Optimal. Leaf size=20 \[ \frac {x}{2 a}+\frac {\sin (x) \cos (x)}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3175, 2635, 8} \[ \frac {x}{2 a}+\frac {\sin (x) \cos (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3175
Rubi steps
\begin {align*} \int \frac {\cos ^4(x)}{a-a \sin ^2(x)} \, dx &=\frac {\int \cos ^2(x) \, dx}{a}\\ &=\frac {\cos (x) \sin (x)}{2 a}+\frac {\int 1 \, dx}{2 a}\\ &=\frac {x}{2 a}+\frac {\cos (x) \sin (x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 18, normalized size = 0.90 \[ \frac {\frac {x}{2}+\frac {1}{4} \sin (2 x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 12, normalized size = 0.60 \[ \frac {\cos \relax (x) \sin \relax (x) + x}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 24, normalized size = 1.20 \[ \frac {\arctan \left (\tan \relax (x)\right )}{2 \, a} + \frac {\tan \relax (x)}{2 \, {\left (\tan \relax (x)^{2} + 1\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 25, normalized size = 1.25 \[ \frac {\tan \relax (x )}{2 a \left (\tan ^{2}\relax (x )+1\right )}+\frac {\arctan \left (\tan \relax (x )\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 21, normalized size = 1.05 \[ \frac {x}{2 \, a} + \frac {\tan \relax (x)}{2 \, {\left (a \tan \relax (x)^{2} + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.81, size = 13, normalized size = 0.65 \[ \frac {2\,x+\sin \left (2\,x\right )}{4\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.92, size = 153, normalized size = 7.65 \[ \frac {x \tan ^{4}{\left (\frac {x}{2} \right )}}{2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {2 x \tan ^{2}{\left (\frac {x}{2} \right )}}{2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {x}{2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {2 \tan ^{3}{\left (\frac {x}{2} \right )}}{2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {2 \tan {\left (\frac {x}{2} \right )}}{2 a \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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