Integrand size = 10, antiderivative size = 79 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=-\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {14 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{15 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}} \]
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Time = 0.06 (sec) , antiderivative size = 79, 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, 3854, 3856, 2719} \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=-\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {2 \sin ^2(x) \cos (x)}{9 a \sqrt {a \csc ^3(x)}}-\frac {14 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{15 a \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]
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Rule 2719
Rule 3854
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = -\frac {(-\csc (x))^{3/2} \int \frac {1}{(-\csc (x))^{9/2}} \, dx}{a \sqrt {a \csc ^3(x)}} \\ & = -\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}}-\frac {\left (7 (-\csc (x))^{3/2}\right ) \int \frac {1}{(-\csc (x))^{5/2}} \, dx}{9 a \sqrt {a \csc ^3(x)}} \\ & = -\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}}-\frac {\left (7 (-\csc (x))^{3/2}\right ) \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{15 a \sqrt {a \csc ^3(x)}} \\ & = -\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}}+\frac {7 \int \sqrt {\sin (x)} \, dx}{15 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {14 \cos (x)}{45 a \sqrt {a \csc ^3(x)}}-\frac {14 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{15 a \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)}-\frac {2 \cos (x) \sin ^2(x)}{9 a \sqrt {a \csc ^3(x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\frac {-84 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right )+(-33 \cos (x)+5 \cos (3 x)) \sin ^{\frac {3}{2}}(x)}{90 \left (a \csc ^3(x)\right )^{3/2} \sin ^{\frac {9}{2}}(x)} \]
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Result contains complex when optimal does not.
Time = 1.29 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.85
method | result | size |
default | \(-\frac {\csc \left (x \right )^{2} \left (5 \cos \left (x \right )^{5} \sqrt {2}+42 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )-21 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \cos \left (x \right )-17 \cos \left (x \right )^{3} \sqrt {2}+42 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-21 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}+33 \cos \left (x \right ) \sqrt {2}-21 \sqrt {2}\right ) \sqrt {8}}{90 \sqrt {a \csc \left (x \right )^{3}}\, a}\) | \(304\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=-\frac {2 \, {\left (5 \, \cos \left (x\right )^{5} - 17 \, \cos \left (x\right )^{3} + 12 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} \sin \left (x\right ) - 21 \, \sqrt {2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - 21 \, \sqrt {-2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right )}{45 \, a^{2}} \]
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\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \csc ^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \csc \left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a \csc ^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{3/2}} \,d x \]
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