Integrand size = 22, antiderivative size = 81 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\frac {\arcsin (\cos (a+b x)-\sin (a+b x))}{4 b}-\frac {\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{4 b}+\frac {\sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \]
[Out]
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 = {4379, 4393, 4390} \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\frac {\arcsin (\cos (a+b x)-\sin (a+b x))}{4 b}+\frac {\sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\frac {\log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]
[In]
[Out]
Rule 4379
Rule 4390
Rule 4393
Rubi steps \begin{align*} \text {integral}& = \frac {\sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\frac {1}{4} \int \csc (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx \\ & = \frac {\sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)}}-\frac {1}{2} \int \frac {\cos (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = \frac {\arcsin (\cos (a+b x)-\sin (a+b x))}{4 b}-\frac {\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{4 b}+\frac {\sin (a+b x)}{b \sqrt {\sin (2 a+2 b x)}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.89 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\frac {\arcsin (\cos (a+b x)-\sin (a+b x))-\log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )+2 \sec (a+b x) \sqrt {\sin (2 (a+b x))}}{4 b} \]
[In]
[Out]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 28.41 (sec) , antiderivative size = 178923370, normalized size of antiderivative = 2208930.49
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (73) = 146\).
Time = 0.26 (sec) , antiderivative size = 296, normalized size of antiderivative = 3.65 \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=-\frac {2 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) \cos \left (b x + a\right ) - 2 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) \cos \left (b x + a\right ) - \cos \left (b x + a\right ) \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right ) - 8 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - 8 \, \cos \left (b x + a\right )}{16 \, b \cos \left (b x + a\right )} \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\int { \frac {\sin \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sin ^3(a+b x)}{\sin ^{\frac {3}{2}}(2 a+2 b x)} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{{\sin \left (2\,a+2\,b\,x\right )}^{3/2}} \,d x \]
[In]
[Out]