Integrand size = 18, antiderivative size = 65 \[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=-\frac {12 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{25 b^2}+\frac {4 \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Time = 0.05 (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, 2715, 2719} \[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=-\frac {12 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2}+\frac {4 \sin ^{\frac {3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b} \]
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Rule 2715
Rule 2719
Rule 3524
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {2 \int \sin ^{\frac {5}{2}}(a+b x) \, dx}{5 b} \\ & = \frac {4 \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b}-\frac {6 \int \sqrt {\sin (a+b x)} \, dx}{25 b} \\ & = -\frac {12 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{25 b^2}+\frac {4 \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x)}{25 b^2}+\frac {2 x \sin ^{\frac {5}{2}}(a+b x)}{5 b} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.16 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.66 \[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=\frac {\sqrt {\sin (a+b x)} \left (5 b x-5 b x \cos (2 (a+b x))+2 \sin (2 (a+b x))-12 \tan \left (\frac {1}{2} (a+b x)\right )+4 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right ) \sqrt {\sec ^2\left (\frac {1}{2} (a+b x)\right )} \tan \left (\frac {1}{2} (a+b x)\right )\right )}{25 b^2} \]
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\[\int x \cos \left (x b +a \right ) \sin \left (x b +a \right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=\text {Exception raised: TypeError} \]
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\[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=\int x \sin ^{\frac {3}{2}}{\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \]
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\[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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\[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=\int { x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int x \cos (a+b x) \sin ^{\frac {3}{2}}(a+b x) \, dx=\int x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^{3/2} \,d x \]
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