Optimal. Leaf size=113 \[ -\frac {(5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac {(5 A+4 C) \sin (c+d x)}{5 d}+\frac {B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 B \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 B x}{8}+\frac {C \sin (c+d x) \cos ^4(c+d x)}{5 d} \]
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Rubi [A] time = 0.11, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3023, 2748, 2633, 2635, 8} \[ -\frac {(5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac {(5 A+4 C) \sin (c+d x)}{5 d}+\frac {B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 B \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 B x}{8}+\frac {C \sin (c+d x) \cos ^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 3023
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^3(c+d x) (5 A+4 C+5 B \cos (c+d x)) \, dx\\ &=\frac {C \cos ^4(c+d x) \sin (c+d x)}{5 d}+B \int \cos ^4(c+d x) \, dx+\frac {1}{5} (5 A+4 C) \int \cos ^3(c+d x) \, dx\\ &=\frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {C \cos ^4(c+d x) \sin (c+d x)}{5 d}+\frac {1}{4} (3 B) \int \cos ^2(c+d x) \, dx-\frac {(5 A+4 C) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac {(5 A+4 C) \sin (c+d x)}{5 d}+\frac {3 B \cos (c+d x) \sin (c+d x)}{8 d}+\frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {C \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {(5 A+4 C) \sin ^3(c+d x)}{15 d}+\frac {1}{8} (3 B) \int 1 \, dx\\ &=\frac {3 B x}{8}+\frac {(5 A+4 C) \sin (c+d x)}{5 d}+\frac {3 B \cos (c+d x) \sin (c+d x)}{8 d}+\frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {C \cos ^4(c+d x) \sin (c+d x)}{5 d}-\frac {(5 A+4 C) \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 87, normalized size = 0.77 \[ \frac {60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x))}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 73, normalized size = 0.65 \[ \frac {45 \, B d x + {\left (24 \, C \cos \left (d x + c\right )^{4} + 30 \, B \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{2} + 45 \, B \cos \left (d x + c\right ) + 80 \, A + 64 \, C\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 89, normalized size = 0.79 \[ \frac {3}{8} \, B x + \frac {C \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {B \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, A + 5 \, C\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {B \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {{\left (6 \, A + 5 \, C\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 89, normalized size = 0.79 \[ \frac {\frac {C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 89, normalized size = 0.79 \[ -\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 104, normalized size = 0.92 \[ \frac {3\,B\,x}{8}+\frac {A\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,C\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {C\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {3\,A\,\sin \left (c+d\,x\right )}{4\,d}+\frac {5\,C\,\sin \left (c+d\,x\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.95, size = 209, normalized size = 1.85 \[ \begin {cases} \frac {2 A \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 B x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 B x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 B \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 B \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 C \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right ) \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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