Integrand size = 22, antiderivative size = 127 \[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=-\frac {3 \arcsin (\cos (a+b x)-\sin (a+b x))}{b}+\frac {3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{b}-\frac {6 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{b}+\frac {4 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}+\frac {\csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{b} \]
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Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4385, 4393, 4386, 4387, 4390} \[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=-\frac {3 \arcsin (\cos (a+b x)-\sin (a+b x))}{b}+\frac {4 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}-\frac {6 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{b}+\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac {3 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{b} \]
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Rule 4385
Rule 4386
Rule 4387
Rule 4390
Rule 4393
Rubi steps \begin{align*} \text {integral}& = \frac {\csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{b}+8 \int \csc (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx \\ & = \frac {\csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{b}+16 \int \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx \\ & = \frac {4 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}+\frac {\csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{b}+12 \int \sin (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx \\ & = -\frac {6 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{b}+\frac {4 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}+\frac {\csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{b}+6 \int \frac {\cos (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx \\ & = -\frac {3 \arcsin (\cos (a+b x)-\sin (a+b x))}{b}+\frac {3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{b}-\frac {6 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{b}+\frac {4 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}+\frac {\csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x)}{b} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.55 \[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\frac {-3 \arcsin (\cos (a+b x)-\sin (a+b x))+3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right )+\csc (a+b x) \sin ^{\frac {3}{2}}(2 (a+b x))}{b} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 107.88 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.91
method | result | size |
default | \(\frac {16 \sqrt {-\frac {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}{\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-1}}\, \left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \operatorname {EllipticF}\left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+2}\, \sqrt {-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \operatorname {EllipticF}\left (\sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )\right )}{3 \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}-\tan \left (\frac {a}{2}+\frac {x b}{2}\right )}\, \sqrt {\tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \left (\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-1\right )}\, b}\) | \(243\) |
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (119) = 238\).
Time = 0.27 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.11 \[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\frac {8 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \cos \left (b x + a\right ) + 6 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 6 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) - 3 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{4 \, b} \]
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Timed out. \[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\text {Timed out} \]
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\[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\int { \csc \left (b x + a\right )^{3} \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}} \,d x } \]
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\[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\int { \csc \left (b x + a\right )^{3} \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \csc ^3(a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx=\int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{5/2}}{{\sin \left (a+b\,x\right )}^3} \,d x \]
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