Optimal. Leaf size=20 \[ \frac {x}{2 a^2}+\frac {\cos (x) \sin (x)}{2 a^2} \]
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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^2}+\frac {\sin (x) \cos (x)}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3254
Rubi steps
\begin {align*} \int \frac {\cos ^6(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac {\int \cos ^2(x) \, dx}{a^2}\\ &=\frac {\cos (x) \sin (x)}{2 a^2}+\frac {\int 1 \, dx}{2 a^2}\\ &=\frac {x}{2 a^2}+\frac {\cos (x) \sin (x)}{2 a^2}\\ \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^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 23, normalized size = 1.15
method | result | size |
risch | \(\frac {x}{2 a^{2}}+\frac {\sin \left (2 x \right )}{4 a^{2}}\) | \(17\) |
default | \(\frac {\frac {\tan \left (x \right )}{2 \left (\tan ^{2}\left (x \right )\right )+2}+\frac {\arctan \left (\tan \left (x \right )\right )}{2}}{a^{2}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 25, normalized size = 1.25 \begin {gather*} \frac {\tan \left (x\right )}{2 \, {\left (a^{2} \tan \left (x\right )^{2} + a^{2}\right )}} + \frac {x}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 12, normalized size = 0.60 \begin {gather*} \frac {\cos \left (x\right ) \sin \left (x\right ) + x}{2 \, a^{2}} \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. 178 vs.
\(2 (17) = 34\).
time = 10.24, size = 178, normalized size = 8.90 \begin {gather*} \frac {x \tan ^{4}{\left (\frac {x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a^{2}} + \frac {2 x \tan ^{2}{\left (\frac {x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a^{2}} + \frac {x}{2 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a^{2}} - \frac {2 \tan ^{3}{\left (\frac {x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a^{2}} + \frac {2 \tan {\left (\frac {x}{2} \right )}}{2 a^{2} \tan ^{4}{\left (\frac {x}{2} \right )} + 4 a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 22, normalized size = 1.10 \begin {gather*} \frac {x}{2 \, a^{2}} + \frac {\tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.93, size = 13, normalized size = 0.65 \begin {gather*} \frac {2\,x+\sin \left (2\,x\right )}{4\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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