Optimal. Leaf size=20 \[ \tanh ^{-1}\left (\sqrt{\sin ^2(x)}\right )-\sqrt{\sin ^2(x)} \]
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Rubi [A] time = 0.0505617, antiderivative size = 20, normalized size of antiderivative = 1., 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 3176
Rule 3205
Rule 50
Rule 63
Rule 206
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*}
Mathematica [B] time = 0.0292728, 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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Maple [A] time = 0.413, size = 17, normalized size = 0.9 \begin{align*} -\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}+{\it Artanh} \left ({\frac{1}{\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.49755, size = 63, normalized size = 3.15 \begin{align*} \frac{1}{2} \, \left (-1\right )^{2 \, \sin \left (x\right )} \log \left (-\frac{\sin \left (x\right )}{\sin \left (x\right ) + 1}\right ) + \frac{1}{2} \, \left (-1\right )^{2 \, \sin \left (x\right )} \log \left (-\frac{\sin \left (x\right )}{\sin \left (x\right ) - 1}\right ) - \sqrt{\sin \left (x\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68795, size = 72, normalized size = 3.6 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) - \sin \left (x\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 \sqrt{- \left (\cos{\left (x \right )} - 1\right ) \left (\cos{\left (x \right )} + 1\right )} \tan{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15621, size = 61, normalized size = 3.05 \begin{align*} -\sqrt{-\cos \left (x\right )^{2} + 1} + \frac{1}{2} \, \log \left (\sqrt{-\cos \left (x\right )^{2} + 1} + 1\right ) - \frac{1}{2} \, \log \left (-\sqrt{-\cos \left (x\right )^{2} + 1} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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