Integrand size = 22, antiderivative size = 106 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {30 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{77 b}-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \]
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Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4385, 2716, 2720} \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {30 \operatorname {EllipticF}\left (a+b x-\frac {\pi }{4},2\right )}{77 b}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)} \]
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Rule 2716
Rule 2720
Rule 4385
Rubi steps \begin{align*} \text {integral}& = -\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {18}{11} \int \frac {1}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx \\ & = -\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {90}{77} \int \frac {1}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx \\ & = -\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {30}{77} \int \frac {1}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = \frac {30 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )}{77 b}-\frac {18 \cos (2 a+2 b x)}{77 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {\csc ^2(a+b x)}{11 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {30 \cos (2 a+2 b x)}{77 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \\ \end{align*}
Time = 0.84 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.81 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {480 \operatorname {EllipticF}\left (a-\frac {\pi }{4}+b x,2\right )+\left (-141 \csc ^2(a+b x)-32 \csc ^4(a+b x)-7 \csc ^6(a+b x)+11 \sec ^2(a+b x) \left (9+\sec ^2(a+b x)\right )\right ) \sqrt {\sin (2 (a+b x))}}{1232 b} \]
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Timed out.
\[\int \frac {\csc \left (x b +a \right )^{2}}{\sin \left (2 x b +2 a \right )^{\frac {9}{2}}}d x\]
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Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.22 \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {240 \, \sqrt {2 i} {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} F(\arcsin \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\,|\,-1) + 240 \, \sqrt {-2 i} {\left (\cos \left (b x + a\right )^{10} - 3 \, \cos \left (b x + a\right )^{8} + 3 \, \cos \left (b x + a\right )^{6} - \cos \left (b x + a\right )^{4}\right )} F(\arcsin \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\,|\,-1) - \sqrt {2} {\left (240 \, \cos \left (b x + a\right )^{8} - 600 \, \cos \left (b x + a\right )^{6} + 444 \, \cos \left (b x + a\right )^{4} - 66 \, \cos \left (b x + a\right )^{2} - 11\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{1232 \, {\left (b \cos \left (b x + a\right )^{10} - 3 \, b \cos \left (b x + a\right )^{8} + 3 \, b \cos \left (b x + a\right )^{6} - b \cos \left (b x + a\right )^{4}\right )}} \]
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Timed out. \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}} \,d x } \]
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\[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int { \frac {\csc \left (b x + a\right )^{2}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\csc ^2(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int \frac {1}{{\sin \left (a+b\,x\right )}^2\,{\sin \left (2\,a+2\,b\,x\right )}^{9/2}} \,d x \]
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