\(\int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx\) [391]

Optimal result

Integrand size = 29, antiderivative size = 218 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}-\frac {3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d} \] Output:

-9/256*a^2*arctanh(cos(d*x+c))/d-2/5*a^2*cot(d*x+c)^5/d-4/7*a^2*cot(d*x+c) 
^7/d-2/9*a^2*cot(d*x+c)^9/d-9/256*a^2*cot(d*x+c)*csc(d*x+c)/d-3/128*a^2*co 
t(d*x+c)*csc(d*x+c)^3/d+9/160*a^2*cot(d*x+c)*csc(d*x+c)^5/d-1/8*a^2*cot(d* 
x+c)^3*csc(d*x+c)^5/d+3/80*a^2*cot(d*x+c)*csc(d*x+c)^7/d-1/10*a^2*cot(d*x+ 
c)^3*csc(d*x+c)^7/d
 

Mathematica [A] (verified)

Time = 7.00 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.62 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \csc ^{10}(c+d x) \left (3219300 \cos (c+d x)+1237320 \cos (3 (c+d x))-278712 \cos (5 (c+d x))-54810 \cos (7 (c+d x))+5670 \cos (9 (c+d x))+357210 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-595350 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+340200 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-127575 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+28350 \cos (8 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-2835 \cos (10 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-357210 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+595350 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-340200 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+127575 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-28350 \cos (8 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+2835 \cos (10 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+1720320 \sin (2 (c+d x))+1228800 \sin (4 (c+d x))+184320 \sin (6 (c+d x))-40960 \sin (8 (c+d x))+4096 \sin (10 (c+d x))\right )}{41287680 d} \] Input:

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]
 

Output:

-1/41287680*(a^2*Csc[c + d*x]^10*(3219300*Cos[c + d*x] + 1237320*Cos[3*(c 
+ d*x)] - 278712*Cos[5*(c + d*x)] - 54810*Cos[7*(c + d*x)] + 5670*Cos[9*(c 
 + d*x)] + 357210*Log[Cos[(c + d*x)/2]] - 595350*Cos[2*(c + d*x)]*Log[Cos[ 
(c + d*x)/2]] + 340200*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 127575*Cos 
[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 28350*Cos[8*(c + d*x)]*Log[Cos[(c + 
d*x)/2]] - 2835*Cos[10*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 357210*Log[Sin[( 
c + d*x)/2]] + 595350*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 340200*Cos[ 
4*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 127575*Cos[6*(c + d*x)]*Log[Sin[(c + 
d*x)/2]] - 28350*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 2835*Cos[10*(c + 
 d*x)]*Log[Sin[(c + d*x)/2]] + 1720320*Sin[2*(c + d*x)] + 1228800*Sin[4*(c 
 + d*x)] + 184320*Sin[6*(c + d*x)] - 40960*Sin[8*(c + d*x)] + 4096*Sin[10* 
(c + d*x)]))/d
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 218, 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.

\(\displaystyle \int \cot ^4(c+d x) \csc ^7(c+d x) (a \sin (c+d x)+a)^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^2}{\sin (c+d x)^{11}}dx\)

\(\Big \downarrow \) 3352

\(\displaystyle \int \left (a^2 \cot ^4(c+d x) \csc ^7(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^6(c+d x)+a^2 \cot ^4(c+d x) \csc ^5(c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {9 a^2 \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {4 a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x) \csc ^7(c+d x)}{10 d}-\frac {a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac {3 a^2 \cot (c+d x) \csc ^7(c+d x)}{80 d}+\frac {9 a^2 \cot (c+d x) \csc ^5(c+d x)}{160 d}-\frac {3 a^2 \cot (c+d x) \csc ^3(c+d x)}{128 d}-\frac {9 a^2 \cot (c+d x) \csc (c+d x)}{256 d}\)

Input:

Int[Cot[c + d*x]^4*Csc[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]
 

Output:

