Optimal. Leaf size=71 \[ \frac {2 \left (a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {\sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {a b \sin (c+d x) \cos (c+d x)}{3 d}+a b x \]
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Rubi [A] time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2753, 2734} \[ \frac {2 \left (a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {\sin (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {a b \sin (c+d x) \cos (c+d x)}{3 d}+a b x \]
Antiderivative was successfully verified.
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Rule 2734
Rule 2753
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^2 \, dx &=\frac {(a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac {1}{3} \int (2 b+2 a \cos (c+d x)) (a+b \cos (c+d x)) \, dx\\ &=a b x+\frac {2 \left (a^2+b^2\right ) \sin (c+d x)}{3 d}+\frac {a b \cos (c+d x) \sin (c+d x)}{3 d}+\frac {(a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 59, normalized size = 0.83 \[ \frac {3 \left (4 a^2+3 b^2\right ) \sin (c+d x)+b (12 a (c+d x)+6 a \sin (2 (c+d x))+b \sin (3 (c+d x)))}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 52, normalized size = 0.73 \[ \frac {3 \, a b d x + {\left (b^{2} \cos \left (d x + c\right )^{2} + 3 \, a b \cos \left (d x + c\right ) + 3 \, a^{2} + 2 \, b^{2}\right )} \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 60, normalized size = 0.85 \[ a b x + \frac {b^{2} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (4 \, a^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 63, normalized size = 0.89 \[ \frac {\frac {b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 a b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{2} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 60, normalized size = 0.85 \[ \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b - 2 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} b^{2} + 6 \, a^{2} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 72, normalized size = 1.01 \[ \frac {a^2\,\sin \left (c+d\,x\right )}{d}+\frac {2\,b^2\,\sin \left (c+d\,x\right )}{3\,d}+a\,b\,x+\frac {b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {a\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 107, normalized size = 1.51 \[ \begin {cases} \frac {a^{2} \sin {\left (c + d x \right )}}{d} + a b x \sin ^{2}{\left (c + d x \right )} + a b x \cos ^{2}{\left (c + d x \right )} + \frac {a b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {2 b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\relax (c )}\right )^{2} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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