Optimal. Leaf size=12 \[ -\frac{\log (\cos (a+b x))}{b} \]
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Rubi [A] time = 0.0040621, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3475} \[ -\frac{\log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin{align*} \int \tan (a+b x) \, dx &=-\frac{\log (\cos (a+b x))}{b}\\ \end{align*}
Mathematica [A] time = 0.0088248, size = 12, normalized size = 1. \[ -\frac{\log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 17, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( bx+a \right ) \right ) ^{2} \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.936232, size = 15, normalized size = 1.25 \begin{align*} \frac{\log \left (\sec \left (b x + a\right )\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80857, size = 49, normalized size = 4.08 \begin{align*} -\frac{\log \left (\frac{1}{\tan \left (b x + a\right )^{2} + 1}\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.123411, size = 19, normalized size = 1.58 \begin{align*} \begin{cases} \frac{\log{\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b} & \text{for}\: b \neq 0 \\x \tan{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07097, size = 18, normalized size = 1.5 \begin{align*} -\frac{\log \left ({\left | \cos \left (b x + a\right ) \right |}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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