Optimal. Leaf size=53 \[ \frac {\tan ^7(a+b x)}{7 b}+\frac {3 \tan ^5(a+b x)}{5 b}+\frac {\tan ^3(a+b x)}{b}+\frac {\tan (a+b x)}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3767} \[ \frac {\tan ^7(a+b x)}{7 b}+\frac {3 \tan ^5(a+b x)}{5 b}+\frac {\tan ^3(a+b x)}{b}+\frac {\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \int \sec ^8(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (a+b x)\right )}{b}\\ &=\frac {\tan (a+b x)}{b}+\frac {\tan ^3(a+b x)}{b}+\frac {3 \tan ^5(a+b x)}{5 b}+\frac {\tan ^7(a+b x)}{7 b}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 43, normalized size = 0.81 \[ \frac {\frac {1}{7} \tan ^7(a+b x)+\frac {3}{5} \tan ^5(a+b x)+\tan ^3(a+b x)+\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 51, normalized size = 0.96 \[ \frac {{\left (16 \, \cos \left (b x + a\right )^{6} + 8 \, \cos \left (b x + a\right )^{4} + 6 \, \cos \left (b x + a\right )^{2} + 5\right )} \sin \left (b x + a\right )}{35 \, b \cos \left (b x + a\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.78, size = 44, normalized size = 0.83 \[ \frac {5 \, \tan \left (b x + a\right )^{7} + 21 \, \tan \left (b x + a\right )^{5} + 35 \, \tan \left (b x + a\right )^{3} + 35 \, \tan \left (b x + a\right )}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 44, normalized size = 0.83 \[ -\frac {\left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (b x +a \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (b x +a \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (b x +a \right )\right )}{35}\right ) \tan \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 44, normalized size = 0.83 \[ \frac {5 \, \tan \left (b x + a\right )^{7} + 21 \, \tan \left (b x + a\right )^{5} + 35 \, \tan \left (b x + a\right )^{3} + 35 \, \tan \left (b x + a\right )}{35 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 39, normalized size = 0.74 \[ \frac {\frac {{\mathrm {tan}\left (a+b\,x\right )}^7}{7}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^5}{5}+{\mathrm {tan}\left (a+b\,x\right )}^3+\mathrm {tan}\left (a+b\,x\right )}{b} \]
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 \[ \int \sec ^{8}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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