Integrand size = 29, antiderivative size = 124 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\text {arctanh}(\cos (c+d x))}{16 a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}+\frac {\cot (c+d x) \csc (c+d x)}{16 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d} \]
1/16*arctanh(cos(d*x+c))/a/d+1/3*cot(d*x+c)^3/a/d+1/5*cot(d*x+c)^5/a/d+1/1 6*cot(d*x+c)*csc(d*x+c)/a/d+1/24*cot(d*x+c)*csc(d*x+c)^3/a/d-1/6*cot(d*x+c )*csc(d*x+c)^5/a/d
Time = 1.13 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.85 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^6(c+d x) \left (1140 \cos (c+d x)+170 \cos (3 (c+d x))-30 \cos (5 (c+d x))-150 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+225 \cos (2 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-90 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+15 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+150 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-225 \cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+90 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-15 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-480 \sin (2 (c+d x))-192 \sin (4 (c+d x))+32 \sin (6 (c+d x))\right )}{7680 a d} \]
-1/7680*(Csc[c + d*x]^6*(1140*Cos[c + d*x] + 170*Cos[3*(c + d*x)] - 30*Cos [5*(c + d*x)] - 150*Log[Cos[(c + d*x)/2]] + 225*Cos[2*(c + d*x)]*Log[Cos[( c + d*x)/2]] - 90*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 15*Cos[6*(c + d *x)]*Log[Cos[(c + d*x)/2]] + 150*Log[Sin[(c + d*x)/2]] - 225*Cos[2*(c + d* x)]*Log[Sin[(c + d*x)/2]] + 90*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 15 *Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 480*Sin[2*(c + d*x)] - 192*Sin[4 *(c + d*x)] + 32*Sin[6*(c + d*x)]))/(a*d)
Time = 0.70 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {3042, 3318, 3042, 3087, 244, 2009, 3091, 3042, 4255, 3042, 4255, 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.
| \(\displaystyle \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^4}{\sin (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) \csc ^5(c+d x)dx}{a}-\frac {\int \cot ^2(c+d x) \csc ^4(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^4 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}\) |
| \(\Big \downarrow \) 3087 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\int \cot ^2(c+d x) \left (\cot ^2(c+d x)+1\right )d(-\cot (c+d x))}{a d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {\int \left (\cot ^4(c+d x)+\cot ^2(c+d x)\right )d(-\cot (c+d x))}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right )^5 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {1}{6} \int \csc ^5(c+d x)dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{6} \int \csc (c+d x)^5dx-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \int \csc ^3(c+d x)dx\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \int \csc (c+d x)^3dx\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )\right )-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 d}}{a}-\frac {-\frac {1}{5} \cot ^5(c+d x)-\frac {1}{3} \cot ^3(c+d x)}{a d}\) |
-((-1/3*Cot[c + d*x]^3 - Cot[c + d*x]^5/5)/(a*d)) + (-1/6*(Cot[c + d*x]*Cs c[c + d*x]^5)/d + ((Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) - (3*(-1/2*ArcTanh[ Cos[c + d*x]]/d - (Cot[c + d*x]*Csc[c + d*x])/(2*d)))/4)/6)/a
3.5.19.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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[g^2/a Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[g^2/(b*d) Int [(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.35
| method | result | size |
| risch | \(-\frac {15 \,{\mathrm e}^{11 i \left (d x +c \right )}-480 i {\mathrm e}^{8 i \left (d x +c \right )}-85 \,{\mathrm e}^{9 i \left (d x +c \right )}+320 i {\mathrm e}^{6 i \left (d x +c \right )}-570 \,{\mathrm e}^{7 i \left (d x +c \right )}-570 \,{\mathrm e}^{5 i \left (d x +c \right )}+192 i {\mathrm e}^{2 i \left (d x +c \right )}-85 \,{\mathrm e}^{3 i \left (d x +c \right )}-32 i+15 \,{\mathrm e}^{i \left (d x +c \right )}}{120 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d a}\) | \(168\) |
| parallelrisch | \(\frac {5 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-120 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )-120 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1920 d a}\) | \(174\) |
| derivativedivides | \(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d a}\) | \(176\) |
| default | \(\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-4 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {4}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{64 d a}\) | \(176\) |
