Integrand size = 28, antiderivative size = 279 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {4 a b^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {b^4 \sin ^9(c+d x)}{9 d} \]
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Time = 0.29 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 2713, 2645, 30, 2644, 276, 14} \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}-\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}-\frac {4 a b^3 \cos ^7(c+d x)}{7 d}+\frac {b^4 \sin ^9(c+d x)}{9 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {b^4 \sin ^5(c+d x)}{5 d} \]
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Rule 14
Rule 30
Rule 276
Rule 2644
Rule 2645
Rule 2713
Rule 3169
Rubi steps \begin{align*} \text {integral}& = \int \left (a^4 \cos ^9(c+d x)+4 a^3 b \cos ^8(c+d x) \sin (c+d x)+6 a^2 b^2 \cos ^7(c+d x) \sin ^2(c+d x)+4 a b^3 \cos ^6(c+d x) \sin ^3(c+d x)+b^4 \cos ^5(c+d x) \sin ^4(c+d x)\right ) \, dx \\ & = a^4 \int \cos ^9(c+d x) \, dx+\left (4 a^3 b\right ) \int \cos ^8(c+d x) \sin (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \cos ^7(c+d x) \sin ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+b^4 \int \cos ^5(c+d x) \sin ^4(c+d x) \, dx \\ & = -\frac {a^4 \text {Subst}\left (\int \left (1-4 x^2+6 x^4-4 x^6+x^8\right ) \, dx,x,-\sin (c+d x)\right )}{d}-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int x^8 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int x^2 \left (1-x^2\right )^3 \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \left (x^2-3 x^4+3 x^6-x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {4 a b^3 \cos ^7(c+d x)}{7 d}-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {4 a b^3 \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {18 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {18 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {2 b^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}-\frac {2 a^2 b^2 \sin ^9(c+d x)}{3 d}+\frac {b^4 \sin ^9(c+d x)}{9 d} \\ \end{align*}
Time = 6.16 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.33 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {4 a^3 b \cos ^9(c+d x)}{9 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {6 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^7(c+d x)}{7 d}+\frac {a^4 \sin ^9(c+d x)}{9 d}+\frac {2 a^2 b^2 \left (105 \sin ^3(c+d x)-189 \sin ^5(c+d x)+135 \sin ^7(c+d x)-35 \sin ^9(c+d x)\right )}{105 d}+\frac {b^4 \left (63 \sin ^5(c+d x)-90 \sin ^7(c+d x)+35 \sin ^9(c+d x)\right )}{315 d}+\frac {4 a b^3 \cos (c+d x) \sin ^8(c+d x) \left (2 \csc ^8(c+d x)+7 \sqrt {1-\sin ^2(c+d x)}-19 \csc ^2(c+d x) \sqrt {1-\sin ^2(c+d x)}+15 \csc ^4(c+d x) \sqrt {1-\sin ^2(c+d x)}-\csc ^6(c+d x) \sqrt {1-\sin ^2(c+d x)}-2 \csc ^8(c+d x) \sqrt {1-\sin ^2(c+d x)}\right )}{63 d \sqrt {\cos ^2(c+d x)}} \]
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Time = 1.73 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.69
method | result | size |
parts | \(\frac {a^{4} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9 d}+\frac {b^{4} \left (\frac {\sin \left (d x +c \right )^{9}}{9}-\frac {2 \sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {4 a \,b^{3} \left (\frac {\cos \left (d x +c \right )^{9}}{9}-\frac {\cos \left (d x +c \right )^{7}}{7}\right )}{d}-\frac {4 a^{3} b \cos \left (d x +c \right )^{9}}{9 d}-\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{9}-\frac {3 \sin \left (d x +c \right )^{7}}{7}+\frac {3 \sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}\right )}{d}\) | \(193\) |
derivativedivides | \(\frac {\frac {a^{4} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}-\frac {4 a^{3} b \cos \left (d x +c \right )^{9}}{9}+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+4 a \,b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{7}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )}{d}\) | \(236\) |
default | \(\frac {\frac {a^{4} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}-\frac {4 a^{3} b \cos \left (d x +c \right )^{9}}{9}+6 a^{2} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+4 a \,b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{7}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )}{d}\) | \(236\) |
risch | \(-\frac {7 a^{3} b \cos \left (d x +c \right )}{32 d}-\frac {3 a \,b^{3} \cos \left (d x +c \right )}{32 d}+\frac {63 a^{4} \sin \left (d x +c \right )}{128 d}+\frac {21 a^{2} b^{2} \sin \left (d x +c \right )}{64 d}+\frac {3 b^{4} \sin \left (d x +c \right )}{128 d}-\frac {a^{3} b \cos \left (9 d x +9 c \right )}{576 d}+\frac {a \,b^{3} \cos \left (9 d x +9 c \right )}{576 d}+\frac {\sin \left (9 d x +9 c \right ) a^{4}}{2304 d}-\frac {\sin \left (9 d x +9 c \right ) a^{2} b^{2}}{384 d}+\frac {\sin \left (9 d x +9 c \right ) b^{4}}{2304 d}-\frac {a^{3} b \cos \left (7 d x +7 c \right )}{64 d}+\frac {3 a \,b^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {9 \sin \left (7 d x +7 c \right ) a^{4}}{1792 d}-\frac {15 \sin \left (7 d x +7 c \right ) a^{2} b^{2}}{896 d}+\frac {\sin \left (7 d x +7 c \right ) b^{4}}{1792 d}-\frac {a^{3} b \cos \left (5 d x +5 c \right )}{16 d}+\frac {9 \sin \left (5 d x +5 c \right ) a^{4}}{320 d}-\frac {3 \sin \left (5 d x +5 c \right ) a^{2} b^{2}}{80 d}-\frac {\sin \left (5 d x +5 c \right ) b^{4}}{320 d}-\frac {7 a^{3} b \cos \left (3 d x +3 c \right )}{48 d}-\frac {a \,b^{3} \cos \left (3 d x +3 c \right )}{24 d}+\frac {7 \sin \left (3 d x +3 c \right ) a^{4}}{64 d}-\frac {\sin \left (3 d x +3 c \right ) b^{4}}{192 d}\) | \(399\) |
parallelrisch | \(\frac {630 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}-2520 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a^{3} b +\left (1680 a^{4}+5040 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}-5040 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} a \,b^{3}+\left (9576 a^{4}-6048 a^{2} b^{2}+2016 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (-23520 a^{3} b +8400 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (10224 a^{4}+34128 a^{2} b^{2}-3456 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-25200 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a \,b^{3}+\left (21316 a^{4}-17088 a^{2} b^{2}+6976 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (-35280 a^{3} b +15120 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (10224 a^{4}+34128 a^{2} b^{2}-3456 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-15120 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{3}+\left (9576 a^{4}-6048 a^{2} b^{2}+2016 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-10080 a^{3} b +2160 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (1680 a^{4}+5040 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-720 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{3}+630 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-280 a^{3} b -80 a \,b^{3}}{315 d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(431\) |
