Optimal. Leaf size=6 \[ \frac {\tan (x)}{a^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 6, 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, 3852, 8}
\begin {gather*} \frac {\tan (x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3254
Rule 3852
Rubi steps
\begin {align*} \int \frac {\cos ^2(x)}{\left (a-a \sin ^2(x)\right )^2} \, dx &=\frac {\int \sec ^2(x) \, dx}{a^2}\\ &=-\frac {\text {Subst}(\int 1 \, dx,x,-\tan (x))}{a^2}\\ &=\frac {\tan (x)}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} \frac {\tan (x)}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 7, normalized size = 1.17
method | result | size |
default | \(\frac {\tan \left (x \right )}{a^{2}}\) | \(7\) |
risch | \(\frac {2 i}{a^{2} \left ({\mathrm e}^{2 i x}+1\right )}\) | \(16\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {4 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}}{a \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 6, normalized size = 1.00 \begin {gather*} \frac {\tan \left (x\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 10, normalized size = 1.67 \begin {gather*} \frac {\sin \left (x\right )}{a^{2} \cos \left (x\right )} \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. 20 vs.
\(2 (5) = 10\).
time = 1.66, size = 20, normalized size = 3.33 \begin {gather*} - \frac {2 \tan {\left (\frac {x}{2} \right )}}{a^{2} \tan ^{2}{\left (\frac {x}{2} \right )} - a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 6, normalized size = 1.00 \begin {gather*} \frac {\tan \left (x\right )}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.78, size = 6, normalized size = 1.00 \begin {gather*} \frac {\mathrm {tan}\left (x\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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