Optimal. Leaf size=105 \[ -\frac {(a C+b B) \sin ^3(c+d x)}{3 d}+\frac {(a C+b B) \sin (c+d x)}{d}+\frac {(4 a B+3 b C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (4 a B+3 b C)+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.21, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3029, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac {(a C+b B) \sin ^3(c+d x)}{3 d}+\frac {(a C+b B) \sin (c+d x)}{d}+\frac {(4 a B+3 b C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (4 a B+3 b C)+\frac {b C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2968
Rule 3023
Rule 3029
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (a+b \cos (c+d x)) (B+C \cos (c+d x)) \, dx\\ &=\int \cos ^2(c+d x) \left (a B+(b B+a C) \cos (c+d x)+b 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 ^2(c+d x) (4 a B+3 b C+4 (b B+a C) \cos (c+d x)) \, dx\\ &=\frac {b C \cos ^3(c+d x) \sin (c+d x)}{4 d}+(b B+a C) \int \cos ^3(c+d x) \, dx+\frac {1}{4} (4 a B+3 b C) \int \cos ^2(c+d x) \, dx\\ &=\frac {(4 a B+3 b C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} (4 a B+3 b C) \int 1 \, dx-\frac {(b B+a C) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {1}{8} (4 a B+3 b C) x+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {(4 a B+3 b C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {(b B+a C) \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 91, normalized size = 0.87 \[ \frac {-32 (a C+b B) \sin ^3(c+d x)+96 (a C+b B) \sin (c+d x)+24 (a B+b C) \sin (2 (c+d x))+48 a B c+48 a B d x+3 b C \sin (4 (c+d x))+36 b c C+36 b C d x}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 81, normalized size = 0.77 \[ \frac {3 \, {\left (4 \, B a + 3 \, C b\right )} d x + {\left (6 \, C b \cos \left (d x + c\right )^{3} + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + 16 \, C a + 16 \, B b + 3 \, {\left (4 \, B a + 3 \, C b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 89, normalized size = 0.85 \[ \frac {1}{8} \, {\left (4 \, B a + 3 \, C b\right )} x + \frac {C b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (C a + B b\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (B a + C b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {3 \, {\left (C a + B b\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 107, normalized size = 1.02 \[ \frac {C b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 101, normalized size = 0.96 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.96, size = 117, normalized size = 1.11 \[ \frac {B\,a\,x}{2}+\frac {3\,C\,b\,x}{8}+\frac {3\,B\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,C\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.14, size = 255, normalized size = 2.43 \[ \begin {cases} \frac {B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 B b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right ) \left (B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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