norman | \(\frac {-\frac {56 a^{3} b +16 a \,b^{3}}{63 d}+\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{d}-\frac {16 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}-\frac {48 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {16 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{7 d}-\frac {80 a \,b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {8 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{d}+\frac {8 \left (19 a^{4}-12 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {8 \left (19 a^{4}-12 a^{2} b^{2}+4 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{5 d}+\frac {16 \left (71 a^{4}+237 a^{2} b^{2}-24 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {16 \left (71 a^{4}+237 a^{2} b^{2}-24 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{35 d}+\frac {4 \left (5329 a^{4}-4272 a^{2} b^{2}+1744 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{315 d}-\frac {2 \left (56 a^{3} b -24 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {4 \left (56 a^{3} b -20 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d}-\frac {4 \left (56 a^{3} b -12 a \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{7 d}+\frac {16 a^{2} \left (a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {16 a^{2} \left (a^{2}+3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(490\) |
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Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.63 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {180 \, a b^{3} \cos \left (d x + c\right )^{7} + 140 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{9} - {\left (35 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (4 \, a^{4} + 3 \, a^{2} b^{2} - 5 \, b^{4}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + 128 \, a^{4} + 96 \, a^{2} b^{2} + 8 \, b^{4} + 4 \, {\left (16 \, a^{4} + 12 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \]
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Time = 1.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.32 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\begin {cases} \frac {128 a^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {64 a^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {4 a^{3} b \cos ^{9}{\left (c + d x \right )}}{9 d} + \frac {32 a^{2} b^{2} \sin ^{9}{\left (c + d x \right )}}{105 d} + \frac {48 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {12 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {2 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {4 a b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {8 a b^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {8 b^{4} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {4 b^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{4} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.67 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {140 \, a^{3} b \cos \left (d x + c\right )^{9} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{4} + 6 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{2} b^{2} - 20 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a b^{3} - {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} b^{4}}{315 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.96 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {a^{3} b \cos \left (5 \, d x + 5 \, c\right )}{16 \, d} - \frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (9 \, d x + 9 \, c\right )}{576 \, d} - \frac {{\left (7 \, a^{3} b - 3 \, a b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {{\left (7 \, a^{3} b + 2 \, a b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {{\left (7 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )}{32 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (9 \, a^{4} - 30 \, a^{2} b^{2} + b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (9 \, a^{4} - 12 \, a^{2} b^{2} - b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (21 \, a^{4} - b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {3 \, {\left (21 \, a^{4} + 14 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )}{128 \, d} \]
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Time = 24.67 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.20 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {\frac {b^4\,\sin \left (3\,c+3\,d\,x\right )}{192}-\frac {3\,b^4\,\sin \left (c+d\,x\right )}{128}-\frac {7\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{64}-\frac {9\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {9\,a^4\,\sin \left (7\,c+7\,d\,x\right )}{1792}-\frac {a^4\,\sin \left (9\,c+9\,d\,x\right )}{2304}-\frac {63\,a^4\,\sin \left (c+d\,x\right )}{128}+\frac {b^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {b^4\,\sin \left (7\,c+7\,d\,x\right )}{1792}-\frac {b^4\,\sin \left (9\,c+9\,d\,x\right )}{2304}+\frac {a\,b^3\,\cos \left (3\,c+3\,d\,x\right )}{24}+\frac {7\,a^3\,b\,\cos \left (3\,c+3\,d\,x\right )}{48}+\frac {a^3\,b\,\cos \left (5\,c+5\,d\,x\right )}{16}-\frac {3\,a\,b^3\,\cos \left (7\,c+7\,d\,x\right )}{448}+\frac {a^3\,b\,\cos \left (7\,c+7\,d\,x\right )}{64}-\frac {a\,b^3\,\cos \left (9\,c+9\,d\,x\right )}{576}+\frac {a^3\,b\,\cos \left (9\,c+9\,d\,x\right )}{576}-\frac {21\,a^2\,b^2\,\sin \left (c+d\,x\right )}{64}+\frac {3\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{80}+\frac {15\,a^2\,b^2\,\sin \left (7\,c+7\,d\,x\right )}{896}+\frac {a^2\,b^2\,\sin \left (9\,c+9\,d\,x\right )}{384}+\frac {3\,a\,b^3\,\cos \left (c+d\,x\right )}{32}+\frac {7\,a^3\,b\,\cos \left (c+d\,x\right )}{32}}{d} \]
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