\(\int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx\) [715]

Optimal result

Integrand size = 29, antiderivative size = 134 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {x}{a}+\frac {15 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cos (c+d x)}{a d}-\frac {\cot (c+d x)}{a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\cot ^5(c+d x)}{5 a d}-\frac {9 \cot (c+d x) \csc (c+d x)}{8 a d}+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \] Output:

-x/a+15/8*arctanh(cos(d*x+c))/a/d-cos(d*x+c)/a/d-cot(d*x+c)/a/d+1/3*cot(d* 
x+c)^3/a/d-1/5*cot(d*x+c)^5/a/d-9/8*cot(d*x+c)*csc(d*x+c)/a/d+1/4*cot(d*x+ 
c)*csc(d*x+c)^3/a/d
 

Mathematica [A] (verified)

Time = 2.58 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.97 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\csc ^5(c+d x) \left (400 \cos (c+d x)-200 \cos (3 (c+d x))+184 \cos (5 (c+d x))+1200 c \sin (c+d x)+1200 d x \sin (c+d x)-2250 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+2250 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+600 \sin (2 (c+d x))-600 c \sin (3 (c+d x))-600 d x \sin (3 (c+d x))+1125 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-1125 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-510 \sin (4 (c+d x))+120 c \sin (5 (c+d x))+120 d x \sin (5 (c+d x))-225 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+225 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (5 (c+d x))+60 \sin (6 (c+d x))\right )}{1920 a d} \] Input:

Integrate[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]
 

Output:

-1/1920*(Csc[c + d*x]^5*(400*Cos[c + d*x] - 200*Cos[3*(c + d*x)] + 184*Cos 
[5*(c + d*x)] + 1200*c*Sin[c + d*x] + 1200*d*x*Sin[c + d*x] - 2250*Log[Cos 
[(c + d*x)/2]]*Sin[c + d*x] + 2250*Log[Sin[(c + d*x)/2]]*Sin[c + d*x] + 60 
0*Sin[2*(c + d*x)] - 600*c*Sin[3*(c + d*x)] - 600*d*x*Sin[3*(c + d*x)] + 1 
125*Log[Cos[(c + d*x)/2]]*Sin[3*(c + d*x)] - 1125*Log[Sin[(c + d*x)/2]]*Si 
n[3*(c + d*x)] - 510*Sin[4*(c + d*x)] + 120*c*Sin[5*(c + d*x)] + 120*d*x*S 
in[5*(c + d*x)] - 225*Log[Cos[(c + d*x)/2]]*Sin[5*(c + d*x)] + 225*Log[Sin 
[(c + d*x)/2]]*Sin[5*(c + d*x)] + 60*Sin[6*(c + d*x)]))/(a*d)
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.01, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3318, 3042, 25, 3072, 252, 252, 262, 219, 3954, 3042, 3954, 3042, 3954, 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.

\(\displaystyle \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^8}{\sin (c+d x)^6 (a \sin (c+d x)+a)}dx\)

\(\Big \downarrow \) 3318

\(\displaystyle \frac {\int \cot ^6(c+d x)dx}{a}-\frac {\int \cos (c+d x) \cot ^5(c+d x)dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}-\frac {\int -\sin \left (c+d x+\frac {\pi }{2}\right ) \tan \left (c+d x+\frac {\pi }{2}\right )^5dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}+\frac {\int \sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )^5dx}{a}\)

\(\Big \downarrow \) 3072

\(\displaystyle \frac {\int \frac {\cos ^6(c+d x)}{\left (1-\cos ^2(c+d x)\right )^3}d\cos (c+d x)}{a d}+\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \int \frac {\cos ^4(c+d x)}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)}{a d}+\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} \int \frac {\cos ^2(c+d x)}{1-\cos ^2(c+d x)}d\cos (c+d x)\right )}{a d}+\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} \left (\int \frac {1}{1-\cos ^2(c+d x)}d\cos (c+d x)-\cos (c+d x)\right )\right )}{a d}+\frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^6dx}{a}+\frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{a d}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {-\int \cot ^4(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}}{a}+\frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\int \tan \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x)}{5 d}}{a}+\frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{a d}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {\int \cot ^2(c+d x)dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}}{a}+\frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}}{a}+\frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{a d}\)

\(\Big \downarrow \) 3954

\(\displaystyle \frac {-\int 1dx-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}}{a}+\frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{a d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {\cos ^5(c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}-\frac {5}{4} \left (\frac {\cos ^3(c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}-\frac {3}{2} (\text {arctanh}(\cos (c+d x))-\cos (c+d x))\right )}{a d}+\frac {-\frac {\cot ^5(c+d x)}{5 d}+\frac {\cot ^3(c+d x)}{3 d}-\frac {\cot (c+d x)}{d}-x}{a}\)

