Optimal. Leaf size=191 \[ -\frac {a^2 (9 A+8 B) \sin ^3(c+d x)}{15 d}+\frac {a^2 (9 A+8 B) \sin (c+d x)}{5 d}+\frac {a^2 (6 A+7 B) \sin (c+d x) \cos ^4(c+d x)}{30 d}+\frac {a^2 (12 A+11 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a^2 (12 A+11 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^2 x (12 A+11 B)+\frac {B \sin (c+d x) \cos ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{6 d} \]
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Rubi [A] time = 0.31, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2976, 2968, 3023, 2748, 2633, 2635, 8} \[ -\frac {a^2 (9 A+8 B) \sin ^3(c+d x)}{15 d}+\frac {a^2 (9 A+8 B) \sin (c+d x)}{5 d}+\frac {a^2 (6 A+7 B) \sin (c+d x) \cos ^4(c+d x)}{30 d}+\frac {a^2 (12 A+11 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a^2 (12 A+11 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^2 x (12 A+11 B)+\frac {B \sin (c+d x) \cos ^4(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2968
Rule 2976
Rule 3023
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx &=\frac {B \cos ^4(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^3(c+d x) (a+a \cos (c+d x)) (2 a (3 A+2 B)+a (6 A+7 B) \cos (c+d x)) \, dx\\ &=\frac {B \cos ^4(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^3(c+d x) \left (2 a^2 (3 A+2 B)+\left (2 a^2 (3 A+2 B)+a^2 (6 A+7 B)\right ) \cos (c+d x)+a^2 (6 A+7 B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 (6 A+7 B) \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{30} \int \cos ^3(c+d x) \left (6 a^2 (9 A+8 B)+5 a^2 (12 A+11 B) \cos (c+d x)\right ) \, dx\\ &=\frac {a^2 (6 A+7 B) \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{5} \left (a^2 (9 A+8 B)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{6} \left (a^2 (12 A+11 B)\right ) \int \cos ^4(c+d x) \, dx\\ &=\frac {a^2 (12 A+11 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 (6 A+7 B) \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}+\frac {1}{8} \left (a^2 (12 A+11 B)\right ) \int \cos ^2(c+d x) \, dx-\frac {\left (a^2 (9 A+8 B)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac {a^2 (9 A+8 B) \sin (c+d x)}{5 d}+\frac {a^2 (12 A+11 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (12 A+11 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 (6 A+7 B) \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}-\frac {a^2 (9 A+8 B) \sin ^3(c+d x)}{15 d}+\frac {1}{16} \left (a^2 (12 A+11 B)\right ) \int 1 \, dx\\ &=\frac {1}{16} a^2 (12 A+11 B) x+\frac {a^2 (9 A+8 B) \sin (c+d x)}{5 d}+\frac {a^2 (12 A+11 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (12 A+11 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^2 (6 A+7 B) \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {B \cos ^4(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{6 d}-\frac {a^2 (9 A+8 B) \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 134, normalized size = 0.70 \[ \frac {a^2 (120 (11 A+10 B) \sin (c+d x)+15 (32 A+31 B) \sin (2 (c+d x))+180 A \sin (3 (c+d x))+60 A \sin (4 (c+d x))+12 A \sin (5 (c+d x))+720 A d x+200 B \sin (3 (c+d x))+75 B \sin (4 (c+d x))+24 B \sin (5 (c+d x))+5 B \sin (6 (c+d x))+660 B c+660 B d x)}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 130, normalized size = 0.68 \[ \frac {15 \, {\left (12 \, A + 11 \, B\right )} a^{2} d x + {\left (40 \, B a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (12 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 16 \, {\left (9 \, A + 8 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (12 \, A + 11 \, B\right )} a^{2} \cos \left (d x + c\right ) + 32 \, {\left (9 \, A + 8 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 166, normalized size = 0.87 \[ \frac {B a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {1}{16} \, {\left (12 \, A a^{2} + 11 \, B a^{2}\right )} x + \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (4 \, A a^{2} + 5 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (9 \, A a^{2} + 10 \, B a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (32 \, A a^{2} + 31 \, B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (11 \, A a^{2} + 10 \, B a^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 217, normalized size = 1.14 \[ \frac {\frac {a^{2} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+2 a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 B \,a^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 216, normalized size = 1.13 \[ \frac {64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 128 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{2} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.59, size = 315, normalized size = 1.65 \[ \frac {\left (\frac {3\,A\,a^2}{2}+\frac {11\,B\,a^2}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {17\,A\,a^2}{2}+\frac {187\,B\,a^2}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {107\,A\,a^2}{5}+\frac {331\,B\,a^2}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {117\,A\,a^2}{5}+\frac {501\,B\,a^2}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {31\,A\,a^2}{2}+\frac {87\,B\,a^2}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a^2}{2}+\frac {53\,B\,a^2}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^2\,\left (12\,A+11\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,A+11\,B\right )}{8\,\left (\frac {3\,A\,a^2}{2}+\frac {11\,B\,a^2}{8}\right )}\right )\,\left (12\,A+11\,B\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.34, size = 600, normalized size = 3.14 \[ \begin {cases} \frac {3 A a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {8 A a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {2 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 A a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {5 B a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 B a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 B a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 B a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {16 B a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {5 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {8 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {11 B a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {2 B a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 B a^{2} \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 )}\right ) \left (a \cos {\relax (c )} + a\right )^{2} \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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