\(\int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) [303]

Optimal result

Integrand size = 21, antiderivative size = 278 \[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\log (\cos (c+d x))}{a^2 d}+\frac {\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 (a+b)^4 d}+\frac {\left (4 a^2-13 a b+12 b^2\right ) \log (1+\sec (c+d x))}{8 (a-b)^4 d}-\frac {b^6 \left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^4 d}-\frac {1}{16 (a+b)^2 d (1-\sec (c+d x))^2}-\frac {5 a+9 b}{16 (a+b)^3 d (1-\sec (c+d x))}-\frac {1}{16 (a-b)^2 d (1+\sec (c+d x))^2}-\frac {5 a-9 b}{16 (a-b)^3 d (1+\sec (c+d x))}+\frac {b^6}{a \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \] Output:

ln(cos(d*x+c))/a^2/d+1/8*(4*a^2+13*a*b+12*b^2)*ln(1-sec(d*x+c))/(a+b)^4/d+ 
1/8*(4*a^2-13*a*b+12*b^2)*ln(1+sec(d*x+c))/(a-b)^4/d-b^6*(7*a^2-b^2)*ln(a+ 
b*sec(d*x+c))/a^2/(a^2-b^2)^4/d-1/16/(a+b)^2/d/(1-sec(d*x+c))^2-1/16*(5*a+ 
9*b)/(a+b)^3/d/(1-sec(d*x+c))-1/16/(a-b)^2/d/(1+sec(d*x+c))^2-1/16*(5*a-9* 
b)/(a-b)^3/d/(1+sec(d*x+c))+b^6/a/(a^2-b^2)^3/d/(a+b*sec(d*x+c))
 

Mathematica [A] (verified)

Time = 6.39 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {b^6 \left (-\frac {\log (\cos (c+d x))}{a^2 b^6}-\frac {\left (4 a^2+13 a b+12 b^2\right ) \log (1-\sec (c+d x))}{8 b^6 (a+b)^4}-\frac {\left (4 a^2-13 a b+12 b^2\right ) \log (1+\sec (c+d x))}{8 (a-b)^4 b^6}+\frac {\left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 (a-b)^4 (a+b)^4}+\frac {1}{16 b^4 (a+b)^2 (b-b \sec (c+d x))^2}+\frac {5 a+9 b}{16 b^5 (a+b)^3 (b-b \sec (c+d x))}-\frac {1}{a (a-b)^3 (a+b)^3 (a+b \sec (c+d x))}+\frac {1}{16 (a-b)^2 b^4 (b+b \sec (c+d x))^2}+\frac {5 a-9 b}{16 (a-b)^3 b^5 (b+b \sec (c+d x))}\right )}{d} \] Input:

Integrate[Cot[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]
 

Output:

-((b^6*(-(Log[Cos[c + d*x]]/(a^2*b^6)) - ((4*a^2 + 13*a*b + 12*b^2)*Log[1 
- Sec[c + d*x]])/(8*b^6*(a + b)^4) - ((4*a^2 - 13*a*b + 12*b^2)*Log[1 + Se 
c[c + d*x]])/(8*(a - b)^4*b^6) + ((7*a^2 - b^2)*Log[a + b*Sec[c + d*x]])/( 
a^2*(a - b)^4*(a + b)^4) + 1/(16*b^4*(a + b)^2*(b - b*Sec[c + d*x])^2) + ( 
5*a + 9*b)/(16*b^5*(a + b)^3*(b - b*Sec[c + d*x])) - 1/(a*(a - b)^3*(a + b 
)^3*(a + b*Sec[c + d*x])) + 1/(16*(a - b)^2*b^4*(b + b*Sec[c + d*x])^2) + 
(5*a - 9*b)/(16*(a - b)^3*b^5*(b + b*Sec[c + d*x]))))/d)
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 25, 4373, 615, 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.

