Optimal. Leaf size=26 \[ \frac{\tan ^3(a+b x)}{3 b}+\frac{\tan (a+b x)}{b} \]
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Rubi [A] time = 0.0112607, antiderivative size = 26, normalized size of antiderivative = 1., 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} \[ \frac{\tan ^3(a+b x)}{3 b}+\frac{\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rubi steps
\begin{align*} \int \sec ^4(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (a+b x)\right )}{b}\\ &=\frac{\tan (a+b x)}{b}+\frac{\tan ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0409817, size = 23, normalized size = 0.88 \[ \frac{\frac{1}{3} \tan ^3(a+b x)+\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 24, normalized size = 0.9 \begin{align*} -{\frac{\tan \left ( bx+a \right ) }{b} \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16608, size = 30, normalized size = 1.15 \begin{align*} \frac{\tan \left (b x + a\right )^{3} + 3 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34596, size = 81, normalized size = 3.12 \begin{align*} \frac{{\left (2 \, \cos \left (b x + a\right )^{2} + 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec ^{4}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32413, size = 30, normalized size = 1.15 \begin{align*} \frac{\tan \left (b x + a\right )^{3} + 3 \, \tan \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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