Integrand size = 22, antiderivative size = 107 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{2}}(2 a+2 b x)} \, dx=\frac {\sin ^3(a+b x)}{9 b \sin ^{\frac {9}{2}}(2 a+2 b x)}+\frac {\sin (a+b x)}{15 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{45 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {8 \sin (a+b x)}{45 b \sqrt {\sin (2 a+2 b x)}} \]
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Time = 0.13 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4381, 4389, 4388, 4377} \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{2}}(2 a+2 b x)} \, dx=\frac {\sin (a+b x)}{15 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {\sin ^3(a+b x)}{9 b \sin ^{\frac {9}{2}}(2 a+2 b x)}+\frac {8 \sin (a+b x)}{45 b \sqrt {\sin (2 a+2 b x)}}-\frac {4 \cos (a+b x)}{45 b \sin ^{\frac {3}{2}}(2 a+2 b x)} \]
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Rule 4377
Rule 4381
Rule 4388
Rule 4389
Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^3(a+b x)}{9 b \sin ^{\frac {9}{2}}(2 a+2 b x)}+\frac {1}{3} \int \frac {\sin (a+b x)}{\sin ^{\frac {7}{2}}(2 a+2 b x)} \, dx \\ & = \frac {\sin ^3(a+b x)}{9 b \sin ^{\frac {9}{2}}(2 a+2 b x)}+\frac {\sin (a+b x)}{15 b \sin ^{\frac {5}{2}}(2 a+2 b x)}+\frac {4}{15} \int \frac {\cos (a+b x)}{\sin ^{\frac {5}{2}}(2 a+2 b x)} \, dx \\ & = \frac {\sin ^3(a+b x)}{9 b \sin ^{\frac {9}{2}}(2 a+2 b x)}+\frac {\sin (a+b x)}{15 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{45 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {8}{45} \int \frac {\sin (a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx \\ & = \frac {\sin ^3(a+b x)}{9 b \sin ^{\frac {9}{2}}(2 a+2 b x)}+\frac {\sin (a+b x)}{15 b \sin ^{\frac {5}{2}}(2 a+2 b x)}-\frac {4 \cos (a+b x)}{45 b \sin ^{\frac {3}{2}}(2 a+2 b x)}+\frac {8 \sin (a+b x)}{45 b \sqrt {\sin (2 a+2 b x)}} \\ \end{align*}
Time = 0.88 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.58 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{2}}(2 a+2 b x)} \, dx=\frac {\left (-15 \cot (a+b x) \csc (a+b x)+113 \sec (a+b x)+17 \sec ^3(a+b x)+5 \sec ^5(a+b x)\right ) \sqrt {\sin (2 (a+b x))}}{1440 b} \]
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Timed out.
\[\int \frac {\sin \left (x b +a \right )^{3}}{\sin \left (2 x b +2 a \right )^{\frac {11}{2}}}d x\]
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none
Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{2}}(2 a+2 b x)} \, dx=\frac {128 \, \cos \left (b x + a\right )^{7} - 128 \, \cos \left (b x + a\right )^{5} + \sqrt {2} {\left (128 \, \cos \left (b x + a\right )^{6} - 96 \, \cos \left (b x + a\right )^{4} - 12 \, \cos \left (b x + a\right )^{2} - 5\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )}}{1440 \, {\left (b \cos \left (b x + a\right )^{7} - b \cos \left (b x + a\right )^{5}\right )}} \]
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Timed out. \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{2}}(2 a+2 b x)} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {11}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]
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Time = 25.91 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.58 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {11}{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}}\,1{}\mathrm {i}}{60\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^3}-\frac {2\,{\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}}}{9\,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}}\,\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}}{9\,b\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^5}+\frac {{\mathrm {e}}^{a\,3{}\mathrm {i}+b\,x\,3{}\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}}\,8{}\mathrm {i}}{45\,b\,\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 )}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}\,\left (\frac {49}{180\,b}-\frac {19\,{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}}{180\,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 )}^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}+1{}\mathrm {i}\right )}^2} \]
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