Integrand size = 29, antiderivative size = 182 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {45 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d} \] Output:
45/128*a^2*arctanh(cos(d*x+c))/d-2/7*a^2*cot(d*x+c)^7/d-35/128*a^2*cot(d*x +c)*csc(d*x+c)/d+5/24*a^2*cot(d*x+c)^3*csc(d*x+c)/d-1/6*a^2*cot(d*x+c)^5*c sc(d*x+c)/d-5/64*a^2*cot(d*x+c)*csc(d*x+c)^3/d+5/48*a^2*cot(d*x+c)^3*csc(d *x+c)^3/d-1/8*a^2*cot(d*x+c)^5*csc(d*x+c)^3/d
Leaf count is larger than twice the leaf count of optimal. \(401\) vs. \(2(182)=364\).
Time = 0.52 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.20 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=a^2 \left (\frac {\cot \left (\frac {1}{2} (c+d x)\right )}{7 d}-\frac {83 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {19 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {17 \csc ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}+\frac {5 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {\csc ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^6\left (\frac {1}{2} (c+d x)\right )}{448 d}-\frac {\csc ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}+\frac {45 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}-\frac {45 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{128 d}+\frac {83 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{512 d}-\frac {17 \sec ^4\left (\frac {1}{2} (c+d x)\right )}{1024 d}-\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right )}{512 d}+\frac {\sec ^8\left (\frac {1}{2} (c+d x)\right )}{2048 d}-\frac {\tan \left (\frac {1}{2} (c+d x)\right )}{7 d}+\frac {19 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{224 d}-\frac {5 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{224 d}+\frac {\sec ^6\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{448 d}\right ) \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]
Output:
a^2*(Cot[(c + d*x)/2]/(7*d) - (83*Csc[(c + d*x)/2]^2)/(512*d) - (19*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(224*d) + (17*Csc[(c + d*x)/2]^4)/(1024*d) + (5*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^4)/(224*d) + Csc[(c + d*x)/2]^6/(51 2*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^6)/(448*d) - Csc[(c + d*x)/2]^8/ (2048*d) + (45*Log[Cos[(c + d*x)/2]])/(128*d) - (45*Log[Sin[(c + d*x)/2]]) /(128*d) + (83*Sec[(c + d*x)/2]^2)/(512*d) - (17*Sec[(c + d*x)/2]^4)/(1024 *d) - Sec[(c + d*x)/2]^6/(512*d) + Sec[(c + d*x)/2]^8/(2048*d) - Tan[(c + d*x)/2]/(7*d) + (19*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(224*d) - (5*Sec[ (c + d*x)/2]^4*Tan[(c + d*x)/2])/(224*d) + (Sec[(c + d*x)/2]^6*Tan[(c + d* x)/2])/(448*d))
Time = 0.57 (sec) , antiderivative size = 182, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^3(c+d x) (a \sin (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)^2}{\sin (c+d x)^9}dx\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \int \left (a^2 \cot ^6(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^6(c+d x) \csc ^2(c+d x)+a^2 \cot ^6(c+d x) \csc (c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {45 a^2 \text {arctanh}(\cos (c+d x))}{128 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x) \csc ^3(c+d x)}{8 d}-\frac {a^2 \cot ^5(c+d x) \csc (c+d x)}{6 d}+\frac {5 a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{48 d}+\frac {5 a^2 \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {5 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {35 a^2 \cot (c+d x) \csc (c+d x)}{128 d}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]
Output:
(45*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (3 5*a^2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d *x])/(24*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x])/(6*d) - (5*a^2*Cot[c + d*x ]*Csc[c + d*x]^3)/(64*d) + (5*a^2*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*d) - (a^2*Cot[c + d*x]^5*Csc[c + d*x]^3)/(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 = 0.39 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(\frac {a^{2} \left (581 \,{\mathrm e}^{15 i \left (d x +c \right )}-2065 \,{\mathrm e}^{13 i \left (d x +c \right )}-8960 i {\mathrm e}^{8 i \left (d x +c \right )}+21 \,{\mathrm e}^{11 i \left (d x +c \right )}+8960 i {\mathrm e}^{10 i \left (d x +c \right )}-5705 \,{\mathrm e}^{9 i \left (d x +c \right )}+1792 i {\mathrm e}^{14 i \left (d x +c \right )}-5705 \,{\mathrm e}^{7 i \left (d x +c \right )}+5376 i {\mathrm e}^{6 i \left (d x +c \right )}+21 \,{\mathrm e}^{5 i \left (d x +c \right )}-1792 i {\mathrm e}^{12 i \left (d x +c \right )}-2065 \,{\mathrm e}^{3 i \left (d x +c \right )}+256 i {\mathrm e}^{2 i \left (d x +c \right )}+581 \,{\mathrm e}^{i \left (d x +c \right )}-5376 i {\mathrm e}^{4 i \left (d x +c \right )}-256 i\right )}{448 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{8}}-\frac {45 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {45 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\) | \(238\) |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(255\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {2 a^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}+a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\) | \(255\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^3*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/448*a^2*(581*exp(15*I*(d*x+c))-2065*exp(13*I*(d*x+c))-8960*I*exp(8*I*(d* x+c))+21*exp(11*I*(d*x+c))+8960*I*exp(10*I*(d*x+c))-5705*exp(9*I*(d*x+c))+ 1792*I*exp(14*I*(d*x+c))-5705*exp(7*I*(d*x+c))+5376*I*exp(6*I*(d*x+c))+21* exp(5*I*(d*x+c))-1792*I*exp(12*I*(d*x+c))-2065*exp(3*I*(d*x+c))+256*I*exp( 2*I*(d*x+c))+581*exp(I*(d*x+c))-5376*I*exp(4*I*(d*x+c))-256*I)/d/(exp(2*I* (d*x+c))-1)^8-45/128*a^2/d*ln(exp(I*(d*x+c))-1)+45/128*a^2/d*ln(exp(I*(d*x +c))+1)
