Integrand size = 25, antiderivative size = 128 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-a x+\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot (c+d x)}{d}+\frac {a \cot ^3(c+d x)}{3 d}-\frac {a \cot ^5(c+d x)}{5 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d} \] Output:
-a*x+5/16*a*arctanh(cos(d*x+c))/d-a*cot(d*x+c)/d+1/3*a*cot(d*x+c)^3/d-1/5* a*cot(d*x+c)^5/d-5/16*a*cot(d*x+c)*csc(d*x+c)/d+5/24*a*cot(d*x+c)^3*csc(d* x+c)/d-1/6*a*cot(d*x+c)^5*csc(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.51 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {11 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \cot ^5(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\tan ^2(c+d x)\right )}{5 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {11 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]
Output:
(-11*a*Csc[(c + d*x)/2]^2)/(64*d) + (a*Csc[(c + d*x)/2]^4)/(32*d) - (a*Csc [(c + d*x)/2]^6)/(384*d) - (a*Cot[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3 /2, -Tan[c + d*x]^2])/(5*d) + (5*a*Log[Cos[(c + d*x)/2]])/(16*d) - (5*a*Lo g[Sin[(c + d*x)/2]])/(16*d) + (11*a*Sec[(c + d*x)/2]^2)/(64*d) - (a*Sec[(c + d*x)/2]^4)/(32*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)
Time = 0.95 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 3317, 3042, 3091, 3042, 3091, 3042, 3091, 3042, 3954, 3042, 3954, 3042, 3954, 24, 4257}
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.
| \(\displaystyle \int \cot ^6(c+d x) \csc (c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)}{\sin (c+d x)^7}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cot ^6(c+d x)dx+a \int \cot ^6(c+d x) \csc (c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx+a \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx+a \left (-\frac {5}{6} \int \cot ^4(c+d x) \csc (c+d x)dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx+a \left (-\frac {5}{6} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx+a \left (-\frac {5}{6} \left (-\frac {3}{4} \int \cot ^2(c+d x) \csc (c+d x)dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx+a \left (-\frac {5}{6} \left (-\frac {3}{4} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx+a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx+a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle a \left (-\int \cot ^4(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}\right )+a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x)}{5 d}\right )+a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle a \left (\int \cot ^2(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}\right )+a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\int \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}\right )+a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle a \left (-\int 1dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}\right )+a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )+a \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )+a \left (-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x\right )\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]*(a + a*Sin[c + d*x]),x]
Output:
a*(-x - Cot[c + d*x]/d + Cot[c + d*x]^3/(3*d) - Cot[c + d*x]^5/(5*d)) + a* (-1/6*(Cot[c + d*x]^5*Csc[c + d*x])/d - (5*(-1/4*(Cot[c + d*x]^3*Csc[c + d *x])/d - (3*(ArcTanh[Cos[c + d*x]]/(2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2* d)))/4))/6)
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.15
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(147\) |
| default | \(\frac {a \left (-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )+a \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )}{d}\) | \(147\) |
| risch | \(-a x +\frac {a \left (165 \,{\mathrm e}^{11 i \left (d x +c \right )}+25 \,{\mathrm e}^{9 i \left (d x +c \right )}-720 i {\mathrm e}^{10 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}+2160 i {\mathrm e}^{8 i \left (d x +c \right )}+450 \,{\mathrm e}^{5 i \left (d x +c \right )}-3680 i {\mathrm e}^{6 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}+3360 i {\mathrm e}^{4 i \left (d x +c \right )}+165 \,{\mathrm e}^{i \left (d x +c \right )}-1488 i {\mathrm e}^{2 i \left (d x +c \right )}+368 i\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\) | \(190\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(-1/5*cot(d*x+c)^5+1/3*cot(d*x+c)^3-cot(d*x+c)-d*x-c)+a*(-1/6/sin(d *x+c)^6*cos(d*x+c)^7+1/24/sin(d*x+c)^4*cos(d*x+c)^7-1/16/sin(d*x+c)^2*cos( d*x+c)^7-1/16*cos(d*x+c)^5-5/48*cos(d*x+c)^3-5/16*cos(d*x+c)-5/16*ln(csc(d *x+c)-cot(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (116) = 232\).
