Optimal. Leaf size=17 \[ -\frac {\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt {-\sin ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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*}
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Mathematica [A] time = 0.01, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\cos \relax (x)^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 34, normalized size = 2.00 \[ -\frac {\sin \relax (x ) \sqrt {-\left (\cos ^{2}\relax (x )\right )}\, \arctan \left (\frac {1}{\sqrt {-\left (\cos ^{2}\relax (x )\right )}}\right )}{\cos \relax (x ) \sqrt {-\left (\sin ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 17, normalized size = 1.00 \[ -\arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) + \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\sqrt {{\cos \relax (x)}^2-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\cos ^{2}{\relax (x )} - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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