Optimal. Leaf size=345 \[ \frac {a b \left (6 a^2 C+126 A b^2+103 b^2 C\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac {a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right )+\frac {\left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac {\left (4 a^4 C+3 a^2 b^2 (63 A+50 C)+4 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac {2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d} \]
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Rubi [A] time = 0.86, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3050, 3049, 3033, 3023, 2734} \[ \frac {\left (84 a^2 b^2 (5 A+4 C)+35 a^4 (3 A+2 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^2}{35 d}+\frac {a b \left (6 a^2 C+126 A b^2+103 b^2 C\right ) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac {\left (3 a^2 b^2 (63 A+50 C)+4 a^4 C+4 b^4 (7 A+6 C)\right ) \sin (c+d x) \cos ^2(c+d x)}{105 d}+\frac {a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \sin (c+d x) \cos (c+d x)}{4 d}+\frac {1}{4} a b x \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac {2 a C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{21 d} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3023
Rule 3033
Rule 3049
Rule 3050
Rubi steps
\begin {align*} \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{7} \int \cos (c+d x) (a+b \cos (c+d x))^3 \left (a (7 A+2 C)+b (7 A+6 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{42} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (2 a^2 (21 A+10 C)+4 a b (21 A+17 C) \cos (c+d x)+6 \left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{210} \int \cos (c+d x) (a+b \cos (c+d x)) \left (2 a \left (6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+2 b \left (12 b^2 (7 A+6 C)+a^2 (315 A+244 C)\right ) \cos (c+d x)+4 a \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {1}{840} \int \cos (c+d x) \left (8 a^2 \left (6 b^2 (7 A+6 C)+a^2 (105 A+62 C)\right )+420 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x)+24 \left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {\left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac {a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}+\frac {\int \cos (c+d x) \left (24 \left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right )+1260 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x)\right ) \, dx}{2520}\\ &=\frac {1}{4} a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) x+\frac {\left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {\left (4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac {a b \left (126 A b^2+6 a^2 C+103 b^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{210 d}+\frac {\left (2 a^2 C+b^2 (7 A+6 C)\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{35 d}+\frac {2 a C \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{21 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.85, size = 351, normalized size = 1.02 \[ \frac {560 a^4 C \sin (3 (c+d x))+13440 a^3 A b c+13440 a^3 A b d x+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+420 a b \left (16 a^2 (A+C)+b^2 (16 A+15 C)\right ) \sin (2 (c+d x))+3360 a^2 A b^2 \sin (3 (c+d x))+4200 a^2 b^2 C \sin (3 (c+d x))+504 a^2 b^2 C \sin (5 (c+d x))+105 \left (16 a^4 (4 A+3 C)+48 a^2 b^2 (6 A+5 C)+5 b^4 (8 A+7 C)\right ) \sin (c+d x)+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+1260 a b^3 C \sin (4 (c+d x))+140 a b^3 C \sin (6 (c+d x))+8400 a b^3 c C+8400 a b^3 C d x+700 A b^4 \sin (3 (c+d x))+84 A b^4 \sin (5 (c+d x))+735 b^4 C \sin (3 (c+d x))+147 b^4 C \sin (5 (c+d x))+15 b^4 C \sin (7 (c+d x))}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 251, normalized size = 0.73 \[ \frac {105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} d x + {\left (60 \, C b^{4} \cos \left (d x + c\right )^{6} + 280 \, C a b^{3} \cos \left (d x + c\right )^{5} + 140 \, {\left (3 \, A + 2 \, C\right )} a^{4} + 336 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \, {\left (7 \, A + 6 \, C\right )} b^{4} + 12 \, {\left (42 \, C a^{2} b^{2} + {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, C a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (35 \, C a^{4} + 42 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 4 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.46, size = 290, normalized size = 0.84 \[ \frac {C b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {C a b^{3} \sin \left (6 \, d x + 6 \, c\right )}{48 \, d} + \frac {1}{4} \, {\left (8 \, A a^{3} b + 6 \, C a^{3} b + 6 \, A a b^{3} + 5 \, C a b^{3}\right )} x + \frac {{\left (24 \, C a^{2} b^{2} + 4 \, A b^{4} + 7 \, C b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (2 \, C a^{3} b + 2 \, A a b^{3} + 3 \, C a b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{16 \, d} + \frac {{\left (16 \, C a^{4} + 96 \, A a^{2} b^{2} + 120 \, C a^{2} b^{2} + 20 \, A b^{4} + 21 \, C b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{3} b + 16 \, C a^{3} b + 16 \, A a b^{3} + 15 \, C a