Optimal. Leaf size=37 \[ \frac {\tan ^7(x)}{7 a^4}+\frac {3 \tan ^5(x)}{5 a^4}+\frac {\tan ^3(x)}{a^4}+\frac {\tan (x)}{a^4} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3175, 3767} \[ \frac {\tan ^7(x)}{7 a^4}+\frac {3 \tan ^5(x)}{5 a^4}+\frac {\tan ^3(x)}{a^4}+\frac {\tan (x)}{a^4} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{\left (a-a \sin ^2(x)\right )^4} \, dx &=\frac {\int \sec ^8(x) \, dx}{a^4}\\ &=-\frac {\operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (x)\right )}{a^4}\\ &=\frac {\tan (x)}{a^4}+\frac {\tan ^3(x)}{a^4}+\frac {3 \tan ^5(x)}{5 a^4}+\frac {\tan ^7(x)}{7 a^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 1.11 \[ \frac {\frac {16 \tan (x)}{35}+\frac {1}{7} \tan (x) \sec ^6(x)+\frac {6}{35} \tan (x) \sec ^4(x)+\frac {8}{35} \tan (x) \sec ^2(x)}{a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 31, normalized size = 0.84 \[ \frac {{\left (16 \, \cos \relax (x)^{6} + 8 \, \cos \relax (x)^{4} + 6 \, \cos \relax (x)^{2} + 5\right )} \sin \relax (x)}{35 \, a^{4} \cos \relax (x)^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 28, normalized size = 0.76 \[ \frac {5 \, \tan \relax (x)^{7} + 21 \, \tan \relax (x)^{5} + 35 \, \tan \relax (x)^{3} + 35 \, \tan \relax (x)}{35 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 24, normalized size = 0.65 \[ \frac {\frac {\left (\tan ^{7}\relax (x )\right )}{7}+\frac {3 \left (\tan ^{5}\relax (x )\right )}{5}+\tan ^{3}\relax (x )+\tan \relax (x )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 28, normalized size = 0.76 \[ \frac {5 \, \tan \relax (x)^{7} + 21 \, \tan \relax (x)^{5} + 35 \, \tan \relax (x)^{3} + 35 \, \tan \relax (x)}{35 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.41, size = 33, normalized size = 0.89 \[ \frac {\mathrm {tan}\relax (x)}{a^4}+\frac {{\mathrm {tan}\relax (x)}^3}{a^4}+\frac {3\,{\mathrm {tan}\relax (x)}^5}{5\,a^4}+\frac {{\mathrm {tan}\relax (x)}^7}{7\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 22.20, size = 675, normalized size = 18.24 \[ - \frac {70 \tan ^{13}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {140 \tan ^{11}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {602 \tan ^{9}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {424 \tan ^{7}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {602 \tan ^{5}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} + \frac {140 \tan ^{3}{\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} - \frac {70 \tan {\left (\frac {x}{2} \right )}}{35 a^{4} \tan ^{14}{\left (\frac {x}{2} \right )} - 245 a^{4} \tan ^{12}{\left (\frac {x}{2} \right )} + 735 a^{4} \tan ^{10}{\left (\frac {x}{2} \right )} - 1225 a^{4} \tan ^{8}{\left (\frac {x}{2} \right )} + 1225 a^{4} \tan ^{6}{\left (\frac {x}{2} \right )} - 735 a^{4} \tan ^{4}{\left (\frac {x}{2} \right )} + 245 a^{4} \tan ^{2}{\left (\frac {x}{2} \right )} - 35 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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