Integrand size = 10, antiderivative size = 132 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}+\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}} \]
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Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 2715, 8} \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {\sin ^7(x) \cos (x)}{10 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \sin ^5(x) \cos (x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin ^3(x) \cos (x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \sin (x) \cos (x)}{128 a^2 \sqrt {a \csc ^4(x)}} \]
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Rule 8
Rule 2715
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\csc ^2(x) \int \sin ^{10}(x) \, dx}{a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (9 \csc ^2(x)\right ) \int \sin ^8(x) \, dx}{10 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int \sin ^6(x) \, dx}{80 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (21 \csc ^2(x)\right ) \int \sin ^4(x) \, dx}{32 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int \sin ^2(x) \, dx}{128 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}}+\frac {\left (63 \csc ^2(x)\right ) \int 1 \, dx}{256 a^2 \sqrt {a \csc ^4(x)}} \\ & = -\frac {63 \cot (x)}{256 a^2 \sqrt {a \csc ^4(x)}}+\frac {63 x \csc ^2(x)}{256 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \csc ^4(x)}}-\frac {21 \cos (x) \sin ^3(x)}{160 a^2 \sqrt {a \csc ^4(x)}}-\frac {9 \cos (x) \sin ^5(x)}{80 a^2 \sqrt {a \csc ^4(x)}}-\frac {\cos (x) \sin ^7(x)}{10 a^2 \sqrt {a \csc ^4(x)}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.42 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\frac {\sqrt {a \csc ^4(x)} \sin ^2(x) (2520 x-2100 \sin (2 x)+600 \sin (4 x)-150 \sin (6 x)+25 \sin (8 x)-2 \sin (10 x))}{10240 a^3} \]
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Time = 0.89 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.46
method | result | size |
default | \(-\frac {\left (128 \cos \left (x \right )^{8} \cot \left (x \right )-656 \cos \left (x \right )^{6} \cot \left (x \right )+1368 \cos \left (x \right )^{4} \cot \left (x \right )-1490 \cos \left (x \right )^{2} \cot \left (x \right )+965 \cot \left (x \right )-315 \csc \left (x \right )^{2} x \right ) \sqrt {16}}{5120 \sqrt {a \csc \left (x \right )^{4}}\, a^{2}}\) | \(61\) |
risch | \(-\frac {63 \,{\mathrm e}^{2 i x} x}{256 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {i {\mathrm e}^{12 i x}}{10240 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {5 i {\mathrm e}^{10 i x}}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {105 i {\mathrm e}^{4 i x}}{1024 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {105 i}{1024 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {15 i {\mathrm e}^{-2 i x}}{512 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {15 i {\mathrm e}^{-4 i x}}{2048 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {37 i \cos \left (8 x \right )}{5120 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {19 \sin \left (8 x \right )}{2560 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}+\frac {115 i \cos \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {125 \sin \left (6 x \right )}{4096 a^{2} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}\) | \(409\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {{\left (315 \, x \cos \left (x\right )^{2} - {\left (128 \, \cos \left (x\right )^{11} - 784 \, \cos \left (x\right )^{9} + 2024 \, \cos \left (x\right )^{7} - 2858 \, \cos \left (x\right )^{5} + 2455 \, \cos \left (x\right )^{3} - 965 \, \cos \left (x\right )\right )} \sin \left (x\right ) - 315 \, x\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{1280 \, a^{3}} \]
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\[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \csc ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=-\frac {965 \, \tan \left (x\right )^{9} + 2370 \, \tan \left (x\right )^{7} + 2688 \, \tan \left (x\right )^{5} + 1470 \, \tan \left (x\right )^{3} + 315 \, \tan \left (x\right )}{1280 \, {\left (a^{\frac {5}{2}} \tan \left (x\right )^{10} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{8} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{6} + 10 \, a^{\frac {5}{2}} \tan \left (x\right )^{4} + 5 \, a^{\frac {5}{2}} \tan \left (x\right )^{2} + a^{\frac {5}{2}}\right )}} + \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \]
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Exception generated. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\left (a \csc ^4(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^4}\right )}^{5/2}} \,d x \]
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