Optimal. Leaf size=29 \[ \frac {\tan (x)}{a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan ^5(x)}{5 a} \]
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Rubi [A]
time = 0.03, antiderivative size = 29, 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 ^5(x)}{5 a}+\frac {2 \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 ^4(x)}{a-a \sin ^2(x)} \, dx &=\frac {\int \sec ^6(x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (x)\right )}{a}\\ &=\frac {\tan (x)}{a}+\frac {2 \tan ^3(x)}{3 a}+\frac {\tan ^5(x)}{5 a}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 31, normalized size = 1.07 \begin {gather*} \frac {\frac {8 \tan (x)}{15}+\frac {4}{15} \sec ^2(x) \tan (x)+\frac {1}{5} \sec ^4(x) \tan (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 20, normalized size = 0.69
method | result | size |
default | \(\frac {\frac {\left (\tan ^{5}\left (x \right )\right )}{5}+\frac {2 \left (\tan ^{3}\left (x \right )\right )}{3}+\tan \left (x \right )}{a}\) | \(20\) |
risch | \(\frac {16 i \left (10 \,{\mathrm e}^{4 i x}+5 \,{\mathrm e}^{2 i x}+1\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5} a}\) | \(32\) |
norman | \(\frac {-\frac {2 \tan \left (\frac {x}{2}\right )}{a}+\frac {8 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {116 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{15 a}+\frac {8 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {2 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{5}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 22, normalized size = 0.76 \begin {gather*} \frac {3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 25, normalized size = 0.86 \begin {gather*} \frac {{\left (8 \, \cos \left (x\right )^{4} + 4 \, \cos \left (x\right )^{2} + 3\right )} \sin \left (x\right )}{15 \, a \cos \left (x\right )^{5}} \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 ^{4}{\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 = 22, normalized size = 0.76 \begin {gather*} \frac {3 \, \tan \left (x\right )^{5} + 10 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.92, size = 21, normalized size = 0.72 \begin {gather*} \frac {\mathrm {tan}\left (x\right )\,\left (3\,{\mathrm {tan}\left (x\right )}^4+10\,{\mathrm {tan}\left (x\right )}^2+15\right )}{15\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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