Integrand size = 29, antiderivative size = 176 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {27 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d} \] Output:
-27/128*a^3*arctanh(cos(d*x+c))/d-4/5*a^3*cot(d*x+c)^5/d-3/7*a^3*cot(d*x+c )^7/d-27/128*a^3*cot(d*x+c)*csc(d*x+c)/d+23/64*a^3*cot(d*x+c)*csc(d*x+c)^3 /d-1/2*a^3*cot(d*x+c)^3*csc(d*x+c)^3/d+1/16*a^3*cot(d*x+c)*csc(d*x+c)^5/d- 1/8*a^3*cot(d*x+c)^3*csc(d*x+c)^5/d
Time = 10.17 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.78 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \left (10 \csc ^8\left (\frac {1}{2} (c+d x)\right ) (24+7 \csc (c+d x))+8 \csc ^6\left (\frac {1}{2} (c+d x)\right ) (-76+105 \csc (c+d x))+8 \csc ^2\left (\frac {1}{2} (c+d x)\right ) (1664+945 \csc (c+d x))-4 \csc ^4\left (\frac {1}{2} (c+d x)\right ) (856+1715 \csc (c+d x))-4 \left (-7560 \csc (c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+(703+1056 \cos (c+d x)+517 \cos (2 (c+d x))+104 \cos (3 (c+d x))) \sec ^8\left (\frac {1}{2} (c+d x)\right )+7560 \csc ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )-27440 \csc ^5(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+13440 \csc ^7(c+d x) \sin ^6\left (\frac {1}{2} (c+d x)\right )+4480 \csc ^9(c+d x) \sin ^8\left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x) (1+\sin (c+d x))^3}{143360 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]
Output:
-1/143360*(a^3*(10*Csc[(c + d*x)/2]^8*(24 + 7*Csc[c + d*x]) + 8*Csc[(c + d *x)/2]^6*(-76 + 105*Csc[c + d*x]) + 8*Csc[(c + d*x)/2]^2*(1664 + 945*Csc[c + d*x]) - 4*Csc[(c + d*x)/2]^4*(856 + 1715*Csc[c + d*x]) - 4*(-7560*Csc[c + d*x]*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + (703 + 1056*Cos[ c + d*x] + 517*Cos[2*(c + d*x)] + 104*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^8 + 7560*Csc[c + d*x]^3*Sin[(c + d*x)/2]^2 - 27440*Csc[c + d*x]^5*Sin[(c + d*x)/2]^4 + 13440*Csc[c + d*x]^7*Sin[(c + d*x)/2]^6 + 4480*Csc[c + d*x]^9* Sin[(c + d*x)/2]^8))*Sin[c + d*x]*(1 + Sin[c + d*x])^3)/(d*(Cos[(c + d*x)/ 2] + Sin[(c + d*x)/2])^6)
Time = 0.58 (sec) , antiderivative size = 176, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^5(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^4(c+d x) \csc ^5(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^3(c+d x)+a^3 \cot ^4(c+d x) \csc ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {27 a^3 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {3 a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}+\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {23 a^3 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {27 a^3 \cot (c+d x) \csc (c+d x)}{128 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]
Output:
(-27*a^3*ArcTanh[Cos[c + d*x]])/(128*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - ( 3*a^3*Cot[c + d*x]^7)/(7*d) - (27*a^3*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (23*a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3)/(2*d) + (a^3*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^3*Cot[c + d *x]^3*Csc[c + d*x]^5)/(8*d)
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]
Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.35
| method | result | size |
| risch | \(\frac {a^{3} \left (945 \,{\mathrm e}^{15 i \left (d x +c \right )}+10675 \,{\mathrm e}^{13 i \left (d x +c \right )}+58240 i {\mathrm e}^{8 i \left (d x +c \right )}-15295 \,{\mathrm e}^{11 i \left (d x +c \right )}-22400 i {\mathrm e}^{10 i \left (d x +c \right )}-32165 \,{\mathrm e}^{9 i \left (d x +c \right )}-4480 i {\mathrm e}^{14 i \left (d x +c \right )}-32165 \,{\mathrm e}^{7 i \left (d x +c \right )}-70784 i {\mathrm e}^{6 i \left (d x +c \right )}-15295 \,{\mathrm e}^{5 i \left (d x +c \right )}+40320 i {\mathrm e}^{12 i \left (d x +c \right )}+10675 \,{\mathrm e}^{3 i \left (d x +c \right )}-8832 i {\mathrm e}^{2 i \left (d x +c \right )}+945 \,{\mathrm e}^{i \left (d x +c \right )}+6272 i {\mathrm e}^{4 i \left (d x +c \right )}+1664 i\right )}{2240 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}+\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}-\frac {27 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\) | \(238\) |
| derivativedivides | \(\frac {-\frac {a^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(278\) |
| default | \(\frac {-\frac {a^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{5}}{16 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{64 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{128 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{128}+\frac {3 \cos \left (d x +c \right )}{128}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(278\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/2240*a^3*(945*exp(15*I*(d*x+c))+10675*exp(13*I*(d*x+c))+58240*I*exp(8*I* (d*x+c))-15295*exp(11*I*(d*x+c))-22400*I*exp(10*I*(d*x+c))-32165*exp(9*I*( d*x+c))-4480*I*exp(14*I*(d*x+c))-32165*exp(7*I*(d*x+c))-70784*I*exp(6*I*(d *x+c))-15295*exp(5*I*(d*x+c))+40320*I*exp(12*I*(d*x+c))+10675*exp(3*I*(d*x +c))-8832*I*exp(2*I*(d*x+c))+945*exp(I*(d*x+c))+6272*I*exp(4*I*(d*x+c))+16 64*I)/d/(exp(2*I*(d*x+c))-1)^8+27/128*a^3/d*ln(exp(I*(d*x+c))-1)-27/128*a^ 3/d*ln(exp(I*(d*x+c))+1)
