Optimal. Leaf size=9 \[ \tanh ^{-1}\left (\sqrt{\sin ^2(x)}\right ) \]
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Rubi [A] time = 0.0529505, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3176, 3205, 63, 206} \[ \tanh ^{-1}\left (\sqrt{\sin ^2(x)}\right ) \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3205
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\tan (x)}{\sqrt{1-\cos ^2(x)}} \, dx &=\int \frac{\tan (x)}{\sqrt{\sin ^2(x)}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{x}} \, dx,x,\sin ^2(x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\sin ^2(x)}\right )\\ &=\tanh ^{-1}\left (\sqrt{\sin ^2(x)}\right )\\ \end{align*}
Mathematica [B] time = 0.0211138, size = 44, normalized size = 4.89 \[ \frac{\sin (x) \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right )}{\sqrt{\sin ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.21, size = 8, normalized size = 0.9 \begin{align*}{\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 = 1.46874, size = 53, normalized size = 5.89 \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7012, size = 59, normalized size = 6.56 \begin{align*} \frac{1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (-\sin \left (x\right ) + 1\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{\tan{\left (x \right )}}{\sqrt{- \left (\cos{\left (x \right )} - 1\right ) \left (\cos{\left (x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17142, size = 45, normalized size = 5. \begin{align*} \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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