Optimal. Leaf size=16 \[ \frac{\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \cos ^2(x)}} \]
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Rubi [A] time = 0.0306739, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3176, 3207, 3770} \[ \frac{\cos (x) \tanh ^{-1}(\sin (x))}{\sqrt{a \cos ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a-a \sin ^2(x)}} \, dx &=\int \frac{1}{\sqrt{a \cos ^2(x)}} \, dx\\ &=\frac{\cos (x) \int \sec (x) \, dx}{\sqrt{a \cos ^2(x)}}\\ &=\frac{\tanh ^{-1}(\sin (x)) \cos (x)}{\sqrt{a \cos ^2(x)}}\\ \end{align*}
Mathematica [B] time = 0.0194808, size = 46, normalized size = 2.88 \[ \frac{\cos (x) \left (\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )\right )}{\sqrt{a \cos ^2(x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.11, size = 20, normalized size = 1.3 \begin{align*}{\frac{\cos \left ( x \right ){\it InverseJacobiAM} \left ( x,1 \right ) }{{\it csgn} \left ( \cos \left ( x \right ) \right ) }{\frac{1}{\sqrt{a \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58033, size = 51, normalized size = 3.19 \begin{align*} \frac{\log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) - \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right )}{2 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65344, size = 182, normalized size = 11.38 \begin{align*} \left [-\frac{\sqrt{a \cos \left (x\right )^{2}} \log \left (-\frac{\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right )}{2 \, a \cos \left (x\right )}, -\frac{\sqrt{-a} \arctan \left (\frac{\sqrt{a \cos \left (x\right )^{2}} \sqrt{-a} \sin \left (x\right )}{a \cos \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 )} + a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-a \sin \left (x\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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