Optimal. Leaf size=17 \[ a x-\frac{b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0071293, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3475} \[ a x-\frac{b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rubi steps
\begin{align*} \int (a+b \tan (c+d x)) \, dx &=a x+b \int \tan (c+d x) \, dx\\ &=a x-\frac{b \log (\cos (c+d x))}{d}\\ \end{align*}
Mathematica [A] time = 0.0046165, size = 17, normalized size = 1. \[ a x-\frac{b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 1.3 \begin{align*} ax+{\frac{b\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05051, size = 22, normalized size = 1.29 \begin{align*} a x + \frac{b \log \left (\sec \left (d x + c\right )\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72243, size = 66, normalized size = 3.88 \begin{align*} \frac{2 \, a d x - b \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.130422, size = 24, normalized size = 1.41 \begin{align*} a x + b \left (\begin{cases} \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{for}\: d \neq 0 \\x \tan{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25719, size = 24, normalized size = 1.41 \begin{align*} a x - \frac{b \log \left ({\left | \cos \left (d x + c\right ) \right |}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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