Integrand size = 21, antiderivative size = 182 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {85 a^3 x}{16}+\frac {a^3 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {a^3 \sin (c+d x)}{d}+\frac {43 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 a^3 \sin ^3(c+d x)}{3 d}-\frac {3 a^3 \sin ^5(c+d x)}{5 d}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d} \]
-85/16*a^3*x+1/2*a^3*arctanh(sin(d*x+c))/d-a^3*sin(d*x+c)/d+43/16*a^3*cos( d*x+c)*sin(d*x+c)/d-5/24*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-2/3*a^3*sin(d*x+c)^3/d-3/5*a^3*sin(d*x+c)^5/d+3*a^3*tan(d*x +c)/d+1/2*a^3*sec(d*x+c)*tan(d*x+c)/d
Time = 0.57 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.75 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {a^3 \sec ^2(c+d x) \left (10200 c+10200 d x-1920 \text {arctanh}(\sin (c+d x)) \cos ^2(c+d x)+10200 (c+d x) \cos (2 (c+d x))-460 \sin (c+d x)-8145 \sin (2 (c+d x))+1156 \sin (3 (c+d x))-1120 \sin (4 (c+d x))-268 \sin (5 (c+d x))+55 \sin (6 (c+d x))+36 \sin (7 (c+d x))+5 \sin (8 (c+d x))\right )}{3840 d} \]
-1/3840*(a^3*Sec[c + d*x]^2*(10200*c + 10200*d*x - 1920*ArcTanh[Sin[c + d* x]]*Cos[c + d*x]^2 + 10200*(c + d*x)*Cos[2*(c + d*x)] - 460*Sin[c + d*x] - 8145*Sin[2*(c + d*x)] + 1156*Sin[3*(c + d*x)] - 1120*Sin[4*(c + d*x)] - 2 68*Sin[5*(c + d*x)] + 55*Sin[6*(c + d*x)] + 36*Sin[7*(c + d*x)] + 5*Sin[8* (c + d*x)]))/d
Time = 0.59 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4360, 25, 25, 3042, 3351, 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.
| \(\displaystyle \int \sin ^6(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^6 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^3(c+d x) \tan ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \sin ^3(c+d x) \tan ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^3(c+d x) \tan ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos \left (c+d x+\frac {\pi }{2}\right )^6 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (-\cos ^6(c+d x) a^9-3 \cos ^5(c+d x) a^9+8 \cos ^3(c+d x) a^9+\sec ^3(c+d x) a^9+6 \cos ^2(c+d x) a^9+3 \sec ^2(c+d x) a^9-6 \cos (c+d x) a^9-8 a^9\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^9 \text {arctanh}(\sin (c+d x))}{2 d}-\frac {3 a^9 \sin ^5(c+d x)}{5 d}-\frac {2 a^9 \sin ^3(c+d x)}{3 d}-\frac {a^9 \sin (c+d x)}{d}+\frac {3 a^9 \tan (c+d x)}{d}-\frac {a^9 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {5 a^9 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {43 a^9 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {a^9 \tan (c+d x) \sec (c+d x)}{2 d}-\frac {85 a^9 x}{16}}{a^6}\) |
((-85*a^9*x)/16 + (a^9*ArcTanh[Sin[c + d*x]])/(2*d) - (a^9*Sin[c + d*x])/d + (43*a^9*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (5*a^9*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (a^9*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (2*a^9*Sin[c + d*x]^3)/(3*d) - (3*a^9*Sin[c + d*x]^5)/(5*d) + (3*a^9*Tan[c + d*x])/d + ( a^9*Sec[c + d*x]*Tan[c + d*x])/(2*d))/a^6
3.1.49.3.1 Defintions of rubi rules used
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[1/a^p Int[Expan dTrig[(d*sin[e + f*x])^n*(a - b*sin[e + f*x])^(p/2)*(a + b*sin[e + f*x])^(m + p/2), x], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && In tegersQ[m, n, p/2] && ((GtQ[m, 0] && GtQ[p, 0] && LtQ[-m - p, n, -1]) || (G tQ[m, 2] && LtQ[p, 0] && GtQ[m + p/2, 0]))
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Time = 2.52 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.09
