Optimal. Leaf size=54 \[ \frac {B \sin (c+d x) \cos (c+d x)}{2 d}+\frac {B x}{2}-\frac {C \sin ^3(c+d x)}{3 d}+\frac {C \sin (c+d x)}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3010, 2748, 2635, 8, 2633} \[ \frac {B \sin (c+d x) \cos (c+d x)}{2 d}+\frac {B x}{2}-\frac {C \sin ^3(c+d x)}{3 d}+\frac {C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 3010
Rubi steps
\begin {align*} \int \cos (c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^2(c+d x) (B+C \cos (c+d x)) \, dx\\ &=B \int \cos ^2(c+d x) \, dx+C \int \cos ^3(c+d x) \, dx\\ &=\frac {B \cos (c+d x) \sin (c+d x)}{2 d}+\frac {1}{2} B \int 1 \, dx-\frac {C \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {B x}{2}+\frac {C \sin (c+d x)}{d}+\frac {B \cos (c+d x) \sin (c+d x)}{2 d}-\frac {C \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 1.06 \[ \frac {B (c+d x)}{2 d}+\frac {B \sin (2 (c+d x))}{4 d}-\frac {C \sin ^3(c+d x)}{3 d}+\frac {C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 42, normalized size = 0.78 \[ \frac {3 \, B d x + {\left (2 \, C \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 4 \, C\right )} \sin \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 47, normalized size = 0.87 \[ \frac {1}{2} \, B x + \frac {C \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {B \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {3 \, C \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 49, normalized size = 0.91 \[ \frac {\frac {C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 46, normalized size = 0.85 \[ \frac {3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B - 4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 55, normalized size = 1.02 \[ \frac {B\,x}{2}+\frac {2\,C\,\sin \left (c+d\,x\right )}{3\,d}+\frac {B\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {C\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 95, normalized size = 1.76 \[ \begin {cases} \frac {B x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {B x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {B \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 C \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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