(-9*a^2*ArcTanh[Cos[c + d*x]])/(256*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (4 
*a^2*Cot[c + d*x]^7)/(7*d) - (2*a^2*Cot[c + d*x]^9)/(9*d) - (9*a^2*Cot[c + 
 d*x]*Csc[c + d*x])/(256*d) - (3*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(128*d) 
+ (9*a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(160*d) - (a^2*Cot[c + d*x]^3*Csc[c 
+ d*x]^5)/(8*d) + (3*a^2*Cot[c + d*x]*Csc[c + d*x]^7)/(80*d) - (a^2*Cot[c 
+ d*x]^3*Csc[c + d*x]^7)/(10*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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.96 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {a^{2} \left (2835 \,{\mathrm e}^{19 i \left (d x +c \right )}-27405 \,{\mathrm e}^{17 i \left (d x +c \right )}-139356 \,{\mathrm e}^{15 i \left (d x +c \right )}+618660 \,{\mathrm e}^{13 i \left (d x +c \right )}+1290240 i {\mathrm e}^{8 i \left (d x +c \right )}+1609650 \,{\mathrm e}^{11 i \left (d x +c \right )}-860160 i {\mathrm e}^{14 i \left (d x +c \right )}+1609650 \,{\mathrm e}^{9 i \left (d x +c \right )}+368640 i {\mathrm e}^{6 i \left (d x +c \right )}+618660 \,{\mathrm e}^{7 i \left (d x +c \right )}-430080 i {\mathrm e}^{12 i \left (d x +c \right )}-139356 \,{\mathrm e}^{5 i \left (d x +c \right )}-40960 i {\mathrm e}^{2 i \left (d x +c \right )}-27405 \,{\mathrm e}^{3 i \left (d x +c \right )}-516096 i {\mathrm e}^{10 i \left (d x +c \right )}+2835 \,{\mathrm e}^{i \left (d x +c \right )}+184320 i {\mathrm e}^{4 i \left (d x +c \right )}+4096 i\right )}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}-\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}\)	\(260\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{256 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(310\)
default	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{9 \sin \left (d x +c \right )^{9}}-\frac {4 \cos \left (d x +c \right )^{5}}{63 \sin \left (d x +c \right )^{7}}-\frac {8 \cos \left (d x +c \right )^{5}}{315 \sin \left (d x +c \right )^{5}}\right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{5}}{10 \sin \left (d x +c \right )^{10}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{32 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{256 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{256}+\frac {3 \cos \left (d x +c \right )}{256}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )}{d}\)	\(310\)

Input:

int(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/40320*a^2*(2835*exp(19*I*(d*x+c))-27405*exp(17*I*(d*x+c))-139356*exp(15* 
I*(d*x+c))+618660*exp(13*I*(d*x+c))+1290240*I*exp(8*I*(d*x+c))+1609650*exp 
(11*I*(d*x+c))-860160*I*exp(14*I*(d*x+c))+1609650*exp(9*I*(d*x+c))+368640* 
I*exp(6*I*(d*x+c))+618660*exp(7*I*(d*x+c))-430080*I*exp(12*I*(d*x+c))-1393 
56*exp(5*I*(d*x+c))-40960*I*exp(2*I*(d*x+c))-27405*exp(3*I*(d*x+c))-516096 
*I*exp(10*I*(d*x+c))+2835*exp(I*(d*x+c))+184320*I*exp(4*I*(d*x+c))+4096*I) 
/d/(exp(2*I*(d*x+c))-1)^10+9/256*a^2/d*ln(exp(I*(d*x+c))-1)-9/256*a^2/d*ln 
(exp(I*(d*x+c))+1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5670 \, a^{2} \cos \left (d x + c\right )^{9} - 26460 \, a^{2} \cos \left (d x + c\right )^{7} + 16128 \, a^{2} \cos \left (d x + c\right )^{5} + 26460 \, a^{2} \cos \left (d x + c\right )^{3} - 5670 \, a^{2} \cos \left (d x + c\right ) - 2835 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2835 \, {\left (a^{2} \cos \left (d x + c\right )^{10} - 5 \, a^{2} \cos \left (d x + c\right )^{8} + 10 \, a^{2} \cos \left (d x + c\right )^{6} - 10 \, a^{2} \cos \left (d x + c\right )^{4} + 5 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1024 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{9} - 36 \, a^{2} \cos \left (d x + c\right )^{7} + 63 \, a^{2} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:

integrate(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="frica 
s")
 

Output:

1/161280*(5670*a^2*cos(d*x + c)^9 - 26460*a^2*cos(d*x + c)^7 + 16128*a^2*c 
os(d*x + c)^5 + 26460*a^2*cos(d*x + c)^3 - 5670*a^2*cos(d*x + c) - 2835*(a 
^2*cos(d*x + c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 - 10*a^2 
*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(1/2*cos(d*x + c) + 1/2) 
+ 2835*(a^2*cos(d*x + c)^10 - 5*a^2*cos(d*x + c)^8 + 10*a^2*cos(d*x + c)^6 
 - 10*a^2*cos(d*x + c)^4 + 5*a^2*cos(d*x + c)^2 - a^2)*log(-1/2*cos(d*x + 
c) + 1/2) + 1024*(8*a^2*cos(d*x + c)^9 - 36*a^2*cos(d*x + c)^7 + 63*a^2*co 
s(d*x + c)^5)*sin(d*x + c))/(d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d 
*cos(d*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**4*csc(d*x+c)**7*(a+a*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.30 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} + 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 630 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1024 \, {\left (63 \, \tan \left (d x + c\right )^{4} + 90 \, \tan \left (d x + c\right )^{2} + 35\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \] Input:

integrate(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="maxim 
a")
 

Output:

1/161280*(63*a^2*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 + 128*cos(d*x + 
 c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))/(cos(d*x + c)^10 - 5*cos(d*x 
+ c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1) - 1 
5*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 630*a^2*(2*(3*cos(d* 
x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*cos(d*x + c))/(cos(d* 
x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 3 
*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1024*(63*tan(d*x + c)^ 
4 + 90*tan(d*x + c)^2 + 35)*a^2/tan(d*x + c)^9)/d
 

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.64 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {126 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 945 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4032 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45360 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {132858 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 30240 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1260 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 6720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 7560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 4032 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 630 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 720 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 945 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 560 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 126 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10}}}{1290240 \, d} \] Input:

integrate(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x, algorithm="giac" 
)
 

