Integrand size = 22, antiderivative size = 81 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{21 b \sqrt {\sin (2 a+2 b x)}} \]
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Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4381, 4389, 4376} \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{21 b \sqrt {\sin (2 a+2 b x)}} \]
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Rule 4376
Rule 4381
Rule 4389
Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2}{7} \int \frac {\sin (a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx \\ & = \frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {4}{21} \int \frac {\cos (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx \\ & = \frac {\sin ^3(a+b x)}{7 b \sin ^{\frac {7}{2}}(2 a+2 b x)}+\frac {2 \sin (a+b x)}{21 b \sin ^{\frac {3}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{21 b \sqrt {\sin (2 a+2 b x)}} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.68 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {(5+12 \cos (2 (a+b x))+4 \cos (4 (a+b x))) \csc (a+b x) \sec ^4(a+b x) \sqrt {\sin (2 (a+b x))}}{336 b} \]
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Timed out.
\[\int \frac {\sin \left (x b +a \right )^{3}}{\sin \left (2 x b +2 a \right )^{\frac {9}{2}}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {32 \, \cos \left (b x + a\right )^{4} \sin \left (b x + a\right ) + \sqrt {2} {\left (32 \, \cos \left (b x + a\right )^{4} - 8 \, \cos \left (b x + a\right )^{2} - 3\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{336 \, b \cos \left (b x + a\right )^{4} \sin \left (b x + a\right )} \]
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Timed out. \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {9}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]
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Time = 24.64 (sec) , antiderivative size = 300, normalized size of antiderivative = 3.70 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {9}{2}}(2 a+2 b x)} \, dx=-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,5{}\mathrm {i}}{84\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^2}+\frac {3\,{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{14\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^3}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}\,1{}\mathrm {i}}{7\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^4}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (\frac {5}{84\,b}+\frac {4\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}}{21\,b}\right )\,\sqrt {\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}}}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )} \]
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