Integrand size = 29, antiderivative size = 168 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 x-\frac {19 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d} \] Output:
a^3*x-19/16*a^3*arctanh(cos(d*x+c))/d+a^3*cot(d*x+c)/d-1/3*a^3*cot(d*x+c)^ 3/d-3/5*a^3*cot(d*x+c)^5/d+17/16*a^3*cot(d*x+c)*csc(d*x+c)/d-3/4*a^3*cot(d *x+c)^3*csc(d*x+c)/d+1/8*a^3*cot(d*x+c)*csc(d*x+c)^3/d-1/6*a^3*cot(d*x+c)^ 3*csc(d*x+c)^3/d
Time = 0.82 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.29 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \left (1920 c+1920 d x+704 \cot \left (\frac {1}{2} (c+d x)\right )+870 \csc ^2\left (\frac {1}{2} (c+d x)\right )-2280 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2280 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-870 \sec ^2\left (\frac {1}{2} (c+d x)\right )+60 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right )-1376 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )-\csc ^6\left (\frac {1}{2} (c+d x)\right ) (5+18 \sin (c+d x))+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (-60+86 \sin (c+d x))-704 \tan \left (\frac {1}{2} (c+d x)\right )+36 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{1920 d} \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
(a^3*(1920*c + 1920*d*x + 704*Cot[(c + d*x)/2] + 870*Csc[(c + d*x)/2]^2 - 2280*Log[Cos[(c + d*x)/2]] + 2280*Log[Sin[(c + d*x)/2]] - 870*Sec[(c + d*x )/2]^2 + 60*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 1376*Csc[c + d*x]^ 3*Sin[(c + d*x)/2]^4 - Csc[(c + d*x)/2]^6*(5 + 18*Sin[c + d*x]) + Csc[(c + d*x)/2]^4*(-60 + 86*Sin[c + d*x]) - 704*Tan[(c + d*x)/2] + 36*Sec[(c + d* x)/2]^4*Tan[(c + d*x)/2]))/(1920*d)
Time = 0.52 (sec) , antiderivative size = 168, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^4(c+d x)+a^3 \cot ^4(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^4(c+d x) \csc (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {19 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {3 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {3 a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac {17 a^3 \cot (c+d x) \csc (c+d x)}{16 d}+a^3 x\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]
Output:
a^3*x - (19*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (a^3*Cot[c + d*x])/d - (a^ 3*Cot[c + d*x]^3)/(3*d) - (3*a^3*Cot[c + d*x]^5)/(5*d) + (17*a^3*Cot[c + d *x]*Csc[c + d*x])/(16*d) - (3*a^3*Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (a^ 3*Cot[c + d*x]*Csc[c + d*x]^3)/(8*d) - (a^3*Cot[c + d*x]^3*Csc[c + d*x]^3) /(6*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 = 0.37 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.17
| method | result | size |
| risch | \(a^{3} x -\frac {a^{3} \left (435 \,{\mathrm e}^{11 i \left (d x +c \right )}-865 \,{\mathrm e}^{9 i \left (d x +c \right )}+240 i {\mathrm e}^{10 i \left (d x +c \right )}-210 \,{\mathrm e}^{7 i \left (d x +c \right )}+1200 i {\mathrm e}^{8 i \left (d x +c \right )}-210 \,{\mathrm e}^{5 i \left (d x +c \right )}-1760 i {\mathrm e}^{6 i \left (d x +c \right )}-865 \,{\mathrm e}^{3 i \left (d x +c \right )}+1440 i {\mathrm e}^{4 i \left (d x +c \right )}+435 \,{\mathrm e}^{i \left (d x +c \right )}-1296 i {\mathrm e}^{2 i \left (d x +c \right )}+176 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {19 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {19 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\) | \(197\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(225\) |
| default | \(\frac {a^{3} \left (-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(225\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
a^3*x-1/120*a^3*(435*exp(11*I*(d*x+c))-865*exp(9*I*(d*x+c))+240*I*exp(10*I *(d*x+c))-210*exp(7*I*(d*x+c))+1200*I*exp(8*I*(d*x+c))-210*exp(5*I*(d*x+c) )-1760*I*exp(6*I*(d*x+c))-865*exp(3*I*(d*x+c))+1440*I*exp(4*I*(d*x+c))+435 *exp(I*(d*x+c))-1296*I*exp(2*I*(d*x+c))+176*I)/d/(exp(2*I*(d*x+c))-1)^6+19 /16*a^3/d*ln(exp(I*(d*x+c))-1)-19/16*a^3/d*ln(exp(I*(d*x+c))+1)
Time = 0.09 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.73 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {480 \, a^{3} d x \cos \left (d x + c\right )^{6} - 1440 \, a^{3} d x \cos \left (d x + c\right )^{4} - 870 \, a^{3} \cos \left (d x + c\right )^{5} + 1440 \, a^{3} d x \cos \left (d x + c\right )^{2} + 1520 \, a^{3} \cos \left (d x + c\right )^{3} - 480 \, a^{3} d x - 570 \, a^{3} \cos \left (d x + c\right ) - 285 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 285 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (11 \, a^{3} \cos \left (d x + c\right )^{5} - 35 \, a^{3} \cos \left (d x + c\right )^{3} + 15 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
