Optimal. Leaf size=171 \[ \frac{\left (16 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d}+\frac{b \left (6 a^2 C+20 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (12 a^2 b C+8 a^3 B+12 a b^2 B+3 b^3 C\right )+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.261676, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.079, Rules used = {3029, 2753, 2734} \[ \frac{\left (16 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d}+\frac{b \left (6 a^2 C+20 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (12 a^2 b C+8 a^3 B+12 a b^2 B+3 b^3 C\right )+\frac{(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 3029
Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \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))^3 (B+C \cos (c+d x)) \, dx\\ &=\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{4} \int (a+b \cos (c+d x))^2 (4 a B+3 b C+(4 b B+3 a C) \cos (c+d x)) \, dx\\ &=\frac{(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{1}{12} \int (a+b \cos (c+d x)) \left (12 a^2 B+8 b^2 B+15 a b C+\left (20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x)\right ) \, dx\\ &=\frac{1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b C+3 b^3 C\right ) x+\frac{\left (16 a^2 b B+4 b^3 B+3 a^3 C+12 a b^2 C\right ) \sin (c+d x)}{6 d}+\frac{b \left (20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.400583, size = 140, normalized size = 0.82 \[ \frac{12 (c+d x) \left (12 a^2 b C+8 a^3 B+12 a b^2 B+3 b^3 C\right )+24 b \left (3 a^2 C+3 a b B+b^2 C\right ) \sin (2 (c+d x))+24 \left (12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right ) \sin (c+d x)+8 b^2 (3 a C+b B) \sin (3 (c+d x))+3 b^3 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 180, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( C{b}^{3} \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{{b}^{3}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Ca{b}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +3\,a{b}^{2}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{2}bC \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{2}bB\sin \left ( dx+c \right ) +{a}^{3}C\sin \left ( dx+c \right ) +{a}^{3}B \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04951, size = 231, normalized size = 1.35 \begin{align*} \frac{96 \,{\left (d x + c\right )} B a^{3} + 72 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b + 72 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} - 96 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{2} - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{3} + 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} + 96 \, C a^{3} \sin \left (d x + c\right ) + 288 \, B a^{2} b \sin \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48211, size = 321, normalized size = 1.88 \begin{align*} \frac{3 \,{\left (8 \, B a^{3} + 12 \, C a^{2} b + 12 \, B a b^{2} + 3 \, C b^{3}\right )} d x +{\left (6 \, C b^{3} \cos \left (d x + c\right )^{3} + 24 \, C a^{3} + 72 \, B a^{2} b + 48 \, C a b^{2} + 16 \, B b^{3} + 8 \,{\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{2} + 9 \,{\left (4 \, C a^{2} b + 4 \, B a b^{2} + C b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, 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 = 1.57755, size = 724, normalized size = 4.23 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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