Optimal. Leaf size=20 \[ \frac {x}{2 a}+\frac {\cos (x) \sin (x)}{2 a} \]
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Rubi [A]
time = 0.03, 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 = {3254, 2715, 8}
\begin {gather*} \frac {x}{2 a}+\frac {\sin (x) \cos (x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3254
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 \begin {gather*} \frac {\frac {x}{2}+\frac {1}{4} \sin (2 x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 23, normalized size = 1.15
method | result | size |
risch | \(\frac {x}{2 a}+\frac {\sin \left (2 x \right )}{4 a}\) | \(17\) |
default | \(\frac {\frac {\tan \left (x \right )}{2 \left (\tan ^{2}\left (x \right )\right )+2}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}}{a}\) | \(23\) |
norman | \(\frac {\frac {x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {\tan \left (\frac {x}{2}\right )}{a}+\frac {2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {\tan ^{9}\left (\frac {x}{2}\right )}{a}-\frac {x}{2 a}-\frac {3 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {3 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}+\frac {x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{4} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 21, normalized size = 1.05 \begin {gather*} \frac {x}{2 \, a} + \frac {\tan \left (x\right )}{2 \, {\left (a \tan \left (x\right )^{2} + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 12, normalized size = 0.60 \begin {gather*} \frac {\cos \left (x\right ) \sin \left (x\right ) + x}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs.
\(2 (14) = 28\).
time = 1.46, size = 153, normalized size = 7.65 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 24, normalized size = 1.20 \begin {gather*} \frac {\arctan \left (\tan \left (x\right )\right )}{2 \, a} + \frac {\tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )} a} \end {gather*}
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 \begin {gather*} \frac {2\,x+\sin \left (2\,x\right )}{4\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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