Optimal. Leaf size=17 \[ a x-\frac{b \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.0257225, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3086, 3475} \[ a x-\frac{b \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3086
Rule 3475
Rubi steps
\begin{align*} \int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx &=\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 = 0.0138661, 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 = 0.065, size = 24, normalized size = 1.4 \begin{align*} ax-{\frac{b\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{ac}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.23574, size = 41, normalized size = 2.41 \begin{align*} \frac{2 \,{\left (d x + c\right )} a - b \log \left (-\sin \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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Fricas [A] time = 0.489085, size = 46, normalized size = 2.71 \begin{align*} \frac{a d x - b \log \left (-\cos \left (d x + c\right )\right )}{d} \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 \left (a \cos{\left (c + d x \right )} + b \sin{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12282, size = 36, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (d x + c\right )} a + b \log \left (\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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