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (-10200 d x \cos \left (2 d x +2 c \right )-960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \cos \left (2 d x +2 c \right )+960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \cos \left (2 d x +2 c \right )-10200 d x +460 \sin \left (d x +c \right )+8145 \sin \left (2 d x +2 c \right )+1120 \sin \left (4 d x +4 c \right )-55 \sin \left (6 d x +6 c \right )-5 \sin \left (8 d x +8 c \right )-36 \sin \left (7 d x +7 c \right )+268 \sin \left (5 d x +5 c \right )-1156 \sin \left (3 d x +3 c \right )-960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+960 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )\right )}{1920 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(199\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(232\) |
| default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(232\) |
| parts | \(\frac {a^{3} \left (-\frac {\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{5}}{2}+\frac {5 \sin \left (d x +c \right )^{3}}{6}+\frac {5 \sin \left (d x +c \right )}{2}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}-\frac {\sin \left (d x +c \right )^{3}}{3}-\sin \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{7}}{\cos \left (d x +c \right )}+\left (\sin \left (d x +c \right )^{5}+\frac {5 \sin \left (d x +c \right )^{3}}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )-\frac {15 d x}{8}-\frac {15 c}{8}\right )}{d}\) | \(240\) |
| risch | \(-\frac {85 a^{3} x}{16}+\frac {17 i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{96 d}+\frac {81 i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {81 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {15 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {i a^{3} \left ({\mathrm e}^{3 i \left (d x +c \right )}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}-6\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {17 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{96 d}-\frac {15 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}-\frac {3 a^{3} \sin \left (5 d x +5 c \right )}{80 d}-\frac {3 a^{3} \sin \left (4 d x +4 c \right )}{64 d}\) | \(264\) |
| norman | \(\frac {-\frac {85 a^{3} x}{16}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {425 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{4}-\frac {85 a^{3} x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{16}+\frac {77 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {277 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 d}+\frac {997 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40 d}+\frac {3933 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{40 d}+\frac {6169 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}-\frac {4319 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{120 d}-\frac {1039 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{24 d}-\frac {93 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{8 d}}{\left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2} \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(366\) |
1/1920*a^3*(-10200*d*x*cos(2*d*x+2*c)-960*ln(tan(1/2*d*x+1/2*c)-1)*cos(2*d *x+2*c)+960*ln(tan(1/2*d*x+1/2*c)+1)*cos(2*d*x+2*c)-10200*d*x+460*sin(d*x+ c)+8145*sin(2*d*x+2*c)+1120*sin(4*d*x+4*c)-55*sin(6*d*x+6*c)-5*sin(8*d*x+8 *c)-36*sin(7*d*x+7*c)+268*sin(5*d*x+5*c)-1156*sin(3*d*x+3*c)-960*ln(tan(1/ 2*d*x+1/2*c)-1)+960*ln(tan(1/2*d*x+1/2*c)+1))/d/(1+cos(2*d*x+2*c))
Time = 0.30 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.97 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {1275 \, a^{3} d x \cos \left (d x + c\right )^{2} - 60 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 60 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (40 \, a^{3} \cos \left (d x + c\right )^{7} + 144 \, a^{3} \cos \left (d x + c\right )^{6} + 50 \, a^{3} \cos \left (d x + c\right )^{5} - 448 \, a^{3} \cos \left (d x + c\right )^{4} - 645 \, a^{3} \cos \left (d x + c\right )^{3} + 544 \, a^{3} \cos \left (d x + c\right )^{2} - 720 \, a^{3} \cos \left (d x + c\right ) - 120 \, a^{3}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{2}} \]
