Optimal. Leaf size=52 \[ \frac{\sin (x) \tan ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\sin (x) \tanh ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}} \]
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Rubi [A] time = 0.0780825, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {4397, 4400, 2601, 2565, 329, 298, 203, 206} \[ \frac{\sin (x) \tan ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\sin (x) \tanh ^{-1}\left (\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}} \]
Antiderivative was successfully verified.
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Rule 4397
Rule 4400
Rule 2601
Rule 2565
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-\cos (x)+\sec (x)}} \, dx &=\int \frac{1}{\sqrt{\sin (x) \tan (x)}} \, dx\\ &=\frac{\left (\sqrt{\sin (x)} \sqrt{\tan (x)}\right ) \int \frac{1}{\sqrt{\sin (x)} \sqrt{\tan (x)}} \, dx}{\sqrt{\sin (x) \tan (x)}}\\ &=\frac{\sin (x) \int \sqrt{\cos (x)} \csc (x) \, dx}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=-\frac{\sin (x) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-x^2} \, dx,x,\cos (x)\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=-\frac{(2 \sin (x)) \operatorname{Subst}\left (\int \frac{x^2}{1-x^4} \, dx,x,\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=-\frac{\sin (x) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}+\frac{\sin (x) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\cos (x)}\right )}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ &=\frac{\tan ^{-1}\left (\sqrt{\cos (x)}\right ) \sin (x)}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}-\frac{\tanh ^{-1}\left (\sqrt{\cos (x)}\right ) \sin (x)}{\sqrt{\cos (x)} \sqrt{\sin (x) \tan (x)}}\\ \end{align*}
Mathematica [A] time = 0.245932, size = 43, normalized size = 0.83 \[ \frac{\cos (x) \cot (x) \sqrt{\sin (x) \tan (x)} \left (\tan ^{-1}\left (\sqrt [4]{\cos ^2(x)}\right )-\tanh ^{-1}\left (\sqrt [4]{\cos ^2(x)}\right )\right )}{\cos ^2(x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.108, size = 105, normalized size = 2. \begin{align*} -{\frac{1+\cos \left ( x \right ) }{2\,\sin \left ( x \right ) } \left ( \arctan \left ({\frac{1}{2}{\frac{1}{\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}}}} \right ) +\ln \left ( -{\frac{1}{ \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}- \left ( \cos \left ( x \right ) \right ) ^{2}+2\,\cos \left ( x \right ) -2\,\sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}-1 \right ) } \right ) \right ) \sqrt{-{\frac{\cos \left ( x \right ) }{ \left ( 1+\cos \left ( x \right ) \right ) ^{2}}}}\sqrt{{\frac{1- \left ( \cos \left ( x \right ) \right ) ^{2}}{\cos \left ( x \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{-\cos \left (x\right ) + \sec \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.32496, size = 228, normalized size = 4.38 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) + \frac{1}{2} \, \log \left (\frac{{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 2 \, \sqrt{-\frac{\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\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{- \cos{\left (x \right )} + \sec{\left (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{-\cos \left (x\right ) + \sec \left (x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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