Optimal. Leaf size=41 \[ \frac{\log (a \cos (c+d x)+b \sin (c+d x))}{b d}-\frac{\log (\cos (c+d x))}{b d} \]
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Rubi [A] time = 0.0816556, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {3102, 3475, 3133} \[ \frac{\log (a \cos (c+d x)+b \sin (c+d x))}{b d}-\frac{\log (\cos (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Rule 3102
Rule 3475
Rule 3133
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx &=\frac{\int \frac{b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b}+\frac{\int \tan (c+d x) \, dx}{b}\\ &=-\frac{\log (\cos (c+d x))}{b d}+\frac{\log (a \cos (c+d x)+b \sin (c+d x))}{b d}\\ \end{align*}
Mathematica [A] time = 0.0168481, size = 18, normalized size = 0.44 \[ \frac{\log (a+b \tan (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.138, size = 19, normalized size = 0.5 \begin{align*}{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{db}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.09116, size = 139, normalized size = 3.39 \begin{align*} \frac{\frac{\log \left (-a - \frac{2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{b} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b} - \frac{\log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.512886, size = 144, normalized size = 3.51 \begin{align*} \frac{\log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - \log \left (\cos \left (d x + c\right )^{2}\right )}{2 \, b 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 \frac{\sec{\left (c + d x \right )}}{a \cos{\left (c + d x \right )} + b \sin{\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 = 1.15861, size = 26, normalized size = 0.63 \begin{align*} \frac{\log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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