Integrand size = 28, antiderivative size = 337 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {b^5 \cos ^5(c+d x)}{5 d}-\frac {10 a^2 b^3 \cos ^7(c+d x)}{7 d}+\frac {2 b^5 \cos ^7(c+d x)}{7 d}-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}+\frac {10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac {b^5 \cos ^9(c+d x)}{9 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac {a b^4 \sin ^5(c+d x)}{d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac {10 a b^4 \sin ^7(c+d x)}{7 d}+\frac {a^5 \sin ^9(c+d x)}{9 d}-\frac {10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac {5 a b^4 \sin ^9(c+d x)}{9 d} \]
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Time = 0.39 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3169, 2713, 2645, 30, 2644, 276, 14} \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {a^5 \sin ^9(c+d x)}{9 d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}-\frac {10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac {30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac {6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac {10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac {10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac {10 a^2 b^3 \cos ^7(c+d x)}{7 d}+\frac {5 a b^4 \sin ^9(c+d x)}{9 d}-\frac {10 a b^4 \sin ^7(c+d x)}{7 d}+\frac {a b^4 \sin ^5(c+d x)}{d}-\frac {b^5 \cos ^9(c+d x)}{9 d}+\frac {2 b^5 \cos ^7(c+d x)}{7 d}-\frac {b^5 \cos ^5(c+d x)}{5 d} \]
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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^5 \cos ^9(c+d x)+5 a^4 b \cos ^8(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^7(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^6(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^5(c+d x) \sin ^4(c+d x)+b^5 \cos ^4(c+d x) \sin ^5(c+d x)\right ) \, dx \\ & = a^5 \int \cos ^9(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^8(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^7(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^5(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx \\ & = -\frac {a^5 \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 (5 a^4 b\right ) \text {Subst}\left (\int x^8 \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (10 a^3 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 (10 a^2 b^3\right ) \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (5 a b^4\right ) \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^5 \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {5 a^4 b \cos ^9(c+d x)}{9 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {a^5 \sin ^9(c+d x)}{9 d}+\frac {\left (10 a^3 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 (10 a^2 b^3\right ) \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (5 a b^4\right ) \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sin (c+d x)\right )}{d}-\frac {b^5 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {b^5 \cos ^5(c+d x)}{5 d}-\frac {10 a^2 b^3 \cos ^7(c+d x)}{7 d}+\frac {2 b^5 \cos ^7(c+d x)}{7 d}-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}+\frac {10 a^2 b^3 \cos ^9(c+d x)}{9 d}-\frac {b^5 \cos ^9(c+d x)}{9 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {10 a^3 b^2 \sin ^3(c+d x)}{3 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {6 a^3 b^2 \sin ^5(c+d x)}{d}+\frac {a b^4 \sin ^5(c+d x)}{d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {30 a^3 b^2 \sin ^7(c+d x)}{7 d}-\frac {10 a b^4 \sin ^7(c+d x)}{7 d}+\frac {a^5 \sin ^9(c+d x)}{9 d}-\frac {10 a^3 b^2 \sin ^9(c+d x)}{9 d}+\frac {5 a b^4 \sin ^9(c+d x)}{9 d} \\ \end{align*}
Time = 6.21 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.61 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {5 a^4 b \cos ^9(c+d x)}{9 d}+\frac {a^5 \sin (c+d x)}{d}-\frac {4 a^5 \sin ^3(c+d x)}{3 d}+\frac {6 a^5 \sin ^5(c+d x)}{5 d}-\frac {4 a^5 \sin ^7(c+d x)}{7 d}+\frac {a^5 \sin ^9(c+d x)}{9 d}+\frac {2 a^3 b^2 \left (105 \sin ^3(c+d x)-189 \sin ^5(c+d x)+135 \sin ^7(c+d x)-35 \sin ^9(c+d x)\right )}{63 d}+\frac {a b^4 \left (63 \sin ^5(c+d x)-90 \sin ^7(c+d x)+35 \sin ^9(c+d x)\right )}{63 d}+\frac {b^5 \cos (c+d x) \sin ^8(c+d x) \left (8 \csc ^8(c+d x)-35 \sqrt {1-\sin ^2(c+d x)}+50 \csc ^2(c+d x) \sqrt {1-\sin ^2(c+d x)}-3 \csc ^4(c+d x) \sqrt {1-\sin ^2(c+d x)}-4 \csc ^6(c+d x) \sqrt {1-\sin ^2(c+d x)}-8 \csc ^8(c+d x) \sqrt {1-\sin ^2(c+d x)}\right )}{315 d \sqrt {\cos ^2(c+d x)}}+\frac {10 a^2 b^3 \cos (c+d x) \sin ^8(c+d x) \left (2 \csc ^8(c+d x)+7 \sqrt {1-\sin ^2(c+d x)}-19 \csc ^2(c+d x) \sqrt {1-\sin ^2(c+d x)}+15 \csc ^4(c+d x) \sqrt {1-\sin ^2(c+d x)}-\csc ^6(c+d x) \sqrt {1-\sin ^2(c+d x)}-2 \csc ^8(c+d x) \sqrt {1-\sin ^2(c+d x)}\right )}{63 d \sqrt {\cos ^2(c+d x)}} \]
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Time = 1.87 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.70
method | result | size |
parts | \(\frac {a^{5} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9 d}-\frac {b^{5} \left (\frac {\cos \left (d x +c \right )^{9}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{7}+\frac {\cos \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {\cos \left (d x +c \right )^{9}}{9}-\frac {\cos \left (d x +c \right )^{7}}{7}\right )}{d}-\frac {5 a^{4} b \cos \left (d x +c \right )^{9}}{9 d}-\frac {10 a^{3} b^{2} \left (\frac {\sin \left (d x +c \right )^{9}}{9}-\frac {3 \sin \left (d x +c \right )^{7}}{7}+\frac {3 \sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {5 a \,b^{4} \left (\frac {\sin \left (d x +c \right )^{9}}{9}-\frac {2 \sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{5}}{5}\right )}{d}\) | \(236\) |
