Optimal. Leaf size=47 \[ \frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (A+2 B)+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.0864057, antiderivative size = 47, 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} \[ \frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (A+2 B)+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d} \]
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+a \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 (A+B)-a (A+2 B) \sec (c+d x)) \, dx\\ &=\frac{a A \cos (c+d x) \sin (c+d x)}{2 d}+(a (A+B)) \int \cos (c+d x) \, dx+\frac{1}{2} (a (A+2 B)) \int 1 \, dx\\ &=\frac{1}{2} a (A+2 B) x+\frac{a (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 = 0.0962903, size = 44, normalized size = 0.94 \[ \frac{a (4 (A+B) \sin (c+d x)+A \sin (2 (c+d x))+2 A c+2 A d x+4 B d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 57, normalized size = 1.2 \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 ) +Aa\sin \left ( dx+c \right ) +Ba\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 = 0.979916, size = 74, normalized size = 1.57 \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 a + 4 \, A a \sin \left (d x + c\right ) + 4 \, B a \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 = 0.458909, size = 99, normalized size = 2.11 \begin{align*} \frac{{\left (A + 2 \, B\right )} a d x +{\left (A a \cos \left (d x + c\right ) + 2 \,{\left (A + B\right )} a\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*} a \left (\int A \cos ^{2}{\left (c + d x \right )}\, dx + \int A \cos ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec{\left (c + d x \right )}\, dx + \int B \cos ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.26993, size = 126, normalized size = 2.68 \begin{align*} \frac{{\left (A a + 2 \, B a\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} + 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 )\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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