Integrand size = 10, antiderivative size = 118 \[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=-\frac {4}{3} a^2 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-\frac {6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-a^2 \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]
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Time = 0.03 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4208, 3852} \[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=-\frac {1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {4}{3} a^2 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-a^2 \sin (x) \cos (x) \sqrt {a \csc ^4(x)} \]
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Rule 3852
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \left (a^2 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^{10}(x) \, dx \\ & = -\left (\left (a^2 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \text {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {4}{3} a^2 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-\frac {6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-a^2 \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=-\frac {1}{315} a^2 \cos (x) \sqrt {a \csc ^4(x)} \left (128+64 \csc ^2(x)+48 \csc ^4(x)+40 \csc ^6(x)+35 \csc ^8(x)\right ) \sin (x) \]
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Time = 0.80 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.42
method | result | size |
default | \(-\frac {\cot \left (x \right ) \csc \left (x \right )^{6} \sqrt {a \csc \left (x \right )^{4}}\, a^{2} \left (128 \cos \left (x \right )^{8}-576 \cos \left (x \right )^{6}+1008 \cos \left (x \right )^{4}-840 \cos \left (x \right )^{2}+315\right ) \sqrt {16}}{1260}\) | \(49\) |
risch | \(\frac {256 i a^{2} \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (126 \,{\mathrm e}^{6 i x}-84 \,{\mathrm e}^{4 i x}-9+37 \cos \left (2 x \right )+35 i \sin \left (2 x \right )\right )}{315 \left ({\mathrm e}^{2 i x}-1\right )^{7}}\) | \(63\) |
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Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75 \[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=\frac {{\left (128 \, a^{2} \cos \left (x\right )^{9} - 576 \, a^{2} \cos \left (x\right )^{7} + 1008 \, a^{2} \cos \left (x\right )^{5} - 840 \, a^{2} \cos \left (x\right )^{3} + 315 \, a^{2} \cos \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{315 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
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\[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=\int \left (a \csc ^{4}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.41 \[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=-\frac {315 \, a^{\frac {5}{2}} \tan \left (x\right )^{8} + 420 \, a^{\frac {5}{2}} \tan \left (x\right )^{6} + 378 \, a^{\frac {5}{2}} \tan \left (x\right )^{4} + 180 \, a^{\frac {5}{2}} \tan \left (x\right )^{2} + 35 \, a^{\frac {5}{2}}}{315 \, \tan \left (x\right )^{9}} \]
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.43 \[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=-\frac {{\left (315 \, a^{2} \tan \left (x\right )^{8} + 420 \, a^{2} \tan \left (x\right )^{6} + 378 \, a^{2} \tan \left (x\right )^{4} + 180 \, a^{2} \tan \left (x\right )^{2} + 35 \, a^{2}\right )} \sqrt {a}}{315 \, \tan \left (x\right )^{9}} \]
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Time = 19.51 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \left (a \csc ^4(x)\right )^{5/2} \, dx=\frac {128\,a^{5/2}\,\left ({\mathrm {e}}^{x\,46{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{x\,48{}\mathrm {i}}\,9{}\mathrm {i}+{\mathrm {e}}^{x\,50{}\mathrm {i}}\,36{}\mathrm {i}-{\mathrm {e}}^{x\,52{}\mathrm {i}}\,84{}\mathrm {i}+{\mathrm {e}}^{x\,54{}\mathrm {i}}\,126{}\mathrm {i}\right )}{315\,\left (\frac {{\mathrm {e}}^{-x\,2{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}}{2}-1\right )\,\left ({\mathrm {e}}^{x\,48{}\mathrm {i}}-7\,{\mathrm {e}}^{x\,50{}\mathrm {i}}+21\,{\mathrm {e}}^{x\,52{}\mathrm {i}}-35\,{\mathrm {e}}^{x\,54{}\mathrm {i}}+35\,{\mathrm {e}}^{x\,56{}\mathrm {i}}-21\,{\mathrm {e}}^{x\,58{}\mathrm {i}}+7\,{\mathrm {e}}^{x\,60{}\mathrm {i}}-{\mathrm {e}}^{x\,62{}\mathrm {i}}\right )} \]
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