Integrand size = 28, antiderivative size = 426 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {35 a^5 x}{128}+\frac {25}{64} a^3 b^2 x+\frac {15}{128} a b^4 x-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d} \]
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Time = 0.51 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3169, 2715, 8, 2645, 30, 2648, 14, 2644} \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {a^5 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^5 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^5 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^5 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^5 x}{128}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}-\frac {5 a^3 b^2 \sin (c+d x) \cos ^7(c+d x)}{4 d}+\frac {5 a^3 b^2 \sin (c+d x) \cos ^5(c+d x)}{24 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos ^3(c+d x)}{96 d}+\frac {25 a^3 b^2 \sin (c+d x) \cos (c+d x)}{64 d}+\frac {25}{64} a^3 b^2 x+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a b^4 \sin ^3(c+d x) \cos ^5(c+d x)}{8 d}-\frac {5 a b^4 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {5 a b^4 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {15 a b^4 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {15}{128} a b^4 x-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d} \]
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Rule 8
Rule 14
Rule 30
Rule 2644
Rule 2645
Rule 2648
Rule 2715
Rule 3169
Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 \cos ^8(c+d x)+5 a^4 b \cos ^7(c+d x) \sin (c+d x)+10 a^3 b^2 \cos ^6(c+d x) \sin ^2(c+d x)+10 a^2 b^3 \cos ^5(c+d x) \sin ^3(c+d x)+5 a b^4 \cos ^4(c+d x) \sin ^4(c+d x)+b^5 \cos ^3(c+d x) \sin ^5(c+d x)\right ) \, dx \\ & = a^5 \int \cos ^8(c+d x) \, dx+\left (5 a^4 b\right ) \int \cos ^7(c+d x) \sin (c+d x) \, dx+\left (10 a^3 b^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\left (10 a^2 b^3\right ) \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx+\left (5 a b^4\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+b^5 \int \cos ^3(c+d x) \sin ^5(c+d x) \, dx \\ & = \frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{8} \left (7 a^5\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{4} \left (5 a^3 b^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} \left (15 a b^4\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (5 a^4 b\right ) \text {Subst}\left (\int x^7 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {1}{48} \left (35 a^5\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{24} \left (25 a^3 b^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (5 a b^4\right ) \int \cos ^4(c+d x) \, dx-\frac {\left (10 a^2 b^3\right ) \text {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {b^5 \text {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {1}{64} \left (35 a^5\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{32} \left (25 a^3 b^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{64} \left (15 a b^4\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d}+\frac {1}{128} \left (35 a^5\right ) \int 1 \, dx+\frac {1}{64} \left (25 a^3 b^2\right ) \int 1 \, dx+\frac {1}{128} \left (15 a b^4\right ) \int 1 \, dx \\ & = \frac {35 a^5 x}{128}+\frac {25}{64} a^3 b^2 x+\frac {15}{128} a b^4 x-\frac {5 a^2 b^3 \cos ^6(c+d x)}{3 d}-\frac {5 a^4 b \cos ^8(c+d x)}{8 d}+\frac {5 a^2 b^3 \cos ^8(c+d x)}{4 d}+\frac {35 a^5 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {25 a^3 b^2 \cos (c+d x) \sin (c+d x)}{64 d}+\frac {15 a b^4 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^5 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {25 a^3 b^2 \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {5 a b^4 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^5 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {5 a^3 b^2 \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^5 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {5 a^3 b^2 \cos ^7(c+d x) \sin (c+d x)}{4 d}-\frac {5 a b^4 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}+\frac {b^5 \sin ^6(c+d x)}{6 d}-\frac {b^5 \sin ^8(c+d x)}{8 d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.57 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.61 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {120 a (a-i b) (a+i b) \left (7 a^2+3 b^2\right ) (c+d x)-24 b \left (35 a^4+30 a^2 b^2+3 b^4\right ) \cos (2 (c+d x))+12 b \left (-35 a^4-10 a^2 b^2+b^4\right ) \cos (4 (c+d x))+8 b \left (-15 a^4+10 a^2 b^2+b^4\right ) \cos (6 (c+d x))-3 b \left (5 a^4-10 a^2 b^2+b^4\right ) \cos (8 (c+d x))+96 a^3 \left (7 a^2+5 b^2\right ) \sin (2 (c+d x))+24 a \left (7 a^4-10 a^2 b^2-5 b^4\right ) \sin (4 (c+d x))+32 a^3 \left (a^2-5 b^2\right ) \sin (6 (c+d x))+3 a \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (8 (c+d x))}{3072 d} \]
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Time = 1.74 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(\frac {24 \left (-35 a^{4} b -30 a^{2} b^{3}-3 b^{5}\right ) \cos \left (2 d x +2 c \right )+12 \left (-35 a^{4} b -10 a^{2} b^{3}+b^{5}\right ) \cos \left (4 d x +4 c \right )+8 \left (-15 a^{4} b +10 a^{2} b^{3}+b^{5}\right ) \cos \left (6 d x +6 c \right )+3 \left (-5 a^{4} b +10 a^{2} b^{3}-b^{5}\right ) \cos \left (8 d x +8 c \right )+24 \left (7 a^{5}-10 a^{3} b^{2}-5 a \,b^{4}\right ) \sin \left (4 d x +4 c \right )+3 \left (a^{5}-10 a^{3} b^{2}+5 a \,b^{4}\right ) \sin \left (8 d x +8 c \right )+96 \left (7 a^{5}+5 a^{3} b^{2}\right ) \sin \left (2 d x +2 c \right )+32 \left (a^{5}-5 a^{3} b^{2}\right ) \sin \left (6 d x +6 c \right )+840 a^{5} d x +1200 a^{3} b^{2} d x +360 a \,b^{4} d x +1395 a^{4} b +730 a^{2} b^{3}+55 b^{5}}{3072 d}\) | \(279\) |
