Integrand size = 21, antiderivative size = 170 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {9}{8} a^2 b x+\frac {5 b^3 x}{16}+\frac {a \left (a^2+3 b^2\right ) \sin (c+d x)}{d}+\frac {b \left (18 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (18 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {b^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {a \left (a^2+6 b^2\right ) \sin ^3(c+d x)}{3 d}+\frac {3 a b^2 \sin ^5(c+d x)}{5 d} \]
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Time = 0.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2872, 3102, 2827, 2713, 2715, 8} \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=-\frac {a \left (5 a^2+12 b^2\right ) \sin ^3(c+d x)}{15 d}+\frac {a \left (5 a^2+12 b^2\right ) \sin (c+d x)}{5 d}+\frac {b \left (18 a^2+5 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {b \left (18 a^2+5 b^2\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} b x \left (18 a^2+5 b^2\right )+\frac {b^2 \sin (c+d x) \cos ^4(c+d x) (a+b \cos (c+d x))}{6 d}+\frac {13 a b^2 \sin (c+d x) \cos ^4(c+d x)}{30 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 2872
Rule 3102
Rubi steps \begin{align*} \text {integral}& = \frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^3(c+d x) \left (2 a \left (3 a^2+2 b^2\right )+b \left (18 a^2+5 b^2\right ) \cos (c+d x)+13 a b^2 \cos ^2(c+d x)\right ) \, dx \\ & = \frac {13 a b^2 \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{30} \int \cos ^3(c+d x) \left (6 a \left (5 a^2+12 b^2\right )+5 b \left (18 a^2+5 b^2\right ) \cos (c+d x)\right ) \, dx \\ & = \frac {13 a b^2 \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{6} \left (b \left (18 a^2+5 b^2\right )\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (a \left (5 a^2+12 b^2\right )\right ) \int \cos ^3(c+d x) \, dx \\ & = \frac {b \left (18 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {13 a b^2 \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{6 d}+\frac {1}{8} \left (b \left (18 a^2+5 b^2\right )\right ) \int \cos ^2(c+d x) \, dx-\frac {\left (a \left (5 a^2+12 b^2\right )\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {a \left (5 a^2+12 b^2\right ) \sin (c+d x)}{5 d}+\frac {b \left (18 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (18 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {13 a b^2 \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{6 d}-\frac {a \left (5 a^2+12 b^2\right ) \sin ^3(c+d x)}{15 d}+\frac {1}{16} \left (b \left (18 a^2+5 b^2\right )\right ) \int 1 \, dx \\ & = \frac {1}{16} b \left (18 a^2+5 b^2\right ) x+\frac {a \left (5 a^2+12 b^2\right ) \sin (c+d x)}{5 d}+\frac {b \left (18 a^2+5 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {b \left (18 a^2+5 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {13 a b^2 \cos ^4(c+d x) \sin (c+d x)}{30 d}+\frac {b^2 \cos ^4(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{6 d}-\frac {a \left (5 a^2+12 b^2\right ) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {1080 a^2 b c+300 b^3 c+1080 a^2 b d x+300 b^3 d x+360 a \left (2 a^2+5 b^2\right ) \sin (c+d x)+45 \left (16 a^2 b+5 b^3\right ) \sin (2 (c+d x))+80 a^3 \sin (3 (c+d x))+300 a b^2 \sin (3 (c+d x))+90 a^2 b \sin (4 (c+d x))+45 b^3 \sin (4 (c+d x))+36 a b^2 \sin (5 (c+d x))+5 b^3 \sin (6 (c+d x))}{960 d} \]
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Time = 4.43 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.79
method | result | size |
parallelrisch | \(\frac {\left (720 a^{2} b +225 b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (80 a^{3}+300 a \,b^{2}\right ) \sin \left (3 d x +3 c \right )+\left (90 a^{2} b +45 b^{3}\right ) \sin \left (4 d x +4 c \right )+36 a \,b^{2} \sin \left (5 d x +5 c \right )+5 b^{3} \sin \left (6 d x +6 c \right )+\left (720 a^{3}+1800 a \,b^{2}\right ) \sin \left (d x +c \right )+1080 b d \left (a^{2}+\frac {5 b^{2}}{18}\right ) x}{960 d}\) | \(135\) |
derivativedivides | \(\frac {\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 a \,b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(145\) |
default | \(\frac {\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+3 a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 a \,b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(145\) |
parts | \(\frac {a^{3} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {b^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {3 a \,b^{2} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {3 a^{2} b \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(153\) |
risch | \(\frac {9 a^{2} b x}{8}+\frac {5 b^{3} x}{16}+\frac {3 a^{3} \sin \left (d x +c \right )}{4 d}+\frac {15 a \,b^{2} \sin \left (d x +c \right )}{8 d}+\frac {b^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a \,b^{2} \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{2} b}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) b^{3}}{64 d}+\frac {a^{3} \sin \left (3 d x +3 c \right )}{12 d}+\frac {5 \sin \left (3 d x +3 c \right ) a \,b^{2}}{16 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) b^{3}}{64 d}\) | \(184\) |