\(\displaystyle \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^5 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^2}dx\)

\(\Big \downarrow \) 4373

\(\displaystyle -\frac {b^6 \int \frac {\cos (c+d x)}{b (a+b \sec (c+d x))^2 \left (b^2-b^2 \sec ^2(c+d x)\right )^3}d(b \sec (c+d x))}{d}\)

\(\Big \downarrow \) 615

\(\displaystyle -\frac {b^6 \int \left (\frac {9 b-5 a}{16 (a-b)^3 b^5 (\sec (c+d x) b+b)^2}+\frac {\cos (c+d x)}{a^2 b^7}+\frac {4 a^2+13 b a+12 b^2}{8 b^6 (a+b)^4 (b-b \sec (c+d x))}+\frac {7 a^2-b^2}{a^2 (a-b)^4 (a+b)^4 (a+b \sec (c+d x))}+\frac {-4 a^2+13 b a-12 b^2}{8 (a-b)^4 b^6 (\sec (c+d x) b+b)}+\frac {5 a+9 b}{16 b^5 (a+b)^3 (b-b \sec (c+d x))^2}+\frac {1}{a (a-b)^3 (a+b)^3 (a+b \sec (c+d x))^2}+\frac {1}{8 b^4 (a+b)^2 (b-b \sec (c+d x))^3}-\frac {1}{8 (a-b)^2 b^4 (\sec (c+d x) b+b)^3}\right )d(b \sec (c+d x))}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b^6 \left (\frac {\log (b \sec (c+d x))}{a^2 b^6}-\frac {1}{a \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac {\left (7 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^4}-\frac {\left (4 a^2+13 a b+12 b^2\right ) \log (b-b \sec (c+d x))}{8 b^6 (a+b)^4}-\frac {\left (4 a^2-13 a b+12 b^2\right ) \log (b \sec (c+d x)+b)}{8 b^6 (a-b)^4}+\frac {5 a-9 b}{16 b^5 (a-b)^3 (b \sec (c+d x)+b)}+\frac {5 a+9 b}{16 b^5 (a+b)^3 (b-b \sec (c+d x))}+\frac {1}{16 b^4 (a+b)^2 (b-b \sec (c+d x))^2}+\frac {1}{16 b^4 (a-b)^2 (b \sec (c+d x)+b)^2}\right )}{d}\)

Input:

Int[Cot[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]
 

Output:

-((b^6*(Log[b*Sec[c + d*x]]/(a^2*b^6) - ((4*a^2 + 13*a*b + 12*b^2)*Log[b - 
 b*Sec[c + d*x]])/(8*b^6*(a + b)^4) + ((7*a^2 - b^2)*Log[a + b*Sec[c + d*x 
]])/(a^2*(a^2 - b^2)^4) - ((4*a^2 - 13*a*b + 12*b^2)*Log[b + b*Sec[c + d*x 
]])/(8*(a - b)^4*b^6) + 1/(16*b^4*(a + b)^2*(b - b*Sec[c + d*x])^2) + (5*a 
 + 9*b)/(16*b^5*(a + b)^3*(b - b*Sec[c + d*x])) - 1/(a*(a^2 - b^2)^3*(a + 
b*Sec[c + d*x])) + 1/(16*(a - b)^2*b^4*(b + b*Sec[c + d*x])^2) + (5*a - 9* 
b)/(16*(a - b)^3*b^5*(b + b*Sec[c + d*x]))))/d)
 

Defintions of rubi rules used

rule 25

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

rule 615

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
 x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] 
 /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4373

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n 
_), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1))   Subst[Int[(b^2 - x^ 
2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, 
 d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
 

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {-\frac {1}{16 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-7 a +11 b}{16 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (4 a^{2}-13 a b +12 b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{8 \left (a -b \right )^{4}}-\frac {b^{7}}{a^{2} \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{6} \left (7 a^{2}-b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} a^{2}}-\frac {1}{16 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {7 a +11 b}{16 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (4 a^{2}+13 a b +12 b^{2}\right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{8 \left (a +b \right )^{4}}}{d}\)	\(230\)
default	\(\frac {-\frac {1}{16 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-7 a +11 b}{16 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (4 a^{2}-13 a b +12 b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{8 \left (a -b \right )^{4}}-\frac {b^{7}}{a^{2} \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}-\frac {b^{6} \left (7 a^{2}-b^{2}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} a^{2}}-\frac {1}{16 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {7 a +11 b}{16 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (4 a^{2}+13 a b +12 b^{2}\right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{8 \left (a +b \right )^{4}}}{d}\)	\(230\)
risch	\(\text {Expression too large to display}\)	\(1577\)