Time = 0.10 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.98 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {480 \, a d x \cos \left (d x + c\right )^{6} - 1440 \, a d x \cos \left (d x + c\right )^{4} - 330 \, a \cos \left (d x + c\right )^{5} + 1440 \, a d x \cos \left (d x + c\right )^{2} + 400 \, a \cos \left (d x + c\right )^{3} - 480 \, a d x - 150 \, a \cos \left (d x + c\right ) - 75 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 75 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (23 \, a \cos \left (d x + c\right )^{5} - 35 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/480*(480*a*d*x*cos(d*x + c)^6 - 1440*a*d*x*cos(d*x + c)^4 - 330*a*cos(d *x + c)^5 + 1440*a*d*x*cos(d*x + c)^2 + 400*a*cos(d*x + c)^3 - 480*a*d*x - 150*a*cos(d*x + c) - 75*(a*cos(d*x + c)^6 - 3*a*cos(d*x + c)^4 + 3*a*cos( d*x + c)^2 - a)*log(1/2*cos(d*x + c) + 1/2) + 75*(a*cos(d*x + c)^6 - 3*a*c os(d*x + c)^4 + 3*a*cos(d*x + c)^2 - a)*log(-1/2*cos(d*x + c) + 1/2) - 32* (23*a*cos(d*x + c)^5 - 35*a*cos(d*x + c)^3 + 15*a*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)
\[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=a \left (\int \cot ^{6}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin {\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx\right ) \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)*(a+a*sin(d*x+c)),x)
Output:
a*(Integral(cot(c + d*x)**6*csc(c + d*x), x) + Integral(sin(c + d*x)*cot(c + d*x)**6*csc(c + d*x), x))
Time = 0.13 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.07 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=-\frac {32 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a - 5 \, a {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/480*(32*(15*d*x + 15*c + (15*tan(d*x + c)^4 - 5*tan(d*x + c)^2 + 3)/tan (d*x + c)^5)*a - 5*a*(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*lo g(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d
Time = 0.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.62 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1920 \, {\left (d x + c\right )} a - 600 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 1320 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {1470 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1320 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/1920*(5*a*tan(1/2*d*x + 1/2*c)^6 + 12*a*tan(1/2*d*x + 1/2*c)^5 - 45*a*ta n(1/2*d*x + 1/2*c)^4 - 140*a*tan(1/2*d*x + 1/2*c)^3 + 225*a*tan(1/2*d*x + 1/2*c)^2 - 1920*(d*x + c)*a - 600*a*log(abs(tan(1/2*d*x + 1/2*c))) + 1320* a*tan(1/2*d*x + 1/2*c) + (1470*a*tan(1/2*d*x + 1/2*c)^6 - 1320*a*tan(1/2*d *x + 1/2*c)^5 - 225*a*tan(1/2*d*x + 1/2*c)^4 + 140*a*tan(1/2*d*x + 1/2*c)^ 3 + 45*a*tan(1/2*d*x + 1/2*c)^2 - 12*a*tan(1/2*d*x + 1/2*c) - 5*a)/tan(1/2 *d*x + 1/2*c)^6)/d
Time = 34.32 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.23 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {11\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {5\,a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{16\,d}-\frac {11\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {2\,a\,\mathrm {atan}\left (\frac {16\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+5\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-16\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {15\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {3\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x)))/sin(c + d*x),x)
Output:
(11*a*tan(c/2 + (d*x)/2))/(16*d) - (5*a*log(sin(c/2 + (d*x)/2)/cos(c/2 + ( d*x)/2)))/(16*d) - (11*a*cot(c/2 + (d*x)/2))/(16*d) - (2*a*atan((16*cos(c/ 2 + (d*x)/2) + 5*sin(c/2 + (d*x)/2))/(5*cos(c/2 + (d*x)/2) - 16*sin(c/2 + (d*x)/2))))/d - (15*a*cot(c/2 + (d*x)/2)^2)/(128*d) + (7*a*cot(c/2 + (d*x) /2)^3)/(96*d) + (3*a*cot(c/2 + (d*x)/2)^4)/(128*d) - (a*cot(c/2 + (d*x)/2) ^5)/(160*d) - (a*cot(c/2 + (d*x)/2)^6)/(384*d) + (15*a*tan(c/2 + (d*x)/2)^ 2)/(128*d) - (7*a*tan(c/2 + (d*x)/2)^3)/(96*d) - (3*a*tan(c/2 + (d*x)/2)^4 )/(128*d) + (a*tan(c/2 + (d*x)/2)^5)/(160*d) + (a*tan(c/2 + (d*x)/2)^6)/(3 84*d)
Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04 \[ \int \cot ^6(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (-368 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-165 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-48 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40 \cos \left (d x +c \right )-75 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}-240 \sin \left (d x +c \right )^{6} d x \right )}{240 \sin \left (d x +c \right )^{6} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)*(a+a*sin(d*x+c)),x)
Output:
(a*( - 368*cos(c + d*x)*sin(c + d*x)**5 - 165*cos(c + d*x)*sin(c + d*x)**4 + 176*cos(c + d*x)*sin(c + d*x)**3 + 130*cos(c + d*x)*sin(c + d*x)**2 - 4 8*cos(c + d*x)*sin(c + d*x) - 40*cos(c + d*x) - 75*log(tan((c + d*x)/2))*s in(c + d*x)**6 - 240*sin(c + d*x)**6*d*x))/(240*sin(c + d*x)**6*d)