Optimal. Leaf size=91 \[ \frac {a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac {1}{2} a x (2 A+B+C)+\frac {a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d} \]
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Rubi [A] time = 0.08, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {3023, 2734} \[ \frac {a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac {1}{2} a x (2 A+B+C)+\frac {a (3 B-C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3023
Rubi steps
\begin {align*} \int (a+a \cos (c+d x)) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d}+\frac {\int (a+a \cos (c+d x)) (a (3 A+2 C)+a (3 B-C) \cos (c+d x)) \, dx}{3 a}\\ &=\frac {1}{2} a (2 A+B+C) x+\frac {a (3 A+3 B+C) \sin (c+d x)}{3 d}+\frac {a (3 B-C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 65, normalized size = 0.71 \[ \frac {a (3 (4 A+4 B+3 C) \sin (c+d x)+12 A d x+3 (B+C) \sin (2 (c+d x))+6 B d x+C \sin (3 (c+d x))+6 C d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 62, normalized size = 0.68 \[ \frac {3 \, {\left (2 \, A + B + C\right )} a d x + {\left (2 \, C a \cos \left (d x + c\right )^{2} + 3 \, {\left (B + C\right )} a \cos \left (d x + c\right ) + 2 \, {\left (3 \, A + 3 \, B + 2 \, C\right )} a\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 76, normalized size = 0.84 \[ \frac {1}{2} \, {\left (2 \, A a + B a + C a\right )} x + \frac {C a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (B a + C a\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (4 \, A a + 4 \, B a + 3 \, C a\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 102, normalized size = 1.12 \[ \frac {\frac {a C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a A \sin \left (d x +c \right )+a B \sin \left (d x +c \right )+a A \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 98, normalized size = 1.08 \[ \frac {12 \, {\left (d x + c\right )} A a + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 12 \, A a \sin \left (d x + c\right ) + 12 \, B a \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 100, normalized size = 1.10 \[ A\,a\,x+\frac {B\,a\,x}{2}+\frac {C\,a\,x}{2}+\frac {A\,a\,\sin \left (c+d\,x\right )}{d}+\frac {B\,a\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 189, normalized size = 2.08 \[ \begin {cases} A a x + \frac {A a \sin {\left (c + d x \right )}}{d} + \frac {B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {B a \sin {\left (c + d x \right )}}{d} + \frac {C a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {C a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {C a \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + a\right ) \left (A + B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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