Optimal. Leaf size=44 \[ -\frac{2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \csc ^2(a+b x)}{b} \]
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Rubi [A] time = 0.0361731, antiderivative size = 44, 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, 2639} \[ -\frac{2 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \csc ^2(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 2639
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx &=-\frac{\csc ^2(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{b}-2 \int \sqrt{\sin (2 a+2 b x)} \, dx\\ &=-\frac{2 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{b}-\frac{\csc ^2(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.12709, size = 37, normalized size = 0.84 \[ -\frac{2 \left (E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )+\sqrt{\sin (2 (a+b x))} \cot (a+b x)\right )}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.096, size = 176, normalized size = 4. \begin{align*}{\frac{1}{b\cos \left ( 2\,bx+2\,a \right ) } \left ( 2\,\sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},1/2\,\sqrt{2} \right ) -\sqrt{\sin \left ( 2\,bx+2\,a \right ) +1}\sqrt{-2\,\sin \left ( 2\,bx+2\,a \right ) +2}\sqrt{-\sin \left ( 2\,bx+2\,a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( 2\,bx+2\,a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) -2\, \left ( \cos \left ( 2\,bx+2\,a \right ) \right ) ^{2}-2\,\cos \left ( 2\,bx+2\,a \right ) \right ){\frac{1}{\sqrt{\sin \left ( 2\,bx+2\,a \right ) }}}} \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 )^{2} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\csc \left (b x + a\right )^{2} \sqrt{\sin \left (2 \, b x + 2 \, a\right )}, x\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 )^{2} \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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