Optimal. Leaf size=213 \[ \frac {1}{16} a^2 (11 A+12 B+14 C) x+\frac {a^2 (8 A+9 B+10 C) \sin (c+d x)}{5 d}+\frac {a^2 (11 A+12 B+14 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (9 A+12 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(A+3 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^2 (8 A+9 B+10 C) \sin ^3(c+d x)}{15 d} \]
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Rubi [A]
time = 0.31, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4171, 4102,
4081, 3872, 2713, 2715, 8} \begin {gather*} -\frac {a^2 (8 A+9 B+10 C) \sin ^3(c+d x)}{15 d}+\frac {a^2 (8 A+9 B+10 C) \sin (c+d x)}{5 d}+\frac {a^2 (9 A+12 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {a^2 (11 A+12 B+14 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(A+3 B) \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{15 d}+\frac {1}{16} a^2 x (11 A+12 B+14 C)+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4081
Rule 4102
Rule 4171
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (2 a (A+3 B)+3 a (A+2 C) \sec (c+d x)) \, dx}{6 a}\\ &=\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(A+3 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^2 (9 A+12 B+10 C)+3 a^2 (7 A+6 B+10 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac {a^2 (9 A+12 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(A+3 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {\int \cos ^3(c+d x) \left (-24 a^3 (8 A+9 B+10 C)-15 a^3 (11 A+12 B+14 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^2 (9 A+12 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(A+3 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{5} \left (a^2 (8 A+9 B+10 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{8} \left (a^2 (11 A+12 B+14 C)\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {a^2 (11 A+12 B+14 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (9 A+12 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(A+3 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^2 (11 A+12 B+14 C)\right ) \int 1 \, dx-\frac {\left (a^2 (8 A+9 B+10 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac {1}{16} a^2 (11 A+12 B+14 C) x+\frac {a^2 (8 A+9 B+10 C) \sin (c+d x)}{5 d}+\frac {a^2 (11 A+12 B+14 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (9 A+12 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(A+3 B) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^2 (8 A+9 B+10 C) \sin ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A]
time = 1.04, size = 170, normalized size = 0.80 \begin {gather*} \frac {a^2 (240 A c+720 B c+660 A d x+720 B d x+840 C d x+120 (10 A+11 B+12 C) \sin (c+d x)+15 (31 A+32 (B+C)) \sin (2 (c+d x))+200 A \sin (3 (c+d x))+180 B \sin (3 (c+d x))+160 C \sin (3 (c+d x))+75 A \sin (4 (c+d x))+60 B \sin (4 (c+d x))+30 C \sin (4 (c+d x))+24 A \sin (5 (c+d x))+12 B \sin (5 (c+d x))+5 A \sin (6 (c+d x)))}{960 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.02, size = 304, normalized size = 1.43
method | result | size |
risch | \(\frac {11 a^{2} A x}{16}+\frac {3 a^{2} B x}{4}+\frac {7 a^{2} x C}{8}+\frac {5 a^{2} A \sin \left (d x +c \right )}{4 d}+\frac {11 a^{2} B \sin \left (d x +c \right )}{8 d}+\frac {3 \sin \left (d x +c \right ) a^{2} C}{2 d}+\frac {a^{2} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{2} A \sin \left (5 d x +5 c \right )}{40 d}+\frac {\sin \left (5 d x +5 c \right ) a^{2} B}{80 d}+\frac {5 a^{2} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2} B}{16 d}+\frac {\sin \left (4 d x +4 c \right ) a^{2} C}{32 d}+\frac {5 a^{2} A \sin \left (3 d x +3 c \right )}{24 d}+\frac {3 \sin \left (3 d x +3 c \right ) a^{2} B}{16 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} C}{6 d}+\frac {31 a^{2} A \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} B}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2} C}{2 d}\) | \(284\) |
derivativedivides | \(\frac {a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 a^{2} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{2} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(304\) |
default | \(\frac {a^{2} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{2} B \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {2 a^{2} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{2} B \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 a^{2} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{2} A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{2} B \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{2} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(304\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 296, normalized size = 1.39 \begin {gather*} \frac {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 )} A 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 )} 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} + 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} - 640 \, {\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} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2}}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.29, size = 145, normalized size = 0.68 \begin {gather*} \frac {15 \, {\left (11 \, A + 12 \, B + 14 \, C\right )} a^{2} d x + {\left (40 \, A a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (11 \, A + 12 \, B + 6 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 16 \, {\left (8 \, A + 9 \, B + 10 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (11 \, A + 12 \, B + 14 \, C\right )} a^{2} \cos \left (d x + c\right ) + 32 \, {\left (8 \, A + 9 \, B + 10 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.55, size = 350, normalized size = 1.64 \begin {gather*} \frac {15 \, {\left (11 \, A a^{2} + 12 \, B a^{2} + 14 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (165 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 180 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 210 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 935 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1020 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1190 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1986 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2568 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2580 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3006 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2808 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3180 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1305 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1860 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2330 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 795 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 780 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 750 \, C a^{2} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.86, size = 333, normalized size = 1.56 \begin {gather*} \frac {\left (\frac {11\,A\,a^2}{8}+\frac {3\,B\,a^2}{2}+\frac {7\,C\,a^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {187\,A\,a^2}{24}+\frac {17\,B\,a^2}{2}+\frac {119\,C\,a^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {331\,A\,a^2}{20}+\frac {107\,B\,a^2}{5}+\frac {43\,C\,a^2}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {501\,A\,a^2}{20}+\frac {117\,B\,a^2}{5}+\frac {53\,C\,a^2}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {87\,A\,a^2}{8}+\frac {31\,B\,a^2}{2}+\frac {233\,C\,a^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^2}{8}+\frac {13\,B\,a^2}{2}+\frac {25\,C\,a^2}{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 {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (11\,A+12\,B+14\,C\right )}{8\,\left (\frac {11\,A\,a^2}{8}+\frac {3\,B\,a^2}{2}+\frac {7\,C\,a^2}{4}\right )}\right )\,\left (11\,A+12\,B+14\,C\right )}{8\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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