Optimal. Leaf size=43 \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{c \sin (a+b x)}} \]
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Rubi [A] time = 0.0179524, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2642, 2641} \[ \frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b \sqrt{c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{c \sin (a+b x)}} \, dx &=\frac{\sqrt{\sin (a+b x)} \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{\sqrt{c \sin (a+b x)}}\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{b \sqrt{c \sin (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0265183, size = 42, normalized size = 0.98 \[ -\frac{2 \sqrt{\sin (a+b x)} F\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )}{b \sqrt{c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 74, normalized size = 1.7 \begin{align*} -{\frac{1}{b\cos \left ( bx+a \right ) }\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{1}{\sqrt{c\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{1}{\sqrt{c \sin \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{\sqrt{c \sin \left (b x + a\right )}}{c \sin \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{1}{\sqrt{c \sin{\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{1}{\sqrt{c \sin \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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