Integrand size = 21, antiderivative size = 117 \[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {x}{a}+\frac {\cot ^3(c+d x) (35-24 \sec (c+d x))}{105 a d}-\frac {\cot (c+d x) (35-16 \sec (c+d x))}{35 a d}-\frac {\cot ^5(c+d x) (7-6 \sec (c+d x))}{35 a d}+\frac {\cot ^7(c+d x) (1-\sec (c+d x))}{7 a d} \] Output:
-x/a+1/105*cot(d*x+c)^3*(35-24*sec(d*x+c))/a/d-1/35*cot(d*x+c)*(35-16*sec( d*x+c))/a/d-1/35*cot(d*x+c)^5*(7-6*sec(d*x+c))/a/d+1/7*cot(d*x+c)^7*(1-sec (d*x+c))/a/d
Leaf count is larger than twice the leaf count of optimal. \(359\) vs. \(2(117)=234\).
Time = 1.56 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.07 \[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\csc \left (\frac {c}{2}\right ) \csc ^5(c+d x) \sec \left (\frac {c}{2}\right ) \sec (c+d x) (-16800 d x \cos (d x)+16800 d x \cos (2 c+d x)-4200 d x \cos (c+2 d x)+4200 d x \cos (3 c+2 d x)+8400 d x \cos (2 c+3 d x)-8400 d x \cos (4 c+3 d x)+3360 d x \cos (3 c+4 d x)-3360 d x \cos (5 c+4 d x)-1680 d x \cos (4 c+5 d x)+1680 d x \cos (6 c+5 d x)-840 d x \cos (5 c+6 d x)+840 d x \cos (7 c+6 d x)+3136 \sin (c)+30112 \sin (d x)-22860 \sin (c+d x)-5715 \sin (2 (c+d x))+11430 \sin (3 (c+d x))+4572 \sin (4 (c+d x))-2286 \sin (5 (c+d x))-1143 \sin (6 (c+d x))+26208 \sin (2 c+d x)+14080 \sin (c+2 d x)-16400 \sin (2 c+3 d x)-11760 \sin (4 c+3 d x)-7904 \sin (3 c+4 d x)-3360 \sin (5 c+4 d x)+3952 \sin (4 c+5 d x)+1680 \sin (6 c+5 d x)+2816 \sin (5 c+6 d x))}{107520 a d (1+\sec (c+d x))} \] Input:
Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x]),x]
Output:
(Csc[c/2]*Csc[c + d*x]^5*Sec[c/2]*Sec[c + d*x]*(-16800*d*x*Cos[d*x] + 1680 0*d*x*Cos[2*c + d*x] - 4200*d*x*Cos[c + 2*d*x] + 4200*d*x*Cos[3*c + 2*d*x] + 8400*d*x*Cos[2*c + 3*d*x] - 8400*d*x*Cos[4*c + 3*d*x] + 3360*d*x*Cos[3* c + 4*d*x] - 3360*d*x*Cos[5*c + 4*d*x] - 1680*d*x*Cos[4*c + 5*d*x] + 1680* d*x*Cos[6*c + 5*d*x] - 840*d*x*Cos[5*c + 6*d*x] + 840*d*x*Cos[7*c + 6*d*x] + 3136*Sin[c] + 30112*Sin[d*x] - 22860*Sin[c + d*x] - 5715*Sin[2*(c + d*x )] + 11430*Sin[3*(c + d*x)] + 4572*Sin[4*(c + d*x)] - 2286*Sin[5*(c + d*x) ] - 1143*Sin[6*(c + d*x)] + 26208*Sin[2*c + d*x] + 14080*Sin[c + 2*d*x] - 16400*Sin[2*c + 3*d*x] - 11760*Sin[4*c + 3*d*x] - 7904*Sin[3*c + 4*d*x] - 3360*Sin[5*c + 4*d*x] + 3952*Sin[4*c + 5*d*x] + 1680*Sin[6*c + 5*d*x] + 28 16*Sin[5*c + 6*d*x]))/(107520*a*d*(1 + Sec[c + d*x]))
Time = 0.66 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4376, 25, 3042, 4370, 25, 3042, 4370, 25, 3042, 4370, 27, 3042, 4370, 24}
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.
