Optimal. Leaf size=84 \[ \frac{(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a C+b B)+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.16633, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4072, 3996, 3787, 2635, 8, 2637} \[ \frac{(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a C+b B)+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 3996
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^3(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx\\ &=\frac{a B \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{3} \int \cos ^2(c+d x) (-3 (b B+a C)-(2 a B+3 b C) \sec (c+d x)) \, dx\\ &=\frac{a B \cos ^2(c+d x) \sin (c+d x)}{3 d}-(-b B-a C) \int \cos ^2(c+d x) \, dx-\frac{1}{3} (-2 a B-3 b C) \int \cos (c+d x) \, dx\\ &=\frac{(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac{(b B+a C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a B \cos ^2(c+d x) \sin (c+d x)}{3 d}-\frac{1}{2} (-b B-a C) \int 1 \, dx\\ &=\frac{1}{2} (b B+a C) x+\frac{(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac{(b B+a C) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{a B \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.161308, size = 75, normalized size = 0.89 \[ \frac{3 (3 a B+4 b C) \sin (c+d x)+3 (a C+b B) \sin (2 (c+d x))+a B \sin (3 (c+d x))+6 a c C+6 a C d x+6 b B c+6 b B d x}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 85, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{Ba \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Bb \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aC \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +C\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 = 0.957332, 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 )} B a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b - 12 \, C 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 = 0.489494, size = 149, normalized size = 1.77 \begin{align*} \frac{3 \,{\left (C a + B b\right )} d x +{\left (2 \, B a \cos \left (d x + c\right )^{2} + 4 \, B a + 6 \, C b + 3 \,{\left (C a + B 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 = 1.17514, size = 243, normalized size = 2.89 \begin{align*} \frac{3 \,{\left (C a + B b\right )}{\left (d x + c\right )} + \frac{2 \,{\left (6 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 6 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 4 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 12 \, C b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, C a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 6 \, 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 )}^{3}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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