Optimal. Leaf size=10 \[ \frac{\tan (a+b x)}{b} \]
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Rubi [A] time = 0.0084205, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3767, 8} \[ \frac{\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(a+b x) \, dx &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,-\tan (a+b x))}{b}\\ &=\frac{\tan (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0029589, size = 10, normalized size = 1. \[ \frac{\tan (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 11, normalized size = 1.1 \begin{align*}{\frac{\tan \left ( bx+a \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934283, size = 14, normalized size = 1.4 \begin{align*} \frac{\tan \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7253, size = 42, normalized size = 4.2 \begin{align*} \frac{\sin \left (b x + a\right )}{b \cos \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.27562, size = 58, normalized size = 5.8 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: \left (a = - \frac{\pi }{2} \vee a = - b x - \frac{\pi }{2}\right ) \wedge \left (a = - b x - \frac{\pi }{2} \vee b = 0\right ) \\\frac{x}{\cos ^{2}{\left (a \right )}} & \text{for}\: b = 0 \\- \frac{2 \tan{\left (\frac{a}{2} + \frac{b x}{2} \right )}}{b \tan ^{2}{\left (\frac{a}{2} + \frac{b x}{2} \right )} - b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08762, size = 14, normalized size = 1.4 \begin{align*} \frac{\tan \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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