Integrand size = 31, antiderivative size = 176 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {1}{8} a^3 (13 A+15 B) x+\frac {a^3 (38 A+45 B) \sin (c+d x)}{15 d}+\frac {a^3 (13 A+15 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 A+45 B) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 A+5 B) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d} \]
[Out]
Time = 0.44 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4102, 4081, 3872, 2715, 8, 2717} \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (38 A+45 B) \sin (c+d x)}{15 d}+\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac {a^3 (13 A+15 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{20 d}+\frac {1}{8} a^3 x (13 A+15 B)+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d} \]
[In]
[Out]
Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4081
Rule 4102
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 (a (7 A+5 B)+a (2 A+5 B) \sec (c+d x)) \, dx \\ & = \frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 A+5 B) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}+\frac {1}{20} \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (a^2 (43 A+45 B)+2 a^2 (11 A+15 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (43 A+45 B) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 A+5 B) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}-\frac {1}{60} \int \cos ^2(c+d x) \left (-15 a^3 (13 A+15 B)-4 a^3 (38 A+45 B) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (43 A+45 B) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 A+5 B) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}+\frac {1}{4} \left (a^3 (13 A+15 B)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{15} \left (a^3 (38 A+45 B)\right ) \int \cos (c+d x) \, dx \\ & = \frac {a^3 (38 A+45 B) \sin (c+d x)}{15 d}+\frac {a^3 (13 A+15 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 A+45 B) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 A+5 B) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}+\frac {1}{8} \left (a^3 (13 A+15 B)\right ) \int 1 \, dx \\ & = \frac {1}{8} a^3 (13 A+15 B) x+\frac {a^3 (38 A+45 B) \sin (c+d x)}{15 d}+\frac {a^3 (13 A+15 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 A+45 B) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 A+5 B) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (780 A c+780 A d x+900 B d x+60 (23 A+26 B) \sin (c+d x)+480 (A+B) \sin (2 (c+d x))+170 A \sin (3 (c+d x))+120 B \sin (3 (c+d x))+45 A \sin (4 (c+d x))+15 B \sin (4 (c+d x))+6 A \sin (5 (c+d x)))}{480 d} \]
[In]
[Out]
Time = 3.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53
method | result | size |
parallelrisch | \(\frac {3 \left (\frac {32 \left (A +B \right ) \sin \left (2 d x +2 c \right )}{3}+\frac {2 \left (\frac {17 A}{3}+4 B \right ) \sin \left (3 d x +3 c \right )}{3}+\left (A +\frac {B}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {2 A \sin \left (5 d x +5 c \right )}{15}+\frac {4 \left (23 A +26 B \right ) \sin \left (d x +c \right )}{3}+\frac {52 d \left (A +\frac {15 B}{13}\right ) x}{3}\right ) a^{3}}{32 d}\) | \(93\) |
risch | \(\frac {13 a^{3} A x}{8}+\frac {15 a^{3} x B}{8}+\frac {23 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {13 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {a^{3} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {17 a^{3} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}\) | \(170\) |
derivativedivides | \(\frac {a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \sin \left (d x +c \right )+a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
default | \(\frac {a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \sin \left (d x +c \right )+a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
norman | \(\frac {-\frac {a^{3} \left (13 A +15 B \right ) x}{8}-\frac {37 a^{3} \left (13 A +15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{60 d}+\frac {5 a^{3} \left (13 A +15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}+\frac {a^{3} \left (13 A +15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}-\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4}+\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {3 a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}-\frac {3 a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}-\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{4}+\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{4}+\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{8}+\frac {a^{3} \left (31 A +1365 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 d}-\frac {a^{3} \left (51 A +49 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (193 A +75 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {a^{3} \left (857 A -45 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{60 d}-\frac {a^{3} \left (1127 A +1005 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{60 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(420\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.62 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (13 \, A + 15 \, B\right )} a^{3} d x + {\left (24 \, A a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (19 \, A + 15 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (13 \, A + 15 \, B\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (38 \, A + 45 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \]
[In]
[Out]
Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.21 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 480 \, B a^{3} \sin \left (d x + c\right )}{480 \, d} \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.19 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (13 \, A a^{3} + 15 \, B a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (195 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 225 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 910 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1050 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1664 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1920 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1330 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1830 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 765 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
[In]
[Out]
Time = 16.07 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.40 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (\frac {13\,A\,a^3}{4}+\frac {15\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {91\,A\,a^3}{6}+\frac {35\,B\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {416\,A\,a^3}{15}+32\,B\,a^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {133\,A\,a^3}{6}+\frac {61\,B\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {49\,B\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,A+15\,B\right )}{4\,\left (\frac {13\,A\,a^3}{4}+\frac {15\,B\,a^3}{4}\right )}\right )\,\left (13\,A+15\,B\right )}{4\,d} \]
[In]
[Out]