Integrand size = 23, antiderivative size = 150 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {7}{8} a^4 (5 A+4 B) x+\frac {8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac {27 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac {4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2830, 2724, 2717, 2715, 8, 2713} \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=-\frac {4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac {8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac {a^4 (5 A+4 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac {27 a^4 (5 A+4 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac {7}{8} a^4 x (5 A+4 B)+\frac {B \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]
[In]
[Out]
Rule 8
Rule 2713
Rule 2715
Rule 2717
Rule 2724
Rule 2830
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} (5 A+4 B) \int (a+a \cos (c+d x))^4 \, dx \\ & = \frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} (5 A+4 B) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx \\ & = \frac {1}{5} a^4 (5 A+4 B) x+\frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} \left (a^4 (5 A+4 B)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (4 a^4 (5 A+4 B)\right ) \int \cos (c+d x) \, dx+\frac {1}{5} \left (4 a^4 (5 A+4 B)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{5} \left (6 a^4 (5 A+4 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{5} a^4 (5 A+4 B) x+\frac {4 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac {3 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{5 d}+\frac {a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{20} \left (3 a^4 (5 A+4 B)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (5 A+4 B)\right ) \int 1 \, dx-\frac {\left (4 a^4 (5 A+4 B)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {4}{5} a^4 (5 A+4 B) x+\frac {8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac {27 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac {4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d}+\frac {1}{40} \left (3 a^4 (5 A+4 B)\right ) \int 1 \, dx \\ & = \frac {7}{8} a^4 (5 A+4 B) x+\frac {8 a^4 (5 A+4 B) \sin (c+d x)}{5 d}+\frac {27 a^4 (5 A+4 B) \cos (c+d x) \sin (c+d x)}{40 d}+\frac {a^4 (5 A+4 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {B (a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac {4 a^4 (5 A+4 B) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.89 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {a^4 \sin (c+d x) \left (210 (5 A+4 B) \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (800 A+664 B+15 (27 A+28 B) \cos (c+d x)+16 (10 A+17 B) \cos ^2(c+d x)+30 (A+4 B) \cos ^3(c+d x)+24 B \cos ^4(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{120 d \sqrt {\sin ^2(c+d x)}} \]
[In]
[Out]
Time = 3.67 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {\left (\left (56 A +64 B \right ) \sin \left (2 d x +2 c \right )+\left (\frac {32 A}{3}+\frac {58 B}{3}\right ) \sin \left (3 d x +3 c \right )+\left (A +4 B \right ) \sin \left (4 d x +4 c \right )+\frac {2 B \sin \left (5 d x +5 c \right )}{5}+\left (224 A +196 B \right ) \sin \left (d x +c \right )+140 \left (A +\frac {4 B}{5}\right ) x d \right ) a^{4}}{32 d}\) | \(94\) |
risch | \(\frac {35 a^{4} x A}{8}+\frac {7 a^{4} B x}{2}+\frac {7 \sin \left (d x +c \right ) a^{4} A}{d}+\frac {49 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} A}{32 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} A}{3 d}+\frac {29 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} A}{4 d}+\frac {2 \sin \left (2 d x +2 c \right ) B \,a^{4}}{d}\) | \(172\) |
parts | \(a^{4} x A +\frac {\left (a^{4} A +4 B \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {\left (4 a^{4} A +B \,a^{4}\right ) \sin \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (6 a^{4} A +4 B \,a^{4}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(187\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )}{d}\) | \(248\) |
default | \(\frac {a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+4 a^{4} A \sin \left (d x +c \right )+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} A \left (d x +c \right )+B \,a^{4} \sin \left (d x +c \right )}{d}\) | \(248\) |
norman | \(\frac {\frac {7 a^{4} \left (5 A +4 B \right ) x}{8}+\frac {79 a^{4} \left (5 A +4 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {224 a^{4} \left (5 A +4 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {49 a^{4} \left (5 A +4 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {7 a^{4} \left (5 A +4 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {35 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{4} \left (5 A +4 B \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{4} \left (93 A +100 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(279\) |
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.73 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {105 \, {\left (5 \, A + 4 \, B\right )} a^{4} d x + {\left (24 \, B a^{4} \cos \left (d x + c\right )^{4} + 30 \, {\left (A + 4 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \, {\left (10 \, A + 17 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (27 \, A + 28 \, B\right )} a^{4} \cos \left (d x + c\right ) + 8 \, {\left (100 \, A + 83 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{120 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (141) = 282\).
Time = 0.31 (sec) , antiderivative size = 544, normalized size of antiderivative = 3.63 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\begin {cases} \frac {3 A a^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + 3 A a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 A a^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + 3 A a^{4} x \cos ^{2}{\left (c + d x \right )} + A a^{4} x + \frac {3 A a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {8 A a^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {5 A a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {4 A a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {3 A a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {4 A a^{4} \sin {\left (c + d x \right )}}{d} + \frac {3 B a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 B a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 B a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac {3 B a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 B a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac {8 B a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 B a^{4} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{2 d} + \frac {4 B a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac {B a^{4} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 B a^{4} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {6 B a^{4} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {2 B a^{4} \sin {\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {B a^{4} \sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )}\right ) \left (a \cos {\left (c \right )} + a\right )^{4} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.57 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=-\frac {640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 480 \, {\left (d x + c\right )} A a^{4} - 32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1920 \, A a^{4} \sin \left (d x + c\right ) - 480 \, B a^{4} \sin \left (d x + c\right )}{480 \, d} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {B a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {7}{8} \, {\left (5 \, A a^{4} + 4 \, B a^{4}\right )} x + \frac {{\left (A a^{4} + 4 \, B a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {{\left (16 \, A a^{4} + 29 \, B a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (7 \, A a^{4} + 8 \, B a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {7 \, {\left (8 \, A a^{4} + 7 \, B a^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \]
[In]
[Out]
Time = 1.64 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.85 \[ \int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)) \, dx=\frac {\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {245\,A\,a^4}{6}+\frac {98\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {224\,A\,a^4}{3}+\frac {896\,B\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {395\,A\,a^4}{6}+\frac {158\,B\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {93\,A\,a^4}{4}+25\,B\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {7\,a^4\,\left (5\,A+4\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{4\,d}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A+4\,B\right )}{4\,\left (\frac {35\,A\,a^4}{4}+7\,B\,a^4\right )}\right )\,\left (5\,A+4\,B\right )}{4\,d} \]
[In]
[Out]