Integrand size = 27, antiderivative size = 124 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {7 \text {arctanh}(\cos (c+d x))}{16 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d} \] Output:
7/16*arctanh(cos(d*x+c))/a^3/d+4/3*cot(d*x+c)^3/a^3/d+3/5*cot(d*x+c)^5/a^3 /d+7/16*cot(d*x+c)*csc(d*x+c)/a^3/d-17/24*cot(d*x+c)*csc(d*x+c)^3/a^3/d-1/ 6*cot(d*x+c)*csc(d*x+c)^5/a^3/d
Time = 3.46 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.95 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (-704 \cot \left (\frac {1}{2} (c+d x)\right )+210 \csc ^2\left (\frac {1}{2} (c+d x)\right )+840 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-840 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-210 \sec ^2\left (\frac {1}{2} (c+d x)\right )+90 \sec ^4\left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right )-544 \csc ^3(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )+\csc ^6\left (\frac {1}{2} (c+d x)\right ) (-5+18 \sin (c+d x))+\csc ^4\left (\frac {1}{2} (c+d x)\right ) (-90+34 \sin (c+d x))+704 \tan \left (\frac {1}{2} (c+d x)\right )-36 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{1920 a^3 d (1+\sin (c+d x))^3} \] Input:
Integrate[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]
Output:
((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*(-704*Cot[(c + d*x)/2] + 210*Csc[ (c + d*x)/2]^2 + 840*Log[Cos[(c + d*x)/2]] - 840*Log[Sin[(c + d*x)/2]] - 2 10*Sec[(c + d*x)/2]^2 + 90*Sec[(c + d*x)/2]^4 + 5*Sec[(c + d*x)/2]^6 - 544 *Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6*(-5 + 18*Sin[c + d *x]) + Csc[(c + d*x)/2]^4*(-90 + 34*Sin[c + d*x]) + 704*Tan[(c + d*x)/2] - 36*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(1920*a^3*d*(1 + Sin[c + d*x])^3 )
Time = 0.64 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3354, 3042, 3352, 2009}
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.
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^7 (a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \cot ^2(c+d x) \csc ^5(c+d x) (a-a \sin (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x)^2 (a-a \sin (c+d x))^3}{\sin (c+d x)^7}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (a^3 \cot ^2(c+d x) \csc ^5(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)-a^3 \cot ^2(c+d x) \csc ^2(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {7 a^3 \text {arctanh}(\cos (c+d x))}{16 d}+\frac {3 a^3 \cot ^5(c+d x)}{5 d}+\frac {4 a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac {17 a^3 \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {7 a^3 \cot (c+d x) \csc (c+d x)}{16 d}}{a^6}\) |
Input:
Int[(Cos[c + d*x]*Cot[c + d*x]^7)/(a + a*Sin[c + d*x])^3,x]
Output:
((7*a^3*ArcTanh[Cos[c + d*x]])/(16*d) + (4*a^3*Cot[c + d*x]^3)/(3*d) + (3* a^3*Cot[c + d*x]^5)/(5*d) + (7*a^3*Cot[c + d*x]*Csc[c + d*x])/(16*d) - (17 *a^3*Cot[c + d*x]*Csc[c + d*x]^3)/(24*d) - (a^3*Cot[c + d*x]*Csc[c + d*x]^ 5)/(6*d))/a^6
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Time = 8.75 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.42
| method | result | size |
| derivativedivides | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {7}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {20}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d \,a^{3}}\) | \(176\) |
| default | \(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{6}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2}+20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {7}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-28 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {20}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {1}{6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}+\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d \,a^{3}}\) | \(176\) |
| risch | \(-\frac {240 i {\mathrm e}^{10 i \left (d x +c \right )}+105 \,{\mathrm e}^{11 i \left (d x +c \right )}-2160 i {\mathrm e}^{8 i \left (d x +c \right )}+365 \,{\mathrm e}^{9 i \left (d x +c \right )}+1760 i {\mathrm e}^{6 i \left (d x +c \right )}-1110 \,{\mathrm e}^{7 i \left (d x +c \right )}-480 i {\mathrm e}^{4 i \left (d x +c \right )}-1110 \,{\mathrm e}^{5 i \left (d x +c \right )}+816 i {\mathrm e}^{2 i \left (d x +c \right )}+365 \,{\mathrm e}^{3 i \left (d x +c \right )}-176 i+105 \,{\mathrm e}^{i \left (d x +c \right )}}{120 a^{3} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{6}}+\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{16 d \,a^{3}}-\frac {7 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{16 d \,a^{3}}\) | \(192\) |
Input:
int(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/64/d/a^3*(1/6*tan(1/2*d*x+1/2*c)^6-6/5*tan(1/2*d*x+1/2*c)^5+7/2*tan(1/2* d*x+1/2*c)^4-14/3*tan(1/2*d*x+1/2*c)^3-1/2*tan(1/2*d*x+1/2*c)^2+20*tan(1/2 *d*x+1/2*c)-7/2/tan(1/2*d*x+1/2*c)^4+6/5/tan(1/2*d*x+1/2*c)^5-28*ln(tan(1/ 2*d*x+1/2*c))-20/tan(1/2*d*x+1/2*c)+1/2/tan(1/2*d*x+1/2*c)^2-1/6/tan(1/2*d *x+1/2*c)^6+14/3/tan(1/2*d*x+1/2*c)^3)
Time = 0.12 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.58 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {210 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (11 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )}{480 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
-1/480*(210*cos(d*x + c)^5 - 80*cos(d*x + c)^3 - 105*(cos(d*x + c)^6 - 3*c os(d*x + c)^4 + 3*cos(d*x + c)^2 - 1)*log(1/2*cos(d*x + c) + 1/2) + 105*(c os(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*(11*cos(d*x + c)^5 - 20*cos(d*x + c)^3)*sin(d*x + c) - 21 0*cos(d*x + c))/(a^3*d*cos(d*x + c)^6 - 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*c os(d*x + c)^2 - a^3*d)
\[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\int \frac {\cos {\left (c + d x \right )} \cot ^{7}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)**7/(a+a*sin(d*x+c))**3,x)
Output:
Integral(cos(c + d*x)*cot(c + d*x)**7/(sin(c + d*x)**3 + 3*sin(c + d*x)**2 + 3*sin(c + d*x) + 1), x)/a**3
Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (112) = 224\).
