Optimal. Leaf size=61 \[ \frac{\sec (a+b x)}{b \sqrt{\csc (a+b x)}}+\frac{\sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b} \]
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Rubi [A] time = 0.0484402, antiderivative size = 61, 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 = {2626, 3771, 2641} \[ \frac{\sec (a+b x)}{b \sqrt{\csc (a+b x)}}+\frac{\sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2626
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\csc (a+b x)} \sec ^2(a+b x) \, dx &=\frac{\sec (a+b x)}{b \sqrt{\csc (a+b x)}}+\frac{1}{2} \int \sqrt{\csc (a+b x)} \, dx\\ &=\frac{\sec (a+b x)}{b \sqrt{\csc (a+b x)}}+\frac{1}{2} \left (\sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx\\ &=\frac{\sec (a+b x)}{b \sqrt{\csc (a+b x)}}+\frac{\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)}}{b}\\ \end{align*}
Mathematica [A] time = 0.147641, size = 49, normalized size = 0.8 \[ \frac{\sec (a+b x)+\frac{F\left (\left .\frac{1}{4} (2 a+2 b x-\pi )\right |2\right )}{\sqrt{\sin (a+b x)}}}{b \sqrt{\csc (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.664, size = 123, normalized size = 2. \begin{align*}{\frac{1}{2\,\cos \left ( bx+a \right ) b}\sqrt{ \left ( \cos \left ( bx+a \right ) \right ) ^{2}\sin \left ( bx+a \right ) } \left ( \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 ) +2\,\sin \left ( bx+a \right ) \right ){\frac{1}{\sqrt{-\sin \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) -1 \right ) \left ( \sin \left ( bx+a \right ) +1 \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 \sqrt{\csc \left (b x + a\right )} \sec \left (b x + a\right )^{2}\,{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 (\sqrt{\csc \left (b x + a\right )} \sec \left (b x + a\right )^{2}, 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 \sqrt{\csc{\left (a + b x \right )}} \sec ^{2}{\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 \sqrt{\csc \left (b x + a\right )} \sec \left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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