\(\int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx\) [599]

Optimal result

Integrand size = 29, antiderivative size = 162 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-a^2 x+\frac {5 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d} \] Output:

-a^2*x+5/8*a^2*arctanh(cos(d*x+c))/d-a^2*cot(d*x+c)/d+1/3*a^2*cot(d*x+c)^3 
/d-1/5*a^2*cot(d*x+c)^5/d-1/7*a^2*cot(d*x+c)^7/d-5/8*a^2*cot(d*x+c)*csc(d* 
x+c)/d+5/12*a^2*cot(d*x+c)^3*csc(d*x+c)/d-1/3*a^2*cot(d*x+c)^5*csc(d*x+c)/ 
d
 

Mathematica [A] (verified)

Time = 1.86 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.62 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 \left (-13440 c-13440 d x-9344 \cot \left (\frac {1}{2} (c+d x)\right )-4620 \csc ^2\left (\frac {1}{2} (c+d x)\right )+8400 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8400 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4620 \sec ^2\left (\frac {1}{2} (c+d x)\right )-840 \sec ^4\left (\frac {1}{2} (c+d x)\right )+70 \sec ^6\left (\frac {1}{2} (c+d x)\right )-4624 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-\frac {15}{2} \csc ^8\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (-70+33 \sin (c+d x))+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (840+289 \sin (c+d x))+9344 \tan \left (\frac {1}{2} (c+d x)\right )-66 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )+15 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{13440 d} \] Input:

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]
 

Output:

(a^2*(-13440*c - 13440*d*x - 9344*Cot[(c + d*x)/2] - 4620*Csc[(c + d*x)/2] 
^2 + 8400*Log[Cos[(c + d*x)/2]] - 8400*Log[Sin[(c + d*x)/2]] + 4620*Sec[(c 
 + d*x)/2]^2 - 840*Sec[(c + d*x)/2]^4 + 70*Sec[(c + d*x)/2]^6 - 4624*Csc[c 
 + d*x]^3*Sin[(c + d*x)/2]^4 - (15*Csc[(c + d*x)/2]^8*Sin[c + d*x])/2 + Cs 
c[(c + d*x)/2]^6*(-70 + 33*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(840 + 289*S 
in[c + d*x]) + 9344*Tan[(c + d*x)/2] - 66*Sec[(c + d*x)/2]^4*Tan[(c + d*x) 
/2] + 15*Sec[(c + d*x)/2]^6*Tan[(c + d*x)/2]))/(13440*d)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {3042, 3352, 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.

\(\displaystyle \int \cot ^6(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a)^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^2}{\sin (c+d x)^8}dx\)

\(\Big \downarrow \) 3352

\(\displaystyle \int \left (a^2 \cot ^6(c+d x)+a^2 \cot ^6(c+d x) \csc ^2(c+d x)+2 a^2 \cot ^6(c+d x) \csc (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 a^2 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{3 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{12 d}-\frac {5 a^2 \cot (c+d x) \csc (c+d x)}{8 d}-a^2 x\)

Input:

Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]
 

Output:

-(a^2*x) + (5*a^2*ArcTanh[Cos[c + d*x]])/(8*d) - (a^2*Cot[c + d*x])/d + (a 
^2*Cot[c + d*x]^3)/(3*d) - (a^2*Cot[c + d*x]^5)/(5*d) - (a^2*Cot[c + d*x]^ 
7)/(7*d) - (5*a^2*Cot[c + d*x]*Csc[c + d*x])/(8*d) + (5*a^2*Cot[c + d*x]^3 
*Csc[c + d*x])/(12*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(3*d)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3352