Input:

int(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/16/(a-b)^2/(1+cos(d*x+c))^2-1/16*(-7*a+11*b)/(a-b)^3/(1+cos(d*x+c) 
)+1/8*(4*a^2-13*a*b+12*b^2)/(a-b)^4*ln(1+cos(d*x+c))-b^7/a^2/(a+b)^3/(a-b) 
^3/(b+a*cos(d*x+c))-b^6*(7*a^2-b^2)/(a+b)^4/(a-b)^4/a^2*ln(b+a*cos(d*x+c)) 
-1/16/(a+b)^2/(-1+cos(d*x+c))^2-1/16*(7*a+11*b)/(a+b)^3/(-1+cos(d*x+c))+1/ 
8*(4*a^2+13*a*b+12*b^2)/(a+b)^4*ln(-1+cos(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1378 vs. \(2 (262) = 524\).

Time = 0.47 (sec) , antiderivative size = 1378, normalized size of antiderivative = 4.96 \[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \] Input:

integrate(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="fricas")
 

Output:

1/8*(6*a^8*b - 18*a^6*b^3 + 2*a^4*b^5 + 2*a^2*b^7 + 8*b^9 + 2*(5*a^8*b - 1 
8*a^6*b^3 + 13*a^4*b^5 - 4*a^2*b^7 + 4*b^9)*cos(d*x + c)^4 - 2*(4*a^9 - 15 
*a^7*b^2 + 18*a^5*b^4 - 7*a^3*b^6)*cos(d*x + c)^3 - 2*(7*a^8*b - 24*a^6*b^ 
3 + 11*a^4*b^5 - 2*a^2*b^7 + 8*b^9)*cos(d*x + c)^2 + 6*(a^9 - 4*a^7*b^2 + 
5*a^5*b^4 - 2*a^3*b^6)*cos(d*x + c) - 8*(7*a^2*b^7 - b^9 + (7*a^3*b^6 - a* 
b^8)*cos(d*x + c)^5 + (7*a^2*b^7 - b^9)*cos(d*x + c)^4 - 2*(7*a^3*b^6 - a* 
b^8)*cos(d*x + c)^3 - 2*(7*a^2*b^7 - b^9)*cos(d*x + c)^2 + (7*a^3*b^6 - a* 
b^8)*cos(d*x + c))*log(a*cos(d*x + c) + b) + (4*a^8*b + 3*a^7*b^2 - 16*a^6 
*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12*a^2*b^7 + (4*a^9 + 3*a^8* 
b - 16*a^7*b^2 - 14*a^6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d* 
x + c)^5 + (4*a^8*b + 3*a^7*b^2 - 16*a^6*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 3 
5*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)^4 - 2*(4*a^9 + 3*a^8*b - 16*a^7*b^2 - 
 14*a^6*b^3 + 24*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^3 - 2*(4* 
a^8*b + 3*a^7*b^2 - 16*a^6*b^3 - 14*a^5*b^4 + 24*a^4*b^5 + 35*a^3*b^6 + 12 
*a^2*b^7)*cos(d*x + c)^2 + (4*a^9 + 3*a^8*b - 16*a^7*b^2 - 14*a^6*b^3 + 24 
*a^5*b^4 + 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1 
/2) + (4*a^8*b - 3*a^7*b^2 - 16*a^6*b^3 + 14*a^5*b^4 + 24*a^4*b^5 - 35*a^3 
*b^6 + 12*a^2*b^7 + (4*a^9 - 3*a^8*b - 16*a^7*b^2 + 14*a^6*b^3 + 24*a^5*b^ 
4 - 35*a^4*b^5 + 12*a^3*b^6)*cos(d*x + c)^5 + (4*a^8*b - 3*a^7*b^2 - 16*a^ 
6*b^3 + 14*a^5*b^4 + 24*a^4*b^5 - 35*a^3*b^6 + 12*a^2*b^7)*cos(d*x + c)...
 

