Integrand size = 29, antiderivative size = 354 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1}{128} a \left (8 a^2+9 b^2\right ) x-\frac {b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac {b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d} \]
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Time = 0.64 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2974, 3128, 3112, 3102, 2827, 2715, 8, 2713} \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}-\frac {5 \left (a^2-4 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}-\frac {a \left (20 a^2-87 b^2\right ) \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {a \left (8 a^2+9 b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {1}{128} a x \left (8 a^2+9 b^2\right )-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^4(c+d x) \cos (c+d x)}{2520 b d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^3(c+d x) \cos (c+d x)}{4032 b^2 d}+\frac {5 a \sin ^3(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\sin ^4(c+d x) \cos (c+d x) (a+b \sin (c+d x))^4}{9 b d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 2974
Rule 3102
Rule 3112
Rule 3128
Rubi steps \begin{align*} \text {integral}& = \frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x))^3 \left (3 \left (5 a^2-24 b^2\right )+3 a b \sin (c+d x)-20 \left (a^2-4 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{72 b^2} \\ & = -\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x))^2 \left (3 a \left (15 a^2-88 b^2\right )+6 b \left (a^2-4 b^2\right ) \sin (c+d x)-3 a \left (20 a^2-87 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{504 b^2} \\ & = -\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) (a+b \sin (c+d x)) \left (9 a^2 \left (10 a^2-89 b^2\right )+3 a b \left (2 a^2-141 b^2\right ) \sin (c+d x)-6 \left (20 a^4-93 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{3024 b^2} \\ & = -\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) \left (45 a^3 \left (10 a^2-89 b^2\right )-144 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)-15 a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \sin ^2(c+d x)\right ) \, dx}{15120 b^2} \\ & = -\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}-\frac {\int \sin ^2(c+d x) \left (-945 a b^2 \left (8 a^2+9 b^2\right )-576 b^3 \left (27 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx}{60480 b^2} \\ & = -\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}+\frac {1}{105} \left (b \left (27 a^2+4 b^2\right )\right ) \int \sin ^3(c+d x) \, dx+\frac {1}{64} \left (a \left (8 a^2+9 b^2\right )\right ) \int \sin ^2(c+d x) \, dx \\ & = -\frac {a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d}+\frac {1}{128} \left (a \left (8 a^2+9 b^2\right )\right ) \int 1 \, dx-\frac {\left (b \left (27 a^2+4 b^2\right )\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{105 d} \\ & = \frac {1}{128} a \left (8 a^2+9 b^2\right ) x-\frac {b \left (27 a^2+4 b^2\right ) \cos (c+d x)}{105 d}+\frac {b \left (27 a^2+4 b^2\right ) \cos ^3(c+d x)}{315 d}-\frac {a \left (8 a^2+9 b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}-\frac {a \left (40 a^4-188 a^2 b^2+189 b^4\right ) \cos (c+d x) \sin ^3(c+d x)}{4032 b^2 d}-\frac {\left (20 a^4-93 a^2 b^2+24 b^4\right ) \cos (c+d x) \sin ^4(c+d x)}{2520 b d}-\frac {a \left (20 a^2-87 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^2}{1008 b^2 d}-\frac {5 \left (a^2-4 b^2\right ) \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^3}{126 b^2 d}+\frac {5 a \cos (c+d x) \sin ^3(c+d x) (a+b \sin (c+d x))^4}{72 b^2 d}-\frac {\cos (c+d x) \sin ^4(c+d x) (a+b \sin (c+d x))^4}{9 b d} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.58 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {15120 a b^2 c+10080 a^3 d x+11340 a b^2 d x-3780 b \left (6 a^2+b^2\right ) \cos (c+d x)-840 \left (9 a^2 b+b^3\right ) \cos (3 (c+d x))+1512 a^2 b \cos (5 (c+d x))+504 b^3 \cos (5 (c+d x))+1080 a^2 b \cos (7 (c+d x))+90 b^3 \cos (7 (c+d x))-70 b^3 \cos (9 (c+d x))+2520 a^3 \sin (2 (c+d x))-2520 a^3 \sin (4 (c+d x))-3780 a b^2 \sin (4 (c+d x))-840 a^3 \sin (6 (c+d x))+\frac {945}{2} a b^2 \sin (8 (c+d x))}{161280 d} \]
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Time = 0.97 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(\frac {1680 \left (-9 a^{2} b -b^{3}\right ) \cos \left (3 d x +3 c \right )+1008 \left (3 a^{2} b +b^{3}\right ) \cos \left (5 d x +5 c \right )+180 \left (12 a^{2} b +b^{3}\right ) \cos \left (7 d x +7 c \right )+2520 \left (-2 a^{3}-3 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )-140 b^{3} \cos \left (9 d x +9 c \right )+5040 a^{3} \sin \left (2 d x +2 c \right )-1680 a^{3} \sin \left (6 d x +6 c \right )+945 a \,b^{2} \sin \left (8 d x +8 c \right )+7560 \left (-6 a^{2} b -b^{3}\right ) \cos \left (d x +c \right )+20160 a^{3} d x +22680 a \,b^{2} d x -55296 a^{2} b -8192 b^{3}}{322560 d}\) | \(198\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{2} b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )}{d}\) | \(218\) |
default | \(\frac {a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{2} b \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 a \,b^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )}{d}\) | \(218\) |
risch | \(\frac {a^{3} x}{16}+\frac {9 a \,b^{2} x}{128}-\frac {9 a^{2} b \cos \left (d x +c \right )}{64 d}-\frac {3 b^{3} \cos \left (d x +c \right )}{128 d}-\frac {b^{3} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {3 a \,b^{2} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {3 b \cos \left (7 d x +7 c \right ) a^{2}}{448 d}+\frac {b^{3} \cos \left (7 d x +7 c \right )}{1792 d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 b \cos \left (5 d x +5 c \right ) a^{2}}{320 d}+\frac {b^{3} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {3 \sin \left (4 d x +4 c \right ) a \,b^{2}}{128 d}-\frac {3 b \cos \left (3 d x +3 c \right ) a^{2}}{64 d}-\frac {b^{3} \cos \left (3 d x +3 c \right )}{192 d}+\frac {a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(253\) |
