Integrand size = 18, antiderivative size = 88 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=-\frac {12 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{35 b^2}-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3524, 2716, 2719} \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=-\frac {12 E\left (\left .\frac {1}{2} \left (a+b x-\frac {\pi }{2}\right )\right |2\right )}{35 b^2}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}}-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)} \]
[In]
[Out]
Rule 2716
Rule 2719
Rule 3524
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}+\frac {2 \int \frac {1}{\sin ^{\frac {7}{2}}(a+b x)} \, dx}{7 b} \\ & = -\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}+\frac {6 \int \frac {1}{\sin ^{\frac {3}{2}}(a+b x)} \, dx}{35 b} \\ & = -\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}}-\frac {6 \int \sqrt {\sin (a+b x)} \, dx}{35 b} \\ & = -\frac {12 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b x\right )\right |2\right )}{35 b^2}-\frac {2 x}{7 b \sin ^{\frac {7}{2}}(a+b x)}-\frac {4 \cos (a+b x)}{35 b^2 \sin ^{\frac {5}{2}}(a+b x)}-\frac {12 \cos (a+b x)}{35 b^2 \sqrt {\sin (a+b x)}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.83 \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=-\frac {2 \left (5 b x+6 \cos (a+b x) \sin ^3(a+b x)-6 E\left (\left .\frac {1}{4} (-2 a+\pi -2 b x)\right |2\right ) \sin ^{\frac {7}{2}}(a+b x)+\sin (2 (a+b x))\right )}{35 b^2 \sin ^{\frac {7}{2}}(a+b x)} \]
[In]
[Out]
\[\int \frac {x \cos \left (x b +a \right )}{\sin \left (x b +a \right )^{\frac {9}{2}}}d x\]
[In]
[Out]
Exception generated. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=\int { \frac {x \cos \left (b x + a\right )}{\sin \left (b x + a\right )^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x \cos (a+b x)}{\sin ^{\frac {9}{2}}(a+b x)} \, dx=\int \frac {x\,\cos \left (a+b\,x\right )}{{\sin \left (a+b\,x\right )}^{9/2}} \,d x \]
[In]
[Out]