Integrand size = 29, antiderivative size = 150 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {9 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}-\frac {a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d} \] Output:
-9/16*a^3*arctanh(cos(d*x+c))/d-4/5*a^3*cot(d*x+c)^5/d-1/7*a^3*cot(d*x+c)^ 7/d+3/16*a^3*cot(d*x+c)*csc(d*x+c)/d-1/4*a^3*cot(d*x+c)^3*csc(d*x+c)/d+3/8 *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
Leaf count is larger than twice the leaf count of optimal. \(363\) vs. \(2(150)=300\).
Time = 0.35 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.42 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=a^3 \left (-\frac {23 \cot \left (\frac {1}{2} (c+d x)\right )}{70 d}+\frac {7 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {297 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{2240 d}+\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {31 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{2240 d}-\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{128 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^6\left (\frac {1}{2} (c+d x)\right )}{896 d}-\frac {9 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {9 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {7 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{128 d}+\frac {23 \tan \left (\frac {1}{2} (c+d x)\right )}{70 d}-\frac {297 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{2240 d}+\frac {31 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{2240 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{896 d}\right ) \] Input:
Integrate[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
a^3*((-23*Cot[(c + d*x)/2])/(70*d) + (7*Csc[(c + d*x)/2]^2)/(64*d) + (297* Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(2240*d) + Csc[(c + d*x)/2]^4/(32*d) - (31*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(2240*d) - Csc[(c + d*x)/2]^6/( 128*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(896*d) - (9*Log[Cos[(c + d *x)/2]])/(16*d) + (9*Log[Sin[(c + d*x)/2]])/(16*d) - (7*Sec[(c + d*x)/2]^2 )/(64*d) - Sec[(c + d*x)/2]^4/(32*d) + Sec[(c + d*x)/2]^6/(128*d) + (23*Ta n[(c + d*x)/2])/(70*d) - (297*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(2240*d ) + (31*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(2240*d) + (Sec[(c + d*x)/2]^ 6*Tan[(c + d*x)/2])/(896*d))
Time = 0.51 (sec) , antiderivative size = 150, 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.
| \(\displaystyle \int \cot ^4(c+d x) \csc ^4(c+d x) (a \sin (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4 (a \sin (c+d x)+a)^3}{\sin (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^3 \cot ^4(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^3(c+d x)+3 a^3 \cot ^4(c+d x) \csc ^2(c+d x)+a^3 \cot ^4(c+d x) \csc (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {9 a^3 \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x) \csc ^3(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x) \csc (c+d x)}{4 d}+\frac {3 a^3 \cot (c+d x) \csc ^3(c+d x)}{8 d}+\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{16 d}\) |
Input:
Int[Cot[c + d*x]^4*Csc[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]
Output:
(-9*a^3*ArcTanh[Cos[c + d*x]])/(16*d) - (4*a^3*Cot[c + d*x]^5)/(5*d) - (a^ 3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (a^3* Cot[c + d*x]^3*Csc[c + d*x])/(4*d) + (3*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)/(2*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 = 0.47 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.36
| method | result | size |
| risch | \(-\frac {a^{3} \left (1680 i {\mathrm e}^{12 i \left (d x +c \right )}+245 \,{\mathrm e}^{13 i \left (d x +c \right )}-4480 i {\mathrm e}^{10 i \left (d x +c \right )}-2380 \,{\mathrm e}^{11 i \left (d x +c \right )}+3920 i {\mathrm e}^{8 i \left (d x +c \right )}-455 \,{\mathrm e}^{9 i \left (d x +c \right )}-8960 i {\mathrm e}^{6 i \left (d x +c \right )}+3248 i {\mathrm e}^{4 i \left (d x +c \right )}+455 \,{\mathrm e}^{5 i \left (d x +c \right )}-896 i {\mathrm e}^{2 i \left (d x +c \right )}+2380 \,{\mathrm e}^{3 i \left (d x +c \right )}+368 i-245 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{280 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {9 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}-\frac {9 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}\) | \(204\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )}{d}\) | \(241\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{4 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{8 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{8}+\frac {3 \cos \left (d x +c \right )}{8}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{5}}{5 \sin \left (d x +c \right )^{5}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{6 \sin \left (d x +c \right )^{6}}-\frac {\cos \left (d x +c \right )^{5}}{24 \sin \left (d x +c \right )^{4}}+\frac {\cos \left (d x +c \right )^{5}}{48 \sin \left (d x +c \right )^{2}}+\frac {\cos \left (d x +c \right )^{3}}{48}+\frac {\cos \left (d x +c \right )}{16}+\frac {\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )+a^{3} \left (-\frac {\cos \left (d x +c \right )^{5}}{7 \sin \left (d x +c \right )^{7}}-\frac {2 \cos \left (d x +c \right )^{5}}{35 \sin \left (d x +c \right )^{5}}\right )}{d}\) | \(241\) |
Input:
int(cot(d*x+c)^4*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-1/280*a^3*(1680*I*exp(12*I*(d*x+c))+245*exp(13*I*(d*x+c))-4480*I*exp(10*I *(d*x+c))-2380*exp(11*I*(d*x+c))+3920*I*exp(8*I*(d*x+c))-455*exp(9*I*(d*x+ c))-8960*I*exp(6*I*(d*x+c))+3248*I*exp(4*I*(d*x+c))+455*exp(5*I*(d*x+c))-8 96*I*exp(2*I*(d*x+c))+2380*exp(3*I*(d*x+c))+368*I-245*exp(I*(d*x+c)))/d/(e xp(2*I*(d*x+c))-1)^7+9/16*a^3/d*ln(exp(I*(d*x+c))-1)-9/16*a^3/d*ln(exp(I*( d*x+c))+1)
