Integrand size = 29, antiderivative size = 98 \[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 B x}{8}+\frac {(A+C) \sin (c+d x)}{d}+\frac {3 B \cos (c+d x) \sin (c+d x)}{8 d}+\frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {(2 A+C) \sin ^3(c+d x)}{3 d}+\frac {A \sin ^5(c+d x)}{5 d} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4132, 2715, 8, 4129, 3092, 380} \[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {(2 A+C) \sin ^3(c+d x)}{3 d}+\frac {(A+C) \sin (c+d x)}{d}+\frac {A \sin ^5(c+d x)}{5 d}+\frac {B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 B \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 B x}{8} \]
[In]
[Out]
Rule 8
Rule 380
Rule 2715
Rule 3092
Rule 4129
Rule 4132
Rubi steps \begin{align*} \text {integral}& = B \int \cos ^4(c+d x) \, dx+\int \cos ^5(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} (3 B) \int \cos ^2(c+d x) \, dx+\int \cos ^3(c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx \\ & = \frac {3 B \cos (c+d x) \sin (c+d x)}{8 d}+\frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} (3 B) \int 1 \, dx-\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (A+C-A x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {3 B x}{8}+\frac {3 B \cos (c+d x) \sin (c+d x)}{8 d}+\frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {\text {Subst}\left (\int \left (A \left (1+\frac {C}{A}\right )-(2 A+C) x^2+A x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {3 B x}{8}+\frac {(A+C) \sin (c+d x)}{d}+\frac {3 B \cos (c+d x) \sin (c+d x)}{8 d}+\frac {B \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {(2 A+C) \sin ^3(c+d x)}{3 d}+\frac {A \sin ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {180 B c+180 B d x+60 (5 A+6 C) \sin (c+d x)+120 B \sin (2 (c+d x))+50 A \sin (3 (c+d x))+40 C \sin (3 (c+d x))+15 B \sin (4 (c+d x))+6 A \sin (5 (c+d x))}{480 d} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(\frac {\left (50 A +40 C \right ) \sin \left (3 d x +3 c \right )+120 B \sin \left (2 d x +2 c \right )+15 B \sin \left (4 d x +4 c \right )+6 A \sin \left (5 d x +5 c \right )+\left (300 A +360 C \right ) \sin \left (d x +c \right )+180 B x d}{480 d}\) | \(79\) |
derivativedivides | \(\frac {\frac {A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(89\) |
default | \(\frac {\frac {A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(89\) |
risch | \(\frac {3 B x}{8}+\frac {5 A \sin \left (d x +c \right )}{8 d}+\frac {3 C \sin \left (d x +c \right )}{4 d}+\frac {A \sin \left (5 d x +5 c \right )}{80 d}+\frac {B \sin \left (4 d x +4 c \right )}{32 d}+\frac {5 A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) C}{12 d}+\frac {B \sin \left (2 d x +2 c \right )}{4 d}\) | \(105\) |
norman | \(\frac {-\frac {3 B x}{8}-\frac {3 B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-\frac {15 B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8}+\frac {15 B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {3 B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{2}+\frac {3 B x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8}-\frac {\left (8 A -9 B +40 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {\left (8 A -5 B +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{4 d}-\frac {\left (8 A +5 B +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (8 A +9 B +40 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}-\frac {\left (152 A -15 B +40 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{30 d}+\frac {\left (152 A +15 B +40 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{30 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(266\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.74 \[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {45 \, B d x + {\left (24 \, A \cos \left (d x + c\right )^{4} + 30 \, B \cos \left (d x + c\right )^{3} + 8 \, {\left (4 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 45 \, B \cos \left (d x + c\right ) + 64 \, A + 80 \, C\right )} \sin \left (d x + c\right )}{120 \, d} \]
[In]
[Out]
\[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{5}{\left (c + d x \right )}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91 \[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C}{480 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (88) = 176\).
Time = 0.31 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.27 \[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {45 \, {\left (d x + c\right )} B + \frac {2 \, {\left (120 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 160 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 320 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 400 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 320 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
[In]
[Out]
Time = 14.92 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.06 \[ \int \cos ^5(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3\,B\,x}{8}+\frac {5\,A\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {A\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {B\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {C\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {5\,A\,\sin \left (c+d\,x\right )}{8\,d}+\frac {3\,C\,\sin \left (c+d\,x\right )}{4\,d} \]
[In]
[Out]