parts | \(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {b^{5} \left (-\frac {\sin \left (d x +c \right )^{8}}{8}+\frac {\sin \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}+\frac {10 a^{2} b^{3} \left (\frac {\cos \left (d x +c \right )^{8}}{8}-\frac {\cos \left (d x +c \right )^{6}}{6}\right )}{d}+\frac {10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}-\frac {5 a^{4} b \cos \left (d x +c \right )^{8}}{8 d}\) | \(285\) |
derivativedivides | \(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {5 \cos \left (d x +c \right )^{8} a^{4} b}{8}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )+5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{5} \left (-\frac {\cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{4}}{8}-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{12}-\frac {\cos \left (d x +c \right )^{4}}{24}\right )}{d}\) | \(305\) |
default | \(\frac {a^{5} \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )-\frac {5 \cos \left (d x +c \right )^{8} a^{4} b}{8}+10 a^{3} b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{6}}{8}-\frac {\cos \left (d x +c \right )^{6}}{24}\right )+5 a \,b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{5}}{8}-\frac {\cos \left (d x +c \right )^{5} \sin \left (d x +c \right )}{16}+\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+b^{5} \left (-\frac {\cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{4}}{8}-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{4}}{12}-\frac {\cos \left (d x +c \right )^{4}}{24}\right )}{d}\) | \(305\) |
risch | \(\frac {25 a^{3} b^{2} x}{64}+\frac {15 a \,b^{4} x}{128}+\frac {b^{5} \cos \left (6 d x +6 c \right )}{384 d}-\frac {3 b^{5} \cos \left (2 d x +2 c \right )}{128 d}+\frac {7 a^{5} \sin \left (2 d x +2 c \right )}{32 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right ) b^{2}}{64 d}-\frac {5 a \sin \left (4 d x +4 c \right ) b^{4}}{128 d}-\frac {35 b \cos \left (2 d x +2 c \right ) a^{4}}{128 d}-\frac {15 b^{3} \cos \left (2 d x +2 c \right ) a^{2}}{64 d}+\frac {5 a^{3} \sin \left (2 d x +2 c \right ) b^{2}}{32 d}-\frac {5 a^{4} b \cos \left (8 d x +8 c \right )}{1024 d}+\frac {5 a^{2} b^{3} \cos \left (8 d x +8 c \right )}{512 d}-\frac {5 a^{3} \sin \left (8 d x +8 c \right ) b^{2}}{512 d}+\frac {5 a \sin \left (8 d x +8 c \right ) b^{4}}{1024 d}-\frac {5 b \cos \left (6 d x +6 c \right ) a^{4}}{128 d}+\frac {5 b^{3} \cos \left (6 d x +6 c \right ) a^{2}}{192 d}-\frac {5 a^{3} \sin \left (6 d x +6 c \right ) b^{2}}{96 d}-\frac {35 a^{4} b \cos \left (4 d x +4 c \right )}{256 d}-\frac {5 a^{2} b^{3} \cos \left (4 d x +4 c \right )}{128 d}-\frac {b^{5} \cos \left (8 d x +8 c \right )}{1024 d}+\frac {b^{5} \cos \left (4 d x +4 c \right )}{256 d}+\frac {a^{5} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {35 a^{5} x}{128}+\frac {a^{5} \sin \left (6 d x +6 c \right )}{96 d}+\frac {7 a^{5} \sin \left (4 d x +4 c \right )}{128 d}\) | \(428\) |
norman | \(\frac {\left (\frac {35}{128} a^{5}+\frac {25}{64} a^{3} b^{2}+\frac {15}{128} a \,b^{4}\right ) x +\left (\frac {35}{16} a^{5}+\frac {25}{8} a^{3} b^{2}+\frac {15}{16} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {35}{16} a^{5}+\frac {25}{8} a^{3} b^{2}+\frac {15}{16} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {35}{128} a^{5}+\frac {25}{64} a^{3} b^{2}+\frac {15}{128} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {245}{16} a^{5}+\frac {175}{8} a^{3} b^{2}+\frac {105}{16} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {245}{16} a^{5}+\frac {175}{8} a^{3} b^{2}+\frac {105}{16} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {245}{32} a^{5}+\frac {175}{16} a^{3} b^{2}+\frac {105}{32} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {245}{32} a^{5}+\frac {175}{16} a^{3} b^{2}+\frac {105}{32} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {1225}{64} a^{5}+\frac {875}{32} a^{3} b^{2}+\frac {525}{64} a \,b^{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {40 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}+\frac {40 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {10 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {10 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}+\frac {2 \left (200 a^{2} b^{3}-16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d}+\frac {2 \left (105 a^{4} b -80 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}+\frac {2 \left (105 a^{4} b -80 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d}+\frac {a \left (91 a^{4}+3970 a^{2} b^{2}-345 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 d}-\frac {a \left (91 a^{4}+3970 a^{2} b^{2}-345 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 d}+\frac {a \left (93 a^{4}-50 a^{2} b^{2}-15 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {a \left (93 a^{4}-50 a^{2} b^{2}-15 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 d}-\frac {5 a \left (217 a^{4}-3530 a^{2} b^{2}+2013 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{192 d}+\frac {5 a \left (217 a^{4}-3530 a^{2} b^{2}+2013 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192 d}+\frac {a \left (1799 a^{4}-8950 a^{2} b^{2}+4995 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192 d}-\frac {a \left (1799 a^{4}-8950 a^{2} b^{2}+4995 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{192 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}\) | \(776\) |
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Time = 0.26 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.52 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {96 \, b^{5} \cos \left (d x + c\right )^{4} + 48 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{8} + 128 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{6} - 15 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} d x - {\left (48 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} - 45 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]
