Optimal. Leaf size=82 \[ -\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d} \]
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Rubi [A] time = 0.114625, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2817, 2815} \[ -\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \sqrt{-\tan ^2(c+d x)} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{-2 \cos (c+d x)-3}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2817
Rule 2815
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-2 \cos (c+d x)} \sqrt{\cos (c+d x)}} \, dx &=\frac{\sqrt{-\cos (c+d x)} \int \frac{1}{\sqrt{-3-2 \cos (c+d x)} \sqrt{-\cos (c+d x)}} \, dx}{\sqrt{\cos (c+d x)}}\\ &=-\frac{2 \sqrt{-\cos (c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\left .\sin ^{-1}\left (\frac{\sqrt{-3-2 \cos (c+d x)}}{\sqrt{5} \sqrt{-\cos (c+d x)}}\right )\right |-5\right ) \sqrt{-\tan ^2(c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 1.0306, size = 153, normalized size = 1.87 \[ \frac{4 \sin ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\cot ^2\left (\frac{1}{2} (c+d x)\right )} \csc (c+d x) \sqrt{-\cos (c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right )} \sqrt{(2 \cos (c+d x)+3) \csc ^2\left (\frac{1}{2} (c+d x)\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{5}{3}} \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)-1}}\right )|\frac{6}{5}\right )}{\sqrt{5} d \sqrt{-2 \cos (c+d x)-3} \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.458, size = 137, normalized size = 1.7 \begin{align*}{\frac{-{\frac{i}{5}}\sqrt{5}\sqrt{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}\sqrt{10}}{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{{\frac{3+2\,\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it EllipticF} \left ({\frac{{\frac{i}{5}} \left ( -1+\cos \left ( dx+c \right ) \right ) \sqrt{5}}{\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{- 2 \cos{\left (c + d x \right )} - 3} \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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