derivativedivides | \(\frac {\frac {a^{5} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}-\frac {5 a^{4} b \cos \left (d x +c \right )^{9}}{9}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{7}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )+b^{5} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{5}}{9}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{63}-\frac {8 \cos \left (d x +c \right )^{5}}{315}\right )}{d}\) | \(291\) |
default | \(\frac {\frac {a^{5} \left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}-\frac {5 a^{4} b \cos \left (d x +c \right )^{9}}{9}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{8}}{9}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{7}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{6}}{9}-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{21}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{105}\right )+b^{5} \left (-\frac {\sin \left (d x +c \right )^{4} \cos \left (d x +c \right )^{5}}{9}-\frac {4 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{63}-\frac {8 \cos \left (d x +c \right )^{5}}{315}\right )}{d}\) | \(291\) |
parallelrisch | \(\frac {630 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17} a^{5}-3150 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a^{4} b +\left (1680 a^{5}+8400 a^{3} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}-12600 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} a^{2} b^{3}+\left (9576 a^{5}-10080 a^{3} b^{2}+10080 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}+\left (-29400 a^{4} b +21000 a^{2} b^{3}-3360 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (10224 a^{5}+56880 a^{3} b^{2}-17280 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (-63000 a^{2} b^{3}+5040 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (21316 a^{5}-28480 a^{3} b^{2}+34880 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (-44100 a^{4} b +37800 a^{2} b^{3}-7056 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (10224 a^{5}+56880 a^{3} b^{2}-17280 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-37800 a^{2} b^{3}+2016 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (9576 a^{5}-10080 a^{3} b^{2}+10080 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-12600 a^{4} b +5400 a^{2} b^{3}-576 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (1680 a^{5}+8400 a^{3} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-1800 a^{2} b^{3}-144 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+630 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{5}-350 a^{4} b -200 a^{2} b^{3}-16 b^{5}}{315 d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(493\) |
risch | \(-\frac {5 b \cos \left (9 d x +9 c \right ) a^{4}}{2304 d}+\frac {5 b^{3} \cos \left (9 d x +9 c \right ) a^{2}}{1152 d}-\frac {5 a^{3} \sin \left (9 d x +9 c \right ) b^{2}}{1152 d}+\frac {5 a \sin \left (9 d x +9 c \right ) b^{4}}{2304 d}-\frac {5 b \cos \left (7 d x +7 c \right ) a^{4}}{256 d}+\frac {15 b^{3} \cos \left (7 d x +7 c \right ) a^{2}}{896 d}-\frac {25 a^{3} \sin \left (7 d x +7 c \right ) b^{2}}{896 d}+\frac {5 a \sin \left (7 d x +7 c \right ) b^{4}}{1792 d}-\frac {5 b \cos \left (5 d x +5 c \right ) a^{4}}{64 d}-\frac {a^{3} \sin \left (5 d x +5 c \right ) b^{2}}{16 d}-\frac {a \sin \left (5 d x +5 c \right ) b^{4}}{64 d}-\frac {3 b^{5} \cos \left (d x +c \right )}{128 d}+\frac {b^{5} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {9 a^{5} \sin \left (7 d x +7 c \right )}{1792 d}+\frac {b^{5} \cos \left (5 d x +5 c \right )}{320 d}+\frac {9 a^{5} \sin \left (5 d x +5 c \right )}{320 d}-\frac {b^{5} \cos \left (3 d x +3 c \right )}{192 d}+\frac {7 a^{5} \sin \left (3 d x +3 c \right )}{64 d}-\frac {b^{5} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a^{5} \sin \left (9 d x +9 c \right )}{2304 d}-\frac {15 a^{2} b^{3} \cos \left (d x +c \right )}{64 d}+\frac {15 a \,b^{4} \sin \left (d x +c \right )}{128 d}-\frac {35 a^{4} b \cos \left (d x +c \right )}{128 d}+\frac {35 a^{3} b^{2} \sin \left (d x +c \right )}{64 d}-\frac {35 b \cos \left (3 d x +3 c \right ) a^{4}}{192 d}-\frac {5 b^{3} \cos \left (3 d x +3 c \right ) a^{2}}{48 d}-\frac {5 a \sin \left (3 d x +3 c \right ) b^{4}}{192 d}+\frac {63 a^{5} \sin \left (d x +c \right )}{128 d}\) | \(494\) |
norman | \(\frac {-\frac {350 a^{4} b +200 a^{2} b^{3}+16 b^{5}}{315 d}+\frac {2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{d}-\frac {40 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}-\frac {10 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{d}-\frac {2 \left (100 a^{2} b^{3}-8 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {4 \left (150 a^{2} b^{3}-8 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5 d}-\frac {\left (200 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d}-\frac {4 \left (70 a^{4} b -50 a^{2} b^{3}+8 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d}-\frac {2 \left (350 a^{4} b -300 a^{2} b^{3}+56 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5 d}-\frac {4 \left (350 a^{4} b -150 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35 d}+\frac {8 a \left (19 a^{4}-20 a^{2} b^{2}+20 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {8 a \left (19 a^{4}-20 a^{2} b^{2}+20 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{5 d}+\frac {16 a \left (71 a^{4}+395 a^{2} b^{2}-120 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}+\frac {16 a \left (71 a^{4}+395 a^{2} b^{2}-120 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{35 d}+\frac {4 a \left (5329 a^{4}-7120 a^{2} b^{2}+8720 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{315 d}+\frac {16 a^{3} \left (a^{2}+5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {16 a^{3} \left (a^{2}+5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(555\) |
