Integrand size = 10, antiderivative size = 123 \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+\frac {154}{195} a^2 \sqrt {a \csc ^3(x)} E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sin ^{\frac {3}{2}}(x) \]
[Out]
Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2719} \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=-\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \sin (x) \cos (x) \sqrt {a \csc ^3(x)}+\frac {154}{195} a^2 \sin ^{\frac {3}{2}}(x) E\left (\left .\frac {\pi }{4}-\frac {x}{2}\right |2\right ) \sqrt {a \csc ^3(x)} \]
[In]
[Out]
Rule 2719
Rule 3853
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{15/2} \, dx}{(-\csc (x))^{3/2}} \\ & = -\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (11 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{11/2} \, dx}{13 (-\csc (x))^{3/2}} \\ & = -\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{7/2} \, dx}{117 (-\csc (x))^{3/2}} \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}+\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int (-\csc (x))^{3/2} \, dx}{195 (-\csc (x))^{3/2}} \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {\left (77 a^2 \sqrt {a \csc ^3(x)}\right ) \int \frac {1}{\sqrt {-\csc (x)}} \, dx}{195 (-\csc (x))^{3/2}} \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)-\frac {1}{195} \left (77 a^2 \sqrt {a \csc ^3(x)} \sin ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sin (x)} \, dx \\ & = -\frac {154}{585} a^2 \cot (x) \sqrt {a \csc ^3(x)}-\frac {22}{117} a^2 \cot (x) \csc ^2(x) \sqrt {a \csc ^3(x)}-\frac {2}{13} a^2 \cot (x) \csc ^4(x) \sqrt {a \csc ^3(x)}-\frac {154}{195} a^2 \cos (x) \sqrt {a \csc ^3(x)} \sin (x)+\frac {154}{195} a^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 = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.47 \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=\frac {\left (a \csc ^3(x)\right )^{5/2} \left (29568 E\left (\left .\frac {1}{4} (\pi -2 x)\right |2\right ) \sin ^{\frac {15}{2}}(x)-9414 \sin (2 x)+5346 \sin (4 x)-1694 \sin (6 x)+231 \sin (8 x)\right )}{37440} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 2.02 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {a^{2} \sqrt {a \csc \left (x \right )^{3}}\, \left (\sin \left (x \right ) \left (-231 \cos \left (x \right )-231\right ) \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}+\sin \left (x \right ) \left (462 \cos \left (x \right )+462\right ) \sqrt {-i \left (i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \sqrt {-i \left (-\csc \left (x \right )+\cot \left (x \right )\right )}\, \sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (x \right )-\csc \left (x \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \left (-231 \sin \left (x \right )-77 \cot \left (x \right )-55 \csc \left (x \right )^{2} \cot \left (x \right )-45 \cot \left (x \right ) \csc \left (x \right )^{4}\right )\right ) \sqrt {8}}{1170}\) | \(185\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.31 \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=-\frac {231 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sqrt {2 i \, a} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) + i \, \sin \left (x\right )\right )\right ) + 231 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \sqrt {-2 i \, a} \sin \left (x\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (x\right ) - i \, \sin \left (x\right )\right )\right ) - 2 \, {\left (231 \, a^{2} \cos \left (x\right )^{7} - 770 \, a^{2} \cos \left (x\right )^{5} + 902 \, a^{2} \cos \left (x\right )^{3} - 408 \, a^{2} \cos \left (x\right )\right )} \sqrt {-\frac {a}{{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}}}{585 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]
[In]
[Out]
\[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=\int \left (a \csc ^{3}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
[In]
[Out]
\[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=\int { \left (a \csc \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
[In]
[Out]
\[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=\int { \left (a \csc \left (x\right )^{3}\right )^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (a \csc ^3(x)\right )^{5/2} \, dx=\int {\left (\frac {a}{{\sin \left (x\right )}^3}\right )}^{5/2} \,d x \]
[In]
[Out]