Integrand size = 10, antiderivative size = 50 \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=-\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{3 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)} \]
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Time = 0.04 (sec) , antiderivative size = 50, 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, 3854, 3856, 2720} \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=-\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{3 \sin ^{\frac {3}{2}}(x) \sqrt {a \csc ^3(x)}} \]
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Rule 2720
Rule 3854
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {(-\csc (x))^{3/2} \int \frac {1}{(-\csc (x))^{3/2}} \, dx}{\sqrt {a \csc ^3(x)}} \\ & = -\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}+\frac {(-\csc (x))^{3/2} \int \sqrt {-\csc (x)} \, dx}{3 \sqrt {a \csc ^3(x)}} \\ & = -\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}+\frac {\int \frac {1}{\sqrt {\sin (x)}} \, dx}{3 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)} \\ & = -\frac {2 \cot (x)}{3 \sqrt {a \csc ^3(x)}}-\frac {2 \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {x}{2},2\right )}{3 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\frac {-2 \cot (x)-\frac {2 \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 x),2\right )}{\sin ^{\frac {3}{2}}(x)}}{3 \sqrt {a \csc ^3(x)}} \]
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Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.26
method | result | size |
default | \(\frac {\left (-i \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )-i \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (x \right ) \sin \left (x \right )\right ) \sqrt {8}}{6 \sqrt {a \csc \left (x \right )^{3}}\, \left (\cos \left (x \right )-1\right ) \left (\cos \left (x \right )+1\right )}\) | \(163\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.36 \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}} - i \, \sqrt {2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right ) + i \, \sqrt {-2 i \, a} {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )}{3 \, a} \]
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\[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\int \frac {1}{\sqrt {a \csc ^{3}{\left (x \right )}}}\, dx \]
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\[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \csc \left (x\right )^{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\int { \frac {1}{\sqrt {a \csc \left (x\right )^{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a \csc ^3(x)}} \, dx=\int \frac {1}{\sqrt {\frac {a}{{\sin \left (x\right )}^3}}} \,d x \]
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