Optimal. Leaf size=60 \[ \frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]
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Rubi [A] time = 0.0661629, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2815} \[ \frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right )|-\frac{1}{5}\right )}{\sqrt{5} d} \]
Antiderivative was successfully verified.
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Rule 2815
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx &=\frac{2 \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{3-2 \cos (c+d x)}}{\sqrt{\cos (c+d x)}}\right )|-\frac{1}{5}\right ) \sqrt{-\tan ^2(c+d x)}}{\sqrt{5} d}\\ \end{align*}
Mathematica [B] time = 1.02886, size = 144, normalized size = 2.4 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{(3-2 \cos (c+d x)) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}}{\sqrt{3}}\right )\right |6\right )}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.494, size = 125, normalized size = 2.1 \begin{align*}{\frac{\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{d \left ( -3+2\,\cos \left ( dx+c \right ) \right ) \left ( -1+\cos \left ( dx+c \right ) \right ) ^{2}} \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}}\sqrt{3-2\,\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{-3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }},i\sqrt{5} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \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{-2 \, \cos \left (d x + c\right ) + 3} \sqrt{\cos \left (d x + c\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{-2 \, \cos \left (d x + c\right ) + 3} \sqrt{\cos \left (d x + c\right )}}{2 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\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{3 - 2 \cos{\left (c + d x \right )}} \sqrt{\cos{\left (c + d 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{-2 \, \cos \left (d x + c\right ) + 3} \sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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