Optimal. Leaf size=55 \[ -\frac{\sqrt{\sin (2 a+2 b x)} \csc ^3(a+b x)}{5 b}-\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{5 b} \]
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Rubi [A] time = 0.0533406, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4300, 4292} \[ -\frac{\sqrt{\sin (2 a+2 b x)} \csc ^3(a+b x)}{5 b}-\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 4292
Rubi steps
\begin{align*} \int \frac{\csc ^3(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx &=-\frac{\csc ^3(a+b x) \sqrt{\sin (2 a+2 b x)}}{5 b}+\frac{4}{5} \int \frac{\csc (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{4 \csc (a+b x) \sqrt{\sin (2 a+2 b x)}}{5 b}-\frac{\csc ^3(a+b x) \sqrt{\sin (2 a+2 b x)}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.0919217, size = 35, normalized size = 0.64 \[ -\frac{\sqrt{\sin (2 (a+b x))} \csc (a+b x) \left (\csc ^2(a+b x)+4\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( \csc \left ( bx+a \right ) \right ) ^{3}{\frac{1}{\sqrt{\sin \left ( 2\,bx+2\,a \right ) }}}}\, dx \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{\csc \left (b x + a\right )^{3}}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.498063, size = 198, normalized size = 3.6 \begin{align*} -\frac{\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 )} + 4 \,{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{5 \,{\left (b \cos \left (b x + a\right )^{2} - b\right )} \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 \frac{\csc \left (b x + a\right )^{3}}{\sqrt{\sin \left (2 \, b x + 2 \, a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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