Integrand size = 21, antiderivative size = 234 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\log (\cos (c+d x))}{a d}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sec (c+d x))}{16 (a+b)^3 d}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\sec (c+d x))}{16 (a-b)^3 d}-\frac {b^6 \log (a+b \sec (c+d x))}{a \left (a^2-b^2\right )^3 d}-\frac {1}{16 (a+b) d (1-\sec (c+d x))^2}-\frac {5 a+7 b}{16 (a+b)^2 d (1-\sec (c+d x))}-\frac {1}{16 (a-b) d (1+\sec (c+d x))^2}-\frac {5 a-7 b}{16 (a-b)^2 d (1+\sec (c+d x))} \] Output:
ln(cos(d*x+c))/a/d+1/16*(8*a^2+21*a*b+15*b^2)*ln(1-sec(d*x+c))/(a+b)^3/d+1 /16*(8*a^2-21*a*b+15*b^2)*ln(1+sec(d*x+c))/(a-b)^3/d-b^6*ln(a+b*sec(d*x+c) )/a/(a^2-b^2)^3/d-1/16/(a+b)/d/(1-sec(d*x+c))^2-1/16*(5*a+7*b)/(a+b)^2/d/( 1-sec(d*x+c))-1/16/(a-b)/d/(1+sec(d*x+c))^2-1/16*(5*a-7*b)/(a-b)^2/d/(1+se c(d*x+c))
Time = 4.12 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {b^6 \left (-\frac {16 \log (\cos (c+d x))}{a b^6}-\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\sec (c+d x))}{b^6 (a+b)^3}-\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\sec (c+d x))}{(a-b)^3 b^6}+\frac {16 \log (a+b \sec (c+d x))}{a (a-b)^3 (a+b)^3}+\frac {1}{b^6 (a+b) (-1+\sec (c+d x))^2}+\frac {-5 a-7 b}{b^6 (a+b)^2 (-1+\sec (c+d x))}+\frac {1}{(a-b) b^6 (1+\sec (c+d x))^2}+\frac {5 a-7 b}{(a-b)^2 b^6 (1+\sec (c+d x))}\right )}{16 d} \] Input:
Integrate[Cot[c + d*x]^5/(a + b*Sec[c + d*x]),x]
Output:
-1/16*(b^6*((-16*Log[Cos[c + d*x]])/(a*b^6) - ((8*a^2 + 21*a*b + 15*b^2)*L og[1 - Sec[c + d*x]])/(b^6*(a + b)^3) - ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sec[c + d*x]])/((a - b)^3*b^6) + (16*Log[a + b*Sec[c + d*x]])/(a*(a - b)^ 3*(a + b)^3) + 1/(b^6*(a + b)*(-1 + Sec[c + d*x])^2) + (-5*a - 7*b)/(b^6*( a + b)^2*(-1 + Sec[c + d*x])) + 1/((a - b)*b^6*(1 + Sec[c + d*x])^2) + (5* a - 7*b)/((a - b)^2*b^6*(1 + Sec[c + d*x]))))/d
Time = 0.54 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^5 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cot \left (\frac {1}{2} (2 c+\pi )+d x\right )^5 \left (a+b \csc \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle -\frac {b^6 \int \frac {\cos (c+d x)}{b (a+b \sec (c+d x)) \left (b^2-b^2 \sec ^2(c+d x)\right )^3}d(b \sec (c+d x))}{d}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle -\frac {b^6 \int \left (\frac {7 b-5 a}{16 (a-b)^2 b^5 (\sec (c+d x) b+b)^2}+\frac {\cos (c+d x)}{a b^7}+\frac {8 a^2+21 b a+15 b^2}{16 b^6 (a+b)^3 (b-b \sec (c+d x))}+\frac {1}{a (a-b)^3 (a+b)^3 (a+b \sec (c+d x))}+\frac {8 a^2-21 b a+15 b^2}{16 b^6 (b-a)^3 (\sec (c+d x) b+b)}+\frac {5 a+7 b}{16 b^5 (a+b)^2 (b-b \sec (c+d x))^2}+\frac {1}{8 b^4 (a+b) (b-b \sec (c+d x))^3}+\frac {1}{8 b^4 (b-a) (\sec (c+d x) b+b)^3}\right )d(b \sec (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^6 \left (\frac {\log (a+b \sec (c+d x))}{a \left (a^2-b^2\right )^3}-\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (b-b \sec (c+d x))}{16 b^6 (a+b)^3}-\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (b \sec (c+d x)+b)}{16 b^6 (a-b)^3}+\frac {\log (b \sec (c+d x))}{a b^6}+\frac {5 a-7 b}{16 b^5 (a-b)^2 (b \sec (c+d x)+b)}+\frac {5 a+7 b}{16 b^5 (a+b)^2 (b-b \sec (c+d x))}+\frac {1}{16 b^4 (a+b) (b-b \sec (c+d x))^2}+\frac {1}{16 b^4 (a-b) (b \sec (c+d x)+b)^2}\right )}{d}\) |
Input:
Int[Cot[c + d*x]^5/(a + b*Sec[c + d*x]),x]
Output:
-((b^6*(Log[b*Sec[c + d*x]]/(a*b^6) - ((8*a^2 + 21*a*b + 15*b^2)*Log[b - b *Sec[c + d*x]])/(16*b^6*(a + b)^3) + Log[a + b*Sec[c + d*x]]/(a*(a^2 - b^2 )^3) - ((8*a^2 - 21*a*b + 15*b^2)*Log[b + b*Sec[c + d*x]])/(16*(a - b)^3*b ^6) + 1/(16*b^4*(a + b)*(b - b*Sec[c + d*x])^2) + (5*a + 7*b)/(16*b^5*(a + b)^2*(b - b*Sec[c + d*x])) + 1/(16*(a - b)*b^4*(b + b*Sec[c + d*x])^2) + (5*a - 7*b)/(16*(a - b)^2*b^5*(b + b*Sec[c + d*x]))))/d)
