Optimal. Leaf size=15 \[ -\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{\sin ^2(x)}} \]
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Rubi [A] time = 0.0198256, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 = 0.0153835, size = 28, normalized size = 1.87 \[ \frac{\sin (x) \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{\sqrt{\sin ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.336, size = 14, normalized size = 0.9 \begin{align*} -{{\it Artanh} \left ( \cos \left ( x \right ) \right ) \sin \left ( x \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 [B] time = 1.63955, size = 47, normalized size = 3.13 \begin{align*} \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61243, size = 77, normalized size = 5.13 \begin{align*} -\frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\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{1 - \cos ^{2}{\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 )^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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