Integrand size = 27, antiderivative size = 157 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-2 a^2 x-\frac {25 a^2 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^2 \cos (c+d x)}{d}-\frac {2 a^2 \cot (c+d x)}{d}+\frac {2 a^2 \cot ^3(c+d x)}{3 d}-\frac {2 a^2 \cot ^5(c+d x)}{5 d}+\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac {7 a^2 \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a^2 \cot (c+d x) \csc ^5(c+d x)}{6 d} \] Output:
-2*a^2*x-25/16*a^2*arctanh(cos(d*x+c))/d+a^2*cos(d*x+c)/d-2*a^2*cot(d*x+c) /d+2/3*a^2*cot(d*x+c)^3/d-2/5*a^2*cot(d*x+c)^5/d+7/16*a^2*cot(d*x+c)*csc(d *x+c)/d+7/24*a^2*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^2*cot(d*x+c)*csc(d*x+c)^5 /d
Time = 8.47 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.72 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \left (-1920 \cot (c+d x)+\csc ^2\left (\frac {1}{2} (c+d x)\right ) (1472-210 \csc (c+d x))+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (12+5 \csc (c+d x))-2 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (82+15 \csc (c+d x))+120 \csc (c+d x) \left (32 (c+d x)+25 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-25 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-2 (241+327 \cos (c+d x)+92 \cos (2 (c+d x))) \sec ^6\left (\frac {1}{2} (c+d x)\right )+840 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )+480 \csc ^5(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-320 \csc ^7(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x) (1+\sin (c+d x))^2}{1920 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]
Output:
-1/1920*(a^2*(-1920*Cot[c + d*x] + Csc[(c + d*x)/2]^2*(1472 - 210*Csc[c + d*x]) + Csc[(c + d*x)/2]^6*(12 + 5*Csc[c + d*x]) - 2*Csc[(c + d*x)/2]^4*(8 2 + 15*Csc[c + d*x]) + 120*Csc[c + d*x]*(32*(c + d*x) + 25*Log[Cos[(c + d* x)/2]] - 25*Log[Sin[(c + d*x)/2]]) - 2*(241 + 327*Cos[c + d*x] + 92*Cos[2* (c + d*x)])*Sec[(c + d*x)/2]^6 + 840*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 + 4 80*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 - 320*Csc[c + d*x]^7*Sin[(c + d*x)/2] ^6)*Sin[c + d*x]*(1 + Sin[c + d*x])^2)/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x )/2])^4)
Time = 0.50 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
| \(\displaystyle \int \cot ^6(c+d x) \csc (c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^2}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^7(c+d x) a^8+2 \csc ^6(c+d x) a^8-2 \csc ^5(c+d x) a^8-6 \csc ^4(c+d x) a^8+6 \csc ^2(c+d x) a^8+2 \csc (c+d x) a^8-\sin (c+d x) a^8-2 a^8\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {25 a^8 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^8 \cos (c+d x)}{d}-\frac {2 a^8 \cot ^5(c+d x)}{5 d}+\frac {2 a^8 \cot ^3(c+d x)}{3 d}-\frac {2 a^8 \cot (c+d x)}{d}-\frac {a^8 \cot (c+d x) \csc ^5(c+d x)}{6 d}+\frac {7 a^8 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {7 a^8 \cot (c+d x) \csc (c+d x)}{16 d}-2 a^8 x}{a^6}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x])^2,x]
Output:
(-2*a^8*x - (25*a^8*ArcTanh[Cos[c + d*x]])/(16*d) + (a^8*Cos[c + d*x])/d - (2*a^8*Cot[c + d*x])/d + (2*a^8*Cot[c + d*x]^3)/(3*d) - (2*a^8*Cot[c + d* x]^5)/(5*d) + (7*a^8*Cot[c + d*x]*Csc[c + d*x])/(16*d) + (7*a^8*Cot[c + d* x]*Csc[c + d*x]^3)/(24*d) - (a^8*Cot[c + d*x]*Csc[c + d*x]^5)/(6*d))/a^6
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]))
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.48
| method | result | size |
