Optimal. Leaf size=20 \[ \tanh ^{-1}\left (\sqrt {\sin ^2(x)}\right )-\sqrt {\sin ^2(x)} \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3176, 3205, 50, 63, 206} \[ \tanh ^{-1}\left (\sqrt {\sin ^2(x)}\right )-\sqrt {\sin ^2(x)} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 206
Rule 3176
Rule 3205
Rubi steps
\begin {align*} \int \sqrt {1-\cos ^2(x)} \tan (x) \, dx &=\int \sqrt {\sin ^2(x)} \tan (x) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1-x} \, dx,x,\sin ^2(x)\right )\\ &=-\sqrt {\sin ^2(x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {x}} \, dx,x,\sin ^2(x)\right )\\ &=-\sqrt {\sin ^2(x)}+\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin ^2(x)}\right )\\ &=\tanh ^{-1}\left (\sqrt {\sin ^2(x)}\right )-\sqrt {\sin ^2(x)}\\ \end {align*}
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Mathematica [B] time = 0.04, size = 47, normalized size = 2.35 \[ \sqrt {\sin ^2(x)} (-\csc (x)) \left (\sin (x)+\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 21, normalized size = 1.05 \[ \frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \relax (x) + 1\right ) - \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.37, size = 45, normalized size = 2.25 \[ -\sqrt {-\cos \relax (x)^{2} + 1} + \frac {1}{2} \, \log \left (\sqrt {-\cos \relax (x)^{2} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-\sqrt {-\cos \relax (x)^{2} + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.99, size = 17, normalized size = 0.85 \[ -\sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}+\arctanh \left (\frac {2}{\sqrt {2-2 \cos \left (2 x \right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 47, normalized size = 2.35 \[ \frac {1}{2} \, \left (-1\right )^{2 \, \sin \relax (x)} \log \left (-\frac {\sin \relax (x)}{\sin \relax (x) + 1}\right ) + \frac {1}{2} \, \left (-1\right )^{2 \, \sin \relax (x)} \log \left (-\frac {\sin \relax (x)}{\sin \relax (x) - 1}\right ) - \sqrt {\sin \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \mathrm {tan}\relax (x)\,\sqrt {1-{\cos \relax (x)}^2} \,d x \]
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 \[ \int \sqrt {- \left (\cos {\relax (x )} - 1\right ) \left (\cos {\relax (x )} + 1\right )} \tan {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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