Optimal. Leaf size=176 \[ \frac {b^2 \left (-14 a^2 C+20 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} x \left (-8 a^4 C+8 a^3 b B+12 a b^3 B+3 b^4 C\right )+\frac {b \left (-13 a^3 C+16 a^2 b B+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d}+\frac {b (4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac {b C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.35, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {3015, 2753, 2734} \[ \frac {b \left (16 a^2 b B-13 a^3 C+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d}+\frac {b^2 \left (-14 a^2 C+20 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} x \left (8 a^3 b B-8 a^4 C+12 a b^3 B+3 b^4 C\right )+\frac {b (4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac {b C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rule 3015
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx &=\frac {\int (a+b \cos (c+d x))^3 \left (b^2 (b B-a C)+b^3 C \cos (c+d x)\right ) \, dx}{b^2}\\ &=\frac {b C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {\int (a+b \cos (c+d x))^2 \left (b^2 \left (3 b^2 C+4 a (b B-a C)\right )+b^3 (4 b B-a C) \cos (c+d x)\right ) \, dx}{4 b^2}\\ &=\frac {b (4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {b C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {\int (a+b \cos (c+d x)) \left (b^2 \left (12 a^2 b B+8 b^3 B-12 a^3 C+7 a b^2 C\right )+b^3 \left (20 a b B-14 a^2 C+9 b^2 C\right ) \cos (c+d x)\right ) \, dx}{12 b^2}\\ &=\frac {1}{8} \left (8 a^3 b B+12 a b^3 B-8 a^4 C+3 b^4 C\right ) x+\frac {b \left (16 a^2 b B+4 b^3 B-13 a^3 C+8 a b^2 C\right ) \sin (c+d x)}{6 d}+\frac {b^2 \left (20 a b B-14 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {b (4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {b C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 134, normalized size = 0.76 \[ \frac {-12 (c+d x) \left (8 a^4 C-8 a^3 b B-12 a b^3 B-3 b^4 C\right )+24 b \left (-8 a^3 C+12 a^2 b B+6 a b^2 C+3 b^3 B\right ) \sin (c+d x)+24 b^3 (3 a B+b C) \sin (2 (c+d x))+8 b^3 (2 a C+b B) \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 133, normalized size = 0.76 \[ -\frac {3 \, {\left (8 \, C a^{4} - 8 \, B a^{3} b - 12 \, B a b^{3} - 3 \, C b^{4}\right )} d x - {\left (6 \, C b^{4} \cos \left (d x + c\right )^{3} - 48 \, C a^{3} b + 72 \, B a^{2} b^{2} + 32 \, C a b^{3} + 16 \, B b^{4} + 8 \, {\left (2 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (4 \, B a b^{3} + C b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.59, size = 144, normalized size = 0.82 \[ \frac {C b^{4} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {1}{8} \, {\left (8 \, C a^{4} - 8 \, B a^{3} b - 12 \, B a b^{3} - 3 \, C b^{4}\right )} x + \frac {{\left (2 \, C a b^{3} + B b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (3 \, B a b^{3} + C b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac {{\left (8 \, C a^{3} b - 12 \, B a^{2} b^{2} - 6 \, C a b^{3} - 3 \, B b^{4}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 168, normalized size = 0.95 \[ \frac {C \,b^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,b^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {2 C a \,b^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 B a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b^{2} B \sin \left (d x +c \right )-2 a^{3} b C \sin \left (d x +c \right )+B \left (d x +c \right ) a^{3} b -a^{4} C \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 162, normalized size = 0.92 \[ -\frac {96 \, {\left (d x + c\right )} C a^{4} - 96 \, {\left (d x + c\right )} B a^{3} b - 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} + 64 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{3} + 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{4} - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{4} + 192 \, C a^{3} b \sin \left (d x + c\right ) - 288 \, B a^{2} b^{2} \sin \left (d x + c\right )}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.02, size = 187, normalized size = 1.06 \[ \frac {3\,C\,b^4\,x}{8}-C\,a^4\,x+\frac {3\,B\,a\,b^3\,x}{2}+B\,a^3\,b\,x+\frac {3\,B\,b^4\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,B\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )}{d}+\frac {C\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{6\,d}+\frac {3\,C\,a\,b^3\,\sin \left (c+d\,x\right )}{2\,d}-\frac {2\,C\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 357, normalized size = 2.03 \[ \begin {cases} B a^{3} b x + \frac {3 B a^{2} b^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 B a b^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a b^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a b^{3} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 B b^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - C a^{4} x - \frac {2 C a^{3} b \sin {\left (c + d x \right )}}{d} + \frac {4 C a b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {2 C a b^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C b^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C b^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C b^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C b^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{2} \left (B a b + B b^{2} \cos {\relax (c )} - C a^{2} + C b^{2} \cos ^{2}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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