| risch | \(-2 a^{2} x +\frac {a^{2} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {a^{2} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {a^{2} \left (105 \,{\mathrm e}^{11 i \left (d x +c \right )}-595 \,{\mathrm e}^{9 i \left (d x +c \right )}+1440 i {\mathrm e}^{10 i \left (d x +c \right )}-150 \,{\mathrm e}^{7 i \left (d x +c \right )}-4320 i {\mathrm e}^{8 i \left (d x +c \right )}-150 \,{\mathrm e}^{5 i \left (d x +c \right )}+7360 i {\mathrm e}^{6 i \left (d x +c \right )}-595 \,{\mathrm e}^{3 i \left (d x +c \right )}-6720 i {\mathrm e}^{4 i \left (d x +c \right )}+105 \,{\mathrm e}^{i \left (d x +c \right )}+2976 i {\mathrm e}^{2 i \left (d x +c \right )}-736 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {25 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}+\frac {25 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\) | \(232\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(239\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{4 \sin \left (d x +c \right )^{4}}+\frac {3 \cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{2}}+\frac {3 \cos \left (d x +c \right )^{5}}{8}+\frac {5 \cos \left (d x +c \right )^{3}}{8}+\frac {15 \cos \left (d x +c \right )}{8}+\frac {15 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+2 a^{2} \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(239\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-2*a^2*x+1/2*a^2/d*exp(I*(d*x+c))+1/2*a^2/d*exp(-I*(d*x+c))-1/120*a^2*(105 *exp(11*I*(d*x+c))-595*exp(9*I*(d*x+c))+1440*I*exp(10*I*(d*x+c))-150*exp(7 *I*(d*x+c))-4320*I*exp(8*I*(d*x+c))-150*exp(5*I*(d*x+c))+7360*I*exp(6*I*(d *x+c))-595*exp(3*I*(d*x+c))-6720*I*exp(4*I*(d*x+c))+105*exp(I*(d*x+c))+297 6*I*exp(2*I*(d*x+c))-736*I)/d/(exp(2*I*(d*x+c))-1)^6-25/16*a^2/d*ln(exp(I* (d*x+c))+1)+25/16*a^2/d*ln(exp(I*(d*x+c))-1)
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (145) = 290\).
Time = 0.11 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.93 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {960 \, a^{2} d x \cos \left (d x + c\right )^{6} - 480 \, a^{2} \cos \left (d x + c\right )^{7} - 2880 \, a^{2} d x \cos \left (d x + c\right )^{4} + 1650 \, a^{2} \cos \left (d x + c\right )^{5} + 2880 \, a^{2} d x \cos \left (d x + c\right )^{2} - 2000 \, a^{2} \cos \left (d x + c\right )^{3} - 960 \, a^{2} d x + 750 \, a^{2} \cos \left (d x + c\right ) + 375 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 375 \, {\left (a^{2} \cos \left (d x + c\right )^{6} - 3 \, a^{2} \cos \left (d x + c\right )^{4} + 3 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 64 \, {\left (23 \, a^{2} \cos \left (d x + c\right )^{5} - 35 \, a^{2} \cos \left (d x + c\right )^{3} + 15 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
-1/480*(960*a^2*d*x*cos(d*x + c)^6 - 480*a^2*cos(d*x + c)^7 - 2880*a^2*d*x *cos(d*x + c)^4 + 1650*a^2*cos(d*x + c)^5 + 2880*a^2*d*x*cos(d*x + c)^2 - 2000*a^2*cos(d*x + c)^3 - 960*a^2*d*x + 750*a^2*cos(d*x + c) + 375*(a^2*co s(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) - 375*(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) - 64*(23*a^2*cos(d*x + c)^5 - 35*a^2*cos(d*x + c)^3 + 15*a^2*cos(d*x + c))*sin(d*x + c))/(d*co s(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)
Timed out. \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.40 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {64 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 5 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, a^{2} {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
-1/480*(64*(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 - 5*a^2*(2*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*co s(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) + 1 5*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 30*a^2*(2*(9*cos(d*x + c)^3 - 7*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 16*cos (d*x + c) + 15*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d