1/480*(480*a^3*d*x*cos(d*x + c)^6 - 1440*a^3*d*x*cos(d*x + c)^4 - 870*a^3* cos(d*x + c)^5 + 1440*a^3*d*x*cos(d*x + c)^2 + 1520*a^3*cos(d*x + c)^3 - 4 80*a^3*d*x - 570*a^3*cos(d*x + c) - 285*(a^3*cos(d*x + c)^6 - 3*a^3*cos(d* x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*log(1/2*cos(d*x + c) + 1/2) + 285*( a^3*cos(d*x + c)^6 - 3*a^3*cos(d*x + c)^4 + 3*a^3*cos(d*x + c)^2 - a^3)*lo g(-1/2*cos(d*x + c) + 1/2) - 32*(11*a^3*cos(d*x + c)^5 - 35*a^3*cos(d*x + c)^3 + 15*a^3*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)
Timed out. \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**3*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.28 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {160 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} + 5 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 90 \, a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {288 \, a^{3}}{\tan \left (d x + c\right )^{5}}}{480 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/480*(160*(3*d*x + 3*c + (3*tan(d*x + c)^2 - 1)/tan(d*x + c)^3)*a^3 + 5*a ^3*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(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)) - 90*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(c os(d*x + c) - 1)) - 288*a^3/tan(d*x + c)^5)/d
Time = 0.22 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.42 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 75 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 100 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1920 \, {\left (d x + c\right )} a^{3} + 2280 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {5586 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 735 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 100 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 75 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/1920*(5*a^3*tan(1/2*d*x + 1/2*c)^6 + 36*a^3*tan(1/2*d*x + 1/2*c)^5 + 75* a^3*tan(1/2*d*x + 1/2*c)^4 - 100*a^3*tan(1/2*d*x + 1/2*c)^3 - 735*a^3*tan( 1/2*d*x + 1/2*c)^2 + 1920*(d*x + c)*a^3 + 2280*a^3*log(abs(tan(1/2*d*x + 1 /2*c))) - 840*a^3*tan(1/2*d*x + 1/2*c) - (5586*a^3*tan(1/2*d*x + 1/2*c)^6 - 840*a^3*tan(1/2*d*x + 1/2*c)^5 - 735*a^3*tan(1/2*d*x + 1/2*c)^4 - 100*a^ 3*tan(1/2*d*x + 1/2*c)^3 + 75*a^3*tan(1/2*d*x + 1/2*c)^2 + 36*a^3*tan(1/2* d*x + 1/2*c) + 5*a^3)/tan(1/2*d*x + 1/2*c)^6)/d
Time = 18.09 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.86 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {49\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {5\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {3\,a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {49\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+19\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{19\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {19\,a^3\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{16\,d}+\frac {7\,a^3\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^3,x)
Output:
(49*a^3*cot(c/2 + (d*x)/2)^2)/(128*d) + (5*a^3*cot(c/2 + (d*x)/2)^3)/(96*d ) - (5*a^3*cot(c/2 + (d*x)/2)^4)/(128*d) - (3*a^3*cot(c/2 + (d*x)/2)^5)/(1 60*d) - (a^3*cot(c/2 + (d*x)/2)^6)/(384*d) - (49*a^3*tan(c/2 + (d*x)/2)^2) /(128*d) - (5*a^3*tan(c/2 + (d*x)/2)^3)/(96*d) + (5*a^3*tan(c/2 + (d*x)/2) ^4)/(128*d) + (3*a^3*tan(c/2 + (d*x)/2)^5)/(160*d) + (a^3*tan(c/2 + (d*x)/ 2)^6)/(384*d) + (2*a^3*atan((16*cos(c/2 + (d*x)/2) + 19*sin(c/2 + (d*x)/2) )/(19*cos(c/2 + (d*x)/2) - 16*sin(c/2 + (d*x)/2))))/d + (19*a^3*log(sin(c/ 2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(16*d) + (7*a^3*cot(c/2 + (d*x)/2))/(16* d) - (7*a^3*tan(c/2 + (d*x)/2))/(16*d)
Time = 0.16 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.80 \[ \int \cot ^4(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+435 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+208 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-110 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-144 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40 \cos \left (d x +c \right )+285 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}+240 \sin \left (d x +c \right )^{6} d x \right )}{240 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^3*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*(176*cos(c + d*x)*sin(c + d*x)**5 + 435*cos(c + d*x)*sin(c + d*x)**4 + 208*cos(c + d*x)*sin(c + d*x)**3 - 110*cos(c + d*x)*sin(c + d*x)**2 - 1 44*cos(c + d*x)*sin(c + d*x) - 40*cos(c + d*x) + 285*log(tan((c + d*x)/2)) *sin(c + d*x)**6 + 240*sin(c + d*x)**6*d*x))/(240*sin(c + d*x)**6*d)