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]
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]
Time = 0.70 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{6} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a}-\frac {1}{2 \left (8 a +8 b \right ) \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {7 a +9 b}{16 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (8 a^{2}+21 a b +15 b^{2}\right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-7 a +9 b}{16 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (8 a^{2}-21 a b +15 b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(193\) |
| default | \(\frac {-\frac {b^{6} \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} a}-\frac {1}{2 \left (8 a +8 b \right ) \left (-1+\cos \left (d x +c \right )\right )^{2}}-\frac {7 a +9 b}{16 \left (a +b \right )^{2} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (8 a^{2}+21 a b +15 b^{2}\right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{16 \left (a +b \right )^{3}}-\frac {1}{2 \left (8 a -8 b \right ) \left (1+\cos \left (d x +c \right )\right )^{2}}-\frac {-7 a +9 b}{16 \left (a -b \right )^{2} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (8 a^{2}-21 a b +15 b^{2}\right ) \ln \left (1+\cos \left (d x +c \right )\right )}{16 \left (a -b \right )^{3}}}{d}\) | \(193\) |
| risch | \(-\frac {i a^{2} c}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {i a^{2} x}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}-\frac {15 i b^{2} c}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {21 i a b c}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {21 i a b x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {i a^{2} c}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {21 i a b c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 i b^{6} x}{a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {15 i b^{2} c}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 i b^{2} x}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {i a^{2} x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}+\frac {21 i a b x}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {15 i b^{2} x}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {i x}{a}+\frac {2 i b^{6} c}{a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {5 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}-9 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-16 a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+24 a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}+b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+16 a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-32 a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-16 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+24 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+5 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-9 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} d}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {21 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{8 d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {21 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{8 d \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(998\) |
Input:
int(cot(d*x+c)^5/(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-b^6/(a+b)^3/(a-b)^3/a*ln(b+a*cos(d*x+c))-1/2/(8*a+8*b)/(-1+cos(d*x+c ))^2-1/16*(7*a+9*b)/(a+b)^2/(-1+cos(d*x+c))+1/16*(8*a^2+21*a*b+15*b^2)/(a+ b)^3*ln(-1+cos(d*x+c))-1/2/(8*a-8*b)/(1+cos(d*x+c))^2-1/16*(-7*a+9*b)/(a-b )^2/(1+cos(d*x+c))+1/16*(8*a^2-21*a*b+15*b^2)/(a-b)^3*ln(1+cos(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 576 vs. \(2 (218) = 436\).
