Optimal. Leaf size=52 \[ \frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (2 a B+b C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.065021, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {3029, 2734} \[ \frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (2 a B+b C)+\frac{b C \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx &=\int (a+b \cos (c+d x)) (B+C \cos (c+d x)) \, dx\\ &=\frac{1}{2} (2 a B+b C) x+\frac{(b B+a C) \sin (c+d x)}{d}+\frac{b C \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0811117, size = 51, normalized size = 0.98 \[ \frac{4 (a C+b B) \sin (c+d x)+4 a B d x+b C \sin (2 (c+d x))+2 b c C+2 b C d x}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 57, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( Cb \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +bB\sin \left ( dx+c \right ) +aC\sin \left ( dx+c \right ) +Ba \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02688, size = 74, normalized size = 1.42 \begin{align*} \frac{4 \,{\left (d x + c\right )} B a +{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b + 4 \, C a \sin \left (d x + c\right ) + 4 \, B b \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67619, size = 104, normalized size = 2. \begin{align*} \frac{{\left (2 \, B a + C b\right )} d x +{\left (C b \cos \left (d x + c\right ) + 2 \, C a + 2 \, B b\right )} \sin \left (d x + c\right )}{2 \, 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 (B + C \cos{\left (c + d x \right )}\right ) \left (a + b \cos{\left (c + d x \right )}\right ) \cos{\left (c + d x \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 [B] time = 1.32393, size = 163, normalized size = 3.13 \begin{align*} \frac{{\left (2 \, B a + C b\right )}{\left (d x + c\right )} + \frac{2 \,{\left (2 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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