Optimal. Leaf size=17 \[ -\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{-\sin ^2(x)}} \]
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Rubi [A] time = 0.0184225, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3176, 3207, 3770} \[ -\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{-\sin ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+\cos ^2(x)}} \, dx &=\int \frac{1}{\sqrt{-\sin ^2(x)}} \, dx\\ &=\frac{\sin (x) \int \csc (x) \, dx}{\sqrt{-\sin ^2(x)}}\\ &=-\frac{\tanh ^{-1}(\cos (x)) \sin (x)}{\sqrt{-\sin ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0102452, size = 30, normalized size = 1.76 \[ \frac{\sin (x) \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{\sqrt{-\sin ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.233, size = 34, normalized size = 2. \begin{align*} -{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}}\arctan \left ({\frac{1}{\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}}}} \right ){\frac{1}{\sqrt{- \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58404, size = 23, normalized size = 1.35 \begin{align*} -\arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) + \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 ^{2}{\left (x \right )} - 1}}\, 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 )^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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