Integrand size = 4, antiderivative size = 36 \[ \int \csc ^7(x) \, dx=-\frac {5}{16} \text {arctanh}(\cos (x))-\frac {5}{16} \cot (x) \csc (x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x) \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3853, 3855} \[ \int \csc ^7(x) \, dx=-\frac {5}{16} \text {arctanh}(\cos (x))-\frac {1}{6} \cot (x) \csc ^5(x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {5}{16} \cot (x) \csc (x) \]
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Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6} \cot (x) \csc ^5(x)+\frac {5}{6} \int \csc ^5(x) \, dx \\ & = -\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x)+\frac {5}{8} \int \csc ^3(x) \, dx \\ & = -\frac {5}{16} \cot (x) \csc (x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x)+\frac {5}{16} \int \csc (x) \, dx \\ & = -\frac {5}{16} \text {arctanh}(\cos (x))-\frac {5}{16} \cot (x) \csc (x)-\frac {5}{24} \cot (x) \csc ^3(x)-\frac {1}{6} \cot (x) \csc ^5(x) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(36)=72\).
Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.64 \[ \int \csc ^7(x) \, dx=-\frac {5}{64} \csc ^2\left (\frac {x}{2}\right )-\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {1}{384} \csc ^6\left (\frac {x}{2}\right )-\frac {5}{16} \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {5}{16} \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {5}{64} \sec ^2\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {1}{384} \sec ^6\left (\frac {x}{2}\right ) \]
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Time = 0.46 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89
method | result | size |
default | \(\left (-\frac {\left (\csc ^{5}\left (x \right )\right )}{6}-\frac {5 \left (\csc ^{3}\left (x \right )\right )}{24}-\frac {5 \csc \left (x \right )}{16}\right ) \cot \left (x \right )+\frac {5 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{16}\) | \(32\) |
parallelrisch | \(-\frac {75 \left (\csc ^{6}\left (x \right )\right ) \left (\left (\cos \left (2 x \right )-\frac {2 \cos \left (4 x \right )}{5}+\frac {\cos \left (6 x \right )}{15}-\frac {2}{3}\right ) \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\frac {44 \cos \left (x \right )}{25}-\frac {34 \cos \left (3 x \right )}{45}+\frac {2 \cos \left (5 x \right )}{15}\right )}{512}\) | \(51\) |
norman | \(\frac {-\frac {1}{384}-\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{128}-\frac {15 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{128}+\frac {15 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{128}+\frac {3 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{128}+\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{384}}{\tan \left (\frac {x}{2}\right )^{6}}+\frac {5 \ln \left (\tan \left (\frac {x}{2}\right )\right )}{16}\) | \(58\) |
risch | \(\frac {15 \,{\mathrm e}^{11 i x}-85 \,{\mathrm e}^{9 i x}+198 \,{\mathrm e}^{7 i x}+198 \,{\mathrm e}^{5 i x}-85 \,{\mathrm e}^{3 i x}+15 \,{\mathrm e}^{i x}}{24 \left ({\mathrm e}^{2 i x}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i x}+1\right )}{16}+\frac {5 \ln \left ({\mathrm e}^{i x}-1\right )}{16}\) | \(76\) |
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.58 \[ \int \csc ^7(x) \, dx=\frac {30 \, \cos \left (x\right )^{5} - 80 \, \cos \left (x\right )^{3} - 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (x\right )}{96 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.67 \[ \int \csc ^7(x) \, dx=- \frac {- 15 \cos ^{5}{\left (x \right )} + 40 \cos ^{3}{\left (x \right )} - 33 \cos {\left (x \right )}}{48 \cos ^{6}{\left (x \right )} - 144 \cos ^{4}{\left (x \right )} + 144 \cos ^{2}{\left (x \right )} - 48} + \frac {5 \log {\left (\cos {\left (x \right )} - 1 \right )}}{32} - \frac {5 \log {\left (\cos {\left (x \right )} + 1 \right )}}{32} \]
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none
Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.50 \[ \int \csc ^7(x) \, dx=\frac {15 \, \cos \left (x\right )^{5} - 40 \, \cos \left (x\right )^{3} + 33 \, \cos \left (x\right )}{48 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} - \frac {5}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {5}{32} \, \log \left (\cos \left (x\right ) - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (28) = 56\).
Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.11 \[ \int \csc ^7(x) \, dx=-\frac {{\left (\frac {9 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {45 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) - 1\right )}^{3}} - \frac {15 \, {\left (\cos \left (x\right ) - 1\right )}}{128 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {3 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{128 \, {\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {{\left (\cos \left (x\right ) - 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5}{32} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \]
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Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int \csc ^7(x) \, dx=\frac {\frac {5\,{\cos \left (x\right )}^5}{16}-\frac {5\,{\cos \left (x\right )}^3}{6}+\frac {11\,\cos \left (x\right )}{16}}{{\cos \left (x\right )}^6-3\,{\cos \left (x\right )}^4+3\,{\cos \left (x\right )}^2-1}-\frac {5\,\mathrm {atanh}\left (\cos \left (x\right )\right )}{16} \]
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