Time = 0.09 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.65 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {736 \, a^{3} \cos \left (d x + c\right )^{7} - 896 \, a^{3} \cos \left (d x + c\right )^{5} + 315 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 70 \, {\left (7 \, a^{3} \cos \left (d x + c\right )^{5} - 24 \, a^{3} \cos \left (d x + c\right )^{3} + 9 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1120 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="frica s")
Output:
-1/1120*(736*a^3*cos(d*x + c)^7 - 896*a^3*cos(d*x + c)^5 + 315*(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)*sin(d*x + c) - 315*(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)*sin(d*x + c ) + 70*(7*a^3*cos(d*x + c)^5 - 24*a^3*cos(d*x + c)^3 + 9*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)*sin(d*x + c))
Timed out. \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*csc(d*x+c)**4*(a+a*sin(d*x+c))**3,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.37 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {35 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 70 \, a^{3} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {672 \, a^{3}}{\tan \left (d x + c\right )^{5}} - \frac {32 \, {\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{3}}{\tan \left (d x + c\right )^{7}}}{1120 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxim a")
Output:
1/1120*(35*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)) - 70*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)) - 672*a^3/tan(d*x + c)^5 - 32*(7*tan(d*x + c )^2 + 5)*a^3/tan(d*x + c)^7)/d
Time = 0.22 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.74 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 77 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 665 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2520 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {6534 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 945 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 665 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 455 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 77 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{4480 \, d} \] Input:
integrate(cot(d*x+c)^4*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac" )
Output:
1/4480*(5*a^3*tan(1/2*d*x + 1/2*c)^7 + 35*a^3*tan(1/2*d*x + 1/2*c)^6 + 77* a^3*tan(1/2*d*x + 1/2*c)^5 - 35*a^3*tan(1/2*d*x + 1/2*c)^4 - 455*a^3*tan(1 /2*d*x + 1/2*c)^3 - 665*a^3*tan(1/2*d*x + 1/2*c)^2 + 2520*a^3*log(abs(tan( 1/2*d*x + 1/2*c))) + 945*a^3*tan(1/2*d*x + 1/2*c) - (6534*a^3*tan(1/2*d*x + 1/2*c)^7 + 945*a^3*tan(1/2*d*x + 1/2*c)^6 - 665*a^3*tan(1/2*d*x + 1/2*c) ^5 - 455*a^3*tan(1/2*d*x + 1/2*c)^4 - 35*a^3*tan(1/2*d*x + 1/2*c)^3 + 77*a ^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3*tan(1/2*d*x + 1/2*c) + 5*a^3)/tan(1/2*d *x + 1/2*c)^7)/d
Time = 18.73 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.58 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3\,\left (5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+35\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+77\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-455\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-665\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+665\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+455\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-77\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2520\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{4480\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int((cot(c + d*x)^4*(a + a*sin(c + d*x))^3)/sin(c + d*x)^4,x)
Output:
(a^3*(5*sin(c/2 + (d*x)/2)^14 - 5*cos(c/2 + (d*x)/2)^14 + 35*cos(c/2 + (d* x)/2)*sin(c/2 + (d*x)/2)^13 - 35*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) + 77*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 35*cos(c/2 + (d*x)/2)^3* sin(c/2 + (d*x)/2)^11 - 455*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 - 6 65*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 + 945*cos(c/2 + (d*x)/2)^6*si n(c/2 + (d*x)/2)^8 - 945*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 665*c os(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 + 455*cos(c/2 + (d*x)/2)^10*sin(c /2 + (d*x)/2)^4 + 35*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 - 77*cos(c /2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 2520*log(sin(c/2 + (d*x)/2)/cos(c/ 2 + (d*x)/2))*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7))/(4480*d*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^7)
Time = 0.17 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.93 \[ \int \cot ^4(c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^{3} \left (-368 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}+245 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+656 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+350 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-208 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-280 \cos \left (d x +c \right ) \sin \left (d x +c \right )-80 \cos \left (d x +c \right )+315 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{7}\right )}{560 \sin \left (d x +c \right )^{7} d} \] Input:
int(cot(d*x+c)^4*csc(d*x+c)^4*(a+a*sin(d*x+c))^3,x)
Output:
(a**3*( - 368*cos(c + d*x)*sin(c + d*x)**6 + 245*cos(c + d*x)*sin(c + d*x) **5 + 656*cos(c + d*x)*sin(c + d*x)**4 + 350*cos(c + d*x)*sin(c + d*x)**3 - 208*cos(c + d*x)*sin(c + d*x)**2 - 280*cos(c + d*x)*sin(c + d*x) - 80*co s(c + d*x) + 315*log(tan((c + d*x)/2))*sin(c + d*x)**7))/(560*sin(c + d*x) **7*d)