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Time = 0.27 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {63 \, b^{5} \cos \left (d x + c\right )^{5} + 35 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{9} + 90 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{7} - {\left (35 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (4 \, a^{5} + 5 \, a^{3} b^{2} - 25 \, a b^{4}\right )} \cos \left (d x + c\right )^{6} + 128 \, a^{5} + 160 \, a^{3} b^{2} + 40 \, a b^{4} + 3 \, {\left (16 \, a^{5} + 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (16 \, a^{5} + 20 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d} \]
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Time = 1.03 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.31 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\begin {cases} \frac {128 a^{5} \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {64 a^{5} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {16 a^{5} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {8 a^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {a^{5} \sin {\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {5 a^{4} b \cos ^{9}{\left (c + d x \right )}}{9 d} + \frac {32 a^{3} b^{2} \sin ^{9}{\left (c + d x \right )}}{63 d} + \frac {16 a^{3} b^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {4 a^{3} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {10 a^{3} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {10 a^{2} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {20 a^{2} b^{3} \cos ^{9}{\left (c + d x \right )}}{63 d} + \frac {8 a b^{4} \sin ^{9}{\left (c + d x \right )}}{63 d} + \frac {4 a b^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{7 d} + \frac {a b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {4 b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {8 b^{5} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{5} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {175 \, a^{4} b \cos \left (d x + c\right )^{9} - {\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{5} + 10 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{3} b^{2} - 50 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{2} b^{3} - 5 \, {\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a b^{4} + {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} b^{5}}{315 \, d} \]
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Time = 0.53 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.93 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {{\left (35 \, a^{4} b - 30 \, a^{2} b^{3} - b^{5}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (25 \, a^{4} b - b^{5}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (35 \, a^{4} b + 20 \, a^{2} b^{3} + b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (35 \, a^{4} b + 30 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )}{128 \, d} + \frac {{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {{\left (9 \, a^{5} - 50 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (9 \, a^{5} - 20 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (21 \, a^{5} - 5 \, a b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (63 \, a^{5} + 70 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \sin \left (d x + c\right )}{128 \, d} \]
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Time = 26.80 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.47 \[ \int \cos ^4(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {2\,a^5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {152\,a^5}{5}-32\,a^3\,b^2+32\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {152\,a^5}{5}-32\,a^3\,b^2+32\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {1136\,a^5}{35}+\frac {1264\,a^3\,b^2}{7}-\frac {384\,a\,b^4}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {1136\,a^5}{35}+\frac {1264\,a^3\,b^2}{7}-\frac {384\,a\,b^4}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {21316\,a^5}{315}-\frac {5696\,a^3\,b^2}{63}+\frac {6976\,a\,b^4}{63}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (40\,a^4\,b-\frac {120\,a^2\,b^3}{7}+\frac {64\,b^5}{35}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (140\,a^4\,b-120\,a^2\,b^3+\frac {112\,b^5}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {280\,a^4\,b}{3}-\frac {200\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )-\frac {10\,a^4\,b}{9}-\frac {16\,b^5}{315}-\frac {40\,a^2\,b^3}{63}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {16\,a^5}{3}+\frac {80\,a^3\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {16\,a^5}{3}+\frac {80\,a^3\,b^2}{3}\right )+2\,a^5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {40\,a^2\,b^3}{7}+\frac {16\,b^5}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {32\,b^5}{5}-120\,a^2\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (16\,b^5-200\,a^2\,b^3\right )-40\,a^2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \]
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