Optimal. Leaf size=128 \[ \frac {\sin (c+d x) (3 a A+2 a C+2 b B)}{3 d}+\frac {\sin (c+d x) \cos (c+d x) (4 a B+4 A b+3 b C)}{8 d}+\frac {1}{8} x (4 a B+4 A b+3 b C)+\frac {(a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.14, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {3033, 3023, 2734} \[ \frac {\sin (c+d x) (3 a A+2 a C+2 b B)}{3 d}+\frac {\sin (c+d x) \cos (c+d x) (4 a B+4 A b+3 b C)}{8 d}+\frac {1}{8} x (4 a B+4 A b+3 b C)+\frac {(a C+b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3023
Rule 3033
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {b C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos (c+d x) \left (4 a A+(4 A b+4 a B+3 b C) \cos (c+d x)+4 (b B+a C) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {(b B+a C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {b C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{12} \int \cos (c+d x) (4 (3 a A+2 b B+2 a C)+3 (4 A b+4 a B+3 b C) \cos (c+d x)) \, dx\\ &=\frac {1}{8} (4 A b+4 a B+3 b C) x+\frac {(3 a A+2 b B+2 a C) \sin (c+d x)}{3 d}+\frac {(4 A b+4 a B+3 b C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(b B+a C) \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {b C \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 118, normalized size = 0.92 \[ \frac {24 \sin (c+d x) (4 a A+3 a C+3 b B)+24 \sin (2 (c+d x)) (a B+A b+b C)+48 a B c+48 a B d x+8 a C \sin (3 (c+d x))+48 A b c+48 A b d x+8 b B \sin (3 (c+d x))+3 b C \sin (4 (c+d x))+36 b c C+36 b C d x}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 97, normalized size = 0.76 \[ \frac {3 \, {\left (4 \, B a + {\left (4 \, A + 3 \, C\right )} b\right )} d x + {\left (6 \, C b \cos \left (d x + c\right )^{3} + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, A + 2 \, C\right )} a + 16 \, B b + 3 \, {\left (4 \, B a + {\left (4 \, A + 3 \, C\right )} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.51, size = 102, normalized size = 0.80 \[ \frac {1}{8} \, {\left (4 \, B a + 4 \, A b + 3 \, C b\right )} x + \frac {C b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (C a + B b\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (B a + A b + C b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a + 3 \, C a + 3 \, B b\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 141, normalized size = 1.10 \[ \frac {C b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 132, normalized size = 1.03 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b + 96 \, A a \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.88, size = 150, normalized size = 1.17 \[ \frac {A\,b\,x}{2}+\frac {B\,a\,x}{2}+\frac {3\,C\,b\,x}{8}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,C\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.22, size = 320, normalized size = 2.50 \[ \begin {cases} \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {A b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 B b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right ) \left (A + B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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