Integrand size = 27, antiderivative size = 143 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {5 x}{16 a}-\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {\cos (c+d x)}{a d}+\frac {\cos ^3(c+d x)}{3 a d}+\frac {\cos ^5(c+d x)}{5 a d}-\frac {5 \cos (c+d x) \sin (c+d x)}{16 a d}-\frac {5 \cos ^3(c+d x) \sin (c+d x)}{24 a d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a d} \] Output:
-5/16*x/a-arctanh(cos(d*x+c))/a/d+cos(d*x+c)/a/d+1/3*cos(d*x+c)^3/a/d+1/5* cos(d*x+c)^5/a/d-5/16*cos(d*x+c)*sin(d*x+c)/a/d-5/24*cos(d*x+c)^3*sin(d*x+ c)/a/d-1/6*cos(d*x+c)^5*sin(d*x+c)/a/d
Time = 1.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {300 c+300 d x-1320 \cos (c+d x)-140 \cos (3 (c+d x))-12 \cos (5 (c+d x))+960 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-960 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+225 \sin (2 (c+d x))+45 \sin (4 (c+d x))+5 \sin (6 (c+d x))}{960 a d} \] Input:
Integrate[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
-1/960*(300*c + 300*d*x - 1320*Cos[c + d*x] - 140*Cos[3*(c + d*x)] - 12*Co s[5*(c + d*x)] + 960*Log[Cos[(c + d*x)/2]] - 960*Log[Sin[(c + d*x)/2]] + 2 25*Sin[2*(c + d*x)] + 45*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)])/(a*d)
Time = 0.58 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {3042, 3318, 3042, 25, 3072, 254, 2009, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x) (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cos ^5(c+d x) \cot (c+d x)dx}{a}-\frac {\int \cos ^6(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\sin \left (c+d x+\frac {\pi }{2}\right )^5 \tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \frac {\cos ^6(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}-\frac {\int \left (-\cos ^4(c+d x)-\cos ^2(c+d x)+\frac {1}{1-\cos ^2(c+d x)}-1\right )d\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}-\frac {\text {arctanh}(\cos (c+d x))-\frac {1}{5} \cos ^5(c+d x)-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}-\frac {\text {arctanh}(\cos (c+d x))-\frac {1}{5} \cos ^5(c+d x)-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}-\frac {\text {arctanh}(\cos (c+d x))-\frac {1}{5} \cos ^5(c+d x)-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}-\frac {\text {arctanh}(\cos (c+d x))-\frac {1}{5} \cos ^5(c+d x)-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}-\frac {\text {arctanh}(\cos (c+d x))-\frac {1}{5} \cos ^5(c+d x)-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}}{a}-\frac {\text {arctanh}(\cos (c+d x))-\frac {1}{5} \cos ^5(c+d x)-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)}{a d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {\text {arctanh}(\cos (c+d x))-\frac {1}{5} \cos ^5(c+d x)-\frac {1}{3} \cos ^3(c+d x)-\cos (c+d x)}{a d}-\frac {\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a}\) |
Input:
Int[(Cos[c + d*x]^7*Cot[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
-((ArcTanh[Cos[c + d*x]] - Cos[c + d*x] - Cos[c + d*x]^3/3 - Cos[c + d*x]^ 5/5)/(a*d)) - ((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^3*S in[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6)/ a
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ (ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, a*(Sin[e + f*x]/ff)], x ]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 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]
Result contains complex when optimal does not.
Time = 4.94 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(-\frac {5 x}{16 a}+\frac {11 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a d}+\frac {11 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}-\frac {\sin \left (6 d x +6 c \right )}{192 d a}+\frac {\cos \left (5 d x +5 c \right )}{80 a d}-\frac {3 \sin \left (4 d x +4 c \right )}{64 d a}+\frac {7 \cos \left (3 d x +3 c \right )}{48 a d}-\frac {15 \sin \left (2 d x +2 c \right )}{64 d a}\) | \(166\) |
| derivativedivides | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{16}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{48}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}-\frac {46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48}-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {23}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(191\) |
| default | \(\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {2 \left (-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{16}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{48}-9 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8}-\frac {46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3}+\frac {15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8}-14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48}-\frac {31 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5}+\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16}-\frac {23}{15}\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}-\frac {5 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d a}\) | \(191\) |
Input:
int(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-5/16*x/a+11/16/a/d*exp(I*(d*x+c))+11/16/a/d*exp(-I*(d*x+c))-1/d/a*ln(exp( I*(d*x+c))+1)+1/d/a*ln(exp(I*(d*x+c))-1)-1/192/d/a*sin(6*d*x+6*c)+1/80/a/d *cos(5*d*x+5*c)-3/64/d/a*sin(4*d*x+4*c)+7/48/a/d*cos(3*d*x+3*c)-15/64/d/a* sin(2*d*x+2*c)
Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.73 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {48 \, \cos \left (d x + c\right )^{5} + 80 \, \cos \left (d x + c\right )^{3} - 75 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 240 \, \cos \left (d x + c\right ) - 120 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 120 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{240 \, a d} \] Input:
integrate(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/240*(48*cos(d*x + c)^5 + 80*cos(d*x + c)^3 - 75*d*x - 5*(8*cos(d*x + c)^ 5 + 10*cos(d*x + c)^3 + 15*cos(d*x + c))*sin(d*x + c) + 240*cos(d*x + c) - 120*log(1/2*cos(d*x + c) + 1/2) + 120*log(-1/2*cos(d*x + c) + 1/2))/(a*d)
Timed out. \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**7*cot(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (131) = 262\).
