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 = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.04, 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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fricas [B] time = 0.45, size = 37, normalized size = 1.68 \[ \frac {\cos \relax (x)^{2} \log \left (\sin \relax (x) + 1\right ) - \cos \relax (x)^{2} \log \left (-\sin \relax (x) + 1\right ) + 2 \, \sin \relax (x)}{4 \, a \cos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 38, normalized size = 1.73 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{4 \, a} - \frac {\log \left (-\sin \relax (x) + 1\right )}{4 \, a} - \frac {\sin \relax (x)}{2 \, {\left (\sin \relax (x)^{2} - 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 44, normalized size = 2.00 \[ -\frac {1}{4 a \left (-1+\sin \relax (x )\right )}-\frac {\ln \left (-1+\sin \relax (x )\right )}{4 a}-\frac {1}{4 a \left (1+\sin \relax (x )\right )}+\frac {\ln \left (1+\sin \relax (x )\right )}{4 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 37, normalized size = 1.68 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{4 \, a} - \frac {\log \left (\sin \relax (x) - 1\right )}{4 \, a} - \frac {\sin \relax (x)}{2 \, {\left (a \sin \relax (x)^{2} - a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.87, size = 25, normalized size = 1.14 \[ \frac {\mathrm {atanh}\left (\sin \relax (x)\right )}{2\,a}+\frac {\sin \relax (x)}{2\,\left (a-a\,{\sin \relax (x)}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {\sec {\relax (x )}}{\sin ^{2}{\relax (x )} - 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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