Optimal. Leaf size=52 \[ \frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (a A+2 b B)+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.0959616, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {3996, 3787, 2637, 8} \[ \frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (a A+2 b B)+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 3996
Rule 3787
Rule 2637
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}-\frac{1}{2} \int \cos (c+d x) (-2 (A b+a B)-(a A+2 b B) \sec (c+d x)) \, dx\\ &=\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}-(-A b-a B) \int \cos (c+d x) \, dx-\frac{1}{2} (-a A-2 b B) \int 1 \, dx\\ &=\frac{1}{2} (a A+2 b B) x+\frac{(A b+a B) \sin (c+d x)}{d}+\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0822269, size = 51, normalized size = 0.98 \[ \frac{4 (a B+A b) \sin (c+d x)+a A \sin (2 (c+d x))+2 a A c+2 a A d x+4 b B d x}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 57, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( Aa \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ab\sin \left ( dx+c \right ) +Ba\sin \left ( dx+c \right ) +Bb \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959907, size = 74, normalized size = 1.42 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a + 4 \,{\left (d x + c\right )} B b + 4 \, B a \sin \left (d x + c\right ) + 4 \, A 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 = 0.468127, size = 104, normalized size = 2. \begin{align*} \frac{{\left (A a + 2 \, B b\right )} d x +{\left (A a \cos \left (d x + c\right ) + 2 \, B a + 2 \, A 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 (A + B \sec{\left (c + d x \right )}\right ) \left (a + b \sec{\left (c + d x \right )}\right ) \cos ^{2}{\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 = 1.16937, size = 163, normalized size = 3.13 \begin{align*} \frac{{\left (A a + 2 \, B b\right )}{\left (d x + c\right )} - \frac{2 \,{\left (A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, A 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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