Optimal. Leaf size=104 \[ -\frac{4 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}+\frac{2 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac{2 \log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{b} \]
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Rubi [A] time = 0.102062, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4300, 4308, 4301, 4306} \[ -\frac{4 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}+\frac{2 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac{2 \log \left (\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 4308
Rule 4301
Rule 4306
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx &=-\frac{\csc ^3(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{b}-4 \int \csc (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx\\ &=-\frac{\csc ^3(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{b}-8 \int \cos (a+b x) \sqrt{\sin (2 a+2 b x)} \, dx\\ &=-\frac{4 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}-\frac{\csc ^3(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{b}-4 \int \frac{\sin (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=\frac{2 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}+\frac{2 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}\right )}{b}-\frac{4 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}-\frac{\csc ^3(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0983066, size = 68, normalized size = 0.65 \[ \frac{2 \left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x))-2 \sqrt{\sin (2 (a+b x))} \csc (a+b x)+\log \left (\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right )\right )}{b} \]
Antiderivative was successfully verified.
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Maple [C] time = 6.01, size = 542, normalized size = 5.2 \begin{align*} \text{result 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 \csc \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 = 0.530154, size = 814, normalized size = 7.83 \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 ) \sin \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 ) \sin \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 ) \sin \left (b x + a\right ) + 8 \, \sqrt{2} \sqrt{\cos \left (b x + a\right ) \sin \left (b x + a\right )} + 8 \, \sin \left (b x + a\right )}{2 \, b \sin \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc \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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