Optimal. Leaf size=94 \[ \frac {2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac {a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} a^2 x (3 A+2 B)+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.06, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2751, 2644} \[ \frac {2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac {a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {1}{2} a^2 x (3 A+2 B)+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2644
Rule 2751
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx &=\frac {B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} (3 A+2 B) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac {1}{2} a^2 (3 A+2 B) x+\frac {2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac {a^2 (3 A+2 B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 61, normalized size = 0.65 \[ \frac {a^2 (3 (8 A+7 B) \sin (c+d x)+3 (A+2 B) \sin (2 (c+d x))+18 A d x+B \sin (3 (c+d x))+12 B d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 70, normalized size = 0.74 \[ \frac {3 \, {\left (3 \, A + 2 \, B\right )} a^{2} d x + {\left (2 \, B a^{2} \cos \left (d x + c\right )^{2} + 3 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, {\left (6 \, A + 5 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 85, normalized size = 0.90 \[ \frac {B a^{2} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {1}{2} \, {\left (3 \, A a^{2} + 2 \, B a^{2}\right )} x + \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (8 \, A a^{2} + 7 \, B a^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 116, normalized size = 1.23 \[ \frac {\frac {B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a^{2} A \sin \left (d x +c \right )+B \,a^{2} \sin \left (d x +c \right )+a^{2} A \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 110, normalized size = 1.17 \[ \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 12 \, {\left (d x + c\right )} A a^{2} - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} + 6 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 24 \, A a^{2} \sin \left (d x + c\right ) + 12 \, B a^{2} \sin \left (d x + c\right )}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 98, normalized size = 1.04 \[ \frac {3\,A\,a^2\,x}{2}+B\,a^2\,x+\frac {2\,A\,a^2\,\sin \left (c+d\,x\right )}{d}+\frac {7\,B\,a^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {B\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{12\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 199, normalized size = 2.12 \[ \begin {cases} \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{2} x + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 A a^{2} \sin {\left (c + d x \right )}}{d} + B a^{2} x \sin ^{2}{\left (c + d x \right )} + B a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac {2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\relax (c )}\right ) \left (a \cos {\relax (c )} + a\right )^{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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