Integrand size = 10, antiderivative size = 123 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{195 a^2 \sqrt {a \sin ^3(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2716, 2719} \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {154 \sin (x) \cos (x)}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right )}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}} \]
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Rule 2716
Rule 2719
Rule 3286
Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^{\frac {3}{2}}(x) \int \frac {1}{\sin ^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \sin ^3(x)}} \\ & = -\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}+\frac {\left (11 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sin ^{\frac {11}{2}}(x)} \, dx}{13 a^2 \sqrt {a \sin ^3(x)}} \\ & = -\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}+\frac {\left (77 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sin ^{\frac {7}{2}}(x)} \, dx}{117 a^2 \sqrt {a \sin ^3(x)}} \\ & = -\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}+\frac {\left (77 \sin ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(x)} \, dx}{195 a^2 \sqrt {a \sin ^3(x)}} \\ & = -\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \sin ^3(x)}}-\frac {\left (77 \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx}{195 a^2 \sqrt {a \sin ^3(x)}} \\ & = -\frac {154 \cot (x)}{585 a^2 \sqrt {a \sin ^3(x)}}-\frac {22 \cot (x) \csc ^2(x)}{117 a^2 \sqrt {a \sin ^3(x)}}-\frac {2 \cot (x) \csc ^4(x)}{13 a^2 \sqrt {a \sin ^3(x)}}-\frac {154 \cos (x) \sin (x)}{195 a^2 \sqrt {a \sin ^3(x)}}+\frac {154 E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x)}{195 a^2 \sqrt {a \sin ^3(x)}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {2 \left (\cot (x) \left (77+55 \csc ^2(x)+45 \csc ^4(x)\right )+231 \cos (x) \sin (x)-231 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {3}{2}}(x)\right )}{585 a^2 \sqrt {a \sin ^3(x)}} \]
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Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.55
method | result | size |
default | \(-\frac {\left (231 \sin \left (x \right ) \cos \left (x \right ) \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )-462 \sin \left (x \right ) \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (x \right )+231 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, F\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (x \right )-462 \sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}\, \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {i \left (\csc \left (x \right )-\cot \left (x \right )\right )}\, E\left (\sqrt {-i \left (i-\cot \left (x \right )+\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (x \right )+231 \sin \left (x \right ) \sqrt {2}+77 \cot \left (x \right ) \sqrt {2}+55 \left (\csc ^{2}\left (x \right )\right ) \cot \left (x \right ) \sqrt {2}+45 \cot \left (x \right ) \left (\csc ^{4}\left (x \right )\right ) \sqrt {2}\right ) \sqrt {8}}{1170 \sqrt {a \left (\sin ^{3}\left (x \right )\right )}\, a^{2}}\) | \(314\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.70 \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=-\frac {231 \, {\left (i \, \sqrt {2} \cos \left (x\right )^{8} - 4 i \, \sqrt {2} \cos \left (x\right )^{6} + 6 i \, \sqrt {2} \cos \left (x\right )^{4} - 4 i \, \sqrt {2} \cos \left (x\right )^{2} + i \, \sqrt {2}\right )} \sqrt {-i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 231 \, {\left (-i \, \sqrt {2} \cos \left (x\right )^{8} + 4 i \, \sqrt {2} \cos \left (x\right )^{6} - 6 i \, \sqrt {2} \cos \left (x\right )^{4} + 4 i \, \sqrt {2} \cos \left (x\right )^{2} - i \, \sqrt {2}\right )} \sqrt {i \, a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - 2 \, {\left (231 \, \cos \left (x\right )^{7} - 770 \, \cos \left (x\right )^{5} + 902 \, \cos \left (x\right )^{3} - 408 \, \cos \left (x\right )\right )} \sqrt {-{\left (a \cos \left (x\right )^{2} - a\right )} \sin \left (x\right )}}{585 \, {\left (a^{3} \cos \left (x\right )^{8} - 4 \, a^{3} \cos \left (x\right )^{6} + 6 \, a^{3} \cos \left (x\right )^{4} - 4 \, a^{3} \cos \left (x\right )^{2} + a^{3}\right )}} \]
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\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \sin ^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \sin \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a \sin ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\sin \left (x\right )}^3\right )}^{5/2}} \,d x \]
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