| \(\displaystyle \int \frac {\cot ^6(c+d x)}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int -\cot ^8(c+d x) (a-a \sec (c+d x))dx}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \cot ^8(c+d x) (a-a \sec (c+d x))dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^8}dx}{a^2}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle -\frac {\frac {1}{7} \int -\cot ^6(c+d x) (7 a-6 a \sec (c+d x))dx-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {1}{7} \int \cot ^6(c+d x) (7 a-6 a \sec (c+d x))dx-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{7} \int \frac {7 a-6 a \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^6}dx-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}-\frac {1}{5} \int -\cot ^4(c+d x) (35 a-24 a \sec (c+d x))dx\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \int \cot ^4(c+d x) (35 a-24 a \sec (c+d x))dx+\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \int \frac {35 a-24 a \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int -3 \cot ^2(c+d x) (35 a-16 a \sec (c+d x))dx-\frac {\cot ^3(c+d x) (35 a-24 a \sec (c+d x))}{3 d}\right )+\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (-\int \cot ^2(c+d x) (35 a-16 a \sec (c+d x))dx-\frac {\cot ^3(c+d x) (35 a-24 a \sec (c+d x))}{3 d}\right )+\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (-\int \frac {35 a-16 a \csc \left (c+d x+\frac {\pi }{2}\right )}{\cot \left (c+d x+\frac {\pi }{2}\right )^2}dx-\frac {\cot ^3(c+d x) (35 a-24 a \sec (c+d x))}{3 d}\right )+\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {1}{5} \left (-\int -35 adx-\frac {\cot ^3(c+d x) (35 a-24 a \sec (c+d x))}{3 d}+\frac {\cot (c+d x) (35 a-16 a \sec (c+d x))}{d}\right )+\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\frac {1}{7} \left (\frac {\cot ^5(c+d x) (7 a-6 a \sec (c+d x))}{5 d}+\frac {1}{5} \left (-\frac {\cot ^3(c+d x) (35 a-24 a \sec (c+d x))}{3 d}+\frac {\cot (c+d x) (35 a-16 a \sec (c+d x))}{d}+35 a x\right )\right )-\frac {\cot ^7(c+d x) (a-a \sec (c+d x))}{7 d}}{a^2}\) |
Input:
Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x]),x]
Output:
-((-1/7*(Cot[c + d*x]^7*(a - a*Sec[c + d*x]))/d + ((Cot[c + d*x]^5*(7*a - 6*a*Sec[c + d*x]))/(5*d) + (35*a*x - (Cot[c + d*x]^3*(35*a - 24*a*Sec[c + d*x]))/(3*d) + (Cot[c + d*x]*(35*a - 16*a*Sec[c + d*x]))/d)/5)/7)/a^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-(e*Cot[c + d*x])^(m + 1))*((a + b*Csc[c + d*x])/( d*e*(m + 1))), x] - Simp[1/(e^2*(m + 1)) Int[(e*Cot[c + d*x])^(m + 2)*(a* (m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && L tQ[m, -1]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.95
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {29}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d a}\) | \(111\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {29 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {8}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {29}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d a}\) | \(111\) |
| risch | \(-\frac {x}{a}+\frac {2 i \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}-210 \,{\mathrm e}^{10 i \left (d x +c \right )}-735 \,{\mathrm e}^{9 i \left (d x +c \right )}+1638 \,{\mathrm e}^{7 i \left (d x +c \right )}+196 \,{\mathrm e}^{6 i \left (d x +c \right )}-1882 \,{\mathrm e}^{5 i \left (d x +c \right )}-880 \,{\mathrm e}^{4 i \left (d x +c \right )}+1025 \,{\mathrm e}^{3 i \left (d x +c \right )}+494 \,{\mathrm e}^{2 i \left (d x +c \right )}-247 \,{\mathrm e}^{i \left (d x +c \right )}-176\right )}{105 d a \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\) | \(155\) |
Input:
int(cot(d*x+c)^6/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/64/d/a*(-1/7*tan(1/2*d*x+1/2*c)^7+8/5*tan(1/2*d*x+1/2*c)^5-29/3*tan(1/2* d*x+1/2*c)^3+64*tan(1/2*d*x+1/2*c)-128*arctan(tan(1/2*d*x+1/2*c))-1/5/tan( 1/2*d*x+1/2*c)^5+8/3/tan(1/2*d*x+1/2*c)^3-29/tan(1/2*d*x+1/2*c))
Time = 0.08 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {176 \, \cos \left (d x + c\right )^{6} + 71 \, \cos \left (d x + c\right )^{5} - 335 \, \cos \left (d x + c\right )^{4} - 125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 105 \, {\left (d x \cos \left (d x + c\right )^{5} + d x \cos \left (d x + c\right )^{4} - 2 \, d x \cos \left (d x + c\right )^{3} - 2 \, d x \cos \left (d x + c\right )^{2} + d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) + 57 \, \cos \left (d x + c\right ) - 48}{105 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="fricas")
