Optimal. Leaf size=67 \[ \frac{2 \cos (a+b x)}{5 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b} \]
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Rubi [A] time = 0.0483108, 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, 2639} \[ \frac{2 \cos (a+b x)}{5 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{5 b} \]
Antiderivative was successfully verified.
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Rule 2628
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{\cos ^2(a+b x)}{\sqrt{\csc (a+b x)}} \, dx &=\frac{2 \cos (a+b x)}{5 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{2}{5} \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx\\ &=\frac{2 \cos (a+b x)}{5 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{1}{5} \left (2 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx\\ &=\frac{2 \cos (a+b x)}{5 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{5 b}\\ \end{align*}
Mathematica [A] time = 0.138767, size = 61, normalized size = 0.91 \[ -\frac{2 \sqrt{\csc (a+b x)} \left (2 \sqrt{\sin (a+b x)} E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )-\sin ^2(a+b x) \cos (a+b x)\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.122, size = 142, normalized size = 2.1 \begin{align*}{\frac{1}{\cos \left ( bx+a \right ) b} \left ( -{\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{5}}-{\frac{4}{5}\sqrt{\sin \left ( bx+a \right ) +1}\sqrt{-2\,\sin \left ( bx+a \right ) +2}\sqrt{-\sin \left ( bx+a \right ) }{\it EllipticE} \left ( \sqrt{\sin \left ( bx+a \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }+{\frac{2}{5}\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 ) } \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 \frac{\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 (\frac{\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 \frac{\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 \frac{\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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