Optimal. Leaf size=243 \[ -\frac {2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac {4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac {a^4 (56 A+49 B+44 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {27 a^4 (56 A+49 B+44 C) \sin (c+d x) \cos (c+d x)}{560 d}+\frac {1}{16} a^4 x (56 A+49 B+44 C)+\frac {(42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{210 d}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{42 a d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d} \]
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Rubi [A] time = 0.45, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3045, 2968, 3023, 2751, 2645, 2637, 2635, 8, 2633} \[ -\frac {2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac {4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac {a^4 (56 A+49 B+44 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac {27 a^4 (56 A+49 B+44 C) \sin (c+d x) \cos (c+d x)}{560 d}+\frac {1}{16} a^4 x (56 A+49 B+44 C)+\frac {(42 A-7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^4}{210 d}+\frac {(7 B+4 C) \sin (c+d x) (a \cos (c+d x)+a)^5}{42 a d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a \cos (c+d x)+a)^4}{7 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2637
Rule 2645
Rule 2751
Rule 2968
Rule 3023
Rule 3045
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \cos (c+d x))^4 \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))^4 \sin (c+d x)}{7 d}+\frac {\int \cos (c+d x) (a+a \cos (c+d x))^4 (a (7 A+2 C)+a (7 B+4 C) \cos (c+d x)) \, dx}{7 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\int (a+a \cos (c+d x))^4 \left (a (7 A+2 C) \cos (c+d x)+a (7 B+4 C) \cos ^2(c+d x)\right ) \, dx}{7 a}\\ &=\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac {\int (a+a \cos (c+d x))^4 \left (5 a^2 (7 B+4 C)+a^2 (42 A-7 B+8 C) \cos (c+d x)\right ) \, dx}{42 a^2}\\ &=\frac {(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 C) \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac {(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac {1}{70} (56 A+49 B+44 C) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx\\ &=\frac {1}{70} a^4 (56 A+49 B+44 C) x+\frac {(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac {1}{70} \left (a^4 (56 A+49 B+44 C)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{35} \left (2 a^4 (56 A+49 B+44 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{35} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {1}{70} a^4 (56 A+49 B+44 C) x+\frac {2 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac {3 a^4 (56 A+49 B+44 C) \cos (c+d x) \sin (c+d x)}{70 d}+\frac {a^4 (56 A+49 B+44 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}+\frac {1}{280} \left (3 a^4 (56 A+49 B+44 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{70} \left (3 a^4 (56 A+49 B+44 C)\right ) \int 1 \, dx-\frac {\left (2 a^4 (56 A+49 B+44 C)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac {2}{35} a^4 (56 A+49 B+44 C) x+\frac {4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \cos (c+d x) \sin (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}-\frac {2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}+\frac {1}{560} \left (3 a^4 (56 A+49 B+44 C)\right ) \int 1 \, dx\\ &=\frac {1}{16} a^4 (56 A+49 B+44 C) x+\frac {4 a^4 (56 A+49 B+44 C) \sin (c+d x)}{35 d}+\frac {27 a^4 (56 A+49 B+44 C) \cos (c+d x) \sin (c+d x)}{560 d}+\frac {a^4 (56 A+49 B+44 C) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac {(42 A-7 B+8 C) (a+a \cos (c+d x))^4 \sin (c+d x)}{210 d}+\frac {C \cos ^2(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(7 B+4 C) (a+a \cos (c+d x))^5 \sin (c+d x)}{42 a d}-\frac {2 a^4 (56 A+49 B+44 C) \sin ^3(c+d x)}{105 d}\\ \end {align*}
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Mathematica [A] time = 0.94, size = 204, normalized size = 0.84 \[ \frac {a^4 (105 (392 A+352 B+323 C) \sin (c+d x)+105 (128 A+127 B+124 C) \sin (2 (c+d x))+4060 A \sin (3 (c+d x))+840 A \sin (4 (c+d x))+84 A \sin (5 (c+d x))+23520 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B c+20580 B d x+5495 C \sin (3 (c+d x))+2100 C \sin (4 (c+d x))+651 C \sin (5 (c+d x))+140 C \sin (6 (c+d x))+15 C \sin (7 (c+d x))+11760 c C+18480 C d x)}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 168, normalized size = 0.69 \[ \frac {105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} d x + {\left (240 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (7 \, A + 28 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, A + 41 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (238 \, A + 252 \, B + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (56 \, A + 49 \, B + 44 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (581 \, A + 504 \, B + 454 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 229, normalized size = 0.94 \[ \frac {C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (56 \, A a^{4} + 49 \, B a^{4} + 44 \, C a^{4}\right )} x + \frac {{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (4 \, A a^{4} + 16 \, B a^{4} + 31 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (8 \, A a^{4} + 15 \, B a^{4} + 20 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (116 \, A a^{4} + 144 \, B a^{4} + 157 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (128 \, A a^{4} + 127 \, B a^{4} + 124 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (392 \, A a^{4} + 352 \, B a^{4} + 323 \, C a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.35, size = 490, normalized size = 2.02 \[ \frac {\frac {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}+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 )+\frac {a^{4} C \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}+4 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 {4 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}+4 a^{4} 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 )+2 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 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 )+\frac {6 a^{4} 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}+4 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 {4 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} 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 \,a^{4} \sin \left (d x +c \right )+a^{4} B \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} 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 [B] time = 0.35, size = 483, normalized size = 1.99 \[ \frac {448 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 13440 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 6720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 1792 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{4} + 2688 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} - 140 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 6720 \, A a^{4} \sin \left (d x + c\right )}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.97, size = 410, normalized size = 1.69 \[ \frac {\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {140\,A\,a^4}{3}+\frac {245\,B\,a^4}{6}+\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {13867\,B\,a^4}{120}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1024\,A\,a^4}{5}+\frac {896\,B\,a^4}{5}+\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {19157\,B\,a^4}{120}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {308\,A\,a^4}{3}+\frac {523\,B\,a^4}{6}+70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {207\,B\,a^4}{8}+\frac {53\,C\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,\left (7\,A\,a^4+\frac {49\,B\,a^4}{8}+\frac {11\,C\,a^4}{2}\right )}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,d}-\frac {a^4\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (56\,A+49\,B+44\,C\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.82, size = 1258, normalized size = 5.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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