Optimal. Leaf size=170 \[ \frac {\left (-2 a^2 C+8 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} x \left (4 a^2 C+8 a b B+3 b^2 C\right )+\frac {\left (a^3 (-C)+4 a^2 b B+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 b d}+\frac {(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d} \]
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Rubi [A] time = 0.19, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3023, 2753, 2734} \[ \frac {\left (4 a^2 b B+a^3 (-C)+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 b d}+\frac {\left (-2 a^2 C+8 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} x \left (4 a^2 C+8 a b B+3 b^2 C\right )+\frac {(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 3023
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}+\frac {\int (a+b \cos (c+d x))^2 (3 b C+(4 b B-a C) \cos (c+d x)) \, dx}{4 b}\\ &=\frac {(4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}+\frac {\int (a+b \cos (c+d x)) \left (b (8 b B+7 a C)+\left (8 a b B-2 a^2 C+9 b^2 C\right ) \cos (c+d x)\right ) \, dx}{12 b}\\ &=\frac {1}{8} \left (8 a b B+4 a^2 C+3 b^2 C\right ) x+\frac {\left (4 a^2 b B+4 b^3 B-a^3 C+8 a b^2 C\right ) \sin (c+d x)}{6 b d}+\frac {\left (8 a b B-2 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 118, normalized size = 0.69 \[ \frac {12 (c+d x) \left (4 a^2 C+8 a b B+3 b^2 C\right )+24 \left (4 a^2 B+6 a b C+3 b^2 B\right ) \sin (c+d x)+24 \left (a^2 C+2 a b B+b^2 C\right ) \sin (2 (c+d x))+8 b (2 a C+b B) \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 114, normalized size = 0.67 \[ \frac {3 \, {\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} d x + {\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 24 \, B a^{2} + 32 \, C a b + 16 \, B b^{2} + 8 \, {\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 124, normalized size = 0.73 \[ \frac {C b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {1}{8} \, {\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} x + \frac {{\left (2 \, C a b + B b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (C a^{2} + 2 \, B a b + C b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, B a^{2} + 6 \, C a b + 3 \, B b^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 152, normalized size = 0.89 \[ \frac {a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{2} \sin \left (d x +c \right )+\frac {2 C a b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 B a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {b^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 142, normalized size = 0.84 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 48 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b - 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{2} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 96 \, B a^{2} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.80, size = 169, normalized size = 0.99 \[ \frac {C\,a^2\,x}{2}+\frac {3\,C\,b^2\,x}{8}+\frac {B\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+B\,a\,b\,x+\frac {C\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,C\,a\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {B\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {C\,a\,b\,\sin \left (3\,c+3\,d\,x\right )}{6\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.35, size = 340, normalized size = 2.00 \[ \begin {cases} \frac {B a^{2} \sin {\left (c + d x \right )}}{d} + B a b x \sin ^{2}{\left (c + d x \right )} + B a b x \cos ^{2}{\left (c + d x \right )} + \frac {B a b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {2 B b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 C a b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 C a b \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{2} \left (B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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