Optimal. Leaf size=164 \[ \frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac {5 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (a+b x)}{3 b}+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b}-\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]
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Rubi [A] time = 0.16, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4300, 4308, 4301, 4302, 4306} \[ \frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \sin ^{\frac {3}{2}}(2 a+2 b x) \cos (a+b x)}{3 b}-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}+\frac {\sin ^{\frac {9}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b}-\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 4301
Rule 4302
Rule 4306
Rule 4308
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx &=\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+4 \int \csc (a+b x) \sin ^{\frac {7}{2}}(2 a+2 b x) \, dx\\ &=\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+8 \int \cos (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x) \, dx\\ &=\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+\frac {20}{3} \int \sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x) \, dx\\ &=-\frac {5 \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+5 \int \cos (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx\\ &=\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}+\frac {5}{2} \int \frac {\sin (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{4 b}+\frac {5 \sin (a+b x) \sqrt {\sin (2 a+2 b x)}}{2 b}-\frac {5 \cos (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}+\frac {4 \sin (a+b x) \sin ^{\frac {5}{2}}(2 a+2 b x)}{3 b}+\frac {\csc ^3(a+b x) \sin ^{\frac {9}{2}}(2 a+2 b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 84, normalized size = 0.51 \[ \frac {2 \sqrt {\sin (2 (a+b x))} (6 \sin (a+b x)+\sin (3 (a+b x)))-5 \left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left (\sin (a+b x)+\sqrt {\sin (2 (a+b x))}+\cos (a+b x)\right )\right )}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 280, normalized size = 1.71 \[ \frac {8 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{2} + 5\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \sin \left (b x + a\right ) + 10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) + 5 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{16 \, b} \]
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 = 81.76, size = 973, normalized size = 5.93 \[ \text {result too large to display} \]
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 \[ \int \csc \left (b x + a\right )^{3} \sin \left (2 \, b x + 2 \, a\right )^{\frac {7}{2}}\,{d x} \]
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 \[ \int \frac {{\sin \left (2\,a+2\,b\,x\right )}^{7/2}}{{\sin \left (a+b\,x\right )}^3} \,d x \]
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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