Optimal. Leaf size=42 \[ \frac{\tan (x)}{2 a \sqrt{a \cos ^2(x)}}+\frac{\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt{a \cos ^2(x)}} \]
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Rubi [A] time = 0.0374599, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3176, 3204, 3207, 3770} \[ \frac{\tan (x)}{2 a \sqrt{a \cos ^2(x)}}+\frac{\cos (x) \tanh ^{-1}(\sin (x))}{2 a \sqrt{a \cos ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3204
Rule 3207
Rule 3770
Rubi steps
\begin{align*} \int \frac{1}{\left (a-a \sin ^2(x)\right )^{3/2}} \, dx &=\int \frac{1}{\left (a \cos ^2(x)\right )^{3/2}} \, dx\\ &=\frac{\tan (x)}{2 a \sqrt{a \cos ^2(x)}}+\frac{\int \frac{1}{\sqrt{a \cos ^2(x)}} \, dx}{2 a}\\ &=\frac{\tan (x)}{2 a \sqrt{a \cos ^2(x)}}+\frac{\cos (x) \int \sec (x) \, dx}{2 a \sqrt{a \cos ^2(x)}}\\ &=\frac{\tanh ^{-1}(\sin (x)) \cos (x)}{2 a \sqrt{a \cos ^2(x)}}+\frac{\tan (x)}{2 a \sqrt{a \cos ^2(x)}}\\ \end{align*}
Mathematica [B] time = 0.0651455, size = 91, normalized size = 2.17 \[ -\frac{\cos (x) \left (-2 \sin (x)+\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\cos (2 x) \left (\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )-\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )\right )}{4 \left (a \cos ^2(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.994, size = 70, normalized size = 1.7 \begin{align*}{\frac{1}{2\,\sin \left ( x \right ) \cos \left ( x \right ) }\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( \ln \left ( 2\,{\frac{\sqrt{a}\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}}+a}{\cos \left ( x \right ) }} \right ) \left ( \cos \left ( x \right ) \right ) ^{2}a+\sqrt{a}\sqrt{a \left ( \sin \left ( x \right ) \right ) ^{2}} \right ){a}^{-{\frac{5}{2}}}{\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 = 1.6278, size = 410, normalized size = 9.76 \begin{align*} \frac{4 \,{\left (\sin \left (3 \, x\right ) - \sin \left (x\right )\right )} \cos \left (4 \, x\right ) +{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) + 1\right ) - 4 \,{\left (\cos \left (3 \, x\right ) - \cos \left (x\right )\right )} \sin \left (4 \, x\right ) + 4 \,{\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \sin \left (3 \, x\right ) - 8 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 8 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \sin \left (x\right ) - 4 \, \sin \left (x\right )}{4 \,{\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \,{\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75947, size = 124, normalized size = 2.95 \begin{align*} -\frac{\sqrt{a \cos \left (x\right )^{2}}{\left (\cos \left (x\right )^{2} \log \left (-\frac{\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right ) - 2 \, \sin \left (x\right )\right )}}{4 \, a^{2} \cos \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23198, size = 59, normalized size = 1.4 \begin{align*} -\frac{\sqrt{a} \log \left ({\left | -\sqrt{a} \tan \left (x\right ) + \sqrt{a \tan \left (x\right )^{2} + a} \right |}\right ) - \sqrt{a \tan \left (x\right )^{2} + a} \tan \left (x\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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