Integrand size = 18, antiderivative size = 38 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3524, 2720} \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right ),2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \]
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Rule 2720
Rule 3524
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{b \sqrt {\sin (a+b x)}}+\frac {2 \int \frac {1}{\sqrt {\sin (a+b x)}} \, dx}{b} \\ & = \frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right ),2\right )}{b^2}-\frac {2 x}{b \sqrt {\sin (a+b x)}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \left (-2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 a+\pi -2 b x),2\right )-\frac {b x}{\sqrt {\sin (a+b x)}}\right )}{b^2} \]
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\[\int \frac {x \cos \left (x b +a \right )}{\sin \left (x b +a \right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\sin ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{3/2}} \,d x \]
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