Optimal. Leaf size=216 \[ \frac {1}{16} a^3 (23 A+30 C) x+\frac {a^3 (34 A+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (23 A+30 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (73 A+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {(31 A+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d} \]
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Rubi [A]
time = 0.37, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {4172, 4102,
4081, 3872, 2715, 8, 2717} \begin {gather*} \frac {a^3 (34 A+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac {a^3 (23 A+30 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{120 d}+\frac {1}{16} a^3 x (23 A+30 C)+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{10 a d}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4081
Rule 4102
Rule 4172
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (3 a A+2 a (A+3 C) \sec (c+d x)) \, dx}{6 a}\\ &=\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left (a^2 (31 A+30 C)+2 a^2 (8 A+15 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {(31 A+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (73 A+90 C)+18 a^3 (7 A+10 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^3 (73 A+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {(31 A+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {\int \cos ^2(c+d x) \left (-45 a^4 (23 A+30 C)-24 a^4 (34 A+45 C) \sec (c+d x)\right ) \, dx}{360 a}\\ &=\frac {a^3 (73 A+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {(31 A+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{8} \left (a^3 (23 A+30 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{15} \left (a^3 (34 A+45 C)\right ) \int \cos (c+d x) \, dx\\ &=\frac {a^3 (34 A+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (23 A+30 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (73 A+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {(31 A+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}+\frac {1}{16} \left (a^3 (23 A+30 C)\right ) \int 1 \, dx\\ &=\frac {1}{16} a^3 (23 A+30 C) x+\frac {a^3 (34 A+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (23 A+30 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (73 A+90 C) \cos ^2(c+d x) \sin (c+d x)}{120 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{10 a d}+\frac {(31 A+30 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{120 d}\\ \end {align*}
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Mathematica [A]
time = 0.49, size = 123, normalized size = 0.57 \begin {gather*} \frac {a^3 (900 A c+1380 A d x+1800 C d x+120 (21 A+26 C) \sin (c+d x)+15 (63 A+64 C) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+240 C \sin (3 (c+d x))+135 A \sin (4 (c+d x))+30 C \sin (4 (c+d x))+36 A \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 = 0.88, size = 245, normalized size = 1.13
method | result | size |
risch | \(\frac {23 a^{3} A x}{16}+\frac {15 a^{3} x C}{8}+\frac {21 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {13 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {A \,a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 A \,a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {9 A \,a^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {19 A \,a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} C}{4 d}+\frac {63 \sin \left (2 d x +2 c \right ) A \,a^{3}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{d}\) | \(189\) |
derivativedivides | \(\frac {\frac {A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{3} C \sin \left (d x +c \right )+3 A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {3 A \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{3} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(245\) |
default | \(\frac {\frac {A \,a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+a^{3} C \sin \left (d x +c \right )+3 A \,a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+3 a^{3} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {3 A \,a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+A \,a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+a^{3} C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(245\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 239, normalized size = 1.11 \begin {gather*} \frac {192 \, {\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^{3} - 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^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 90 \, {\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^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 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^{3} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 960 \, C a^{3} \sin \left (d x + c\right )}{960 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.03, size = 126, normalized size = 0.58 \begin {gather*} \frac {15 \, {\left (23 \, A + 30 \, C\right )} a^{3} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 144 \, A a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (23 \, A + 6 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (17 \, A + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (23 \, A + 30 \, C\right )} a^{3} \cos \left (d x + c\right ) + 16 \, {\left (34 \, A + 45 \, C\right )} a^{3}\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 = 0.49, size = 244, normalized size = 1.13 \begin {gather*} \frac {15 \, {\left (23 \, A a^{3} + 30 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (345 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 450 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2550 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5940 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5814 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7500 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5130 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1470 \, C a^{3} \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 = 5.38, size = 285, normalized size = 1.32 \begin {gather*} \frac {\left (\frac {23\,A\,a^3}{8}+\frac {15\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {391\,A\,a^3}{24}+\frac {85\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {759\,A\,a^3}{20}+\frac {99\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {969\,A\,a^3}{20}+\frac {125\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {211\,A\,a^3}{8}+\frac {171\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {105\,A\,a^3}{8}+\frac {49\,C\,a^3}{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^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,A+30\,C\right )}{8\,\left (\frac {23\,A\,a^3}{8}+\frac {15\,C\,a^3}{4}\right )}\right )\,\left (23\,A+30\,C\right )}{8\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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