Optimal. Leaf size=7 \[ \frac{\tanh ^{-1}(\sin (x))}{a^2} \]
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Rubi [A] time = 0.0393184, antiderivative size = 7, normalized size of antiderivative = 1., 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*}
Mathematica [B] time = 0.0035549, 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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Maple [A] time = 0.033, size = 8, normalized size = 1.1 \begin{align*}{\frac{{\it Artanh} \left ( \sin \left ( x \right ) \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.975028, size = 28, normalized size = 4. \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a^{2}} - \frac{\log \left (\sin \left (x\right ) - 1\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92245, size = 62, normalized size = 8.86 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right ) - \log \left (-\sin \left (x\right ) + 1\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 11.15, size = 22, normalized size = 3.14 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14169, size = 31, normalized size = 4.43 \begin{align*} \frac{\log \left (\sin \left (x\right ) + 1\right )}{2 \, a^{2}} - \frac{\log \left (-\sin \left (x\right ) + 1\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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