Time = 0.04 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.21 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {\frac {\frac {600 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {36 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {105 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {600 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{3} \sin \left (d x + c\right )^{6}}}{1920 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
1/1920*((600*sin(d*x + c)/(cos(d*x + c) + 1) - 15*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 140*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 105*sin(d*x + c)^4 /(cos(d*x + c) + 1)^4 - 36*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6)/a^3 - 840*log(sin(d*x + c)/(cos(d*x + c) + 1 ))/a^3 + (36*sin(d*x + c)/(cos(d*x + c) + 1) - 105*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 140*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^4 /(cos(d*x + c) + 1)^4 - 600*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 5)*(cos( d*x + c) + 1)^6/(a^3*sin(d*x + c)^6))/d
Time = 0.23 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.74 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2058 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} - \frac {5 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 36 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 600 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{18}}}{1920 \, d} \] Input:
integrate(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-1/1920*(840*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (2058*tan(1/2*d*x + 1/2* c)^6 - 600*tan(1/2*d*x + 1/2*c)^5 + 15*tan(1/2*d*x + 1/2*c)^4 + 140*tan(1/ 2*d*x + 1/2*c)^3 - 105*tan(1/2*d*x + 1/2*c)^2 + 36*tan(1/2*d*x + 1/2*c) - 5)/(a^3*tan(1/2*d*x + 1/2*c)^6) - (5*a^15*tan(1/2*d*x + 1/2*c)^6 - 36*a^15 *tan(1/2*d*x + 1/2*c)^5 + 105*a^15*tan(1/2*d*x + 1/2*c)^4 - 140*a^15*tan(1 /2*d*x + 1/2*c)^3 - 15*a^15*tan(1/2*d*x + 1/2*c)^2 + 600*a^15*tan(1/2*d*x + 1/2*c))/a^18)/d
Time = 32.36 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.73 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+36\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-36\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \] Input:
int((cos(c + d*x)*cot(c + d*x)^7)/(a + a*sin(c + d*x))^3,x)
Output:
-(5*cos(c/2 + (d*x)/2)^12 - 5*sin(c/2 + (d*x)/2)^12 + 36*cos(c/2 + (d*x)/2 )*sin(c/2 + (d*x)/2)^11 - 36*cos(c/2 + (d*x)/2)^11*sin(c/2 + (d*x)/2) - 10 5*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 140*cos(c/2 + (d*x)/2)^3*si n(c/2 + (d*x)/2)^9 + 15*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 600*co s(c/2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^7 + 600*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 - 140*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^3 + 105*cos(c/2 + (d*x)/2)^10*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)^6*sin(c/2 + (d*x)/2)^6)/(1920*a^3*d*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d* x)/2)^6)
Time = 0.17 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+105 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-170 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+144 \cos \left (d x +c \right ) \sin \left (d x +c \right )-40 \cos \left (d x +c \right )-105 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{6}}{240 \sin \left (d x +c \right )^{6} a^{3} d} \] Input:
int(cos(d*x+c)*cot(d*x+c)^7/(a+a*sin(d*x+c))^3,x)
Output:
( - 176*cos(c + d*x)*sin(c + d*x)**5 + 105*cos(c + d*x)*sin(c + d*x)**4 + 32*cos(c + d*x)*sin(c + d*x)**3 - 170*cos(c + d*x)*sin(c + d*x)**2 + 144*c os(c + d*x)*sin(c + d*x) - 40*cos(c + d*x) - 105*log(tan((c + d*x)/2))*sin (c + d*x)**6)/(240*sin(c + d*x)**6*a**3*d)