Integrand size = 27, antiderivative size = 138 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {3 a^3 x}{2}+\frac {33 a^3 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {3 a^3 \cos (c+d x)}{d}+\frac {2 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{d}-\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {a^3 \cos (c+d x) \sin (c+d x)}{2 d} \] Output:
3/2*a^3*x+33/8*a^3*arctanh(cos(d*x+c))/d-3*a^3*cos(d*x+c)/d+2*a^3*cot(d*x+ c)/d-a^3*cot(d*x+c)^3/d-7/8*a^3*cot(d*x+c)*csc(d*x+c)/d-1/4*a^3*cot(d*x+c) *csc(d*x+c)^3/d-1/2*a^3*cos(d*x+c)*sin(d*x+c)/d
Time = 7.29 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.56 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (1+\sin (c+d x))^3 \left (96 (c+d x)-192 \cos (c+d x)+96 \cot \left (\frac {1}{2} (c+d x)\right )-14 \csc ^2\left (\frac {1}{2} (c+d x)\right )-\csc ^4\left (\frac {1}{2} (c+d x)\right )+264 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-264 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+14 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+64 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-4 \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sin (c+d x)-16 \sin (2 (c+d x))-96 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{64 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1 + Sin[c + d*x])^3*(96*(c + d*x) - 192*Cos[c + d*x] + 96*Cot[(c + d *x)/2] - 14*Csc[(c + d*x)/2]^2 - Csc[(c + d*x)/2]^4 + 264*Log[Cos[(c + d*x )/2]] - 264*Log[Sin[(c + d*x)/2]] + 14*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/ 2]^4 + 64*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 4*Csc[(c + d*x)/2]^4*Sin[c + d*x] - 16*Sin[2*(c + d*x)] - 96*Tan[(c + d*x)/2]))/(64*d*(Cos[(c + d*x)/2 ] + Sin[(c + d*x)/2])^6)
Time = 0.42 (sec) , antiderivative size = 142, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc (c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)^5}dx\) |
| \(\Big \downarrow \) 3351 |
| \(\displaystyle \frac {\int \left (\csc ^5(c+d x) a^7+3 \csc ^4(c+d x) a^7+\csc ^3(c+d x) a^7-5 \csc ^2(c+d x) a^7+\sin ^2(c+d x) a^7-5 \csc (c+d x) a^7+3 \sin (c+d x) a^7+a^7\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {33 a^7 \text {arctanh}(\cos (c+d x))}{8 d}-\frac {3 a^7 \cos (c+d x)}{d}-\frac {a^7 \cot ^3(c+d x)}{d}+\frac {2 a^7 \cot (c+d x)}{d}-\frac {a^7 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a^7 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {7 a^7 \cot (c+d x) \csc (c+d x)}{8 d}+\frac {3 a^7 x}{2}}{a^4}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]*(a + a*Sin[c + d*x])^3,x]
Output:
((3*a^7*x)/2 + (33*a^7*ArcTanh[Cos[c + d*x]])/(8*d) - (3*a^7*Cos[c + d*x]) /d + (2*a^7*Cot[c + d*x])/d - (a^7*Cot[c + d*x]^3)/d - (7*a^7*Cot[c + d*x] *Csc[c + d*x])/(8*d) - (a^7*Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) - (a^7*Cos[ c + d*x]*Sin[c + d*x])/(2*d))/a^4
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 = 1.12 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.61
| method | result | size |
| risch | \(\frac {3 a^{3} x}{2}+\frac {i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {3 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}-\frac {i a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a^{3} \left (7 \,{\mathrm e}^{7 i \left (d x +c \right )}-15 \,{\mathrm e}^{5 i \left (d x +c \right )}+40 i {\mathrm e}^{6 i \left (d x +c \right )}-15 \,{\mathrm e}^{3 i \left (d x +c \right )}-72 i {\mathrm e}^{4 i \left (d x +c \right )}+7 \,{\mathrm e}^{i \left (d x +c \right )}+56 i {\mathrm e}^{2 i \left (d x +c \right )}-24 i\right )}{4 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {33 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}\) | \(222\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(224\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{\sin \left (d x +c \right )}-\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{2 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{3}}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a^{3} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )}{d}\) | \(224\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
3/2*a^3*x+1/8*I*a^3/d*exp(2*I*(d*x+c))-3/2*a^3/d*exp(I*(d*x+c))-3/2*a^3/d* exp(-I*(d*x+c))-1/8*I*a^3/d*exp(-2*I*(d*x+c))+1/4*a^3*(7*exp(7*I*(d*x+c))- 15*exp(5*I*(d*x+c))+40*I*exp(6*I*(d*x+c))-15*exp(3*I*(d*x+c))-72*I*exp(4*I *(d*x+c))+7*exp(I*(d*x+c))+56*I*exp(2*I*(d*x+c))-24*I)/d/(exp(2*I*(d*x+c)) -1)^4+33/8*a^3/d*ln(exp(I*(d*x+c))+1)-33/8*a^3/d*ln(exp(I*(d*x+c))-1)
