Optimal. Leaf size=27 \[ \frac{\sec ^3(a+b x)}{3 b}-\frac{\sec (a+b x)}{b} \]
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Rubi [A] time = 0.0200087, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2606} \[ \frac{\sec ^3(a+b x)}{3 b}-\frac{\sec (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 2606
Rubi steps
\begin{align*} \int \sec (a+b x) \tan ^3(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\sec (a+b x)\right )}{b}\\ &=-\frac{\sec (a+b x)}{b}+\frac{\sec ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0233389, size = 27, normalized size = 1. \[ \frac{\sec ^3(a+b x)}{3 b}-\frac{\sec (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 60, normalized size = 2.2 \begin{align*}{\frac{1}{b} \left ({\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{3\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{3\,\cos \left ( bx+a \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( bx+a \right ) \right ) ^{2} \right ) \cos \left ( bx+a \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01126, size = 34, normalized size = 1.26 \begin{align*} -\frac{3 \, \cos \left (b x + a\right )^{2} - 1}{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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Fricas [A] time = 1.55351, size = 65, normalized size = 2.41 \begin{align*} -\frac{3 \, \cos \left (b x + a\right )^{2} - 1}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16811, size = 34, normalized size = 1.26 \begin{align*} -\frac{3 \, \cos \left (b x + a\right )^{2} - 1}{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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