Output:
-1/105*(176*cos(d*x + c)^6 + 71*cos(d*x + c)^5 - 335*cos(d*x + c)^4 - 125* cos(d*x + c)^3 + 225*cos(d*x + c)^2 + 105*(d*x*cos(d*x + c)^5 + d*x*cos(d* x + c)^4 - 2*d*x*cos(d*x + c)^3 - 2*d*x*cos(d*x + c)^2 + d*x*cos(d*x + c) + d*x)*sin(d*x + c) + 57*cos(d*x + c) - 48)/((a*d*cos(d*x + c)^5 + a*d*cos (d*x + c)^4 - 2*a*d*cos(d*x + c)^3 - 2*a*d*cos(d*x + c)^2 + a*d*cos(d*x + c) + a*d)*sin(d*x + c))
\[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(cot(d*x+c)**6/(a+a*sec(d*x+c)),x)
Output:
Integral(cot(c + d*x)**6/(sec(c + d*x) + 1), x)/a
Time = 0.11 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.51 \[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\frac {\frac {6720 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1015 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {168 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} - \frac {13440 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {7 \, {\left (\frac {40 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {435 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a \sin \left (d x + c\right )^{5}}}{6720 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="maxima")
Output:
1/6720*((6720*sin(d*x + c)/(cos(d*x + c) + 1) - 1015*sin(d*x + c)^3/(cos(d *x + c) + 1)^3 + 168*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 15*sin(d*x + c) ^7/(cos(d*x + c) + 1)^7)/a - 13440*arctan(sin(d*x + c)/(cos(d*x + c) + 1)) /a + 7*(40*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 435*sin(d*x + c)^4/(cos(d *x + c) + 1)^4 - 3)*(cos(d*x + c) + 1)^5/(a*sin(d*x + c)^5))/d
Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {6720 \, {\left (d x + c\right )}}{a} + \frac {7 \, {\left (435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 168 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1015 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6720 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{7}}}{6720 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sec(d*x+c)),x, algorithm="giac")
Output:
-1/6720*(6720*(d*x + c)/a + 7*(435*tan(1/2*d*x + 1/2*c)^4 - 40*tan(1/2*d*x + 1/2*c)^2 + 3)/(a*tan(1/2*d*x + 1/2*c)^5) + (15*a^6*tan(1/2*d*x + 1/2*c) ^7 - 168*a^6*tan(1/2*d*x + 1/2*c)^5 + 1015*a^6*tan(1/2*d*x + 1/2*c)^3 - 67 20*a^6*tan(1/2*d*x + 1/2*c))/a^7)/d
Time = 12.63 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.76 \[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-168\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1015\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3045\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{6720\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \] Input:
int(cot(c + d*x)^6/(a + a/cos(c + d*x)),x)
Output:
-(21*cos(c/2 + (d*x)/2)^12 + 15*sin(c/2 + (d*x)/2)^12 - 168*cos(c/2 + (d*x )/2)^2*sin(c/2 + (d*x)/2)^10 + 1015*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2 )^8 - 6720*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6 + 3045*cos(c/2 + (d*x )/2)^8*sin(c/2 + (d*x)/2)^4 - 280*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2) ^2 + 6720*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5*(c + d*x))/(6720*a*d*c os(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5)
Time = 0.34 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^6(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {-15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+168 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-1015 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+6720 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-6720 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} d x -3045 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+280 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-21}{6720 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a d} \] Input:
int(cot(d*x+c)^6/(a+a*sec(d*x+c)),x)
Output:
( - 15*tan((c + d*x)/2)**12 + 168*tan((c + d*x)/2)**10 - 1015*tan((c + d*x )/2)**8 + 6720*tan((c + d*x)/2)**6 - 6720*tan((c + d*x)/2)**5*d*x - 3045*t an((c + d*x)/2)**4 + 280*tan((c + d*x)/2)**2 - 21)/(6720*tan((c + d*x)/2)* *5*a*d)