Integrand size = 18, antiderivative size = 65 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b^2}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}} \]
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Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3524, 2716, 2719} \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {4 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{3 b^2}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)} \]
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Rule 2716
Rule 2719
Rule 3524
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sin ^{\frac {3}{2}}(a+b x)} \, dx}{3 b} \\ & = -\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}}-\frac {2 \int \sqrt {\sin (a+b x)} \, dx}{3 b} \\ & = -\frac {4 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{3 b^2}-\frac {2 x}{3 b \sin ^{\frac {3}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{3 b^2 \sqrt {\sin (a+b x)}} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=-\frac {2 \left (b x-2 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {3}{2}}(a+b x)+\sin (2 (a+b x))\right )}{3 b^2 \sin ^{\frac {3}{2}}(a+b x)} \]
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\[\int \frac {x \cos \left (x b +a \right )}{\sin \left (x b +a \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\sin ^{\frac {5}{2}}{\left (a + b x \right )}}\, dx \]
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\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {5}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{5/2}} \,d x \]
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