Optimal. Leaf size=241 \[ -\frac {a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac {a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac {a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{15 d}+\frac {a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^4 x (49 A+44 B)+\frac {(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 d}+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
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Rubi [A] time = 0.59, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2976, 2968, 3023, 2748, 2635, 8, 2633} \[ -\frac {a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}+\frac {a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac {a^4 (301 A+276 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {(7 A+10 B) \sin (c+d x) \cos ^3(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{42 d}+\frac {7 (A+B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{15 d}+\frac {a^4 (49 A+44 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^4 x (49 A+44 B)+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d} \]
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 ^2(c+d x) (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx &=\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos ^2(c+d x) (a+a \cos (c+d x))^3 (a (7 A+3 B)+a (7 A+10 B) \cos (c+d x)) \, dx\\ &=\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {1}{42} \int \cos ^2(c+d x) (a+a \cos (c+d x))^2 \left (3 a^2 (21 A+16 B)+98 a^2 (A+B) \cos (c+d x)\right ) \, dx\\ &=\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{210} \int \cos ^2(c+d x) (a+a \cos (c+d x)) \left (3 a^3 (203 A+178 B)+3 a^3 (301 A+276 B) \cos (c+d x)\right ) \, dx\\ &=\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{210} \int \cos ^2(c+d x) \left (3 a^4 (203 A+178 B)+\left (3 a^4 (203 A+178 B)+3 a^4 (301 A+276 B)\right ) \cos (c+d x)+3 a^4 (301 A+276 B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{840} \int \cos ^2(c+d x) \left (105 a^4 (49 A+44 B)+24 a^4 (252 A+227 B) \cos (c+d x)\right ) \, dx\\ &=\frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{8} \left (a^4 (49 A+44 B)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{35} \left (a^4 (252 A+227 B)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac {a^4 (49 A+44 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^4 (49 A+44 B)\right ) \int 1 \, dx-\frac {\left (a^4 (252 A+227 B)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac {1}{16} a^4 (49 A+44 B) x+\frac {a^4 (252 A+227 B) \sin (c+d x)}{35 d}+\frac {a^4 (49 A+44 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (301 A+276 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {a B \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{7 d}+\frac {(7 A+10 B) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac {7 (A+B) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^4 (252 A+227 B) \sin ^3(c+d x)}{105 d}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 156, normalized size = 0.65 \[ \frac {a^4 (105 (352 A+323 B) \sin (c+d x)+105 (127 A+124 B) \sin (2 (c+d x))+5040 A \sin (3 (c+d x))+1575 A \sin (4 (c+d x))+336 A \sin (5 (c+d x))+35 A \sin (6 (c+d x))+20580 A d x+5495 B \sin (3 (c+d x))+2100 B \sin (4 (c+d x))+651 B \sin (5 (c+d x))+140 B \sin (6 (c+d x))+15 B \sin (7 (c+d x))+18480 B c+18480 B d x)}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 150, normalized size = 0.62 \[ \frac {105 \, {\left (49 \, A + 44 \, B\right )} a^{4} d x + {\left (240 \, B a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{5} + 192 \, {\left (7 \, A + 12 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (41 \, A + 44 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (252 \, A + 227 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (49 \, A + 44 \, B\right )} a^{4} \cos \left (d x + c\right ) + 32 \, {\left (252 \, A + 227 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.53, size = 193, normalized size = 0.80 \[ \frac {B a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (49 \, A a^{4} + 44 \, B a^{4}\right )} x + \frac {{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{4} + 31 \, B a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {5 \, {\left (3 \, A a^{4} + 4 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (144 \, A a^{4} + 157 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (127 \, A a^{4} + 124 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (352 \, A a^{4} + 323 \, B a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 358, normalized size = 1.49 \[ \frac {A \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{4} B \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {4 A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} B \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+6 A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 a^{4} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 356, normalized size = 1.48 \[ \frac {1792 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} B a^{4} + 2688 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 140 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4}}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.64, size = 353, normalized size = 1.46 \[ \frac {\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {245\,A\,a^4}{6}+\frac {110\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {13867\,A\,a^4}{120}+\frac {3113\,B\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {896\,A\,a^4}{5}+\frac {5632\,B\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {19157\,A\,a^4}{120}+\frac {1501\,B\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {523\,A\,a^4}{6}+70\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {53\,B\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a^4\,\left (49\,A+44\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (49\,A+44\,B\right )}{8\,\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}\right )}\right )\,\left (49\,A+44\,B\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.01, size = 960, normalized size = 3.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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