Optimal. Leaf size=15 \[ -\frac {\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt {\sin ^2(x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 = 0.02, 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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fricas [A] time = 1.35, size = 19, normalized size = 1.27 \[ -\frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
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 [A] time = 0.71, size = 14, normalized size = 0.93 \[ -\frac {2 \arctanh \left (\cos \relax (x )\right ) \sin \relax (x )}{\sqrt {2-2 \cos \left (2 x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.13, size = 35, normalized size = 2.33 \[ \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{\sqrt {1-{\cos \relax (x)}^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {1 - \cos ^{2}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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