Integrand size = 27, antiderivative size = 96 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a \text {arctanh}(\cos (c+d x))}{16 d}-\frac {a \cot ^7(c+d x)}{7 d}-\frac {5 a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {5 a \cot ^3(c+d x) \csc (c+d x)}{24 d}-\frac {a \cot ^5(c+d x) \csc (c+d x)}{6 d} \] Output:
5/16*a*arctanh(cos(d*x+c))/d-1/7*a*cot(d*x+c)^7/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
Time = 0.03 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.82 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a \cot ^7(c+d x)}{7 d}-\frac {11 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}-\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}+\frac {11 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
-1/7*(a*Cot[c + d*x]^7)/d - (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) + (5*a*Log[Cos[(c + d* x)/2]])/(16*d) - (5*a*Log[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 = 0.67 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3317, 3042, 3087, 15, 3091, 3042, 3091, 3042, 3091, 3042, 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.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^2(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a \sin (c+d x)+a)}{\sin (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cot ^6(c+d x) \csc ^2(c+d x)dx+a \int \cot ^6(c+d x) \csc (c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx+a \int \sec \left (c+d x-\frac {\pi }{2}\right )^2 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {a \int \cot ^6(c+d x)d(-\cot (c+d x))}{d}+a \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle a \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx-\frac {a \cot ^7(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \left (-\frac {5}{6} \int \cot ^4(c+d x) \csc (c+d x)dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )-\frac {a \cot ^7(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {5}{6} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )-\frac {a \cot ^7(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \left (-\frac {5}{6} \left (-\frac {3}{4} \int \cot ^2(c+d x) \csc (c+d x)dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )-\frac {a \cot ^7(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {5}{6} \left (-\frac {3}{4} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )-\frac {a \cot ^7(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )-\frac {a \cot ^7(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )-\frac {a \cot ^7(c+d x)}{7 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle a \left (-\frac {5}{6} \left (-\frac {3}{4} \left (\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}\right )-\frac {a \cot ^7(c+d x)}{7 d}\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
-1/7*(a*Cot[c + d*x]^7)/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_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S ymbol] :> Simp[1/f Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] && !(IntegerQ[(n - 1) /2] && LtQ[0, n, m - 1])
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[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.35 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(128\) |
| default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{7}}{6 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{24 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{16 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{16}-\frac {5 \cos \left (d x +c \right )^{3}}{48}-\frac {5 \cos \left (d x +c \right )}{16}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16}\right )-\frac {a \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(128\) |
| risch | \(\frac {a \left (336 i {\mathrm e}^{12 i \left (d x +c \right )}+231 \,{\mathrm e}^{13 i \left (d x +c \right )}-196 \,{\mathrm e}^{11 i \left (d x +c \right )}+1680 i {\mathrm e}^{8 i \left (d x +c \right )}+595 \,{\mathrm e}^{9 i \left (d x +c \right )}+1008 i {\mathrm e}^{4 i \left (d x +c \right )}-595 \,{\mathrm e}^{5 i \left (d x +c \right )}+196 \,{\mathrm e}^{3 i \left (d x +c \right )}+48 i-231 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{168 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}+\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d}-\frac {5 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d}\) | \(162\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(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)))-1/7*a/sin(d*x+c)^7*cos(d*x+c)^7)
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (86) = 172\).
Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.19 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {96 \, a \cos \left (d x + c\right )^{7} + 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 105 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 14 \, {\left (33 \, a \cos \left (d x + c\right )^{5} - 40 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{672 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/672*(96*a*cos(d*x + c)^7 + 105*(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)*sin(d*x + c) - 105*(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)*sin(d*x + c) + 14*(33*a*cos(d*x + c)^5 - 40*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)*sin(d*x + c))
Timed out. \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**2*(a+a*sin(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.10 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {96 \, a \cot \left (d x + c\right )^{7} - 7 \, a {\left (\frac {2 \, {\left (33 \, \cos \left (d x + c\right )^{5} - 40 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{672 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
-1/672*(96*a*cot(d*x + c)^7 - 7*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*log(cos(d*x + c) + 1) - 15*log(cos(d*x + c) - 1)))/d
Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (86) = 172\).
Time = 0.18 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.38 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 7 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 840 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2178 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 315 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 63 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{2688 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/2688*(3*a*tan(1/2*d*x + 1/2*c)^7 + 7*a*tan(1/2*d*x + 1/2*c)^6 - 21*a*tan (1/2*d*x + 1/2*c)^5 - 63*a*tan(1/2*d*x + 1/2*c)^4 + 63*a*tan(1/2*d*x + 1/2 *c)^3 + 315*a*tan(1/2*d*x + 1/2*c)^2 - 840*a*log(abs(tan(1/2*d*x + 1/2*c)) ) - 105*a*tan(1/2*d*x + 1/2*c) + (2178*a*tan(1/2*d*x + 1/2*c)^7 + 105*a*ta n(1/2*d*x + 1/2*c)^6 - 315*a*tan(1/2*d*x + 1/2*c)^5 - 63*a*tan(1/2*d*x + 1 /2*c)^4 + 63*a*tan(1/2*d*x + 1/2*c)^3 + 21*a*tan(1/2*d*x + 1/2*c)^2 - 7*a* tan(1/2*d*x + 1/2*c) - 3*a)/tan(1/2*d*x + 1/2*c)^7)/d
Time = 34.51 (sec) , antiderivative size = 385, normalized size of antiderivative = 4.01 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {a\,\left (3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+7\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+315\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-63\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\right )}{2688\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int((cot(c + d*x)^6*(a + a*sin(c + d*x)))/sin(c + d*x)^2,x)
Output:
-(a*(3*cos(c/2 + (d*x)/2)^14 - 3*sin(c/2 + (d*x)/2)^14 - 7*cos(c/2 + (d*x) /2)*sin(c/2 + (d*x)/2)^13 + 7*cos(c/2 + (d*x)/2)^13*sin(c/2 + (d*x)/2) + 2 1*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 + 63*cos(c/2 + (d*x)/2)^3*sin (c/2 + (d*x)/2)^11 - 63*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^10 - 315*c os(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9 + 105*cos(c/2 + (d*x)/2)^6*sin(c/ 2 + (d*x)/2)^8 - 105*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 315*cos(c /2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^5 + 63*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 - 63*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2)^3 - 21*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 840*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))/(2688*d*cos(c/2 + (d*x )/2)^7*sin(c/2 + (d*x)/2)^7)
Time = 0.15 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.43 \[ \int \cot ^6(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (48 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-231 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+182 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+144 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-56 \cos \left (d x +c \right ) \sin \left (d x +c \right )-48 \cos \left (d x +c \right )-105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{7}\right )}{336 \sin \left (d x +c \right )^{7} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^2*(a+a*sin(d*x+c)),x)
Output:
(a*(48*cos(c + d*x)*sin(c + d*x)**6 - 231*cos(c + d*x)*sin(c + d*x)**5 - 1 44*cos(c + d*x)*sin(c + d*x)**4 + 182*cos(c + d*x)*sin(c + d*x)**3 + 144*c os(c + d*x)*sin(c + d*x)**2 - 56*cos(c + d*x)*sin(c + d*x) - 48*cos(c + d* x) - 105*log(tan((c + d*x)/2))*sin(c + d*x)**7))/(336*sin(c + d*x)**7*d)