Integrand size = 27, antiderivative size = 142 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x}{a}+\frac {5 \text {arctanh}(\cos (c+d x))}{16 a d}+\frac {\cot (c+d x)}{a d}-\frac {\cot ^3(c+d x)}{3 a d}+\frac {\cot ^5(c+d x)}{5 a d}-\frac {5 \cot (c+d x) \csc (c+d x)}{16 a d}+\frac {5 \cot ^3(c+d x) \csc (c+d x)}{24 a d}-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 a d} \] Output:
x/a+5/16*arctanh(cos(d*x+c))/a/d+cot(d*x+c)/a/d-1/3*cot(d*x+c)^3/a/d+1/5*c ot(d*x+c)^5/a/d-5/16*cot(d*x+c)*csc(d*x+c)/a/d+5/24*cot(d*x+c)^3*csc(d*x+c )/a/d-1/6*cot(d*x+c)^5*csc(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(317\) vs. \(2(142)=284\).
Time = 2.78 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.23 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^6(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (-2400 c-2400 d x+900 \cos (c+d x)+50 \cos (3 (c+d x))-1440 c \cos (4 (c+d x))-1440 d x \cos (4 (c+d x))+330 \cos (5 (c+d x))+240 c \cos (6 (c+d x))+240 d x \cos (6 (c+d x))-750 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-450 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+75 \cos (6 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+225 \cos (2 (c+d x)) \left (16 (c+d x)+5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+750 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+450 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-75 \cos (6 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1200 \sin (2 (c+d x))+768 \sin (4 (c+d x))-368 \sin (6 (c+d x))\right )}{7680 a d (1+\sin (c+d x))} \] Input:
Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]
Output:
-1/7680*(Csc[c + d*x]^6*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(-2400*c - 2400*d*x + 900*Cos[c + d*x] + 50*Cos[3*(c + d*x)] - 1440*c*Cos[4*(c + d*x )] - 1440*d*x*Cos[4*(c + d*x)] + 330*Cos[5*(c + d*x)] + 240*c*Cos[6*(c + d *x)] + 240*d*x*Cos[6*(c + d*x)] - 750*Log[Cos[(c + d*x)/2]] - 450*Cos[4*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 75*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 225*Cos[2*(c + d*x)]*(16*(c + d*x) + 5*Log[Cos[(c + d*x)/2]] - 5*Log[Si n[(c + d*x)/2]]) + 750*Log[Sin[(c + d*x)/2]] + 450*Cos[4*(c + d*x)]*Log[Si n[(c + d*x)/2]] - 75*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1200*Sin[2*( c + d*x)] + 768*Sin[4*(c + d*x)] - 368*Sin[6*(c + d*x)]))/(a*d*(1 + Sin[c + d*x]))
Time = 0.97 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 3318, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^7 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^6(c+d x) \csc (c+d x)dx}{a}-\frac {\int \cot ^6(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {5}{6} \int \cot ^4(c+d x) \csc (c+d x)dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{6} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \int \cot ^2(c+d x) \csc (c+d x)dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {-\int \cot ^4(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {-\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x)}{5 d}}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \cot ^2(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {-\int 1dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {5}{6} \left (-\frac {3}{4} \left (\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}\right )-\frac {\cot ^5(c+d x) \csc (c+d x)}{6 d}}{a}-\frac {-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x}{a}\) |
Input:
Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x]),x]
Output:
-((-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)/a
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[((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 = 0.85 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.32
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-44 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {15}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {44}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}\) | \(188\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}+\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}-44 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {2}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {3}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {15}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {44}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d a}\) | \(188\) |
| risch | \(\frac {x}{a}+\frac {720 i {\mathrm e}^{10 i \left (d x +c \right )}+165 \,{\mathrm e}^{11 i \left (d x +c \right )}-2160 i {\mathrm e}^{8 i \left (d x +c \right )}+25 \,{\mathrm e}^{9 i \left (d x +c \right )}+3680 i {\mathrm e}^{6 i \left (d x +c \right )}+450 \,{\mathrm e}^{7 i \left (d x +c \right )}-3360 i {\mathrm e}^{4 i \left (d x +c \right )}+450 \,{\mathrm e}^{5 i \left (d x +c \right )}+1488 i {\mathrm e}^{2 i \left (d x +c \right )}+25 \,{\mathrm e}^{3 i \left (d x +c \right )}-368 i+165 \,{\mathrm e}^{i \left (d x +c \right )}}{120 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d a}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d a}\) | \(197\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/64/d/a*(1/6*tan(1/2*d*x+1/2*c)^6-2/5*tan(1/2*d*x+1/2*c)^5-3/2*tan(1/2*d* x+1/2*c)^4+14/3*tan(1/2*d*x+1/2*c)^3+15/2*tan(1/2*d*x+1/2*c)^2-44*tan(1/2* d*x+1/2*c)+128*arctan(tan(1/2*d*x+1/2*c))-1/6/tan(1/2*d*x+1/2*c)^6+2/5/tan (1/2*d*x+1/2*c)^5+3/2/tan(1/2*d*x+1/2*c)^4-14/3/tan(1/2*d*x+1/2*c)^3-15/2/ tan(1/2*d*x+1/2*c)^2+44/tan(1/2*d*x+1/2*c)-20*ln(tan(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.66 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {480 \, d x \cos \left (d x + c\right )^{6} - 1440 \, d x \cos \left (d x + c\right )^{4} + 330 \, \cos \left (d x + c\right )^{5} + 1440 \, d x \cos \left (d x + c\right )^{2} - 400 \, \cos \left (d x + c\right )^{3} - 480 \, d x + 75 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 75 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (23 \, \cos \left (d x + c\right )^{5} - 35 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 150 \, \cos \left (d x + c\right )}{480 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - a d\right )}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/480*(480*d*x*cos(d*x + c)^6 - 1440*d*x*cos(d*x + c)^4 + 330*cos(d*x + c) ^5 + 1440*d*x*cos(d*x + c)^2 - 400*cos(d*x + c)^3 - 480*d*x + 75*(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) - 75*(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*(23*cos(d*x + c)^5 - 35*cos(d*x + c)^3 + 15*co s(d*x + c))*sin(d*x + c) + 150*cos(d*x + c))/(a*d*cos(d*x + c)^6 - 3*a*d*c os(d*x + c)^4 + 3*a*d*cos(d*x + c)^2 - a*d)
\[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cos {\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**7/(a+a*sin(d*x+c)),x)
Output:
Integral(cos(c + d*x)*cot(c + d*x)**7/(sin(c + d*x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (130) = 260\).
Time = 0.13 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.10 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {1320 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {225 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {45 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {12 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a} - \frac {3840 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {600 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {{\left (\frac {12 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {45 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {225 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1320 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a \sin \left (d x + c\right )^{6}}}{1920 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/1920*((1320*sin(d*x + c)/(cos(d*x + c) + 1) - 225*sin(d*x + c)^2/(cos(d *x + c) + 1)^2 - 140*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 45*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 - 3840*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 600*log(sin(d*x + c)/(cos(d*x + c) + 1))/a - (12*sin(d*x + c)/( cos(d*x + c) + 1) + 45*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 225*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 132 0*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 = 0.18 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.58 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {1920 \, {\left (d x + c\right )}}{a} - \frac {600 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {5 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 225 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1320 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} + \frac {1470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1320 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/1920*(1920*(d*x + c)/a - 600*log(abs(tan(1/2*d*x + 1/2*c)))/a + (5*a^5*t an(1/2*d*x + 1/2*c)^6 - 12*a^5*tan(1/2*d*x + 1/2*c)^5 - 45*a^5*tan(1/2*d*x + 1/2*c)^4 + 140*a^5*tan(1/2*d*x + 1/2*c)^3 + 225*a^5*tan(1/2*d*x + 1/2*c )^2 - 1320*a^5*tan(1/2*d*x + 1/2*c))/a^6 + (1470*tan(1/2*d*x + 1/2*c)^6 + 1320*tan(1/2*d*x + 1/2*c)^5 - 225*tan(1/2*d*x + 1/2*c)^4 - 140*tan(1/2*d*x + 1/2*c)^3 + 45*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 = 33.10 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.91 \[ \int \frac {\cos (c+d x) \cot ^7(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 )+45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-225\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-1320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+225\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3840\,\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 )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+600\,\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} \] Input:
int((cos(c + d*x)*cot(c + d*x)^7)/(a + a*sin(c + d*x)),x)
Output:
-(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) + 45 *cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 - 140*cos(c/2 + (d*x)/2)^3*sin (c/2 + (d*x)/2)^9 - 225*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 + 1320*c os(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 - 1320*cos(c/2 + (d*x)/2)^7*sin(c /2 + (d*x)/2)^5 + 225*cos(c/2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 + 140*cos( c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 - 45*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 3840*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)))*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6 + 600*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)
Time = 33.47 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {368 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}-165 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}-176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+48 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40 \cos \left (d x +c \right )-75 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}+240 \sin \left (d x +c \right )^{6} d x}{240 \sin \left (d x +c \right )^{6} a d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c)),x)
Output:
(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 + 48*cos( c + d*x)*sin(c + d*x) - 40*cos(c + d*x) - 75*log(tan((c + d*x)/2))*sin(c + d*x)**6 + 240*sin(c + d*x)**6*d*x)/(240*sin(c + d*x)**6*a*d)