Time = 0.21 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.65 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3840 \, {\left (d x + c\right )} a^{2} + 3000 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 2640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {3840 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {7350 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2640 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 255 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^2,x, algorithm="giac")
Output:
1/1920*(5*a^2*tan(1/2*d*x + 1/2*c)^6 + 24*a^2*tan(1/2*d*x + 1/2*c)^5 - 15* a^2*tan(1/2*d*x + 1/2*c)^4 - 280*a^2*tan(1/2*d*x + 1/2*c)^3 - 255*a^2*tan( 1/2*d*x + 1/2*c)^2 - 3840*(d*x + c)*a^2 + 3000*a^2*log(abs(tan(1/2*d*x + 1 /2*c))) + 2640*a^2*tan(1/2*d*x + 1/2*c) + 3840*a^2/(tan(1/2*d*x + 1/2*c)^2 + 1) - (7350*a^2*tan(1/2*d*x + 1/2*c)^6 + 2640*a^2*tan(1/2*d*x + 1/2*c)^5 - 255*a^2*tan(1/2*d*x + 1/2*c)^4 - 280*a^2*tan(1/2*d*x + 1/2*c)^3 - 15*a^ 2*tan(1/2*d*x + 1/2*c)^2 + 24*a^2*tan(1/2*d*x + 1/2*c) + 5*a^2)/tan(1/2*d* x + 1/2*c)^6)/d
Time = 35.02 (sec) , antiderivative size = 657, normalized size of antiderivative = 4.18 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx =\text {Too large to display} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x),x)
Output:
(5*a^2*sin(c/2 + (d*x)/2)^14 - 5*a^2*cos(c/2 + (d*x)/2)^14 + 24*a^2*cos(c/ 2 + (d*x)/2)*sin(c/2 + (d*x)/2)^13 - 24*a^2*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) - 10*a^2*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 256*a^2*c os(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^11 - 270*a^2*cos(c/2 + (d*x)/2)^4*s in(c/2 + (d*x)/2)^10 + 2360*a^2*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 - 255*a^2*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 4095*a^2*cos(c/2 + ( d*x)/2)^8*sin(c/2 + (d*x)/2)^6 - 2360*a^2*cos(c/2 + (d*x)/2)^9*sin(c/2 + ( d*x)/2)^5 + 270*a^2*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 + 256*a^2*c os(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 + 10*a^2*cos(c/2 + (d*x)/2)^12*s in(c/2 + (d*x)/2)^2 + 3000*a^2*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))* cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 3000*a^2*log(sin(c/2 + (d*x)/2 )/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 7680*a^2 *atan((32*cos(c/2 + (d*x)/2) - 25*sin(c/2 + (d*x)/2))/(25*cos(c/2 + (d*x)/ 2) + 32*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 + 7 680*a^2*atan((32*cos(c/2 + (d*x)/2) - 25*sin(c/2 + (d*x)/2))/(25*cos(c/2 + (d*x)/2) + 32*sin(c/2 + (d*x)/2)))*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2 )^6)/(1920*d*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6*(cos(c/2 + (d*x)/2) ^2 + sin(c/2 + (d*x)/2)^2))
Time = 0.16 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.03 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-5888 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+840 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2816 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-768 \cos \left (d x +c \right ) \sin \left (d x +c \right )-320 \cos \left (d x +c \right )+3000 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}-3840 \sin \left (d x +c \right )^{6} d x -2175 \sin \left (d x +c \right )^{6}\right )}{1920 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(1920*cos(c + d*x)*sin(c + d*x)**6 - 5888*cos(c + d*x)*sin(c + d*x)* *5 + 840*cos(c + d*x)*sin(c + d*x)**4 + 2816*cos(c + d*x)*sin(c + d*x)**3 + 560*cos(c + d*x)*sin(c + d*x)**2 - 768*cos(c + d*x)*sin(c + d*x) - 320*c os(c + d*x) + 3000*log(tan((c + d*x)/2))*sin(c + d*x)**6 - 3840*sin(c + d* x)**6*d*x - 2175*sin(c + d*x)**6))/(1920*sin(c + d*x)**6*d)