norman | \(\frac {\frac {a \left (8 a^{2}+9 b^{2}\right ) x}{128}+\frac {3 a \left (8 a^{2}+169 b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {108 a^{2} b +16 b^{3}}{315 d}+\frac {9 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {9 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{18}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}-\frac {12 a^{2} b \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a \left (8 a^{2}+9 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {3 a \left (24 a^{2}+155 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {3 a \left (8 a^{2}+169 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {3 a \left (24 a^{2}+155 b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d}-\frac {a \left (152 a^{2}-117 b^{2}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}+\frac {a \left (8 a^{2}+9 b^{2}\right ) \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {9 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {9 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {21 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {63 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {63 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {21 a \left (8 a^{2}+9 b^{2}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {a \left (152 a^{2}-117 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d}-\frac {2 \left (6 a^{2} b -8 b^{3}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 \left (9 a^{2} b +8 b^{3}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (108 a^{2} b +16 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}-\frac {4 \left (21 a^{2} b -8 b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 \left (78 a^{2} b +56 b^{3}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {4 \left (3 a^{2} b +16 b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(694\) |
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Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.46 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {4480 \, b^{3} \cos \left (d x + c\right )^{9} - 5760 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{7} + 8064 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5} - 315 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} d x - 105 \, {\left (144 \, a b^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{3} + 27 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40320 \, d} \]
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Time = 1.00 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.43 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 a^{2} b \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {6 a^{2} b \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac {9 a b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {9 a b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {27 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {9 a b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {33 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {33 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {9 a b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {8 b^{3} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{3} \sin ^{2}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.40 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {1680 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 27648 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} b + 945 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 1024 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b^{3}}{322560 \, d} \]
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Time = 0.48 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.58 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {b^{3} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {1}{128} \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} x + \frac {{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (3 \, a^{2} b + b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (9 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {3 \, {\left (6 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} \]
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Time = 11.85 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.63 \[ \int \cos ^4(c+d x) \sin ^2(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^2+9\,b^2\right )}{64\,\left (\frac {a^3}{8}+\frac {9\,a\,b^2}{64}\right )}\right )\,\left (8\,a^2+9\,b^2\right )}{64\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {a^3}{8}+\frac {9\,a\,b^2}{64}\right )+\frac {12\,a^2\,b}{35}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {a^3}{8}+\frac {9\,a\,b^2}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {39\,a\,b^2}{32}-\frac {19\,a^3}{12}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {39\,a\,b^2}{32}-\frac {19\,a^3}{12}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,a^3}{4}+\frac {465\,a\,b^2}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {9\,a^3}{4}+\frac {465\,a\,b^2}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {3\,a^3}{4}+\frac {507\,a\,b^2}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {3\,a^3}{4}+\frac {507\,a\,b^2}{32}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (12\,a^2\,b-16\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (12\,a^2\,b+\frac {32\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {84\,a^2\,b}{5}-\frac {32\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {12\,a^2\,b}{35}+\frac {64\,b^3}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {108\,a^2\,b}{35}+\frac {16\,b^3}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {156\,a^2\,b}{5}+\frac {112\,b^3}{5}\right )+\frac {16\,b^3}{315}+12\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,\left (8\,a^2+9\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d} \]
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