Optimal. Leaf size=110 \[ -\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}-\frac {5 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{16 b}+\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{32 b} \]
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Rubi [A] time = 0.08, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {4298, 4302, 4305} \[ -\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}-\frac {5 \sqrt {\sin (2 a+2 b x)} \cos (a+b x)}{16 b}+\frac {5 \log \left (\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}+\cos (a+b x)\right )}{32 b} \]
Antiderivative was successfully verified.
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Rule 4298
Rule 4302
Rule 4305
Rubi steps
\begin {align*} \int \sin ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx &=-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{8} \int \sin (a+b x) \sqrt {\sin (2 a+2 b x)} \, dx\\ &=-\frac {5 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}+\frac {5}{16} \int \frac {\cos (a+b x)}{\sqrt {\sin (2 a+2 b x)}} \, dx\\ &=-\frac {5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}+\frac {5 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 a+2 b x)}\right )}{32 b}-\frac {5 \cos (a+b x) \sqrt {\sin (2 a+2 b x)}}{16 b}-\frac {\sin (a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{8 b}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 86, normalized size = 0.78 \[ \frac {2 \sqrt {\sin (2 (a+b x))} (\cos (3 (a+b x))-6 \cos (a+b x))+5 \left (\log \left (\sin (a+b x)+\sqrt {\sin (2 (a+b x))}+\cos (a+b x)\right )-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))\right )}{32 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 281, normalized size = 2.55 \[ \frac {8 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - 9 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} {\left (\cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) - 1}\right ) - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} - \cos \left (b x + a\right ) - \sin \left (b x + a\right )}{\cos \left (b x + a\right ) - \sin \left (b x + a\right )}\right ) - 5 \, \log \left (-32 \, \cos \left (b x + a\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (b x + a\right )^{3} - {\left (4 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right ) - 5 \, \cos \left (b x + a\right )\right )} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 32 \, \cos \left (b x + a\right )^{2} + 16 \, \cos \left (b x + a\right ) \sin \left (b x + a\right ) + 1\right )}{128 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 23.19, size = 47453403, normalized size = 431394.57 \[ \text {output too large to display} \]
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 \[ \int \sqrt {\sin \left (2 \, b x + 2 \, a\right )} \sin \left (b x + a\right )^{3}\,{d x} \]
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 \[ \int {\sin \left (a+b\,x\right )}^3\,\sqrt {\sin \left (2\,a+2\,b\,x\right )} \,d x \]
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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