Optimal. Leaf size=77 \[ \frac {a (3 A+2 B) \sin (c+d x)}{3 d}+\frac {a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (A+B)+\frac {a B \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2968, 3023, 2734} \[ \frac {a (3 A+2 B) \sin (c+d x)}{3 d}+\frac {a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {1}{2} a x (A+B)+\frac {a B \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x)) (A+B \cos (c+d x)) \, dx &=\int \cos (c+d x) \left (a A+(a A+a B) \cos (c+d x)+a B \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos (c+d x) (a (3 A+2 B)+3 a (A+B) \cos (c+d x)) \, dx\\ &=\frac {1}{2} a (A+B) x+\frac {a (3 A+2 B) \sin (c+d x)}{3 d}+\frac {a (A+B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac {a B \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 65, normalized size = 0.84 \[ \frac {a (3 (4 A+3 B) \sin (c+d x)+3 (A+B) \sin (2 (c+d x))+6 A c+6 A d x+B \sin (3 (c+d x))+6 B c+6 B d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 56, normalized size = 0.73 \[ \frac {3 \, {\left (A + B\right )} a d x + {\left (2 \, B a \cos \left (d x + c\right )^{2} + 3 \, {\left (A + B\right )} a \cos \left (d x + c\right ) + 2 \, {\left (3 \, A + 2 \, B\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 68, normalized size = 0.88 \[ \frac {1}{2} \, {\left (A a + B a\right )} x + \frac {B a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (A a + B a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a + 3 \, B a\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 85, normalized size = 1.10 \[ \frac {\frac {a B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 79, normalized size = 1.03 \[ \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 12 \, A a \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 84, normalized size = 1.09 \[ \frac {A\,a\,x}{2}+\frac {B\,a\,x}{2}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {3\,B\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.53, size = 168, normalized size = 2.18 \[ \begin {cases} \frac {A a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {2 B a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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