Optimal. Leaf size=17 \[ -\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \sin ^2(x)}} \]
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Rubi [A] time = 0.0113021, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3207, 3770} \[ -\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \sin ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \sin ^2(x)}} \, dx &=\frac{\sin (x) \int \csc (x) \, dx}{\sqrt{a \sin ^2(x)}}\\ &=-\frac{\tanh ^{-1}(\cos (x)) \sin (x)}{\sqrt{a \sin ^2(x)}}\\ \end{align*}
Mathematica [A] time = 0.0148339, size = 30, normalized size = 1.76 \[ \frac{\sin (x) \left (\log \left (\sin \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{\sqrt{a \sin ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.599, size = 49, normalized size = 2.9 \begin{align*} -{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}+a}{\sin \left ( x \right ) }} \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59911, size = 35, normalized size = 2.06 \begin{align*} \frac{\sqrt{-a}{\left (\arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71577, size = 193, normalized size = 11.35 \begin{align*} \left [\frac{\sqrt{-a \cos \left (x\right )^{2} + a} \log \left (-\frac{\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{2 \, a \sin \left (x\right )}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{-a \cos \left (x\right )^{2} + a} \sqrt{-a} \cos \left (x\right )}{a \sin \left (x\right )}\right )}{a}\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{1}{\sqrt{a \sin ^{2}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23031, size = 20, normalized size = 1.18 \begin{align*} \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{\sqrt{a} \mathrm{sgn}\left (\sin \left (x\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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