Optimal. Leaf size=48 \[ \frac{2 \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right )}{3 b}-\frac{\sqrt{\sin (2 a+2 b x)} \csc ^2(a+b x)}{3 b} \]
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Rubi [A] time = 0.0367026, antiderivative size = 48, 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, 2641} \[ \frac{2 F\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{3 b}-\frac{\sqrt{\sin (2 a+2 b x)} \csc ^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4300
Rule 2641
Rubi steps
\begin{align*} \int \frac{\csc ^2(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx &=-\frac{\csc ^2(a+b x) \sqrt{\sin (2 a+2 b x)}}{3 b}+\frac{2}{3} \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=\frac{2 F\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{3 b}-\frac{\csc ^2(a+b x) \sqrt{\sin (2 a+2 b x)}}{3 b}\\ \end{align*}
Mathematica [A] time = 0.951406, size = 82, normalized size = 1.71 \[ -\frac{\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) \text{EllipticF}\left (\sin ^{-1}(\cos (a+b x)-\sin (a+b x)),\frac{1}{2}\right )}{\sqrt{\sin (2 (a+b x))+1}}+\sqrt{\sin (2 (a+b x))} \csc ^2(a+b x)}{3 b} \]
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 ) ^{2}{\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 )^{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 (\frac{\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 \frac{\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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