| norman | \(\frac {-\frac {1}{384 a d}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1920 d a}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{640 d a}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}+\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}-\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}+\frac {7 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d a}-\frac {7 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{384 d a}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{640 d a}-\frac {7 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1920 d a}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d a}-\frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a d}\) | \(280\) |
-1/120*(15*exp(11*I*(d*x+c))-480*I*exp(8*I*(d*x+c))-85*exp(9*I*(d*x+c))+32 0*I*exp(6*I*(d*x+c))-570*exp(7*I*(d*x+c))-570*exp(5*I*(d*x+c))+192*I*exp(2 *I*(d*x+c))-85*exp(3*I*(d*x+c))-32*I+15*exp(I*(d*x+c)))/a/d/(exp(2*I*(d*x+ c))-1)^6-1/16/d/a*ln(exp(I*(d*x+c))-1)+1/16/d/a*ln(exp(I*(d*x+c))+1)
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.52 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {30 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 15 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (2 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 30 \, \cos \left (d x + c\right )}{480 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
-1/480*(30*cos(d*x + c)^5 - 80*cos(d*x + c)^3 - 15*(cos(d*x + c)^6 - 3*cos (d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) + 15*(cos( d*x + c)^6 - 3*cos(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(-1/2*cos(d*x + c ) + 1/2) - 32*(2*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*sin(d*x + c) - 30*cos( d*x + c))/(a*d*cos(d*x + c)^6 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^ 2 - a*d)
Timed out. \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (112) = 224\).
Time = 0.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.21 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {120 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {12 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {120 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a \sin \left (d x + c\right )^{6}}}{1920 \, d} \]
1/1920*((120*sin(d*x + c)/(cos(d*x + c) + 1) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 20*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^4/( cos(d*x + c) + 1)^4 - 12*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)/a - 120*log(sin(d*x + c)/(cos(d*x + c) + 1))/a + (12*sin(d*x + c)/(cos(d*x + c) + 1) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 20*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^4/(cos(d* x + c) + 1)^4 - 120*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos(d*x + c) + 1)^6/(a*sin(d*x + c)^6))/d
Time = 0.45 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.74 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {120 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 20 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {294 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
-1/1920*(120*log(abs(tan(1/2*d*x + 1/2*c)))/a - (5*a^5*tan(1/2*d*x + 1/2*c )^6 - 12*a^5*tan(1/2*d*x + 1/2*c)^5 + 15*a^5*tan(1/2*d*x + 1/2*c)^4 - 20*a ^5*tan(1/2*d*x + 1/2*c)^3 - 15*a^5*tan(1/2*d*x + 1/2*c)^2 + 120*a^5*tan(1/ 2*d*x + 1/2*c))/a^6 - (294*tan(1/2*d*x + 1/2*c)^6 - 120*tan(1/2*d*x + 1/2* c)^5 + 15*tan(1/2*d*x + 1/2*c)^4 + 20*tan(1/2*d*x + 1/2*c)^3 - 15*tan(1/2* d*x + 1/2*c)^2 + 12*tan(1/2*d*x + 1/2*c) - 5)/(a*tan(1/2*d*x + 1/2*c)^6))/ d
Time = 10.46 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73 \[ \int \frac {\cot ^4(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+12\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-12\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+20\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-20\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+120\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]
-(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 + 12*cos(c/2 + (d*x)/2 )*sin(c/2 + (d*x)/2)^11 - 12*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 15 *cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 20*cos(c/2 + (d*x)/2)^3*sin( c/2 + (d*x)/2)^9 + 15*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 120*cos( c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 120*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2)^5 - 15*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 20*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 15*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x) /2)^2 + 120*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(c/2 + (d*x)/2)^ 6*sin(c/2 + (d*x)/2)^6)/(1920*a*d*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^ 6)