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig 
[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F 
reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(173\)
default	\(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\)	\(173\)
risch	\(-a^{2} x +\frac {a^{2} \left (-1680 i {\mathrm e}^{12 i \left (d x +c \right )}+1155 \,{\mathrm e}^{13 i \left (d x +c \right )}+10080 i {\mathrm e}^{10 i \left (d x +c \right )}-980 \,{\mathrm e}^{11 i \left (d x +c \right )}-16240 i {\mathrm e}^{8 i \left (d x +c \right )}+2975 \,{\mathrm e}^{9 i \left (d x +c \right )}+24640 i {\mathrm e}^{6 i \left (d x +c \right )}-14448 i {\mathrm e}^{4 i \left (d x +c \right )}-2975 \,{\mathrm e}^{5 i \left (d x +c \right )}+6496 i {\mathrm e}^{2 i \left (d x +c \right )}+980 \,{\mathrm e}^{3 i \left (d x +c \right )}-1168 i-1155 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{420 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}-\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {5 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}\)	\(210\)

Input:

int(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^2*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+2*a^2*(-1/6 
/sin(d*x+c)^6*cos(d*x+c)^7+1/24/sin(d*x+c)^4*cos(d*x+c)^7-1/16/sin(d*x+c)^ 
2*cos(d*x+c)^7-1/16*cos(d*x+c)^5-5/48*cos(d*x+c)^3-5/16*cos(d*x+c)-5/16*ln 
(csc(d*x+c)-cot(d*x+c)))-1/7*a^2/sin(d*x+c)^7*cos(d*x+c)^7)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (148) = 296\).

Time = 0.10 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.99 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2336 \, a^{2} \cos \left (d x + c\right )^{7} - 6496 \, a^{2} \cos \left (d x + c\right )^{5} + 5600 \, a^{2} \cos \left (d x + c\right )^{3} - 1680 \, a^{2} \cos \left (d x + c\right ) - 525 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 525 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (24 \, a^{2} d x \cos \left (d x + c\right )^{6} - 72 \, a^{2} d x \cos \left (d x + c\right )^{4} - 33 \, a^{2} \cos \left (d x + c\right )^{5} + 72 \, a^{2} d x \cos \left (d x + c\right )^{2} + 40 \, a^{2} \cos \left (d x + c\right )^{3} - 24 \, a^{2} d x - 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:

integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x, algorithm="frica 
s")
 

Output:

-1/1680*(2336*a^2*cos(d*x + c)^7 - 6496*a^2*cos(d*x + c)^5 + 5600*a^2*cos( 
d*x + c)^3 - 1680*a^2*cos(d*x + c) - 525*(a^2*cos(d*x + c)^6 - 3*a^2*cos(d 
*x + c)^4 + 3*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d* 
x + c) + 525*(a^2*cos(d*x + c)^6 - 3*a^2*cos(d*x + c)^4 + 3*a^2*cos(d*x + 
c)^2 - a^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 70*(24*a^2*d*x*cos 
(d*x + c)^6 - 72*a^2*d*x*cos(d*x + c)^4 - 33*a^2*cos(d*x + c)^5 + 72*a^2*d 
*x*cos(d*x + c)^2 + 40*a^2*cos(d*x + c)^3 - 24*a^2*d*x - 15*a^2*cos(d*x + 
c))*sin(d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + 
c)^2 - d)*sin(d*x + c))
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**6*csc(d*x+c)**2*(a+a*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.95 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {240 \, a^{2} \cot \left (d x + c\right )^{7} + 112 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 35 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{1680 \, d} \] Input:

integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x, algorithm="maxim 
a")
 

Output:

-1/1680*(240*a^2*cot(d*x + c)^7 + 112*(15*d*x + 15*c + (15*tan(d*x + c)^4 
- 5*tan(d*x + c)^2 + 3)/tan(d*x + c)^5)*a^2 - 35*a^2*(2*(33*cos(d*x + c)^5 
 - 40*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 
 + 3*cos(d*x + c)^2 - 1) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) 
- 1)))/d
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.67 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 665 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3150 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 13440 \, {\left (d x + c\right )} a^{2} - 8400 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 8715 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {21780 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8715 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3150 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 665 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 70 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \] Input:

integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x, algorithm="giac" 
)
 

