Optimal. Leaf size=220 \[ \frac {a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac {a^4 (159 A+176 B) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac {7 a^4 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(73 A+72 B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{120 d}+\frac {7}{16} a^4 x (7 A+8 B)+\frac {(3 A+2 B) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{10 d}+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d} \]
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Rubi [A] time = 0.53, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4017, 3996, 3787, 2635, 8, 2637} \[ \frac {a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac {a^4 (159 A+176 B) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac {7 a^4 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(3 A+2 B) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{10 d}+\frac {(73 A+72 B) \sin (c+d x) \cos ^3(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{120 d}+\frac {7}{16} a^4 x (7 A+8 B)+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2637
Rule 3787
Rule 3996
Rule 4017
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (3 a (3 A+2 B)+2 a (A+3 B) \sec (c+d x)) \, dx\\ &=\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac {1}{30} \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (a^2 (73 A+72 B)+14 a^2 (2 A+3 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac {(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{120} \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (159 A+176 B)+6 a^3 (43 A+52 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac {(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {1}{360} \int \cos ^2(c+d x) \left (-315 a^4 (7 A+8 B)-24 a^4 (72 A+83 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac {(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{8} \left (7 a^4 (7 A+8 B)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{15} \left (a^4 (72 A+83 B)\right ) \int \cos (c+d x) \, dx\\ &=\frac {a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac {7 a^4 (7 A+8 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac {(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{16} \left (7 a^4 (7 A+8 B)\right ) \int 1 \, dx\\ &=\frac {7}{16} a^4 (7 A+8 B) x+\frac {a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac {7 a^4 (7 A+8 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^4 (159 A+176 B) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {(3 A+2 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 d}+\frac {(73 A+72 B) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 134, normalized size = 0.61 \[ \frac {a^4 (120 (44 A+49 B) \sin (c+d x)+15 (127 A+128 B) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+225 A \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+2940 A c+2940 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x)}{960 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 130, normalized size = 0.59 \[ \frac {105 \, {\left (7 \, A + 8 \, B\right )} a^{4} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (41 \, A + 24 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \, {\left (18 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (7 \, A + 8 \, B\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (72 \, A + 83 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 244, normalized size = 1.11 \[ \frac {105 \, {\left (7 \, A a^{4} + 8 \, B a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.00, size = 306, normalized size = 1.39 \[ \frac {A \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{4} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 A \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a^{4} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} B \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 297, normalized size = 1.35 \[ \frac {256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, B a^{4} \sin \left (d x + c\right )}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 286, normalized size = 1.30 \[ \frac {\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {833\,A\,a^4}{24}+\frac {119\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1617\,A\,a^4}{20}+\frac {462\,B\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1967\,A\,a^4}{20}+\frac {562\,B\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1471\,A\,a^4}{24}+\frac {233\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+25\,B\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+8\,B\right )}{8\,\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4\right )}\right )\,\left (7\,A+8\,B\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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