Output:

1/1290240*(126*a^2*tan(1/2*d*x + 1/2*c)^10 + 560*a^2*tan(1/2*d*x + 1/2*c)^ 
9 + 945*a^2*tan(1/2*d*x + 1/2*c)^8 + 720*a^2*tan(1/2*d*x + 1/2*c)^7 - 630* 
a^2*tan(1/2*d*x + 1/2*c)^6 - 4032*a^2*tan(1/2*d*x + 1/2*c)^5 - 7560*a^2*ta 
n(1/2*d*x + 1/2*c)^4 - 6720*a^2*tan(1/2*d*x + 1/2*c)^3 + 1260*a^2*tan(1/2* 
d*x + 1/2*c)^2 + 45360*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 30240*a^2*tan( 
1/2*d*x + 1/2*c) - (132858*a^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a^2*tan(1/2 
*d*x + 1/2*c)^9 + 1260*a^2*tan(1/2*d*x + 1/2*c)^8 - 6720*a^2*tan(1/2*d*x + 
 1/2*c)^7 - 7560*a^2*tan(1/2*d*x + 1/2*c)^6 - 4032*a^2*tan(1/2*d*x + 1/2*c 
)^5 - 630*a^2*tan(1/2*d*x + 1/2*c)^4 + 720*a^2*tan(1/2*d*x + 1/2*c)^3 + 94 
5*a^2*tan(1/2*d*x + 1/2*c)^2 + 560*a^2*tan(1/2*d*x + 1/2*c) + 126*a^2)/tan 
(1/2*d*x + 1/2*c)^10)/d
 

Mupad [B] (verification not implemented)

Time = 18.53 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.81 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}+\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}+\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}-\frac {3\,a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{1024\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192\,d}-\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{512\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2048\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {3\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{4096\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}+\frac {9\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{256\,d}-\frac {3\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {3\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \] Input:

int((cot(c + d*x)^4*(a + a*sin(c + d*x))^2)/sin(c + d*x)^7,x)
 

Output:

(a^2*cot(c/2 + (d*x)/2)^3)/(192*d) - (a^2*cot(c/2 + (d*x)/2)^2)/(1024*d) + 
 (3*a^2*cot(c/2 + (d*x)/2)^4)/(512*d) + (a^2*cot(c/2 + (d*x)/2)^5)/(320*d) 
 + (a^2*cot(c/2 + (d*x)/2)^6)/(2048*d) - (a^2*cot(c/2 + (d*x)/2)^7)/(1792* 
d) - (3*a^2*cot(c/2 + (d*x)/2)^8)/(4096*d) - (a^2*cot(c/2 + (d*x)/2)^9)/(2 
304*d) - (a^2*cot(c/2 + (d*x)/2)^10)/(10240*d) + (a^2*tan(c/2 + (d*x)/2)^2 
)/(1024*d) - (a^2*tan(c/2 + (d*x)/2)^3)/(192*d) - (3*a^2*tan(c/2 + (d*x)/2 
)^4)/(512*d) - (a^2*tan(c/2 + (d*x)/2)^5)/(320*d) - (a^2*tan(c/2 + (d*x)/2 
)^6)/(2048*d) + (a^2*tan(c/2 + (d*x)/2)^7)/(1792*d) + (3*a^2*tan(c/2 + (d* 
x)/2)^8)/(4096*d) + (a^2*tan(c/2 + (d*x)/2)^9)/(2304*d) + (a^2*tan(c/2 + ( 
d*x)/2)^10)/(10240*d) + (9*a^2*log(tan(c/2 + (d*x)/2)))/(256*d) - (3*a^2*c 
ot(c/2 + (d*x)/2))/(128*d) + (3*a^2*tan(c/2 + (d*x)/2))/(128*d)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.86 \[ \int \cot ^4(c+d x) \csc ^7(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (-4096 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9}-2835 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-2048 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-1890 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-1536 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+14616 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+25600 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-17920 \cos \left (d x +c \right ) \sin \left (d x +c \right )-8064 \cos \left (d x +c \right )+2835 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{10}\right )}{80640 \sin \left (d x +c \right )^{10} d} \] Input:

int(cot(d*x+c)^4*csc(d*x+c)^7*(a+a*sin(d*x+c))^2,x)
 

Output:

(a**2*( - 4096*cos(c + d*x)*sin(c + d*x)**9 - 2835*cos(c + d*x)*sin(c + d* 
x)**8 - 2048*cos(c + d*x)*sin(c + d*x)**7 - 1890*cos(c + d*x)*sin(c + d*x) 
**6 - 1536*cos(c + d*x)*sin(c + d*x)**5 + 14616*cos(c + d*x)*sin(c + d*x)* 
*4 + 25600*cos(c + d*x)*sin(c + d*x)**3 + 1008*cos(c + d*x)*sin(c + d*x)** 
2 - 17920*cos(c + d*x)*sin(c + d*x) - 8064*cos(c + d*x) + 2835*log(tan((c 
+ d*x)/2))*sin(c + d*x)**10))/(80640*sin(c + d*x)**10*d)