Optimal. Leaf size=213 \[ -\frac {a^2 (10 A+9 B+8 C) \sin ^3(c+d x)}{15 d}+\frac {a^2 (10 A+9 B+8 C) \sin (c+d x)}{5 d}+\frac {a^2 (10 A+12 B+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {a^2 (14 A+12 B+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^2 x (14 A+12 B+11 C)+\frac {(3 B+C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{15 d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d} \]
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Rubi [A] time = 0.50, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {3045, 2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac {a^2 (10 A+9 B+8 C) \sin ^3(c+d x)}{15 d}+\frac {a^2 (10 A+9 B+8 C) \sin (c+d x)}{5 d}+\frac {a^2 (10 A+12 B+9 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {a^2 (14 A+12 B+11 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^2 x (14 A+12 B+11 C)+\frac {(3 B+C) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{15 d}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 2968
Rule 2976
Rule 3023
Rule 3045
Rubi steps
\begin {align*} \int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x))^2 (3 a (2 A+C)+2 a (3 B+C) \cos (c+d x)) \, dx}{6 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 B+C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^2 (10 A+6 B+7 C)+3 a^2 (10 A+12 B+9 C) \cos (c+d x)\right ) \, dx}{30 a}\\ &=\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 B+C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^2(c+d x) \left (3 a^3 (10 A+6 B+7 C)+\left (3 a^3 (10 A+6 B+7 C)+3 a^3 (10 A+12 B+9 C)\right ) \cos (c+d x)+3 a^3 (10 A+12 B+9 C) \cos ^2(c+d x)\right ) \, dx}{30 a}\\ &=\frac {a^2 (10 A+12 B+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 B+C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^2(c+d x) \left (15 a^3 (14 A+12 B+11 C)+24 a^3 (10 A+9 B+8 C) \cos (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^2 (10 A+12 B+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 B+C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{5} \left (a^2 (10 A+9 B+8 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{8} \left (a^2 (14 A+12 B+11 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {a^2 (14 A+12 B+11 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (10 A+12 B+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 B+C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^2 (14 A+12 B+11 C)\right ) \int 1 \, dx-\frac {\left (a^2 (10 A+9 B+8 C)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac {1}{16} a^2 (14 A+12 B+11 C) x+\frac {a^2 (10 A+9 B+8 C) \sin (c+d x)}{5 d}+\frac {a^2 (14 A+12 B+11 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (10 A+12 B+9 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(3 B+C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^2 (10 A+9 B+8 C) \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 171, normalized size = 0.80 \[ \frac {a^2 (120 (12 A+11 B+10 C) \sin (c+d x)+15 (32 A+32 B+31 C) \sin (2 (c+d x))+160 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+840 A d x+180 B \sin (3 (c+d x))+60 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+720 B c+720 B d x+200 C \sin (3 (c+d x))+75 C \sin (4 (c+d x))+24 C \sin (5 (c+d x))+5 C \sin (6 (c+d x))+420 c C+660 C d x)}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 145, normalized size = 0.68 \[ \frac {15 \, {\left (14 \, A + 12 \, B + 11 \, C\right )} a^{2} d x + {\left (40 \, C a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 12 \, B + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 16 \, {\left (10 \, A + 9 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (14 \, A + 12 \, B + 11 \, C\right )} a^{2} \cos \left (d x + c\right ) + 32 \, {\left (10 \, A + 9 \, B + 8 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 196, normalized size = 0.92 \[ \frac {C a^{2} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {1}{16} \, {\left (14 \, A a^{2} + 12 \, B a^{2} + 11 \, C a^{2}\right )} x + \frac {{\left (B a^{2} + 2 \, C a^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (2 \, A a^{2} + 4 \, B a^{2} + 5 \, C a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (8 \, A a^{2} + 9 \, B a^{2} + 10 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (32 \, A a^{2} + 32 \, B a^{2} + 31 \, C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (12 \, A a^{2} + 11 \, B a^{2} + 10 \, C a^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 304, normalized size = 1.43 \[ \frac {a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} C \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {2 a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 296, normalized size = 1.39 \[ -\frac {640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 128 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{2} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.77, size = 366, normalized size = 1.72 \[ \frac {\left (\frac {7\,A\,a^2}{4}+\frac {3\,B\,a^2}{2}+\frac {11\,C\,a^2}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {119\,A\,a^2}{12}+\frac {17\,B\,a^2}{2}+\frac {187\,C\,a^2}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {43\,A\,a^2}{2}+\frac {107\,B\,a^2}{5}+\frac {331\,C\,a^2}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {53\,A\,a^2}{2}+\frac {117\,B\,a^2}{5}+\frac {501\,C\,a^2}{20}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {233\,A\,a^2}{12}+\frac {31\,B\,a^2}{2}+\frac {87\,C\,a^2}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {25\,A\,a^2}{4}+\frac {13\,B\,a^2}{2}+\frac {53\,C\,a^2}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (14\,A+12\,B+11\,C\right )}{8\,\left (\frac {7\,A\,a^2}{4}+\frac {3\,B\,a^2}{2}+\frac {11\,C\,a^2}{8}\right )}\right )\,\left (14\,A+12\,B+11\,C\right )}{8\,d}-\frac {a^2\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (14\,A+12\,B+11\,C\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.90, size = 821, normalized size = 3.85 \[ \begin {cases} \frac {3 A a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 A a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 A a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 A a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {8 B a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 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 )}}{4 d} + \frac {2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {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 )}}{4 d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {5 C a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {15 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {5 C a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {5 C a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {16 C a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {5 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {8 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 C a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {11 C a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {2 C a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C 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 \cos {\relax (c )} + a\right )^{2} \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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