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Time = 0.77 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.94 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\begin {cases} \frac {35 a^{5} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{5} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 a^{5} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 a^{5} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 a^{5} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{5} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 a^{5} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 a^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 a^{5} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {5 a^{4} b \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {25 a^{3} b^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {25 a^{3} b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {75 a^{3} b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {25 a^{3} b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {25 a^{3} b^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {25 a^{3} b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {275 a^{3} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{192 d} + \frac {365 a^{3} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{192 d} - \frac {25 a^{3} b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} + \frac {5 a^{2} b^{3} \sin ^{8}{\left (c + d x \right )}}{12 d} + \frac {5 a^{2} b^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {5 a^{2} b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{2 d} + \frac {15 a b^{4} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a b^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {15 a b^{4} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {55 a b^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {15 a b^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {b^{5} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {b^{5} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{5} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.25 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.54 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {1920 \, a^{4} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} - 10 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} - 1280 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} a^{2} b^{3} - 15 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{4} + 128 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 4 \, \sin \left (d x + c\right )^{6}\right )} b^{5}}{3072 \, d} \]
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Time = 0.58 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.65 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {5}{128} \, {\left (7 \, a^{5} + 10 \, a^{3} b^{2} + 3 \, a b^{4}\right )} x - \frac {{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (15 \, a^{4} b - 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (35 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (35 \, a^{4} b + 30 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{5} - 5 \, a^{3} b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {{\left (7 \, a^{5} - 10 \, a^{3} b^{2} - 5 \, a b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (7 \, a^{5} + 5 \, a^{3} b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
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Time = 26.06 (sec) , antiderivative size = 650, normalized size of antiderivative = 1.53 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (-\frac {93\,a^5}{64}+\frac {25\,a^3\,b^2}{32}+\frac {15\,a\,b^4}{64}\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (-\frac {93\,a^5}{64}+\frac {25\,a^3\,b^2}{32}+\frac {15\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {91\,a^5}{192}+\frac {1985\,a^3\,b^2}{96}-\frac {115\,a\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {91\,a^5}{192}+\frac {1985\,a^3\,b^2}{96}-\frac {115\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {1799\,a^5}{192}-\frac {4475\,a^3\,b^2}{96}+\frac {1665\,a\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {1799\,a^5}{192}-\frac {4475\,a^3\,b^2}{96}+\frac {1665\,a\,b^4}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {1085\,a^5}{192}-\frac {8825\,a^3\,b^2}{96}+\frac {3355\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {1085\,a^5}{192}-\frac {8825\,a^3\,b^2}{96}+\frac {3355\,a\,b^4}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (70\,a^4\,b-\frac {160\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (70\,a^4\,b-\frac {160\,a^2\,b^3}{3}+\frac {32\,b^5}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {32\,b^5}{3}-\frac {400\,a^2\,b^3}{3}\right )+40\,a^2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+40\,a^2\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+10\,a^4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+10\,a^4\,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 )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {5\,a\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (7\,a^4+10\,a^2\,b^2+3\,b^4\right )}{64\,d}+\frac {5\,a\,\mathrm {atan}\left (\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,a^2+3\,b^2\right )\,\left (a^2+b^2\right )}{64\,\left (\frac {35\,a^5}{64}+\frac {25\,a^3\,b^2}{32}+\frac {15\,a\,b^4}{64}\right )}\right )\,\left (7\,a^2+3\,b^2\right )\,\left (a^2+b^2\right )}{64\,d} \]
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