Optimal. Leaf size=7 \[ \frac {\tanh ^{-1}(\sin (x))}{a^2} \]
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Rubi [A] time = 0.04, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3175, 3770} \[ \frac {\tanh ^{-1}(\sin (x))}{a^2} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos ^3(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac {\int \sec (x) \, dx}{a^2}\\ &=\frac {\tanh ^{-1}(\sin (x))}{a^2}\\ \end {align*}
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Mathematica [B] time = 0.00, size = 37, normalized size = 5.29 \[ \frac {\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 20, normalized size = 2.86 \[ \frac {\log \left (\sin \relax (x) + 1\right ) - \log \left (-\sin \relax (x) + 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 23, normalized size = 3.29 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{2 \, a^{2}} - \frac {\log \left (-\sin \relax (x) + 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 8, normalized size = 1.14 \[ \frac {\arctanh \left (\sin \relax (x )\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 21, normalized size = 3.00 \[ \frac {\log \left (\sin \relax (x) + 1\right )}{2 \, a^{2}} - \frac {\log \left (\sin \relax (x) - 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 7, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (\sin \relax (x)\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.46, size = 22, normalized size = 3.14 \[ - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{a^{2}} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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