Time = 0.10 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.67 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {24 \, a^{3} d x \cos \left (d x + c\right )^{4} - 48 \, a^{3} \cos \left (d x + c\right )^{5} - 48 \, a^{3} d x \cos \left (d x + c\right )^{2} + 110 \, a^{3} \cos \left (d x + c\right )^{3} + 24 \, a^{3} d x - 66 \, a^{3} \cos \left (d x + c\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 8 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 4 \, a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/16*(24*a^3*d*x*cos(d*x + c)^4 - 48*a^3*cos(d*x + c)^5 - 48*a^3*d*x*cos(d *x + c)^2 + 110*a^3*cos(d*x + c)^3 + 24*a^3*d*x - 66*a^3*cos(d*x + c) + 33 *(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(1/2*cos(d*x + c) + 1/2) - 33*(a^3*cos(d*x + c)^4 - 2*a^3*cos(d*x + c)^2 + a^3)*log(-1/2*cos(d *x + c) + 1/2) - 8*(a^3*cos(d*x + c)^5 + 4*a^3*cos(d*x + c)^3 - 3*a^3*cos( d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.51 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {8 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a^{3} - 16 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, a^{3} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{16 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/16*(8*(3*d*x + 3*c + (3*tan(d*x + c)^2 + 2)/(tan(d*x + c)^3 + tan(d*x + c)))*a^3 - 16*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^3 + a^3*(2*(5*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)) - 12*a^3*(2 *cos(d*x + c)/(cos(d*x + c)^2 - 1) - 4*cos(d*x + c) + 3*log(cos(d*x + c) + 1) - 3*log(cos(d*x + c) - 1)))/d
Time = 0.24 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.75 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 96 \, {\left (d x + c\right )} a^{3} - 264 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {64 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {550 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 88 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)*(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/64*(a^3*tan(1/2*d*x + 1/2*c)^4 + 8*a^3*tan(1/2*d*x + 1/2*c)^3 + 16*a^3*t an(1/2*d*x + 1/2*c)^2 + 96*(d*x + c)*a^3 - 264*a^3*log(abs(tan(1/2*d*x + 1 /2*c))) - 88*a^3*tan(1/2*d*x + 1/2*c) + 64*(a^3*tan(1/2*d*x + 1/2*c)^3 - 6 *a^3*tan(1/2*d*x + 1/2*c)^2 - a^3*tan(1/2*d*x + 1/2*c) - 6*a^3)/(tan(1/2*d *x + 1/2*c)^2 + 1)^2 + (550*a^3*tan(1/2*d*x + 1/2*c)^4 + 88*a^3*tan(1/2*d* x + 1/2*c)^3 - 16*a^3*tan(1/2*d*x + 1/2*c)^2 - 8*a^3*tan(1/2*d*x + 1/2*c) - a^3)/tan(1/2*d*x + 1/2*c)^4)/d
Time = 17.72 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.38 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {33\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {3\,a^3\,\mathrm {atan}\left (\frac {9\,a^6}{\frac {99\,a^6}{4}+9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {99\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {99\,a^6}{4}+9\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d}-\frac {11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-38\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+100\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-26\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {417\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}-18\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {9\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x),x)
Output:
(a^3*tan(c/2 + (d*x)/2)^2)/(4*d) + (a^3*tan(c/2 + (d*x)/2)^3)/(8*d) + (a^3 *tan(c/2 + (d*x)/2)^4)/(64*d) - (33*a^3*log(tan(c/2 + (d*x)/2)))/(8*d) - ( 3*a^3*atan((9*a^6)/((99*a^6)/4 + 9*a^6*tan(c/2 + (d*x)/2)) - (99*a^6*tan(c /2 + (d*x)/2))/(4*((99*a^6)/4 + 9*a^6*tan(c/2 + (d*x)/2)))))/d - (11*a^3*t an(c/2 + (d*x)/2))/(8*d) - ((9*a^3*tan(c/2 + (d*x)/2)^2)/2 - 18*a^3*tan(c/ 2 + (d*x)/2)^3 + (417*a^3*tan(c/2 + (d*x)/2)^4)/4 - 26*a^3*tan(c/2 + (d*x) /2)^5 + 100*a^3*tan(c/2 + (d*x)/2)^6 - 38*a^3*tan(c/2 + (d*x)/2)^7 + a^3/4 + 2*a^3*tan(c/2 + (d*x)/2))/(d*(16*tan(c/2 + (d*x)/2)^4 + 32*tan(c/2 + (d *x)/2)^6 + 16*tan(c/2 + (d*x)/2)^8))
Time = 0.16 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.05 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+192 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-56 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-64 \cos \left (d x +c \right ) \sin \left (d x +c \right )-16 \cos \left (d x +c \right )-264 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}+96 \sin \left (d x +c \right )^{4} d x +225 \sin \left (d x +c \right )^{4}\right )}{64 \sin \left (d x +c \right )^{4} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 32*cos(c + d*x)*sin(c + d*x)**5 - 192*cos(c + d*x)*sin(c + d*x)* *4 + 192*cos(c + d*x)*sin(c + d*x)**3 - 56*cos(c + d*x)*sin(c + d*x)**2 - 64*cos(c + d*x)*sin(c + d*x) - 16*cos(c + d*x) - 264*log(tan((c + d*x)/2)) *sin(c + d*x)**4 + 96*sin(c + d*x)**4*d*x + 225*sin(c + d*x)**4))/(64*sin( c + d*x)**4*d)