Integrand size = 10, antiderivative size = 48 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x) \]
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Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2719} \[ \int \sqrt {a \csc ^3(x)} \, dx=2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)} \]
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Rule 2719
Rule 3853
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a \csc ^3(x)} \int (-\csc (x))^{3/2} \, dx}{(-\csc (x))^{3/2}} \\ & = -2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {\sqrt {a \csc ^3(x)} \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{(-\csc (x))^{3/2}} \\ & = -2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\left (\sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx \\ & = -2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x) \]
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Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 5.62
method | result | size |
default | \(-\frac {\sqrt {a \csc \left (x \right )^{3}}\, \left (-2 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )+\sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \cos \left (x \right )-2 \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}+\sqrt {2}\right ) \sin \left (x \right ) \sqrt {8}}{2}\) | \(270\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.33 \[ \int \sqrt {a \csc ^3(x)} \, dx=-2 \, \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} \cos \left (x\right ) \sin \left (x\right ) - \sqrt {2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) - \sqrt {-2 i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) \]
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\[ \int \sqrt {a \csc ^3(x)} \, dx=\int \sqrt {a \csc ^{3}{\left (x \right )}}\, dx \]
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\[ \int \sqrt {a \csc ^3(x)} \, dx=\int { \sqrt {a \csc \left (x\right )^{3}} \,d x } \]
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\[ \int \sqrt {a \csc ^3(x)} \, dx=\int { \sqrt {a \csc \left (x\right )^{3}} \,d x } \]
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Timed out. \[ \int \sqrt {a \csc ^3(x)} \, dx=\int \sqrt {\frac {a}{{\sin \left (x\right )}^3}} \,d x \]
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