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}}+\frac {3 \ln \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{8 b}\) | \(83\) |
risch | \(-\frac {5 \,{\mathrm e}^{7 i \left (b x +a \right )}+3 \,{\mathrm e}^{5 i \left (b x +a \right )}+3 \,{\mathrm e}^{3 i \left (b x +a \right )}+5 \,{\mathrm e}^{i \left (b x +a \right )}}{4 b \left ({\mathrm e}^{2 i \left (b x +a \right )}-1\right )^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}+1\right )}{8 b}+\frac {3 \ln \left ({\mathrm e}^{i \left (b x +a \right )}-1\right )}{8 b}\) | \(99\) |
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (49) = 98\).
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.04 \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=-\frac {10 \, \cos \left (b x + a\right )^{3} + 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (b x + a\right )}{16 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]
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Time = 0.99 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.67 \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=\begin {cases} \frac {3 \log {\left (\tan {\left (\frac {a}{2} + \frac {b x}{2} \right )} \right )}}{8 b} + \frac {\tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{64 b} - \frac {\tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}}{8 b} + \frac {1}{8 b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} - \frac {1}{64 b \tan ^{4}{\left (\frac {a}{2} + \frac {b x}{2} \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{4}{\left (a \right )}}{\sin ^{5}{\left (a \right )}} & \text {otherwise} \end {cases} \]
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none
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.29 \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=-\frac {\frac {2 \, {\left (5 \, \cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} + 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) - 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (49) = 98\).
Time = 0.33 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.53 \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=-\frac {\frac {{\left (\frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {18 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 12 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \]
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Time = 0.24 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.42 \[ \int \cot ^4(a+b x) \csc (a+b x) \, dx=\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4}{64\,b}-\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2}{8\,b}+\frac {3\,\ln \left (\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )\right )}{8\,b}+\frac {{\mathrm {cot}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^4\,\left (\frac {{\mathrm {tan}\left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2}{8}-\frac {1}{64}\right )}{b} \]
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