Optimal. Leaf size=28 \[ -\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.03, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {4292} \[ -\frac {\sin ^{\frac {3}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4292
Rubi steps
\begin {align*} \int \csc ^3(a+b x) \sqrt {\sin (2 a+2 b x)} \, dx &=-\frac {\csc ^3(a+b x) \sin ^{\frac {3}{2}}(2 a+2 b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 27, normalized size = 0.96 \[ -\frac {\sin ^{\frac {3}{2}}(2 (a+b x)) \csc ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 53, normalized size = 1.89 \[ \frac {2 \, {\left (\sqrt {2} \sqrt {\cos \left (b x + a\right ) \sin \left (b x + a\right )} \cos \left (b x + a\right ) + \cos \left (b x + a\right )^{2} - 1\right )}}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (b x + a\right )^{3} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 31.02, size = 192, normalized size = 6.86 \[ \frac {\sqrt {-\frac {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right ) \left (4 \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}\, \sqrt {-2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+2}\, \sqrt {-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, \EllipticF \left (\sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1}, \frac {\sqrt {2}}{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )+\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{3 \tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {\tan \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )-\tan \left (\frac {b x}{2}+\frac {a}{2}\right )}\, b} \]
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 \[ \int \csc \left (b x + a\right )^{3} \sqrt {\sin \left (2 \, b x + 2 \, a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 95, normalized size = 3.39 \[ \frac {4\,\sqrt {\sin \left (2\,a+2\,b\,x\right )}\,\left (4\,{\sin \left (\frac {a}{2}+\frac {b\,x}{2}\right )}^2-6\,{\sin \left (\frac {3\,a}{2}+\frac {3\,b\,x}{2}\right )}^2+2\,{\sin \left (\frac {5\,a}{2}+\frac {5\,b\,x}{2}\right )}^2\right )}{3\,b\,\left (30\,{\sin \left (a+b\,x\right )}^2-12\,{\sin \left (2\,a+2\,b\,x\right )}^2+2\,{\sin \left (3\,a+3\,b\,x\right )}^2\right )} \]
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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