Optimal. Leaf size=76 \[ -\frac {B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 C \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 C x}{8} \]
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Rubi [A] time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3010, 2748, 2633, 2635, 8} \[ -\frac {B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 C \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 C x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 3010
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\int \cos ^3(c+d x) (B+C \cos (c+d x)) \, dx\\ &=B \int \cos ^3(c+d x) \, dx+C \int \cos ^4(c+d x) \, dx\\ &=\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} (3 C) \int \cos ^2(c+d x) \, dx-\frac {B \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {B \sin (c+d x)}{d}+\frac {3 C \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {B \sin ^3(c+d x)}{3 d}+\frac {1}{8} (3 C) \int 1 \, dx\\ &=\frac {3 C x}{8}+\frac {B \sin (c+d x)}{d}+\frac {3 C \cos (c+d x) \sin (c+d x)}{8 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {B \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 73, normalized size = 0.96 \[ -\frac {B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d}+\frac {3 C (c+d x)}{8 d}+\frac {C \sin (2 (c+d x))}{4 d}+\frac {C \sin (4 (c+d x))}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 53, normalized size = 0.70 \[ \frac {9 \, C d x + {\left (6 \, C \cos \left (d x + c\right )^{3} + 8 \, B \cos \left (d x + c\right )^{2} + 9 \, C \cos \left (d x + c\right ) + 16 \, B\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 62, normalized size = 0.82 \[ \frac {3}{8} \, C x + \frac {C \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {B \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {C \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {3 \, B \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 60, normalized size = 0.79 \[ \frac {C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 57, normalized size = 0.75 \[ -\frac {32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 75, normalized size = 0.99 \[ \frac {3\,C\,x}{8}+\frac {2\,B\,\sin \left (c+d\,x\right )}{3\,d}+\frac {3\,C\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {B\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {C\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 150, normalized size = 1.97 \[ \begin {cases} \frac {2 B \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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