Integrand size = 10, antiderivative size = 71 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sqrt {a \csc ^3(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sin ^{\frac {3}{2}}(x) \]
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Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2720} \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sin ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sqrt {a \csc ^3(x)} \]
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Rule 2720
Rule 3853
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{9/2} \, dx}{(-\csc (x))^{3/2}} \\ & = -\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {\left (5 a \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{5/2} \, dx}{7 (-\csc (x))^{3/2}} \\ & = -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {\left (5 a \sqrt {a \csc ^3(x)}\right ) \int \sqrt {-\csc (x)} \, dx}{21 (-\csc (x))^{3/2}} \\ & = -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}+\frac {1}{21} \left (5 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sqrt {\sin (x)}} \, dx \\ & = -\frac {10}{21} a \cos (x) \sqrt {a \csc ^3(x)}-\frac {2}{7} a \cot (x) \csc (x) \sqrt {a \csc ^3(x)}-\frac {10}{21} a \sqrt {a \csc ^3(x)} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right ) \sin ^{\frac {3}{2}}(x) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.65 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {1}{84} \left (a \csc ^3(x)\right )^{3/2} \left (40 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right ) \sin ^{\frac {9}{2}}(x)+22 \sin (2 x)-5 \sin (4 x)\right ) \]
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Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.37
method | result | size |
default | \(\frac {\left (\frac {a \sin \left (x \right )^{3} \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right )^{3}}{\left (1-\cos \left (x \right )\right )^{3}}\right )^{\frac {3}{2}} \left (1-\cos \left (x \right )\right )^{2} \left (40 i \csc \left (x \right )^{5} \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (1-\cos \left (x \right )\right )^{3}+3 \csc \left (x \right )^{10} \left (1-\cos \left (x \right )\right )^{8}+26 \csc \left (x \right )^{8} \left (1-\cos \left (x \right )\right )^{6}-26 \csc \left (x \right )^{4} \left (1-\cos \left (x \right )\right )^{2}-3 \csc \left (x \right )^{2}\right ) \sqrt {8}}{336 \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right )^{4} \sqrt {\csc \left (x \right ) \left (\csc \left (x \right )^{2} \left (1-\cos \left (x \right )\right )^{2}+1\right ) \left (1-\cos \left (x \right )\right )}\, \sqrt {\csc \left (x \right )^{3} \left (1-\cos \left (x \right )\right )^{3}+\csc \left (x \right )-\cot \left (x \right )}}\) | \(239\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.39 \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=-\frac {5 \, {\left (i \, a \cos \left (x\right )^{2} - i \, a\right )} \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + 5 \, {\left (-i \, a \cos \left (x\right )^{2} + i \, a\right )} \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right ) + 2 \, {\left (5 \, a \cos \left (x\right )^{3} - 8 \, a \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{21 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \]
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\[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int \left (a \csc ^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int { \left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int { \left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a \csc ^3(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{3/2} \,d x \]
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