Optimal. Leaf size=81 \[ \frac{\sin (a+b x)}{b \sqrt{\sin (2 a+2 b x)}}+\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]
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Rubi [A] time = 0.0778138, antiderivative size = 81, normalized size of antiderivative = 1., 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 = {4294, 4308, 4305} \[ \frac{\sin (a+b x)}{b \sqrt{\sin (2 a+2 b x)}}+\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{4 b} \]
Antiderivative was successfully verified.
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Rule 4294
Rule 4308
Rule 4305
Rubi steps
\begin{align*} \int \frac{\sin ^3(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx &=\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{\sin ^{-1}(\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*}
Mathematica [A] time = 0.0905944, size = 72, normalized size = 0.89 \[ \frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+2 \sqrt{\sin (2 (a+b x))} \sec (a+b x)-\log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )}{4 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 30.918, size = 149404972, normalized size = 1844505.8 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (b x + a\right )^{3}}{\sin \left (2 \, b x + 2 \, a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.535688, size = 815, normalized size = 10.06 \begin{align*} -\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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