Time = 0.09 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.54 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {1890 \, a^{3} \cos \left (d x + c\right )^{7} + 2030 \, a^{3} \cos \left (d x + c\right )^{5} - 6930 \, a^{3} \cos \left (d x + c\right )^{3} + 1890 \, a^{3} \cos \left (d x + c\right ) - 945 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 945 \, {\left (a^{3} \cos \left (d x + c\right )^{8} - 4 \, a^{3} \cos \left (d x + c\right )^{6} + 6 \, a^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 256 \, {\left (13 \, a^{3} \cos \left (d x + c\right )^{7} - 28 \, a^{3} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{8960 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/8960*(1890*a^3*cos(d*x + c)^7 + 2030*a^3*cos(d*x + c)^5 - 6930*a^3*cos(d *x + c)^3 + 1890*a^3*cos(d*x + c) - 945*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d* x + c)^6 + 6*a^3*cos(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos( d*x + c) + 1/2) + 945*(a^3*cos(d*x + c)^8 - 4*a^3*cos(d*x + c)^6 + 6*a^3*c os(d*x + c)^4 - 4*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d*x + c) + 1/2) + 256*(13*a^3*cos(d*x + c)^7 - 28*a^3*cos(d*x + c)^5)*sin(d*x + c))/(d*cos( d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**5*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.40 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {1792 \, a^{3}}{\tan \left (d x + c\right )^{5}} - \frac {768 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{8960 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/8960*(35*a^3*(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 )) + 280*a^3*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(co s(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1792*a^3/tan(d*x + c)^5 - 768*(7*tan( d*x + c)^2 + 5)*a^3/tan(d*x + c)^7)/d
Time = 0.25 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.66 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15120 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 9520 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {41094 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 9520 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1680 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 3920 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1960 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 240 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{71680 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/71680*(35*a^3*tan(1/2*d*x + 1/2*c)^8 + 240*a^3*tan(1/2*d*x + 1/2*c)^7 + 560*a^3*tan(1/2*d*x + 1/2*c)^6 + 112*a^3*tan(1/2*d*x + 1/2*c)^5 - 1960*a^3 *tan(1/2*d*x + 1/2*c)^4 - 3920*a^3*tan(1/2*d*x + 1/2*c)^3 - 1680*a^3*tan(1 /2*d*x + 1/2*c)^2 + 15120*a^3*log(abs(tan(1/2*d*x + 1/2*c))) + 9520*a^3*ta n(1/2*d*x + 1/2*c) - (41094*a^3*tan(1/2*d*x + 1/2*c)^8 + 9520*a^3*tan(1/2* d*x + 1/2*c)^7 - 1680*a^3*tan(1/2*d*x + 1/2*c)^6 - 3920*a^3*tan(1/2*d*x + 1/2*c)^5 - 1960*a^3*tan(1/2*d*x + 1/2*c)^4 + 112*a^3*tan(1/2*d*x + 1/2*c)^ 3 + 560*a^3*tan(1/2*d*x + 1/2*c)^2 + 240*a^3*tan(1/2*d*x + 1/2*c) + 35*a^3 )/tan(1/2*d*x + 1/2*c)^8)/d
Time = 18.22 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.81 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {7\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{256\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2048\,d}+\frac {27\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{128\,d}-\frac {17\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {17\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^5,x)
Output:
(3*a^3*cot(c/2 + (d*x)/2)^2)/(128*d) + (7*a^3*cot(c/2 + (d*x)/2)^3)/(128*d ) + (7*a^3*cot(c/2 + (d*x)/2)^4)/(256*d) - (a^3*cot(c/2 + (d*x)/2)^5)/(640 *d) - (a^3*cot(c/2 + (d*x)/2)^6)/(128*d) - (3*a^3*cot(c/2 + (d*x)/2)^7)/(8 96*d) - (a^3*cot(c/2 + (d*x)/2)^8)/(2048*d) - (3*a^3*tan(c/2 + (d*x)/2)^2) /(128*d) - (7*a^3*tan(c/2 + (d*x)/2)^3)/(128*d) - (7*a^3*tan(c/2 + (d*x)/2 )^4)/(256*d) + (a^3*tan(c/2 + (d*x)/2)^5)/(640*d) + (a^3*tan(c/2 + (d*x)/2 )^6)/(128*d) + (3*a^3*tan(c/2 + (d*x)/2)^7)/(896*d) + (a^3*tan(c/2 + (d*x) /2)^8)/(2048*d) + (27*a^3*log(tan(c/2 + (d*x)/2)))/(128*d) - (17*a^3*cot(c /2 + (d*x)/2))/(128*d) + (17*a^3*tan(c/2 + (d*x)/2))/(128*d)
Time = 0.16 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.88 \[ \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-1664 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-945 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+1408 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+3850 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-1400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )-560 \cos \left (d x +c \right )+945 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{8}\right )}{4480 \sin \left (d x +c \right )^{8} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^5*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 1664*cos(c + d*x)*sin(c + d*x)**7 - 945*cos(c + d*x)*sin(c + d*x )**6 + 1408*cos(c + d*x)*sin(c + d*x)**5 + 3850*cos(c + d*x)*sin(c + d*x)* *4 + 2176*cos(c + d*x)*sin(c + d*x)**3 - 1400*cos(c + d*x)*sin(c + d*x)**2 - 1920*cos(c + d*x)*sin(c + d*x) - 560*cos(c + d*x) + 945*log(tan((c + d* x)/2))*sin(c + d*x)**8))/(4480*sin(c + d*x)**8*d)