Sympy [F]

\[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \] Input:

integrate(cot(d*x+c)**5/(a+b*sec(d*x+c))**2,x)
 

Output:

Integral(cot(c + d*x)**5/(a + b*sec(c + d*x))**2, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (262) = 524\).

Time = 0.05 (sec) , antiderivative size = 558, normalized size of antiderivative = 2.01 \[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {8 \, {\left (7 \, a^{2} b^{6} - b^{8}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}} - \frac {{\left (4 \, a^{2} - 13 \, a b + 12 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (4 \, a^{2} + 13 \, a b + 12 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {2 \, {\left (3 \, a^{6} b - 6 \, a^{4} b^{3} - 5 \, a^{2} b^{5} - 4 \, b^{7} + {\left (5 \, a^{6} b - 13 \, a^{4} b^{3} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - {\left (4 \, a^{7} - 11 \, a^{5} b^{2} + 7 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - {\left (7 \, a^{6} b - 17 \, a^{4} b^{3} - 6 \, a^{2} b^{5} - 8 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 2 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7} + {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{5} + {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{8} b - 3 \, a^{6} b^{3} + 3 \, a^{4} b^{5} - a^{2} b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )}}{8 \, d} \] Input:

integrate(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="maxima")
 

Output:

-1/8*(8*(7*a^2*b^6 - b^8)*log(a*cos(d*x + c) + b)/(a^10 - 4*a^8*b^2 + 6*a^ 
6*b^4 - 4*a^4*b^6 + a^2*b^8) - (4*a^2 - 13*a*b + 12*b^2)*log(cos(d*x + c) 
+ 1)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (4*a^2 + 13*a*b + 12*b^ 
2)*log(cos(d*x + c) - 1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 2*( 
3*a^6*b - 6*a^4*b^3 - 5*a^2*b^5 - 4*b^7 + (5*a^6*b - 13*a^4*b^3 - 4*b^7)*c 
os(d*x + c)^4 - (4*a^7 - 11*a^5*b^2 + 7*a^3*b^4)*cos(d*x + c)^3 - (7*a^6*b 
 - 17*a^4*b^3 - 6*a^2*b^5 - 8*b^7)*cos(d*x + c)^2 + 3*(a^7 - 3*a^5*b^2 + 2 
*a^3*b^4)*cos(d*x + c))/(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7 + (a^9 - 
3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*cos(d*x + c)^5 + (a^8*b - 3*a^6*b^3 + 3*a 
^4*b^5 - a^2*b^7)*cos(d*x + c)^4 - 2*(a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^ 
6)*cos(d*x + c)^3 - 2*(a^8*b - 3*a^6*b^3 + 3*a^4*b^5 - a^2*b^7)*cos(d*x + 
c)^2 + (a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*cos(d*x + c)))/d
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.68 \[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {{\left (7 \, a^{2} b^{6} - b^{8}\right )} \log \left ({\left | a \cos \left (d x + c\right ) + b \right |}\right )}{a^{10} d - 4 \, a^{8} b^{2} d + 6 \, a^{6} b^{4} d - 4 \, a^{4} b^{6} d + a^{2} b^{8} d} + \frac {{\left (4 \, a^{2} + 13 \, a b + 12 \, b^{2}\right )} \log \left ({\left | -\cos \left (d x + c\right ) + 1 \right |}\right )}{8 \, {\left (a^{4} d + 4 \, a^{3} b d + 6 \, a^{2} b^{2} d + 4 \, a b^{3} d + b^{4} d\right )}} + \frac {{\left (4 \, a^{2} - 13 \, a b + 12 \, b^{2}\right )} \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{8 \, {\left (a^{4} d - 4 \, a^{3} b d + 6 \, a^{2} b^{2} d - 4 \, a b^{3} d + b^{4} d\right )}} - \frac {{\left (4 \, a^{8} - 15 \, a^{6} b^{2} + 18 \, a^{4} b^{4} - 7 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} - \frac {{\left (5 \, a^{8} b - 18 \, a^{6} b^{3} + 13 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + 4 \, b^{9}\right )} \cos \left (d x + c\right )^{4}}{a} - 3 \, {\left (a^{8} - 4 \, a^{6} b^{2} + 5 \, a^{4} b^{4} - 2 \, a^{2} b^{6}\right )} \cos \left (d x + c\right ) + \frac {{\left (7 \, a^{8} b - 24 \, a^{6} b^{3} + 11 \, a^{4} b^{5} - 2 \, a^{2} b^{7} + 8 \, b^{9}\right )} \cos \left (d x + c\right )^{2}}{a} - \frac {3 \, a^{8} b - 9 \, a^{6} b^{3} + a^{4} b^{5} + a^{2} b^{7} + 4 \, b^{9}}{a}}{4 \, {\left (a \cos \left (d x + c\right ) + b\right )} {\left (a + b\right )}^{4} {\left (a - b\right )}^{4} a d {\left (\cos \left (d x + c\right ) + 1\right )}^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} \] Input:

integrate(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x, algorithm="giac")
 

Output:

-(7*a^2*b^6 - b^8)*log(abs(a*cos(d*x + c) + b))/(a^10*d - 4*a^8*b^2*d + 6* 
a^6*b^4*d - 4*a^4*b^6*d + a^2*b^8*d) + 1/8*(4*a^2 + 13*a*b + 12*b^2)*log(a 
bs(-cos(d*x + c) + 1))/(a^4*d + 4*a^3*b*d + 6*a^2*b^2*d + 4*a*b^3*d + b^4* 
d) + 1/8*(4*a^2 - 13*a*b + 12*b^2)*log(abs(-cos(d*x + c) - 1))/(a^4*d - 4* 
a^3*b*d + 6*a^2*b^2*d - 4*a*b^3*d + b^4*d) - 1/4*((4*a^8 - 15*a^6*b^2 + 18 
*a^4*b^4 - 7*a^2*b^6)*cos(d*x + c)^3 - (5*a^8*b - 18*a^6*b^3 + 13*a^4*b^5 
- 4*a^2*b^7 + 4*b^9)*cos(d*x + c)^4/a - 3*(a^8 - 4*a^6*b^2 + 5*a^4*b^4 - 2 
*a^2*b^6)*cos(d*x + c) + (7*a^8*b - 24*a^6*b^3 + 11*a^4*b^5 - 2*a^2*b^7 + 
8*b^9)*cos(d*x + c)^2/a - (3*a^8*b - 9*a^6*b^3 + a^4*b^5 + a^2*b^7 + 4*b^9 
)/a)/((a*cos(d*x + c) + b)*(a + b)^4*(a - b)^4*a*d*(cos(d*x + c) + 1)^2*(c 
os(d*x + c) - 1)^2)
 

Mupad [B] (verification not implemented)

Time = 11.78 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.69 \[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{4\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-13\,a^4+20\,a^3\,b+18\,a^2\,b^2-44\,a\,b^3+19\,b^4\right )}{4\,{\left (a+b\right )}^2}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (3\,a^7-10\,a^6\,b+5\,a^5\,b^2+20\,a^4\,b^3-35\,a^3\,b^4+22\,a^2\,b^5-5\,a\,b^6+32\,b^7\right )}{a\,{\left (a+b\right )}^3\,\left (a-b\right )}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (16\,a^4-64\,a^3\,b+96\,a^2\,b^2-64\,a\,b^3+16\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (16\,a^4-32\,a^3\,b+32\,a\,b^3-16\,b^4\right )\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d\,{\left (a-b\right )}^2}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {16\,a^2+32\,a\,b-48\,b^2}{512\,{\left (a-b\right )}^4}-\frac {7}{32\,{\left (a-b\right )}^2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,a^2+13\,a\,b+12\,b^2\right )}{d\,\left (4\,a^4+16\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3+4\,b^4\right )}-\frac {b^6\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (7\,a^2-b^2\right )}{a^2\,d\,{\left (a^2-b^2\right )}^4} \] Input:

int(cot(c + d*x)^5/(a + b/cos(c + d*x))^2,x)
 