Input:

Int[(Cos[c + d*x]^2*Cot[c + d*x]^6)/(a + a*Sin[c + d*x]),x]
 

Output:

(Cos[c + d*x]^5/(4*(1 - Cos[c + d*x]^2)^2) - (5*((-3*(ArcTanh[Cos[c + d*x] 
] - Cos[c + d*x]))/2 + Cos[c + d*x]^3/(2*(1 - Cos[c + d*x]^2))))/4)/(a*d) 
+ (-x - Cot[c + d*x]/d + Cot[c + d*x]^3/(3*d) - Cot[c + d*x]^5/(5*d))/a
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3072

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]
 

rule 3318

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]
 

rule 3954

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]
 

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {64}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-64 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {7}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {22}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-60 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}\)	\(179\)
default	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+8 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {64}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-64 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {7}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {8}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {22}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-60 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 d a}\)	\(179\)
risch	\(-\frac {x}{a}-\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 a d}-\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 a d}+\frac {-360 i {\mathrm e}^{8 i \left (d x +c \right )}+135 \,{\mathrm e}^{9 i \left (d x +c \right )}+720 i {\mathrm e}^{6 i \left (d x +c \right )}-150 \,{\mathrm e}^{7 i \left (d x +c \right )}-1120 i {\mathrm e}^{4 i \left (d x +c \right )}+560 i {\mathrm e}^{2 i \left (d x +c \right )}+150 \,{\mathrm e}^{3 i \left (d x +c \right )}-184 i-135 \,{\mathrm e}^{i \left (d x +c \right )}}{60 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{5}}-\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}\)	\(198\)

Input:

int(cos(d*x+c)^2*cot(d*x+c)^6/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/32/d/a*(1/5*tan(1/2*d*x+1/2*c)^5-1/2*tan(1/2*d*x+1/2*c)^4-7/3*tan(1/2*d* 
x+1/2*c)^3+8*tan(1/2*d*x+1/2*c)^2+22*tan(1/2*d*x+1/2*c)-64/(1+tan(1/2*d*x+ 
1/2*c)^2)-64*arctan(tan(1/2*d*x+1/2*c))-1/5/tan(1/2*d*x+1/2*c)^5+1/2/tan(1 
/2*d*x+1/2*c)^4+7/3/tan(1/2*d*x+1/2*c)^3-8/tan(1/2*d*x+1/2*c)^2-22/tan(1/2 
*d*x+1/2*c)-60*ln(tan(1/2*d*x+1/2*c)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.57 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {368 \, \cos \left (d x + c\right )^{5} - 560 \, \cos \left (d x + c\right )^{3} - 225 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 225 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (8 \, d x \cos \left (d x + c\right )^{4} + 8 \, \cos \left (d x + c\right )^{5} - 16 \, d x \cos \left (d x + c\right )^{2} - 25 \, \cos \left (d x + c\right )^{3} + 8 \, d x + 15 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 240 \, \cos \left (d x + c\right )}{240 \, {\left (a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{2} + a d\right )} \sin \left (d x + c\right )} \] Input:

integrate(cos(d*x+c)^2*cot(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="fricas" 
)
 

Output:

-1/240*(368*cos(d*x + c)^5 - 560*cos(d*x + c)^3 - 225*(cos(d*x + c)^4 - 2* 
cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 225*(cos(d* 
x + c)^4 - 2*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) 
 + 30*(8*d*x*cos(d*x + c)^4 + 8*cos(d*x + c)^5 - 16*d*x*cos(d*x + c)^2 - 2 
5*cos(d*x + c)^3 + 8*d*x + 15*cos(d*x + c))*sin(d*x + c) + 240*cos(d*x + c 
))/((a*d*cos(d*x + c)^4 - 2*a*d*cos(d*x + c)^2 + a*d)*sin(d*x + c))
 

Sympy [F]

\[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\cos ^{2}{\left (c + d x \right )} \cot ^{6}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:

integrate(cos(d*x+c)**2*cot(d*x+c)**6/(a+a*sin(d*x+c)),x)
 

Output:

Integral(cos(c + d*x)**2*cot(c + d*x)**6/(sin(c + d*x) + 1), x)/a
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 319 vs. \(2 (124) = 248\).

