Optimal. Leaf size=30 \[ \frac {(A+C) \sin (c+d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {3013} \[ \frac {(A+C) \sin (c+d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 3013
Rubi steps
\begin {align*} \int \cos (c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {(A+C) \sin (c+d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.67 \[ \frac {A \sin (c) \cos (d x)}{d}+\frac {A \cos (c) \sin (d x)}{d}-\frac {C \sin ^3(c+d x)}{3 d}+\frac {C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 28, normalized size = 0.93 \[ \frac {{\left (C \cos \left (d x + c\right )^{2} + 3 \, A + 2 \, C\right )} \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 34, normalized size = 1.13 \[ -\frac {{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C - 3 \, A \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 33, normalized size = 1.10 \[ \frac {\frac {C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 34, normalized size = 1.13 \[ -\frac {{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C - 3 \, A \sin \left (d x + c\right )}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 28, normalized size = 0.93 \[ -\frac {\frac {C\,{\sin \left (c+d\,x\right )}^3}{3}-\sin \left (c+d\,x\right )\,\left (A+C\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 56, normalized size = 1.87 \[ \begin {cases} \frac {A \sin {\left (c + d x \right )}}{d} + \frac {2 C \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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