Integrand size = 27, antiderivative size = 102 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{a}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x)}{a d}+\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 a d} \] Output:
-x/a-3/8*arctanh(cos(d*x+c))/a/d-cot(d*x+c)/a/d+1/3*cot(d*x+c)^3/a/d+3/8*c ot(d*x+c)*csc(d*x+c)/a/d-1/4*cot(d*x+c)^3*csc(d*x+c)/a/d
Leaf count is larger than twice the leaf count of optimal. \(232\) vs. \(2(102)=204\).
Time = 1.74 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.27 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^4(c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 \left (72 c+72 d x+18 \cos (c+d x)+30 \cos (3 (c+d x))+24 c \cos (4 (c+d x))+24 d x \cos (4 (c+d x))+27 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+9 \cos (4 (c+d x)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-12 \cos (2 (c+d x)) \left (8 c+8 d x+3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-27 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-9 \cos (4 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 \sin (2 (c+d x))-32 \sin (4 (c+d x))\right )}{192 a d (1+\sin (c+d x))} \] Input:
Integrate[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]
Output:
-1/192*(Csc[c + d*x]^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(72*c + 72* d*x + 18*Cos[c + d*x] + 30*Cos[3*(c + d*x)] + 24*c*Cos[4*(c + d*x)] + 24*d *x*Cos[4*(c + d*x)] + 27*Log[Cos[(c + d*x)/2]] + 9*Cos[4*(c + d*x)]*Log[Co s[(c + d*x)/2]] - 12*Cos[2*(c + d*x)]*(8*c + 8*d*x + 3*Log[Cos[(c + d*x)/2 ]] - 3*Log[Sin[(c + d*x)/2]]) - 27*Log[Sin[(c + d*x)/2]] - 9*Cos[4*(c + d* x)]*Log[Sin[(c + d*x)/2]] + 32*Sin[2*(c + d*x)] - 32*Sin[4*(c + d*x)]))/(a *d*(1 + Sin[c + d*x]))
Time = 0.67 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3318, 3042, 3091, 3042, 3091, 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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6}{\sin (c+d x)^5 (a \sin (c+d x)+a)}dx\) |
| \(\Big \downarrow \) 3318 |
| \(\displaystyle \frac {\int \cot ^4(c+d x) \csc (c+d x)dx}{a}-\frac {\int \cot ^4(c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^4dx}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {3}{4} \int \cot ^2(c+d x) \csc (c+d x)dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3}{4} \int \sec \left (c+d x-\frac {\pi }{2}\right ) \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx}{a}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \frac {-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {-\int \cot ^2(c+d x)dx-\frac {\cot ^3(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {-\int \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {\int 1dx-\frac {\cot ^3(c+d x)}{3 d}+\frac {\cot (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {-\frac {3}{4} \left (-\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {-\frac {\cot ^3(c+d x)}{3 d}+\frac {\cot (c+d x)}{d}+x}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {3}{4} \left (\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )-\frac {\cot ^3(c+d x) \csc (c+d x)}{4 d}}{a}-\frac {-\frac {\cot ^3(c+d x)}{3 d}+\frac {\cot (c+d x)}{d}+x}{a}\) |
Input:
Int[(Cos[c + d*x]*Cot[c + d*x]^5)/(a + a*Sin[c + d*x]),x]
Output:
-((x + Cot[c + d*x]/d - Cot[c + d*x]^3/(3*d))/a) + (-1/4*(Cot[c + d*x]^3*C sc[c + d*x])/d - (3*(ArcTanh[Cos[c + d*x]]/(2*d) - (Cot[c + d*x]*Csc[c + d *x])/(2*d)))/4)/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 = 0.56 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\) | \(136\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-32 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {2}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {10}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d a}\) | \(136\) |
| risch | \(-\frac {x}{a}-\frac {48 i {\mathrm e}^{6 i \left (d x +c \right )}+15 \,{\mathrm e}^{7 i \left (d x +c \right )}-96 i {\mathrm e}^{4 i \left (d x +c \right )}+9 \,{\mathrm e}^{5 i \left (d x +c \right )}+80 i {\mathrm e}^{2 i \left (d x +c \right )}+9 \,{\mathrm e}^{3 i \left (d x +c \right )}-32 i+15 \,{\mathrm e}^{i \left (d x +c \right )}}{12 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}\) | \(152\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/16/d/a*(1/4*tan(1/2*d*x+1/2*c)^4-2/3*tan(1/2*d*x+1/2*c)^3-2*tan(1/2*d*x+ 1/2*c)^2+10*tan(1/2*d*x+1/2*c)-32*arctan(tan(1/2*d*x+1/2*c))-1/4/tan(1/2*d *x+1/2*c)^4+2/3/tan(1/2*d*x+1/2*c)^3+2/tan(1/2*d*x+1/2*c)^2-10/tan(1/2*d*x +1/2*c)+6*ln(tan(1/2*d*x+1/2*c)))
Time = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.68 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {48 \, d x \cos \left (d x + c\right )^{4} - 96 \, d x \cos \left (d x + c\right )^{2} + 30 \, \cos \left (d x + c\right )^{3} + 48 \, d x + 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 9 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (4 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 18 \, \cos \left (d x + c\right )}{48 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/48*(48*d*x*cos(d*x + c)^4 - 96*d*x*cos(d*x + c)^2 + 30*cos(d*x + c)^3 + 48*d*x + 9*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 9*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2) - 16*(4*cos(d*x + c)^3 - 3*cos(d*x + c))*sin(d*x + c) - 18*cos(d*x + c))/(a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)
\[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cos {\left (c + d x \right )} \cot ^{5}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**5/(a+a*sin(d*x+c)),x)
Output:
Integral(cos(c + d*x)*cot(c + d*x)**5/(sin(c + d*x) + 1), x)/a
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (94) = 188\).
