Optimal. Leaf size=18 \[ \frac {\tan (x)}{a}+\frac {\tan ^3(x)}{3 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3254, 3852}
\begin {gather*} \frac {\tan ^3(x)}{3 a}+\frac {\tan (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3254
Rule 3852
Rubi steps
\begin {align*} \int \frac {\sec ^2(x)}{a-a \sin ^2(x)} \, dx &=\frac {\int \sec ^4(x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (x)\right )}{a}\\ &=\frac {\tan (x)}{a}+\frac {\tan ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.17 \begin {gather*} \frac {\frac {2 \tan (x)}{3}+\frac {1}{3} \sec ^2(x) \tan (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 14, normalized size = 0.78
method | result | size |
default | \(\frac {\frac {\left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right )}{a}\) | \(14\) |
risch | \(\frac {4 i \left (3 \,{\mathrm e}^{2 i x}+1\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3} a}\) | \(25\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {4 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {2 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{3}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 14, normalized size = 0.78 \begin {gather*} \frac {\tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 19, normalized size = 1.06 \begin {gather*} \frac {{\left (2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{3 \, a \cos \left (x\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\sec ^{2}{\left (x \right )}}{\sin ^{2}{\left (x \right )} - 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 14, normalized size = 0.78 \begin {gather*} \frac {\tan \left (x\right )^{3} + 3 \, \tan \left (x\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.87, size = 13, normalized size = 0.72 \begin {gather*} \frac {\mathrm {tan}\left (x\right )\,\left ({\mathrm {tan}\left (x\right )}^2+3\right )}{3\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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