Output:

((3*a*b^2 - 3*a^2*b + a^3 - b^3)/(4*(a + b)) + (tan(c/2 + (d*x)/2)^2*(20*a 
^3*b - 44*a*b^3 - 13*a^4 + 19*b^4 + 18*a^2*b^2))/(4*(a + b)^2) + (tan(c/2 
+ (d*x)/2)^4*(3*a^7 - 10*a^6*b - 5*a*b^6 + 32*b^7 + 22*a^2*b^5 - 35*a^3*b^ 
4 + 20*a^4*b^3 + 5*a^5*b^2))/(a*(a + b)^3*(a - b)))/(d*(tan(c/2 + (d*x)/2) 
^6*(16*a^4 - 64*a^3*b - 64*a*b^3 + 16*b^4 + 96*a^2*b^2) - tan(c/2 + (d*x)/ 
2)^4*(32*a*b^3 - 32*a^3*b + 16*a^4 - 16*b^4))) - tan(c/2 + (d*x)/2)^4/(64* 
d*(a - b)^2) - log(tan(c/2 + (d*x)/2)^2 + 1)/(a^2*d) - (tan(c/2 + (d*x)/2) 
^2*((32*a*b + 16*a^2 - 48*b^2)/(512*(a - b)^4) - 7/(32*(a - b)^2)))/d + (l 
og(tan(c/2 + (d*x)/2))*(13*a*b + 4*a^2 + 12*b^2))/(d*(16*a*b^3 + 16*a^3*b 
+ 4*a^4 + 4*b^4 + 24*a^2*b^2)) - (b^6*log(a + b - a*tan(c/2 + (d*x)/2)^2 + 
 b*tan(c/2 + (d*x)/2)^2)*(7*a^2 - b^2))/(a^2*d*(a^2 - b^2)^4)
 

Reduce [B] (verification not implemented)

Time = 4.35 (sec) , antiderivative size = 1590, normalized size of antiderivative = 5.72 \[ \int \frac {\cot ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx =\text {Too large to display} \] Input:

int(cot(d*x+c)^5/(a+b*sec(d*x+c))^2,x)
 

Output:

( - 32*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a**9 + 12 
8*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a**7*b**2 - 19 
2*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a**5*b**4 + 12 
8*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a**3*b**6 - 32 
*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a*b**8 - 224*co 
s(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin( 
c + d*x)**4*a**3*b**6 + 32*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c 
 + d*x)/2)**2*b - a - b)*sin(c + d*x)**4*a*b**8 + 32*cos(c + d*x)*log(tan( 
(c + d*x)/2))*sin(c + d*x)**4*a**9 - 24*cos(c + d*x)*log(tan((c + d*x)/2)) 
*sin(c + d*x)**4*a**8*b - 128*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d 
*x)**4*a**7*b**2 + 112*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4* 
a**6*b**3 + 192*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**5*b* 
*4 - 280*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**4*b**5 + 96 
*cos(c + d*x)*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**3*b**6 - 7*cos(c + 
d*x)*sin(c + d*x)**4*a**9 - 20*cos(c + d*x)*sin(c + d*x)**4*a**8*b + 47*co 
s(c + d*x)*sin(c + d*x)**4*a**7*b**2 + 40*cos(c + d*x)*sin(c + d*x)**4*a** 
6*b**3 - 73*cos(c + d*x)*sin(c + d*x)**4*a**5*b**4 + 44*cos(c + d*x)*sin(c 
 + d*x)**4*a**4*b**5 - 31*cos(c + d*x)*sin(c + d*x)**4*a**3*b**6 + 32*cos( 
c + d*x)*sin(c + d*x)**4*a**2*b**7 - 32*cos(c + d*x)*sin(c + d*x)**4*a*b** 
8 + 32*cos(c + d*x)*sin(c + d*x)**2*a**9 - 120*cos(c + d*x)*sin(c + d*x...