Integrand size = 18, antiderivative size = 108 \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{147 b^2} \]
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Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4298, 3854, 3856, 2720} \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{147 b^2}+\frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)} \]
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Rule 2720
Rule 3854
Rule 3856
Rule 4298
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {7}{2}}(a+b x)} \, dx}{7 b} \\ & = \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}-\frac {10 \int \frac {1}{\csc ^{\frac {3}{2}}(a+b x)} \, dx}{49 b} \\ & = \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {10 \int \sqrt {\csc (a+b x)} \, dx}{147 b} \\ & = \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {\left (10 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{147 b} \\ & = \frac {2 x}{7 b \csc ^{\frac {7}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{49 b^2 \csc ^{\frac {5}{2}}(a+b x)}+\frac {20 \cos (a+b x)}{147 b^2 \sqrt {\csc (a+b x)}}-\frac {20 \sqrt {\csc (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right ) \sqrt {\sin (a+b x)}}{147 b^2} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.86 \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\frac {\sqrt {\csc (a+b x)} \left (63 b x-84 b x \cos (2 (a+b x))+21 b x \cos (4 (a+b x))+80 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right ) \sqrt {\sin (a+b x)}+52 \sin (2 (a+b x))-6 \sin (4 (a+b x))\right )}{588 b^2} \]
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\[\int \frac {x \cos \left (x b +a \right )}{\csc \left (x b +a \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\csc ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
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