Integrand size = 38, antiderivative size = 105 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} (3 a B+4 b C) x+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {(3 a B+4 b C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {(b B+a C) \sin ^3(c+d x)}{3 d} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4157, 4081, 3872, 2713, 2715, 8} \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {(a C+b B) \sin ^3(c+d x)}{3 d}+\frac {(a C+b B) \sin (c+d x)}{d}+\frac {(3 a B+4 b C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x (3 a B+4 b C)+\frac {a B \sin (c+d x) \cos ^3(c+d x)}{4 d} \]
[In]
[Out]
Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4081
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^4(c+d x) (a+b \sec (c+d x)) (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{4} \int \cos ^3(c+d x) (-4 (b B+a C)-(3 a B+4 b C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^3(c+d x) \sin (c+d x)}{4 d}-(-b B-a C) \int \cos ^3(c+d x) \, dx-\frac {1}{4} (-3 a B-4 b C) \int \cos ^2(c+d x) \, dx \\ & = \frac {(3 a B+4 b C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {1}{8} (-3 a B-4 b C) \int 1 \, dx-\frac {(b B+a C) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {1}{8} (3 a B+4 b C) x+\frac {(b B+a C) \sin (c+d x)}{d}+\frac {(3 a B+4 b C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {(b B+a C) \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {36 a B c+48 b c C+36 a B d x+48 b C d x+96 (b B+a C) \sin (c+d x)-32 (b B+a C) \sin ^3(c+d x)+24 (a B+b C) \sin (2 (c+d x))+3 a B \sin (4 (c+d x))}{96 d} \]
[In]
[Out]
Time = 0.56 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {\left (24 a B +24 C b \right ) \sin \left (2 d x +2 c \right )+\left (8 B b +8 C a \right ) \sin \left (3 d x +3 c \right )+3 a B \sin \left (4 d x +4 c \right )+\left (72 B b +72 C a \right ) \sin \left (d x +c \right )+36 x d \left (a B +\frac {4 C b}{3}\right )}{96 d}\) | \(86\) |
derivativedivides | \(\frac {a B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {C a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(107\) |
default | \(\frac {a B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {C a \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(107\) |
risch | \(\frac {3 a B x}{8}+\frac {x C b}{2}+\frac {3 \sin \left (d x +c \right ) B b}{4 d}+\frac {3 \sin \left (d x +c \right ) C a}{4 d}+\frac {a B \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B b}{12 d}+\frac {\sin \left (3 d x +3 c \right ) C a}{12 d}+\frac {a B \sin \left (2 d x +2 c \right )}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C b}{4 d}\) | \(118\) |
norman | \(\frac {\left (\frac {3 a B}{8}+\frac {C b}{2}\right ) x +\left (-\frac {15 a B}{8}-\frac {5 C b}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-\frac {15 a B}{8}-\frac {5 C b}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {3 a B}{8}+\frac {C b}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {3 a B}{8}+\frac {C b}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {3 a B}{8}+\frac {C b}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {9 a B}{8}+\frac {3 C b}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {9 a B}{8}+\frac {3 C b}{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-\frac {8 \left (B b +C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}-\frac {2 \left (3 a B -2 B b -2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {2 \left (3 a B +2 B b +2 C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}+\frac {\left (a B -8 B b -8 C a -12 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {\left (a B +8 B b +8 C a -12 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}-\frac {\left (5 a B -8 B b -8 C a +4 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}+\frac {\left (5 a B +8 B b +8 C a +4 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(407\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.77 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, B a + 4 \, C b\right )} d x + {\left (6 \, B a \cos \left (d x + c\right )^{3} + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{2} + 16 \, C a + 16 \, B b + 3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
[In]
[Out]
Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b}{96 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (97) = 194\).
Time = 0.30 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.59 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, B a + 4 \, C b\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
[In]
[Out]
Time = 17.46 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.11 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3\,B\,a\,x}{8}+\frac {C\,b\,x}{2}+\frac {3\,B\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,C\,a\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {B\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {C\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d} \]
[In]
[Out]