Time = 0.15 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.81 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\frac {165 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1488 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {25 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3360 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {450 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {3680 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {450 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {2160 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {25 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {720 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {165 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 368}{a + \frac {6 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} + \frac {75 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {120 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{120 \, d} \] Input:
integrate(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/120*((165*sin(d*x + c)/(cos(d*x + c) + 1) - 1488*sin(d*x + c)^2/(cos(d* x + c) + 1)^2 - 25*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3360*sin(d*x + c) ^4/(cos(d*x + c) + 1)^4 + 450*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 3680*s in(d*x + c)^6/(cos(d*x + c) + 1)^6 - 450*sin(d*x + c)^7/(cos(d*x + c) + 1) ^7 - 2160*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 25*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 720*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 - 165*sin(d*x + c )^11/(cos(d*x + c) + 1)^11 - 368)/(a + 6*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a*sin(d*x + c)^6/(cos (d*x + c) + 1)^6 + 15*a*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + a*sin(d*x + c)^12/(cos(d*x + c) + 1)^12) + 75*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a - 120*log(sin(d*x + c)/(cos( d*x + c) + 1))/a)/d
Time = 0.14 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.36 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {75 \, {\left (d x + c\right )}}{a} - \frac {240 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {2 \, {\left (165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 720 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 25 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2160 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3680 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 450 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3360 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 25 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1488 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 368\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a}}{240 \, d} \] Input:
integrate(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/240*(75*(d*x + c)/a - 240*log(abs(tan(1/2*d*x + 1/2*c)))/a - 2*(165*tan (1/2*d*x + 1/2*c)^11 + 720*tan(1/2*d*x + 1/2*c)^10 - 25*tan(1/2*d*x + 1/2* c)^9 + 2160*tan(1/2*d*x + 1/2*c)^8 + 450*tan(1/2*d*x + 1/2*c)^7 + 3680*tan (1/2*d*x + 1/2*c)^6 - 450*tan(1/2*d*x + 1/2*c)^5 + 3360*tan(1/2*d*x + 1/2* c)^4 + 25*tan(1/2*d*x + 1/2*c)^3 + 1488*tan(1/2*d*x + 1/2*c)^2 - 165*tan(1 /2*d*x + 1/2*c) + 368)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a))/d
Time = 33.78 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.13 \[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {5\,\mathrm {atan}\left (\frac {25}{64\,\left (\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {5}{4}\right )}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {25\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {5}{4}\right )}\right )}{8\,a\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}+\frac {\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {92\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {62\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {46}{15}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \] Input:
int((cos(c + d*x)^7*cot(c + d*x))/(a + a*sin(c + d*x)),x)
Output:
(5*atan(25/(64*((25*tan(c/2 + (d*x)/2))/64 + 5/4)) - (5*tan(c/2 + (d*x)/2) )/(4*((25*tan(c/2 + (d*x)/2))/64 + 5/4))))/(8*a*d) + log(tan(c/2 + (d*x)/2 ))/(a*d) + ((62*tan(c/2 + (d*x)/2)^2)/5 - (11*tan(c/2 + (d*x)/2))/8 + (5*t an(c/2 + (d*x)/2)^3)/24 + 28*tan(c/2 + (d*x)/2)^4 - (15*tan(c/2 + (d*x)/2) ^5)/4 + (92*tan(c/2 + (d*x)/2)^6)/3 + (15*tan(c/2 + (d*x)/2)^7)/4 + 18*tan (c/2 + (d*x)/2)^8 - (5*tan(c/2 + (d*x)/2)^9)/24 + 6*tan(c/2 + (d*x)/2)^10 + (11*tan(c/2 + (d*x)/2)^11)/8 + 46/15)/(d*(a + 6*a*tan(c/2 + (d*x)/2)^2 + 15*a*tan(c/2 + (d*x)/2)^4 + 20*a*tan(c/2 + (d*x)/2)^6 + 15*a*tan(c/2 + (d *x)/2)^8 + 6*a*tan(c/2 + (d*x)/2)^10 + a*tan(c/2 + (d*x)/2)^12))
\[ \int \frac {\cos ^7(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {\cos \left (d x +c \right )^{7} \cot \left (d x +c \right )}{\sin \left (d x +c \right ) a +a}d x \] Input:
int(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
int(cos(d*x+c)^7*cot(d*x+c)/(a+a*sin(d*x+c)),x)