Optimal. Leaf size=127 \[ \frac {4 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}-\frac {3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac {6 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{b}+\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac {3 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{b} \]
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Rubi [A] time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4300, 4308, 4301, 4302, 4305} \[ \frac {4 \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{b}-\frac {3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac {6 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{b}+\frac {\sin ^{\frac {7}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac {3 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 4301
Rule 4302
Rule 4305
Rule 4308
Rubi steps
\begin {align*} \int \csc ^3(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}+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 \sin ^{-1}(\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*}
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Mathematica [A] time = 0.14, size = 70, normalized size = 0.55 \[ \frac {-3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\sin ^{\frac {3}{2}}(2 (a+b x)) \csc (a+b x)+3 \log \left (\sin (a+b x)+\sqrt {\sin (2 (a+b x))}+\cos (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 268, normalized size = 2.11 \[ \frac {8 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \cos \left (b x + a\right ) + 6 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 6 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) - 3 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 41.22, size = 243, normalized size = 1.91 \[ \frac {16 \sqrt {-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1}}\, \left (\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+2}\, \sqrt {-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+2}\, \sqrt {-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}, \frac {\sqrt {2}}{2}\right )-\left (\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{3} \sin \left (2 \, b x + 2 \, a\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{5/2}}{{\sin \left (a+b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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