Time = 0.11 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.38 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\frac {\frac {660 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {240 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {70 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {6 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a} + \frac {\frac {15 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {64 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {225 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {590 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2160 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {660 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 6}{\frac {a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}} - \frac {1920 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {1800 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a}}{960 \, d} \] Input:

integrate(cos(d*x+c)^2*cot(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="maxima" 
)
 

Output:

1/960*((660*sin(d*x + c)/(cos(d*x + c) + 1) + 240*sin(d*x + c)^2/(cos(d*x 
+ c) + 1)^2 - 70*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 15*sin(d*x + c)^4/( 
cos(d*x + c) + 1)^4 + 6*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a + (15*sin(d 
*x + c)/(cos(d*x + c) + 1) + 64*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 225* 
sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 590*sin(d*x + c)^4/(cos(d*x + c) + 1 
)^4 - 2160*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 660*sin(d*x + c)^6/(cos(d 
*x + c) + 1)^6 - 6)/(a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + a*sin(d*x + c 
)^7/(cos(d*x + c) + 1)^7) - 1920*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a 
 - 1800*log(sin(d*x + c)/(cos(d*x + c) + 1))/a)/d
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.62 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {960 \, {\left (d x + c\right )}}{a} + \frac {1800 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a} + \frac {1920}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a} - \frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} - \frac {4110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 660 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \] Input:

integrate(cos(d*x+c)^2*cot(d*x+c)^6/(a+a*sin(d*x+c)),x, algorithm="giac")
 

Output:

-1/960*(960*(d*x + c)/a + 1800*log(abs(tan(1/2*d*x + 1/2*c)))/a + 1920/((t 
an(1/2*d*x + 1/2*c)^2 + 1)*a) - (6*a^4*tan(1/2*d*x + 1/2*c)^5 - 15*a^4*tan 
(1/2*d*x + 1/2*c)^4 - 70*a^4*tan(1/2*d*x + 1/2*c)^3 + 240*a^4*tan(1/2*d*x 
+ 1/2*c)^2 + 660*a^4*tan(1/2*d*x + 1/2*c))/a^5 - (4110*tan(1/2*d*x + 1/2*c 
)^5 - 660*tan(1/2*d*x + 1/2*c)^4 - 240*tan(1/2*d*x + 1/2*c)^3 + 70*tan(1/2 
*d*x + 1/2*c)^2 + 15*tan(1/2*d*x + 1/2*c) - 6)/(a*tan(1/2*d*x + 1/2*c)^5)) 
/d
 

Mupad [B] (verification not implemented)

Time = 31.98 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.08 \[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,a\,d}-\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a\,d}+\frac {2\,\mathrm {atan}\left (\frac {15\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {15}{2}\right )}+\frac {4}{4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {15}{2}}\right )}{a\,d}-\frac {15\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,a\,d}-\frac {22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+72\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {59\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {1}{5}}{d\,\left (32\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )}+\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,a\,d} \] Input:

int((cos(c + d*x)^2*cot(c + d*x)^6)/(a + a*sin(c + d*x)),x)
 

Output:

tan(c/2 + (d*x)/2)^2/(4*a*d) - (7*tan(c/2 + (d*x)/2)^3)/(96*a*d) - tan(c/2 
 + (d*x)/2)^4/(64*a*d) + tan(c/2 + (d*x)/2)^5/(160*a*d) + (2*atan((15*tan( 
c/2 + (d*x)/2))/(2*(4*tan(c/2 + (d*x)/2) - 15/2)) + 4/(4*tan(c/2 + (d*x)/2 
) - 15/2)))/(a*d) - (15*log(tan(c/2 + (d*x)/2)))/(8*a*d) - ((15*tan(c/2 + 
(d*x)/2)^3)/2 - (32*tan(c/2 + (d*x)/2)^2)/15 - tan(c/2 + (d*x)/2)/2 + (59* 
tan(c/2 + (d*x)/2)^4)/3 + 72*tan(c/2 + (d*x)/2)^5 + 22*tan(c/2 + (d*x)/2)^ 
6 + 1/5)/(d*(32*a*tan(c/2 + (d*x)/2)^5 + 32*a*tan(c/2 + (d*x)/2)^7)) + (11 
*tan(c/2 + (d*x)/2))/(16*a*d)
 

Reduce [F]

\[ \int \frac {\cos ^2(c+d x) \cot ^6(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {\cos \left (d x +c \right )^{2} \cot \left (d x +c \right )^{6}}{\sin \left (d x +c \right ) a +a}d x \] Input:

int(cos(d*x+c)^2*cot(d*x+c)^6/(a+a*sin(d*x+c)),x)
 

Output:

int(cos(d*x+c)^2*cot(d*x+c)^6/(a+a*sin(d*x+c)),x)