Integrand size = 10, antiderivative size = 62 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {2}{3} a \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-\frac {1}{5} a \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-a \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \]
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Time = 0.02 (sec) , antiderivative size = 62, 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} \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {1}{5} a \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {2}{3} a \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-a \sin (x) \cos (x) \sqrt {a \csc ^4(x)} \]
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Rule 3852
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \left (a \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^6(x) \, dx \\ & = -\left (\left (a \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,\cot (x)\right )\right ) \\ & = -\frac {2}{3} a \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-\frac {1}{5} a \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-a \cos (x) \sqrt {a \csc ^4(x)} \sin (x) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.53 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {1}{15} a \cos (x) \sqrt {a \csc ^4(x)} \left (8+4 \csc ^2(x)+3 \csc ^4(x)\right ) \sin (x) \]
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Time = 0.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.56
method | result | size |
default | \(-\frac {\csc \left (x \right )^{2} \cot \left (x \right ) a \sqrt {a \csc \left (x \right )^{4}}\, \left (8 \cos \left (x \right )^{4}-20 \cos \left (x \right )^{2}+15\right ) \sqrt {16}}{60}\) | \(35\) |
risch | \(\frac {16 i a \sqrt {\frac {a \,{\mathrm e}^{4 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}\, \left (-5+11 \cos \left (2 x \right )+9 i \sin \left (2 x \right )\right )}{15 \left ({\mathrm e}^{2 i x}-1\right )^{3}}\) | \(47\) |
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Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.84 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=\frac {{\left (8 \, a \cos \left (x\right )^{5} - 20 \, a \cos \left (x\right )^{3} + 15 \, a \cos \left (x\right )\right )} \sqrt {\frac {a}{\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1}}}{15 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \]
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\[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=\int \left (a \csc ^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.48 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {15 \, a^{\frac {3}{2}} \tan \left (x\right )^{4} + 10 \, a^{\frac {3}{2}} \tan \left (x\right )^{2} + 3 \, a^{\frac {3}{2}}}{15 \, \tan \left (x\right )^{5}} \]
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Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.37 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=-\frac {{\left (15 \, \tan \left (x\right )^{4} + 10 \, \tan \left (x\right )^{2} + 3\right )} a^{\frac {3}{2}}}{15 \, \tan \left (x\right )^{5}} \]
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Time = 17.62 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int \left (a \csc ^4(x)\right )^{3/2} \, dx=\frac {\frac {a^{3/2}\,8{}\mathrm {i}}{15}-\frac {4\,a^{3/2}\,\left (2\,{\sin \left (2\,x\right )}^3-9\,\sin \left (2\,x\right )+3\,\sin \left (4\,x\right )+2{}\mathrm {i}\right )}{15}}{{\left (\cos \left (2\,x\right )-1\right )}^3} \]
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