Integrand size = 21, antiderivative size = 127 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\frac {49 a^4 x}{16}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {49 a^4 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {41 a^4 \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a^4 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {4 a^4 \sin ^3(c+d x)}{d}+\frac {4 a^4 \sin ^5(c+d x)}{5 d} \] Output:
49/16*a^4*x+8*a^4*sin(d*x+c)/d+49/16*a^4*cos(d*x+c)*sin(d*x+c)/d+41/24*a^4 *cos(d*x+c)^3*sin(d*x+c)/d+1/6*a^4*cos(d*x+c)^5*sin(d*x+c)/d-4*a^4*sin(d*x +c)^3/d+4/5*a^4*sin(d*x+c)^5/d
Time = 0.13 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\frac {a^4 (2940 d x+5280 \sin (c+d x)+1905 \sin (2 (c+d x))+720 \sin (3 (c+d x))+225 \sin (4 (c+d x))+48 \sin (5 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \] Input:
Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4,x]
Output:
(a^4*(2940*d*x + 5280*Sin[c + d*x] + 1905*Sin[2*(c + d*x)] + 720*Sin[3*(c + d*x)] + 225*Sin[4*(c + d*x)] + 48*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)]) )/(960*d)
Time = 0.37 (sec) , antiderivative size = 127, 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.
\(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^4 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
\(\Big \downarrow \) 4278 |
\(\displaystyle \int \left (a^4 \cos ^6(c+d x)+4 a^4 \cos ^5(c+d x)+6 a^4 \cos ^4(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^2(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 a^4 \sin ^5(c+d x)}{5 d}-\frac {4 a^4 \sin ^3(c+d x)}{d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {41 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {49 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {49 a^4 x}{16}\) |
Input:
Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4,x]
Output:
(49*a^4*x)/16 + (8*a^4*Sin[c + d*x])/d + (49*a^4*Cos[c + d*x]*Sin[c + d*x] )/(16*d) + (41*a^4*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a^4*Cos[c + d*x] ^5*Sin[c + d*x])/(6*d) - (4*a^4*Sin[c + d*x]^3)/d + (4*a^4*Sin[c + d*x]^5) /(5*d)
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 = 1.75 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.59
method | result | size |
parallelrisch | \(\frac {\left (588 d x +\sin \left (6 d x +6 c \right )+1056 \sin \left (d x +c \right )+381 \sin \left (2 d x +2 c \right )+144 \sin \left (3 d x +3 c \right )+45 \sin \left (4 d x +4 c \right )+\frac {48 \sin \left (5 d x +5 c \right )}{5}\right ) a^{4}}{192 d}\) | \(75\) |
risch | \(\frac {49 a^{4} x}{16}+\frac {11 a^{4} \sin \left (d x +c \right )}{2 d}+\frac {a^{4} \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} \sin \left (5 d x +5 c \right )}{20 d}+\frac {15 a^{4} \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 a^{4} \sin \left (3 d x +3 c \right )}{4 d}+\frac {127 a^{4} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
derivativedivides | \(\frac {a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(169\) |
default | \(\frac {a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(169\) |
Input:
int(cos(d*x+c)^6*(a+a*sec(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/192*(588*d*x+sin(6*d*x+6*c)+1056*sin(d*x+c)+381*sin(2*d*x+2*c)+144*sin(3 *d*x+3*c)+45*sin(4*d*x+4*c)+48/5*sin(5*d*x+5*c))*a^4/d
Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\frac {735 \, a^{4} d x + {\left (40 \, a^{4} \cos \left (d x + c\right )^{5} + 192 \, a^{4} \cos \left (d x + c\right )^{4} + 410 \, a^{4} \cos \left (d x + c\right )^{3} + 576 \, a^{4} \cos \left (d x + c\right )^{2} + 735 \, a^{4} \cos \left (d x + c\right ) + 1152 \, a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4,x, algorithm="fricas")
Output:
1/240*(735*a^4*d*x + (40*a^4*cos(d*x + c)^5 + 192*a^4*cos(d*x + c)^4 + 410 *a^4*cos(d*x + c)^3 + 576*a^4*cos(d*x + c)^2 + 735*a^4*cos(d*x + c) + 1152 *a^4)*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*(a+a*sec(d*x+c))**4,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.30 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\frac {256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4}}{960 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4,x, algorithm="maxima")
Output:
1/960*(256*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^4 - 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^4 - 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*a^4 + 180*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^4 + 240*(2*d*x + 2*c + si n(2*d*x + 2*c))*a^4)/d
Time = 0.21 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\frac {735 \, {\left (d x + c\right )} a^{4} + \frac {2 \, {\left (735 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4165 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11802 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4,x, algorithm="giac")
Output:
1/240*(735*(d*x + c)*a^4 + 2*(735*a^4*tan(1/2*d*x + 1/2*c)^11 + 4165*a^4*t an(1/2*d*x + 1/2*c)^9 + 9702*a^4*tan(1/2*d*x + 1/2*c)^7 + 11802*a^4*tan(1/ 2*d*x + 1/2*c)^5 + 7355*a^4*tan(1/2*d*x + 1/2*c)^3 + 3105*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 12.51 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\frac {49\,a^4\,x}{16}+\frac {\frac {49\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {833\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {1617\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {1967\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {1471\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {207\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:
int(cos(c + d*x)^6*(a + a/cos(c + d*x))^4,x)
Output:
(49*a^4*x)/16 + ((1471*a^4*tan(c/2 + (d*x)/2)^3)/24 + (1967*a^4*tan(c/2 + (d*x)/2)^5)/20 + (1617*a^4*tan(c/2 + (d*x)/2)^7)/20 + (833*a^4*tan(c/2 + ( d*x)/2)^9)/24 + (49*a^4*tan(c/2 + (d*x)/2)^11)/8 + (207*a^4*tan(c/2 + (d*x )/2))/8)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^6)
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx=\frac {a^{4} \left (40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-490 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+1185 \cos \left (d x +c \right ) \sin \left (d x +c \right )+192 \sin \left (d x +c \right )^{5}-960 \sin \left (d x +c \right )^{3}+1920 \sin \left (d x +c \right )+735 d x \right )}{240 d} \] Input:
int(cos(d*x+c)^6*(a+a*sec(d*x+c))^4,x)
Output:
(a**4*(40*cos(c + d*x)*sin(c + d*x)**5 - 490*cos(c + d*x)*sin(c + d*x)**3 + 1185*cos(c + d*x)*sin(c + d*x) + 192*sin(c + d*x)**5 - 960*sin(c + d*x)* *3 + 1920*sin(c + d*x) + 735*d*x))/(240*d)