Optimal. Leaf size=22 \[ \frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a} \]
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Rubi [A] time = 0.0434152, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3175, 3768, 3770} \[ \frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\tan (x) \sec (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec (x)}{a-a \sin ^2(x)} \, dx &=\frac{\int \sec ^3(x) \, dx}{a}\\ &=\frac{\sec (x) \tan (x)}{2 a}+\frac{\int \sec (x) \, dx}{2 a}\\ &=\frac{\tanh ^{-1}(\sin (x))}{2 a}+\frac{\sec (x) \tan (x)}{2 a}\\ \end{align*}
Mathematica [B] time = 0.0410465, size = 45, normalized size = 2.05 \[ \frac{\tan (x) \sec (x)-\log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )+\log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right )}{2 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 44, normalized size = 2. \begin{align*} -{\frac{1}{4\,a \left ( -1+\sin \left ( x \right ) \right ) }}-{\frac{\ln \left ( -1+\sin \left ( x \right ) \right ) }{4\,a}}-{\frac{1}{4\,a \left ( 1+\sin \left ( x \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( x \right ) \right ) }{4\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.995265, size = 50, normalized size = 2.27 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{4 \, a} - \frac{\sin \left (x\right )}{2 \,{\left (a \sin \left (x\right )^{2} - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.97521, size = 113, normalized size = 5.14 \begin{align*} \frac{\cos \left (x\right )^{2} \log \left (\sin \left (x\right ) + 1\right ) - \cos \left (x\right )^{2} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, \sin \left (x\right )}{4 \, a \cos \left (x\right )^{2}} \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*} - \frac{\int \frac{\sec{\left (x \right )}}{\sin ^{2}{\left (x \right )} - 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12035, size = 51, normalized size = 2.32 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{4 \, a} - \frac{\sin \left (x\right )}{2 \,{\left (\sin \left (x\right )^{2} - 1\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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