Integrand size = 18, antiderivative size = 85 \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {12 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{25 b^2} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4298, 3854, 3856, 2719} \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {12 \sqrt {\sin (a+b x)} \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{25 b^2}+\frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)} \]
[In]
[Out]
Rule 2719
Rule 3854
Rule 3856
Rule 4298
Rubi steps \begin{align*} \text {integral}& = \frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}-\frac {2 \int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx}{5 b} \\ & = \frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {6 \int \frac {1}{\sqrt {\csc (a+b x)}} \, dx}{25 b} \\ & = \frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {\left (6 \sqrt {\csc (a+b x)} \sqrt {\sin (a+b x)}\right ) \int \sqrt {\sin (a+b x)} \, dx}{25 b} \\ & = \frac {2 x}{5 b \csc ^{\frac {5}{2}}(a+b x)}+\frac {4 \cos (a+b x)}{25 b^2 \csc ^{\frac {3}{2}}(a+b x)}-\frac {12 \sqrt {\csc (a+b x)} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right ) \sqrt {\sin (a+b x)}}{25 b^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.34 \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\frac {\left (-10+4 \cos (a+b x)+2 \cos (2 (a+b x))+4 \sqrt {2} \sqrt {\frac {1}{1+\cos (a+b x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2\left (\frac {1}{2} (a+b x)\right )\right )+10 b x \sin (a+b x)+5 b x \sin (2 (a+b x))\right ) \tan \left (\frac {1}{2} (a+b x)\right )}{25 b^2 \sqrt {\csc (a+b x)}} \]
[In]
[Out]
\[\int \frac {x \cos \left (x b +a \right )}{\csc \left (x b +a \right )^{\frac {3}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x \cos {\left (a + b x \right )}}{\csc ^{\frac {3}{2}}{\left (a + b x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\csc \left (b x + a\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x \cos (a+b x)}{\csc ^{\frac {3}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\left (\frac {1}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
[In]
[Out]