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.13 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {120 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{a} - \frac {384 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {72 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {{\left (\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {24 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {120 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{a \sin \left (d x + c\right )^{4}}}{192 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
1/192*((120*sin(d*x + c)/(cos(d*x + c) + 1) - 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 8*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^4/(cos (d*x + c) + 1)^4)/a - 384*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 72*l og(sin(d*x + c)/(cos(d*x + c) + 1))/a + (8*sin(d*x + c)/(cos(d*x + c) + 1) + 24*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 120*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3)*(cos(d*x + c) + 1)^4/(a*sin(d*x + c)^4))/d
Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.64 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {192 \, {\left (d x + c\right )}}{a} - \frac {72 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} - \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {150 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
-1/192*(192*(d*x + c)/a - 72*log(abs(tan(1/2*d*x + 1/2*c)))/a - (3*a^3*tan (1/2*d*x + 1/2*c)^4 - 8*a^3*tan(1/2*d*x + 1/2*c)^3 - 24*a^3*tan(1/2*d*x + 1/2*c)^2 + 120*a^3*tan(1/2*d*x + 1/2*c))/a^4 + (150*tan(1/2*d*x + 1/2*c)^4 + 120*tan(1/2*d*x + 1/2*c)^3 - 24*tan(1/2*d*x + 1/2*c)^2 - 8*tan(1/2*d*x + 1/2*c) + 3)/(a*tan(1/2*d*x + 1/2*c)^4))/d
Time = 33.23 (sec) , antiderivative size = 317, normalized size of antiderivative = 3.11 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-120\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+384\,\mathrm {atan}\left (\frac {8\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+8\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+72\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{192\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4} \] Input:
int((cos(c + d*x)*cot(c + d*x)^5)/(a + a*sin(c + d*x)),x)
Output:
(3*sin(c/2 + (d*x)/2)^8 - 3*cos(c/2 + (d*x)/2)^8 - 8*cos(c/2 + (d*x)/2)*si n(c/2 + (d*x)/2)^7 + 8*cos(c/2 + (d*x)/2)^7*sin(c/2 + (d*x)/2) - 24*cos(c/ 2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^6 + 120*cos(c/2 + (d*x)/2)^3*sin(c/2 + ( d*x)/2)^5 - 120*cos(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^3 + 24*cos(c/2 + ( d*x)/2)^6*sin(c/2 + (d*x)/2)^2 + 384*atan((8*cos(c/2 + (d*x)/2) - 3*sin(c/ 2 + (d*x)/2))/(3*cos(c/2 + (d*x)/2) + 8*sin(c/2 + (d*x)/2)))*cos(c/2 + (d* x)/2)^4*sin(c/2 + (d*x)/2)^4 + 72*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2 ))*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^4)/(192*a*d*cos(c/2 + (d*x)/2)^ 4*sin(c/2 + (d*x)/2)^4)
Time = 6.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01 \[ \int \frac {\cos (c+d x) \cot ^5(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+8 \cos \left (d x +c \right ) \sin \left (d x +c \right )-6 \cos \left (d x +c \right )+9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{4}-24 \sin \left (d x +c \right )^{4} d x}{24 \sin \left (d x +c \right )^{4} a d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^5/(a+a*sin(d*x+c)),x)
Output:
( - 32*cos(c + d*x)*sin(c + d*x)**3 + 15*cos(c + d*x)*sin(c + d*x)**2 + 8* cos(c + d*x)*sin(c + d*x) - 6*cos(c + d*x) + 9*log(tan((c + d*x)/2))*sin(c + d*x)**4 - 24*sin(c + d*x)**4*d*x)/(24*sin(c + d*x)**4*a*d)