-1/240*(1275*a^3*d*x*cos(d*x + c)^2 - 60*a^3*cos(d*x + c)^2*log(sin(d*x + c) + 1) + 60*a^3*cos(d*x + c)^2*log(-sin(d*x + c) + 1) + (40*a^3*cos(d*x + c)^7 + 144*a^3*cos(d*x + c)^6 + 50*a^3*cos(d*x + c)^5 - 448*a^3*cos(d*x + c)^4 - 645*a^3*cos(d*x + c)^3 + 544*a^3*cos(d*x + c)^2 - 720*a^3*cos(d*x + c) - 120*a^3)*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.32 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {96 \, {\left (6 \, \sin \left (d x + c\right )^{5} + 10 \, \sin \left (d x + c\right )^{3} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 30 \, \sin \left (d x + c\right )\right )} a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 80 \, {\left (4 \, \sin \left (d x + c\right )^{3} - \frac {6 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 24 \, \sin \left (d x + c\right )\right )} a^{3} + 360 \, {\left (15 \, d x + 15 \, c - \frac {9 \, \tan \left (d x + c\right )^{3} + 7 \, \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1} - 8 \, \tan \left (d x + c\right )\right )} a^{3}}{960 \, d} \]
-1/960*(96*(6*sin(d*x + c)^5 + 10*sin(d*x + c)^3 - 15*log(sin(d*x + c) + 1 ) + 15*log(sin(d*x + c) - 1) + 30*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 - 80*( 4*sin(d*x + c)^3 - 6*sin(d*x + c)/(sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1) + 24*sin(d*x + c))*a^3 + 360*(15*d*x + 15*c - (9*tan(d*x + c)^3 + 7*tan(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c )^2 + 1) - 8*tan(d*x + c))*a^3)/d
Time = 0.42 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.16 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=-\frac {1275 \, {\left (d x + c\right )} a^{3} - 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) + 120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {240 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} + \frac {2 \, {\left (795 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4025 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7614 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5634 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 345 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
-1/240*(1275*(d*x + c)*a^3 - 120*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) + 120*a^3*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 240*(5*a^3*tan(1/2*d*x + 1/2* c)^3 - 7*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^2 + 2*(795 *a^3*tan(1/2*d*x + 1/2*c)^11 + 4025*a^3*tan(1/2*d*x + 1/2*c)^9 + 7614*a^3* tan(1/2*d*x + 1/2*c)^7 + 5634*a^3*tan(1/2*d*x + 1/2*c)^5 - 345*a^3*tan(1/2 *d*x + 1/2*c)^3 - 315*a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 14.53 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.43 \[ \int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx=\frac {a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {85\,a^3\,x}{16}+\frac {-\frac {93\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{8}-\frac {1039\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{24}-\frac {4319\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{120}+\frac {6169\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{120}+\frac {3933\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{40}+\frac {997\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{40}+\frac {277\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8}+\frac {77\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
(a^3*atanh(tan(c/2 + (d*x)/2)))/d - (85*a^3*x)/16 + ((277*a^3*tan(c/2 + (d *x)/2)^3)/8 + (997*a^3*tan(c/2 + (d*x)/2)^5)/40 + (3933*a^3*tan(c/2 + (d*x )/2)^7)/40 + (6169*a^3*tan(c/2 + (d*x)/2)^9)/120 - (4319*a^3*tan(c/2 + (d* x)/2)^11)/120 - (1039*a^3*tan(c/2 + (d*x)/2)^13)/24 - (93*a^3*tan(c/2 + (d *x)/2)^15)/8 + (77*a^3*tan(c/2 + (d*x)/2))/8)/(d*(4*tan(c/2 + (d*x)/2)^2 + 4*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^6 - 10*tan(c/2 + (d*x)/2)^8 - 4*tan(c/2 + (d*x)/2)^10 + 4*tan(c/2 + (d*x)/2)^12 + 4*tan(c/2 + (d*x)/2 )^14 + tan(c/2 + (d*x)/2)^16 + 1))