Output:

1/13440*(15*a^2*tan(1/2*d*x + 1/2*c)^7 + 70*a^2*tan(1/2*d*x + 1/2*c)^6 - 2 
1*a^2*tan(1/2*d*x + 1/2*c)^5 - 630*a^2*tan(1/2*d*x + 1/2*c)^4 - 665*a^2*ta 
n(1/2*d*x + 1/2*c)^3 + 3150*a^2*tan(1/2*d*x + 1/2*c)^2 - 13440*(d*x + c)*a 
^2 - 8400*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 8715*a^2*tan(1/2*d*x + 1/2* 
c) + (21780*a^2*tan(1/2*d*x + 1/2*c)^7 - 8715*a^2*tan(1/2*d*x + 1/2*c)^6 - 
 3150*a^2*tan(1/2*d*x + 1/2*c)^5 + 665*a^2*tan(1/2*d*x + 1/2*c)^4 + 630*a^ 
2*tan(1/2*d*x + 1/2*c)^3 + 21*a^2*tan(1/2*d*x + 1/2*c)^2 - 70*a^2*tan(1/2* 
d*x + 1/2*c) - 15*a^2)/tan(1/2*d*x + 1/2*c)^7)/d
 

Mupad [B] (verification not implemented)

Time = 33.80 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.17 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {19\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {15\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}-\frac {19\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {5\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{8\,d}-\frac {83\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {83\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \] Input:

int((cot(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^2,x)
 

Output:

(19*a^2*cot(c/2 + (d*x)/2)^3)/(384*d) - (15*a^2*cot(c/2 + (d*x)/2)^2)/(64* 
d) + (3*a^2*cot(c/2 + (d*x)/2)^4)/(64*d) + (a^2*cot(c/2 + (d*x)/2)^5)/(640 
*d) - (a^2*cot(c/2 + (d*x)/2)^6)/(192*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(896 
*d) + (15*a^2*tan(c/2 + (d*x)/2)^2)/(64*d) - (19*a^2*tan(c/2 + (d*x)/2)^3) 
/(384*d) - (3*a^2*tan(c/2 + (d*x)/2)^4)/(64*d) - (a^2*tan(c/2 + (d*x)/2)^5 
)/(640*d) + (a^2*tan(c/2 + (d*x)/2)^6)/(192*d) + (a^2*tan(c/2 + (d*x)/2)^7 
)/(896*d) - (2*a^2*atan((8*cos(c/2 + (d*x)/2) + 5*sin(c/2 + (d*x)/2))/(5*c 
os(c/2 + (d*x)/2) - 8*sin(c/2 + (d*x)/2))))/d - (5*a^2*log(sin(c/2 + (d*x) 
/2)/cos(c/2 + (d*x)/2)))/(8*d) - (83*a^2*cot(c/2 + (d*x)/2))/(128*d) + (83 
*a^2*tan(c/2 + (d*x)/2))/(128*d)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-1168 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-1155 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+910 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-280 \cos \left (d x +c \right ) \sin \left (d x +c \right )-120 \cos \left (d x +c \right )-525 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{7}-840 \sin \left (d x +c \right )^{7} d x \right )}{840 \sin \left (d x +c \right )^{7} d} \] Input:

int(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c))^2,x)
 

Output:

(a**2*( - 1168*cos(c + d*x)*sin(c + d*x)**6 - 1155*cos(c + d*x)*sin(c + d* 
x)**5 + 256*cos(c + d*x)*sin(c + d*x)**4 + 910*cos(c + d*x)*sin(c + d*x)** 
3 + 192*cos(c + d*x)*sin(c + d*x)**2 - 280*cos(c + d*x)*sin(c + d*x) - 120 
*cos(c + d*x) - 525*log(tan((c + d*x)/2))*sin(c + d*x)**7 - 840*sin(c + d* 
x)**7*d*x))/(840*sin(c + d*x)**7*d)