Optimal. Leaf size=84 \[ \frac{(2 a A+3 b B) \sin (c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.125352, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {3996, 3787, 2635, 8, 2637} \[ \frac{(2 a A+3 b B) \sin (c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3996
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx &=\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) (-3 (A b+a B)-(2 a A+3 b B) \sec (c+d x)) \, dx\\ &=\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-(-A b-a B) \int \cos ^2(c+d x) \, dx-\frac{1}{3} (-2 a A-3 b B) \int \cos (c+d x) \, dx\\ &=\frac{(2 a A+3 b B) \sin (c+d x)}{3 d}+\frac{(A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{2} (-A b-a B) \int 1 \, dx\\ &=\frac{1}{2} (A b+a B) x+\frac{(2 a A+3 b B) \sin (c+d x)}{3 d}+\frac{(A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a A \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.156339, size = 75, normalized size = 0.89 \[ \frac{3 (3 a A+4 b B) \sin (c+d x)+3 (a B+A b) \sin (2 (c+d x))+a A \sin (3 (c+d x))+6 a B c+6 a B d x+6 A b c+6 A b d x}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 85, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{Aa \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Ab \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ba \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +B\sin \left ( dx+c \right ) b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964429, size = 107, normalized size = 1.27 \begin{align*} -\frac{4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b - 12 \, B b \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.474375, size = 149, normalized size = 1.77 \begin{align*} \frac{3 \,{\left (B a + A b\right )} d x +{\left (2 \, A a \cos \left (d x + c\right )^{2} + 4 \, A a + 6 \, B b + 3 \,{\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16053, size = 243, normalized size = 2.89 \begin{align*} \frac{3 \,{\left (B a + A b\right )}{\left (d x + c\right )} + \frac{2 \,{\left (6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, B 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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