Time = 0.10 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.40 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {512 \, a^{2} \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 1162 \, a^{2} \cos \left (d x + c\right )^{7} + 3066 \, a^{2} \cos \left (d x + c\right )^{5} - 2310 \, a^{2} \cos \left (d x + c\right )^{3} + 630 \, a^{2} \cos \left (d x + c\right ) - 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{1792 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="frica s")
Output:
-1/1792*(512*a^2*cos(d*x + c)^7*sin(d*x + c) - 1162*a^2*cos(d*x + c)^7 + 3 066*a^2*cos(d*x + c)^5 - 2310*a^2*cos(d*x + c)^3 + 630*a^2*cos(d*x + c) - 315*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4* a^2*cos(d*x + c)^2 + a^2)*log(1/2*cos(d*x + c) + 1/2) + 315*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2))/(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. \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**3*(a+a*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.21 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {7 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 56 \, a^{2} {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1536 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{5376 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/5376*(7*a^2*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c) ^3 + 15*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) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) - 56*a^2*(2*(33*cos(d*x + c)^5 - 40*cos(d*x + c)^3 + 15*cos(d*x + c ))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1) + 15*log(cos (d*x + c) + 1) - 15*log(cos(d*x + c) - 1)) + 1536*a^2/tan(d*x + c)^7)/d
Time = 0.23 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {7 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 32 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 224 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1792 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 1120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {13698 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1792 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 672 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 280 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 224 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8}}}{14336 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^3*(a+a*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/14336*(7*a^2*tan(1/2*d*x + 1/2*c)^8 + 32*a^2*tan(1/2*d*x + 1/2*c)^7 - 22 4*a^2*tan(1/2*d*x + 1/2*c)^5 - 280*a^2*tan(1/2*d*x + 1/2*c)^4 + 672*a^2*ta n(1/2*d*x + 1/2*c)^3 + 1792*a^2*tan(1/2*d*x + 1/2*c)^2 - 5040*a^2*log(abs( tan(1/2*d*x + 1/2*c))) - 1120*a^2*tan(1/2*d*x + 1/2*c) + (13698*a^2*tan(1/ 2*d*x + 1/2*c)^8 + 1120*a^2*tan(1/2*d*x + 1/2*c)^7 - 1792*a^2*tan(1/2*d*x + 1/2*c)^6 - 672*a^2*tan(1/2*d*x + 1/2*c)^5 + 280*a^2*tan(1/2*d*x + 1/2*c) ^4 + 224*a^2*tan(1/2*d*x + 1/2*c)^3 - 32*a^2*tan(1/2*d*x + 1/2*c) - 7*a^2) /tan(1/2*d*x + 1/2*c)^8)/d
Time = 36.40 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.13 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2\,\left (7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-32\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+224\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-1792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-1120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+1792\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+672\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-280\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-224\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5040\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\right )}{14336\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x))^2)/sin(c + d*x)^3,x)
Output:
-(a^2*(7*cos(c/2 + (d*x)/2)^16 - 7*sin(c/2 + (d*x)/2)^16 - 32*cos(c/2 + (d *x)/2)*sin(c/2 + (d*x)/2)^15 + 32*cos(c/2 + (d*x)/2)^15*sin(c/2 + (d*x)/2) + 224*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^13 + 280*cos(c/2 + (d*x)/2) ^4*sin(c/2 + (d*x)/2)^12 - 672*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^11 - 1792*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 1120*cos(c/2 + (d*x)/2 )^7*sin(c/2 + (d*x)/2)^9 - 1120*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^7 + 1792*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^6 + 672*cos(c/2 + (d*x)/2) ^11*sin(c/2 + (d*x)/2)^5 - 280*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 - 224*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2)^3 + 5040*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)^8))/(1433 6*d*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8)
Time = 0.17 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.85 \[ \int \cot ^6(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^{2} \left (256 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-581 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+210 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+168 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-256 \cos \left (d x +c \right ) \sin \left (d x +c \right )-112 \cos \left (d x +c \right )-315 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{8}\right )}{896 \sin \left (d x +c \right )^{8} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^3*(a+a*sin(d*x+c))^2,x)
Output:
(a**2*(256*cos(c + d*x)*sin(c + d*x)**7 - 581*cos(c + d*x)*sin(c + d*x)**6 - 768*cos(c + d*x)*sin(c + d*x)**5 + 210*cos(c + d*x)*sin(c + d*x)**4 + 7 68*cos(c + d*x)*sin(c + d*x)**3 + 168*cos(c + d*x)*sin(c + d*x)**2 - 256*c os(c + d*x)*sin(c + d*x) - 112*cos(c + d*x) - 315*log(tan((c + d*x)/2))*si n(c + d*x)**8))/(896*sin(c + d*x)**8*d)