Optimal. Leaf size=88 \[ \frac {(4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (4 A+3 C)-\frac {B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
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Rubi [A] time = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3023, 2748, 2635, 8, 2633} \[ \frac {(4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (4 A+3 C)-\frac {B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^2(c+d x) (4 A+3 C+4 B \cos (c+d x)) \, dx\\ &=\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 d}+B \int \cos ^3(c+d x) \, dx+\frac {1}{4} (4 A+3 C) \int \cos ^2(c+d x) \, dx\\ &=\frac {(4 A+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 {1}{8} (4 A+3 C) \int 1 \, dx-\frac {B \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {1}{8} (4 A+3 C) x+\frac {B \sin (c+d x)}{d}+\frac {(4 A+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 = 0.16, size = 70, normalized size = 0.80 \[ \frac {24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x-32 B \sin ^3(c+d x)+96 B \sin (c+d x)+3 C \sin (4 (c+d x))+36 c C+36 C d x}{96 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 65, normalized size = 0.74 \[ \frac {3 \, {\left (4 \, A + 3 \, C\right )} d x + {\left (6 \, C \cos \left (d x + c\right )^{3} + 8 \, B \cos \left (d x + c\right )^{2} + 3 \, {\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right ) + 16 \, B\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 70, normalized size = 0.80 \[ \frac {1}{8} \, {\left (4 \, A + 3 \, C\right )} x + \frac {C \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {B \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac {{\left (A + C\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {3 \, B \sin \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 84, normalized size = 0.95 \[ \frac {C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 77, normalized size = 0.88 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 81, normalized size = 0.92 \[ \frac {A\,x}{2}+\frac {3\,C\,x}{8}+\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,B\,\sin \left (c+d\,x\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.03, size = 197, normalized size = 2.24 \[ \begin {cases} \frac {A x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {A x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {A \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {2 B \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {B \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 C x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 C x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 C \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 C \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) \cos ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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