Optimal. Leaf size=181 \[ \frac {a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac {a^2 (8 A+7 B+6 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (8 A+7 B+6 C)+\frac {(20 A-5 B+6 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{60 d}+\frac {(5 B+2 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 a d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d} \]
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Rubi [A] time = 0.34, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3045, 2968, 3023, 2751, 2644} \[ \frac {a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac {a^2 (8 A+7 B+6 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac {1}{8} a^2 x (8 A+7 B+6 C)+\frac {(20 A-5 B+6 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{60 d}+\frac {(5 B+2 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{20 a d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 2644
Rule 2751
Rule 2968
Rule 3023
Rule 3045
Rubi steps
\begin {align*} \int \cos (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 ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {\int \cos (c+d x) (a+a \cos (c+d x))^2 (a (5 A+2 C)+a (5 B+2 C) \cos (c+d x)) \, dx}{5 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {\int (a+a \cos (c+d x))^2 \left (a (5 A+2 C) \cos (c+d x)+a (5 B+2 C) \cos ^2(c+d x)\right ) \, dx}{5 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(5 B+2 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 a d}+\frac {\int (a+a \cos (c+d x))^2 \left (3 a^2 (5 B+2 C)+a^2 (20 A-5 B+6 C) \cos (c+d x)\right ) \, dx}{20 a^2}\\ &=\frac {(20 A-5 B+6 C) (a+a \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(5 B+2 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 a d}+\frac {1}{12} (8 A+7 B+6 C) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac {1}{8} a^2 (8 A+7 B+6 C) x+\frac {a^2 (8 A+7 B+6 C) \sin (c+d x)}{6 d}+\frac {a^2 (8 A+7 B+6 C) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(20 A-5 B+6 C) (a+a \cos (c+d x))^2 \sin (c+d x)}{60 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(5 B+2 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{20 a d}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 132, normalized size = 0.73 \[ \frac {a^2 (60 (14 A+12 B+11 C) \sin (c+d x)+240 (A+B+C) \sin (2 (c+d x))+40 A \sin (3 (c+d x))+480 A d x+80 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+420 B c+420 B d x+90 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+6 C \sin (5 (c+d x))+240 c C+360 C d x)}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 122, normalized size = 0.67 \[ \frac {15 \, {\left (8 \, A + 7 \, B + 6 \, C\right )} a^{2} d x + {\left (24 \, C a^{2} \cos \left (d x + c\right )^{4} + 30 \, {\left (B + 2 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (5 \, A + 10 \, B + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (8 \, A + 7 \, B + 6 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (25 \, A + 20 \, B + 18 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 160, normalized size = 0.88 \[ \frac {C a^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{8} \, {\left (8 \, A a^{2} + 7 \, B a^{2} + 6 \, C a^{2}\right )} x + \frac {{\left (B a^{2} + 2 \, C a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (4 \, A a^{2} + 8 \, B a^{2} + 9 \, C a^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (A a^{2} + B a^{2} + C a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \, d} + \frac {{\left (14 \, A a^{2} + 12 \, B a^{2} + 11 \, C a^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 247, normalized size = 1.36 \[ \frac {\frac {a^{2} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 B \,a^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+2 a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{2} A \sin \left (d x +c \right )+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 236, normalized size = 1.30 \[ -\frac {160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{2} + 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} - 480 \, A a^{2} \sin \left (d x + c\right )}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 322, normalized size = 1.78 \[ \frac {\left (2\,A\,a^2+\frac {7\,B\,a^2}{4}+\frac {3\,C\,a^2}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {28\,A\,a^2}{3}+\frac {49\,B\,a^2}{6}+7\,C\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {56\,A\,a^2}{3}+\frac {40\,B\,a^2}{3}+\frac {72\,C\,a^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {52\,A\,a^2}{3}+\frac {79\,B\,a^2}{6}+9\,C\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (6\,A\,a^2+\frac {25\,B\,a^2}{4}+\frac {13\,C\,a^2}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A+7\,B+6\,C\right )}{4\,\left (2\,A\,a^2+\frac {7\,B\,a^2}{4}+\frac {3\,C\,a^2}{2}\right )}\right )\,\left (8\,A+7\,B+6\,C\right )}{4\,d}-\frac {a^2\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (8\,A+7\,B+6\,C\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.80, size = 570, normalized size = 3.15 \[ \begin {cases} A a^{2} x \sin ^{2}{\left (c + d x \right )} + A a^{2} x \cos ^{2}{\left (c + d x \right )} + \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 ^{2}{\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {A a^{2} \sin {\left (c + d x \right )}}{d} + \frac {3 B a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {B a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {4 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 B a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {2 B a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {B a^{2} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {3 C a^{2} x \sin ^{4}{\left (c + d x \right )}}{4} + \frac {3 C a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} + \frac {3 C a^{2} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {8 C a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 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 )}}{4 d} + \frac {2 C a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {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 )}}{4 d} + \frac {C a^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{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 {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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