Optimal. Leaf size=41 \[ \frac {\tan ^5(a+b x)}{5 b}+\frac {2 \tan ^3(a+b x)}{3 b}+\frac {\tan (a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 41, 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} \[ \frac {\tan ^5(a+b x)}{5 b}+\frac {2 \tan ^3(a+b x)}{3 b}+\frac {\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin {align*} \int \sec ^6(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (a+b x)\right )}{b}\\ &=\frac {\tan (a+b x)}{b}+\frac {2 \tan ^3(a+b x)}{3 b}+\frac {\tan ^5(a+b x)}{5 b}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 35, normalized size = 0.85 \[ \frac {\frac {1}{5} \tan ^5(a+b x)+\frac {2}{3} \tan ^3(a+b x)+\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 41, normalized size = 1.00 \[ \frac {{\left (8 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 3\right )} \sin \left (b x + a\right )}{15 \, b \cos \left (b x + a\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 34, normalized size = 0.83 \[ \frac {3 \, \tan \left (b x + a\right )^{5} + 10 \, \tan \left (b x + a\right )^{3} + 15 \, \tan \left (b x + a\right )}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 34, normalized size = 0.83 \[ -\frac {\left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (b x +a \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (b x +a \right )\right )}{15}\right ) \tan \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 34, normalized size = 0.83 \[ \frac {3 \, \tan \left (b x + a\right )^{5} + 10 \, \tan \left (b x + a\right )^{3} + 15 \, \tan \left (b x + a\right )}{15 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 31, normalized size = 0.76 \[ \frac {\frac {{\mathrm {tan}\left (a+b\,x\right )}^5}{5}+\frac {2\,{\mathrm {tan}\left (a+b\,x\right )}^3}{3}+\mathrm {tan}\left (a+b\,x\right )}{b} \]
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 \[ \int \sec ^{6}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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