b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{16 \, d} + \frac {{\left (64 \, A a^{4} + 48 \, C a^{4} + 288 \, A a^{2} b^{2} + 240 \, C a^{2} b^{2} + 40 \, A b^{4} + 35 \, C b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 332, normalized size = 0.96 \[ \frac {A \,a^{4} \sin \left (d x +c \right )+\frac {a^{4} C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{3} b C \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 A \,a^{2} b^{2} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\frac {6 C \,a^{2} b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+4 a A \,b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 C a \,b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {A \,b^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \,b^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 329, normalized size = 0.95 \[ -\frac {560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b + 3360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{2} b^{2} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} - 112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A b^{4} + 48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C b^{4} - 1680 \, A a^{4} \sin \left (d x + c\right )}{1680 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.06, size = 798, normalized size = 2.31 \[ \frac {\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2-5\,A\,a\,b^3-4\,A\,a^3\,b-\frac {11\,C\,a\,b^3}{2}-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (12\,A\,a^4+\frac {20\,A\,b^4}{3}+\frac {28\,C\,a^4}{3}+4\,C\,b^4+56\,A\,a^2\,b^2+40\,C\,a^2\,b^2-12\,A\,a\,b^3-16\,A\,a^3\,b-\frac {14\,C\,a\,b^3}{3}-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}-9\,A\,a\,b^3-20\,A\,a^3\,b-\frac {85\,C\,a\,b^3}{6}-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (40\,A\,a^4+\frac {104\,A\,b^4}{5}+24\,C\,a^4+\frac {424\,C\,b^4}{35}+144\,A\,a^2\,b^2+\frac {624\,C\,a^2\,b^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}+9\,A\,a\,b^3+20\,A\,a^3\,b+\frac {85\,C\,a\,b^3}{6}+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (12\,A\,a^4+\frac {20\,A\,b^4}{3}+\frac {28\,C\,a^4}{3}+4\,C\,b^4+56\,A\,a^2\,b^2+40\,C\,a^2\,b^2+12\,A\,a\,b^3+16\,A\,a^3\,b+\frac {14\,C\,a\,b^3}{3}+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2+5\,A\,a\,b^3+4\,A\,a^3\,b+\frac {11\,C\,a\,b^3}{2}+5\,C\,a^3\,b\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\,b\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{2\,d}+\frac {a\,b\,\mathrm {atan}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{2\,\left (3\,A\,a\,b^3+4\,A\,a^3\,b+\frac {5\,C\,a\,b^3}{2}+3\,C\,a^3\,b\right )}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.04, size = 850, normalized size = 2.46 \[ \begin {cases} \frac {A a^{4} \sin {\left (c + d x \right )}}{d} + 2 A a^{3} b x \sin ^{2}{\left (c + d x \right )} + 2 A a^{3} b x \cos ^{2}{\left (c + d x \right )} + \frac {2 A a^{3} b \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 A a^{2} b^{2} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {6 A a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a b^{3} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 A a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 A a b^{3} x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 A a b^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 A a b^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {8 A b^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A b^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {2 C a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {C a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 C a^{3} b x \sin ^{4}{\left (c + d x \right )}}{2} + 3 C a^{3} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + \frac {3 C a^{3} b x \cos ^{4}{\left (c + d x \right )}}{2} + \frac {3 C a^{3} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {5 C a^{3} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {16 C a^{2} b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {8 C a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {6 C a^{2} b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C a b^{3} x \sin ^{6}{\left (c + d x \right )}}{4} + \frac {15 C a b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {15 C a b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4} + \frac {5 C a b^{3} x \cos ^{6}{\left (c + d x \right )}}{4} + \frac {5 C a b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {10 C a b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} + \frac {11 C a b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{4 d} + \frac {16 C b^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 C b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 C b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {C b^{4} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \left (a + b \cos {\relax (c )}\right )^{4} \cos {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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