norman | \(\frac {\left (\frac {9}{8} a^{2} b +\frac {5}{16} b^{3}\right ) x +\left (\frac {9}{8} a^{2} b +\frac {5}{16} b^{3}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {27}{4} a^{2} b +\frac {15}{8} b^{3}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {27}{4} a^{2} b +\frac {15}{8} b^{3}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {45}{2} a^{2} b +\frac {25}{4} b^{3}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {135}{8} a^{2} b +\frac {75}{16} b^{3}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {135}{8} a^{2} b +\frac {75}{16} b^{3}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (16 a^{3}-30 a^{2} b +48 a \,b^{2}-11 b^{3}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {\left (16 a^{3}+30 a^{2} b +48 a \,b^{2}+11 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {3 \left (80 a^{3}-10 a^{2} b +208 a \,b^{2}-25 b^{3}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {3 \left (80 a^{3}+10 a^{2} b +208 a \,b^{2}+25 b^{3}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {\left (176 a^{3}-126 a^{2} b +336 a \,b^{2}+5 b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {\left (176 a^{3}+126 a^{2} b +336 a \,b^{2}-5 b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(414\) |
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Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.78 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {15 \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} d x + {\left (40 \, b^{3} \cos \left (d x + c\right )^{5} + 144 \, a b^{2} \cos \left (d x + c\right )^{4} + 10 \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 160 \, a^{3} + 384 \, a b^{2} + 16 \, {\left (5 \, a^{3} + 12 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (158) = 316\).
Time = 0.37 (sec) , antiderivative size = 393, normalized size of antiderivative = 2.31 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\begin {cases} \frac {2 a^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {9 a^{2} b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 a^{2} b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {9 a^{2} b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {9 a^{2} b \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {15 a^{2} b \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 a b^{2} \sin ^{5}{\left (c + d x \right )}}{5 d} + \frac {4 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 b^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 b^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 b^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 b^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 b^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 b^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 b^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \cos {\left (c \right )}\right )^{3} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=-\frac {320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} b - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a b^{2} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b^{3}}{960 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.88 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {b^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, a b^{2} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {1}{16} \, {\left (18 \, a^{2} b + 5 \, b^{3}\right )} x + \frac {3 \, {\left (2 \, a^{2} b + b^{3}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (4 \, a^{3} + 15 \, a b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {3 \, {\left (16 \, a^{2} b + 5 \, b^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {3 \, {\left (2 \, a^{3} + 5 \, a b^{2}\right )} \sin \left (d x + c\right )}{8 \, d} \]
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Time = 15.87 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.24 \[ \int \cos ^3(c+d x) (a+b \cos (c+d x))^3 \, dx=\frac {\left (2\,a^3-\frac {15\,a^2\,b}{4}+6\,a\,b^2-\frac {11\,b^3}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {22\,a^3}{3}-\frac {21\,a^2\,b}{4}+14\,a\,b^2+\frac {5\,b^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (12\,a^3-\frac {3\,a^2\,b}{2}+\frac {156\,a\,b^2}{5}-\frac {15\,b^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,a^3+\frac {3\,a^2\,b}{2}+\frac {156\,a\,b^2}{5}+\frac {15\,b^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {22\,a^3}{3}+\frac {21\,a^2\,b}{4}+14\,a\,b^2-\frac {5\,b^3}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^3+\frac {15\,a^2\,b}{4}+6\,a\,b^2+\frac {11\,b^3}{8}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {b\,\mathrm {atan}\left (\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (18\,a^2+5\,b^2\right )}{8\,\left (\frac {9\,a^2\,b}{4}+\frac {5\,b^3}{8}\right )}\right )\,\left (18\,a^2+5\,b^2\right )}{8\,d}-\frac {b\,\left (18\,a^2+5\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \]
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