Optimal. Leaf size=84 \[ \frac {(3 a B+2 b C) \sin (c+d x)}{3 d}+\frac {(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} x (a C+b B)+\frac {b C \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 104, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3023, 2734} \[ \frac {\left (a^2 (-C)+3 a b B+2 b^2 C\right ) \sin (c+d x)}{3 b d}+\frac {(3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} x (a C+b B)+\frac {C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3023
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 b d}+\frac {\int (a+b \cos (c+d x)) (2 b C+(3 b B-a C) \cos (c+d x)) \, dx}{3 b}\\ &=\frac {1}{2} (b B+a C) x+\frac {\left (3 a b B-a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b d}+\frac {(3 b B-a C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 b d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 75, normalized size = 0.89 \[ \frac {3 (4 a B+3 b C) \sin (c+d x)+3 (a C+b B) \sin (2 (c+d x))+6 a c C+6 a C d x+6 b B c+6 b B d x+b C \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 60, normalized size = 0.71 \[ \frac {3 \, {\left (C a + B b\right )} d x + {\left (2 \, C b \cos \left (d x + c\right )^{2} + 6 \, B a + 4 \, C b + 3 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 68, normalized size = 0.81 \[ \frac {1}{2} \, {\left (C a + B b\right )} x + \frac {C b \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (C a + B b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, B a + 3 \, C b\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 85, normalized size = 1.01 \[ \frac {\frac {C b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 79, normalized size = 0.94 \[ \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b + 12 \, B a \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.76, size = 84, normalized size = 1.00 \[ \frac {B\,b\,x}{2}+\frac {C\,a\,x}{2}+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 170, normalized size = 2.02 \[ \begin {cases} \frac {B a \sin {\left (c + d x \right )}}{d} + \frac {B b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {C a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 C b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right ) \left (B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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