Integrand size = 22, antiderivative size = 106 \[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\frac {6 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4385, 2715, 2719} \[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\frac {6 E\left (\left .a+b x-\frac {\pi }{4}\right |2\right )}{5 b}-\frac {2 \sin ^{\frac {7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}-\frac {2 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}+\frac {\sin ^{\frac {11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b} \]
[In]
[Out]
Rule 2715
Rule 2719
Rule 4385
Rubi steps \begin{align*} \text {integral}& = \frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b}+\frac {18}{7} \int \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx \\ & = -\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b}+2 \int \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx \\ & = -\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b}+\frac {6}{5} \int \sqrt {\sin (2 a+2 b x)} \, dx \\ & = \frac {6 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{5 b}-\frac {2 \cos (2 a+2 b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{7 b}+\frac {\csc ^2(a+b x) \sin ^{\frac {11}{2}}(2 a+2 b x)}{7 b} \\ \end{align*}
Time = 0.88 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\frac {84 E\left (\left .a-\frac {\pi }{4}+b x\right |2\right )+\sqrt {\sin (2 (a+b x))} (15 \sin (2 (a+b x))-14 \sin (4 (a+b x))-5 \sin (6 (a+b x)))}{70 b} \]
[In]
[Out]
Time = 22.56 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.92
method | result | size |
default | \(\frac {8 \sqrt {2}\, \left (\frac {\sqrt {2}\, \sin \left (2 x b +2 a \right )^{\frac {7}{2}}}{56}-\frac {\sqrt {2}\, \left (6 \sqrt {\sin \left (2 x b +2 a \right )+1}\, \sqrt {-2 \sin \left (2 x b +2 a \right )+2}\, \sqrt {-\sin \left (2 x b +2 a \right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (2 x b +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {\sin \left (2 x b +2 a \right )+1}\, \sqrt {-2 \sin \left (2 x b +2 a \right )+2}\, \sqrt {-\sin \left (2 x b +2 a \right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (2 x b +2 a \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (2 x b +2 a \right )^{4}+2 \sin \left (2 x b +2 a \right )^{2}\right )}{80 \cos \left (2 x b +2 a \right ) \sqrt {\sin \left (2 x b +2 a \right )}}\right )}{b}\) | \(204\) |
[In]
[Out]
\[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\int { \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\int { \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}} \,d x } \]
[In]
[Out]
\[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\int { \csc \left (b x + a\right )^{2} \sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \csc ^2(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x) \, dx=\int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{9/2}}{{\sin \left (a+b\,x\right )}^2} \,d x \]
[In]
[Out]