Optimal. Leaf size=67 \[ \frac{2 \cos (a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b} \]
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Rubi [A] time = 0.0505877, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2628, 3771, 2641} \[ \frac{2 \cos (a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 2628
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \cos ^2(a+b x) \sqrt{\csc (a+b x)} \, dx &=\frac{2 \cos (a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{2}{3} \int \sqrt{\csc (a+b x)} \, dx\\ &=\frac{2 \cos (a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{1}{3} \left (2 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{2 \cos (a+b x)}{3 b \sqrt{\csc (a+b x)}}+\frac{4 \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{3 b}\\ \end{align*}
Mathematica [A] time = 0.101008, size = 53, normalized size = 0.79 \[ \frac{\sqrt{\csc (a+b x)} \left (\sin (2 (a+b x))-4 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.098, size = 88, normalized size = 1.3 \begin{align*}{\frac{1}{\cos \left ( bx+a \right ) b} \left ({\frac{2}{3}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }+{\frac{2\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) }{3}} \right ){\frac{1}{\sqrt{\sin \left ( bx+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 \cos \left (b x + a\right )^{2} \sqrt{\csc \left (b x + 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 (\cos \left (b x + a\right )^{2} \sqrt{\csc \left (b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos ^{2}{\left (a + b x \right )} \sqrt{\csc{\left (a + b x \right )}}\, dx \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 \cos \left (b x + a\right )^{2} \sqrt{\csc \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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