Integrand size = 21, antiderivative size = 129 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {23 a^3 x}{16}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {23 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {23 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {7 a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^5(c+d x)}{5 d} \]
23/16*a^3*x+4*a^3*sin(d*x+c)/d+23/16*a^3*cos(d*x+c)*sin(d*x+c)/d+23/24*a^3 *cos(d*x+c)^3*sin(d*x+c)/d+1/6*a^3*cos(d*x+c)^5*sin(d*x+c)/d-7/3*a^3*sin(d *x+c)^3/d+3/5*a^3*sin(d*x+c)^5/d
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3 (1380 d x+2520 \sin (c+d x)+945 \sin (2 (c+d x))+380 \sin (3 (c+d x))+135 \sin (4 (c+d x))+36 \sin (5 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \]
(a^3*(1380*d*x + 2520*Sin[c + d*x] + 945*Sin[2*(c + d*x)] + 380*Sin[3*(c + d*x)] + 135*Sin[4*(c + d*x)] + 36*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)])) /(960*d)
Time = 0.34 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4278, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
\(\Big \downarrow \) 4278 |
\(\displaystyle \int \left (a^3 \cos ^6(c+d x)+3 a^3 \cos ^5(c+d x)+3 a^3 \cos ^4(c+d x)+a^3 \cos ^3(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 a^3 \sin ^5(c+d x)}{5 d}-\frac {7 a^3 \sin ^3(c+d x)}{3 d}+\frac {4 a^3 \sin (c+d x)}{d}+\frac {a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {23 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {23 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {23 a^3 x}{16}\) |
(23*a^3*x)/16 + (4*a^3*Sin[c + d*x])/d + (23*a^3*Cos[c + d*x]*Sin[c + d*x] )/(16*d) + (23*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^3*Cos[c + d*x] ^5*Sin[c + d*x])/(6*d) - (7*a^3*Sin[c + d*x]^3)/(3*d) + (3*a^3*Sin[c + d*x ]^5)/(5*d)
3.1.29.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[ExpandTrig[(a + b*csc[e + f*x])^m*(d*csc[e + f *x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && I GtQ[m, 0] && RationalQ[n]
Time = 0.77 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.58
method | result | size |
parallelrisch | \(\frac {\left (276 d x +\sin \left (6 d x +6 c \right )+504 \sin \left (d x +c \right )+189 \sin \left (2 d x +2 c \right )+76 \sin \left (3 d x +3 c \right )+27 \sin \left (4 d x +4 c \right )+\frac {36 \sin \left (5 d x +5 c \right )}{5}\right ) a^{3}}{192 d}\) | \(75\) |
risch | \(\frac {23 a^{3} x}{16}+\frac {21 a^{3} \sin \left (d x +c \right )}{8 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {9 a^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {19 a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {63 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
derivativedivides | \(\frac {\frac {a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(143\) |
default | \(\frac {\frac {a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {3 a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(143\) |
norman | \(\frac {\frac {23 a^{3} x}{16}+\frac {105 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 d}+\frac {353 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 d}-\frac {1303 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{40 d}-\frac {1339 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}+\frac {989 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{120 d}+\frac {253 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24 d}+\frac {23 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8 d}+\frac {23 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4}+\frac {23 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-\frac {23 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}-\frac {115 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}-\frac {23 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}+\frac {23 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{4}+\frac {23 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{4}+\frac {23 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{16}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}\) | \(325\) |
1/192*(276*d*x+sin(6*d*x+6*c)+504*sin(d*x+c)+189*sin(2*d*x+2*c)+76*sin(3*d *x+3*c)+27*sin(4*d*x+4*c)+36/5*sin(5*d*x+5*c))*a^3/d
Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {345 \, a^{3} d x + {\left (40 \, a^{3} \cos \left (d x + c\right )^{5} + 144 \, a^{3} \cos \left (d x + c\right )^{4} + 230 \, a^{3} \cos \left (d x + c\right )^{3} + 272 \, a^{3} \cos \left (d x + c\right )^{2} + 345 \, a^{3} \cos \left (d x + c\right ) + 544 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]
1/240*(345*a^3*d*x + (40*a^3*cos(d*x + c)^5 + 144*a^3*cos(d*x + c)^4 + 230 *a^3*cos(d*x + c)^3 + 272*a^3*cos(d*x + c)^2 + 345*a^3*cos(d*x + c) + 544* a^3)*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{960 \, d} \]
1/960*(192*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^3 - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d* x + 2*c))*a^3 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^3 + 90*(12*d*x + 1 2*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^3)/d
Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.99 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {345 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5814 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
1/240*(345*(d*x + c)*a^3 + 2*(345*a^3*tan(1/2*d*x + 1/2*c)^11 + 1955*a^3*t an(1/2*d*x + 1/2*c)^9 + 4554*a^3*tan(1/2*d*x + 1/2*c)^7 + 5814*a^3*tan(1/2 *d*x + 1/2*c)^5 + 3165*a^3*tan(1/2*d*x + 1/2*c)^3 + 1575*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 15.43 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.94 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {23\,a^3\,x}{16}+\frac {\frac {23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {391\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {759\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {969\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {211\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {105\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]