Integrand size = 10, antiderivative size = 67 \[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {6 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b} \]
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Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856, 2719} \[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\frac {6 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{5 b}-\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)} \]
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Rule 2719
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {3}{5} \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx \\ & = -\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {1}{5} \left (3 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx \\ & = -\frac {2 \cos (a+b x)}{5 b \csc ^{\frac {3}{2}}(a+b x)}+\frac {6 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{5 b} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {2 \sqrt {\csc (a+b x)} \left (3 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sqrt {\sin (a+b x)}+\cos (a+b x) \sin ^2(a+b x)\right )}{5 b} \]
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Time = 0.50 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.12
method | result | size |
default | \(\frac {\frac {2 \sin \left (x b +a \right )^{4}}{5}-\frac {2 \sin \left (x b +a \right )^{2}}{5}-\frac {6 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {\sin \left (x b +a \right )+1}\, \sqrt {-2 \sin \left (x b +a \right )+2}\, \sqrt {-\sin \left (x b +a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (x b +a \right )+1}, \frac {\sqrt {2}}{2}\right )}{5}}{\cos \left (x b +a \right ) \sqrt {\sin \left (x b +a \right )}\, b}\) | \(142\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.27 \[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\frac {3 \, \sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + 3 \, \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right ) + \frac {2 \, {\left (\cos \left (b x + a\right )^{3} - \cos \left (b x + a\right )\right )}}{\sqrt {\sin \left (b x + a\right )}}}{5 \, b} \]
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\[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{\csc ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {1}{\csc \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
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