Integrand size = 8, antiderivative size = 55 \[ \int \csc ^5(a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3853, 3855} \[ \int \csc ^5(a+b x) \, dx=-\frac {3 \text {arctanh}(\cos (a+b x))}{8 b}-\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}-\frac {3 \cot (a+b x) \csc (a+b x)}{8 b} \]
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Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac {3}{4} \int \csc ^3(a+b x) \, dx \\ & = -\frac {3 \cot (a+b x) \csc (a+b x)}{8 b}-\frac {\cot (a+b x) \csc ^3(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 (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 = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.05 \[ \int \csc ^5(a+b x) \, dx=-\frac {3 \csc ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}-\frac {\csc ^4\left (\frac {1}{2} (a+b x)\right )}{64 b}-\frac {3 \log \left (\cos \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \log \left (\sin \left (\frac {1}{2} (a+b x)\right )\right )}{8 b}+\frac {3 \sec ^2\left (\frac {1}{2} (a+b x)\right )}{32 b}+\frac {\sec ^4\left (\frac {1}{2} (a+b x)\right )}{64 b} \]
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Time = 0.54 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91
method | result | size |
derivativedivides | \(\frac {\left (-\frac {\csc \left (x b +a \right )^{3}}{4}-\frac {3 \csc \left (x b +a \right )}{8}\right ) \cot \left (x b +a \right )+\frac {3 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{8}}{b}\) | \(50\) |
default | \(\frac {\left (-\frac {\csc \left (x b +a \right )^{3}}{4}-\frac {3 \csc \left (x b +a \right )}{8}\right ) \cot \left (x b +a \right )+\frac {3 \ln \left (\csc \left (x b +a \right )-\cot \left (x b +a \right )\right )}{8}}{b}\) | \(50\) |
parallelrisch | \(\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}-\cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+8 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-8 \cot \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+24 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{64 b}\) | \(69\) |
norman | \(\frac {-\frac {1}{64 b}-\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6}}{8 b}+\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{8}}{64 b}}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}}+\frac {3 \ln \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{8 b}\) | \(83\) |
risch | \(\frac {3 \,{\mathrm e}^{7 i \left (x b +a \right )}-11 \,{\mathrm e}^{5 i \left (x b +a \right )}-11 \,{\mathrm e}^{3 i \left (x b +a \right )}+3 \,{\mathrm e}^{i \left (x b +a \right )}}{4 b \left ({\mathrm e}^{2 i \left (x b +a \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}-1\right )}{8 b}-\frac {3 \ln \left ({\mathrm e}^{i \left (x b +a \right )}+1\right )}{8 b}\) | \(99\) |
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (49) = 98\).
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.04 \[ \int \csc ^5(a+b x) \, dx=\frac {6 \, \cos \left (b x + a\right )^{3} - 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (b x + a\right ) + \frac {1}{2}\right ) - 10 \, \cos \left (b x + a\right )}{16 \, {\left (b \cos \left (b x + a\right )^{4} - 2 \, b \cos \left (b x + a\right )^{2} + b\right )}} \]
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\[ \int \csc ^5(a+b x) \, dx=\int \csc ^{5}{\left (a + b x \right )}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.29 \[ \int \csc ^5(a+b x) \, dx=\frac {\frac {2 \, {\left (3 \, \cos \left (b x + a\right )^{3} - 5 \, \cos \left (b x + a\right )\right )}}{\cos \left (b x + a\right )^{4} - 2 \, \cos \left (b x + a\right )^{2} + 1} - 3 \, \log \left (\cos \left (b x + a\right ) + 1\right ) + 3 \, \log \left (\cos \left (b x + a\right ) - 1\right )}{16 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (49) = 98\).
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.51 \[ \int \csc ^5(a+b x) \, dx=\frac {\frac {{\left (\frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {18 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - 1\right )} {\left (\cos \left (b x + a\right ) + 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) - 1\right )}^{2}} - \frac {8 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + 12 \, \log \left (\frac {{\left | -\cos \left (b x + a\right ) + 1 \right |}}{{\left | \cos \left (b x + a\right ) + 1 \right |}}\right )}{64 \, b} \]
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Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07 \[ \int \csc ^5(a+b x) \, dx=-\frac {3\,\mathrm {atanh}\left (\cos \left (a+b\,x\right )\right )}{8\,b}-\frac {\frac {5\,\cos \left (a+b\,x\right )}{8}-\frac {3\,{\cos \left (a+b\,x\right )}^3}{8}}{b\,\left ({\cos \left (a+b\,x\right )}^4-2\,{\cos \left (a+b\,x\right )}^2+1\right )} \]
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