Time = 0.27 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.46 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {12 \, a^{6} - 32 \, a^{4} b^{2} + 20 \, a^{2} b^{4} + 2 \, {\left (5 \, a^{5} b - 14 \, a^{3} b^{3} + 9 \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 8 \, {\left (2 \, a^{6} - 5 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (3 \, a^{5} b - 10 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right ) - 16 \, {\left (b^{6} \cos \left (d x + c\right )^{4} - 2 \, b^{6} \cos \left (d x + c\right )^{2} + b^{6}\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + {\left (8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5} + {\left (8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (8 \, a^{6} + 3 \, a^{5} b - 24 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5} + {\left (8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (8 \, a^{6} - 3 \, a^{5} b - 24 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 24 \, a^{2} b^{4} - 15 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{16 \, {\left ({\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d\right )}} \] Input:
integrate(cot(d*x+c)^5/(a+b*sec(d*x+c)),x, algorithm="fricas")
Output:
1/16*(12*a^6 - 32*a^4*b^2 + 20*a^2*b^4 + 2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5 )*cos(d*x + c)^3 - 8*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c)^2 - 2*(3 *a^5*b - 10*a^3*b^3 + 7*a*b^5)*cos(d*x + c) - 16*(b^6*cos(d*x + c)^4 - 2*b ^6*cos(d*x + c)^2 + b^6)*log(a*cos(d*x + c) + b) + (8*a^6 + 3*a^5*b - 24*a ^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5 + (8*a^6 + 3*a^5*b - 24*a^4*b^ 2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cos(d*x + c)^4 - 2*(8*a^6 + 3*a^5* b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cos(d*x + c)^2)*log(1 /2*cos(d*x + c) + 1/2) + (8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a ^2*b^4 - 15*a*b^5 + (8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^ 4 - 15*a*b^5)*cos(d*x + c)^4 - 2*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^ 3 + 24*a^2*b^4 - 15*a*b^5)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/( (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)^4 - 2*(a^7 - 3*a^5*b^ 2 + 3*a^3*b^4 - a*b^6)*d*cos(d*x + c)^2 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a *b^6)*d)
\[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\int \frac {\cot ^{5}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \] Input:
integrate(cot(d*x+c)**5/(a+b*sec(d*x+c)),x)
Output:
Integral(cot(c + d*x)**5/(a + b*sec(c + d*x)), x)
Time = 0.04 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.24 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {\frac {16 \, b^{6} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} - \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, {\left ({\left (5 \, a^{2} b - 9 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 6 \, a^{3} - 10 \, a b^{2} - 4 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, a^{2} b - 7 \, b^{3}\right )} \cos \left (d x + c\right )\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} - 2 \, a^{2} b^{2} + b^{4} - 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}}}{16 \, d} \] Input:
integrate(cot(d*x+c)^5/(a+b*sec(d*x+c)),x, algorithm="maxima")
Output:
-1/16*(16*b^6*log(a*cos(d*x + c) + b)/(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 ) - (8*a^2 - 21*a*b + 15*b^2)*log(cos(d*x + c) + 1)/(a^3 - 3*a^2*b + 3*a*b ^2 - b^3) - (8*a^2 + 21*a*b + 15*b^2)*log(cos(d*x + c) - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - 2*((5*a^2*b - 9*b^3)*cos(d*x + c)^3 + 6*a^3 - 10*a*b^2 - 4*(2*a^3 - 3*a*b^2)*cos(d*x + c)^2 - (3*a^2*b - 7*b^3)*cos(d*x + c))/(( a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^4 + a^4 - 2*a^2*b^2 + b^4 - 2*(a^4 - 2 *a^2*b^2 + b^4)*cos(d*x + c)^2))/d
Time = 0.15 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.30 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=-\frac {b^{6} \log \left ({\left | a \cos \left (d x + c\right ) + b \right |}\right )}{a^{7} d - 3 \, a^{5} b^{2} d + 3 \, a^{3} b^{4} d - a b^{6} d} + \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left ({\left | \cos \left (d x + c\right ) + 1 \right |}\right )}{16 \, {\left (a^{3} d - 3 \, a^{2} b d + 3 \, a b^{2} d - b^{3} d\right )}} + \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left ({\left | \cos \left (d x + c\right ) - 1 \right |}\right )}{16 \, {\left (a^{3} d + 3 \, a^{2} b d + 3 \, a b^{2} d + b^{3} d\right )}} + \frac {6 \, a^{5} - 16 \, a^{3} b^{2} + 10 \, a b^{4} + {\left (5 \, a^{4} b - 14 \, a^{2} b^{3} + 9 \, b^{5}\right )} \cos \left (d x + c\right )^{3} - 4 \, {\left (2 \, a^{5} - 5 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, a^{4} b - 10 \, a^{2} b^{3} + 7 \, b^{5}\right )} \cos \left (d x + c\right )}{8 \, {\left (a + b\right )}^{3} {\left (a - b\right )}^{3} d {\left (\cos \left (d x + c\right ) + 1\right )}^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}} \] Input:
integrate(cot(d*x+c)^5/(a+b*sec(d*x+c)),x, algorithm="giac")
Output:
-b^6*log(abs(a*cos(d*x + c) + b))/(a^7*d - 3*a^5*b^2*d + 3*a^3*b^4*d - a*b ^6*d) + 1/16*(8*a^2 - 21*a*b + 15*b^2)*log(abs(cos(d*x + c) + 1))/(a^3*d - 3*a^2*b*d + 3*a*b^2*d - b^3*d) + 1/16*(8*a^2 + 21*a*b + 15*b^2)*log(abs(c os(d*x + c) - 1))/(a^3*d + 3*a^2*b*d + 3*a*b^2*d + b^3*d) + 1/8*(6*a^5 - 1 6*a^3*b^2 + 10*a*b^4 + (5*a^4*b - 14*a^2*b^3 + 9*b^5)*cos(d*x + c)^3 - 4*( 2*a^5 - 5*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^2 - (3*a^4*b - 10*a^2*b^3 + 7*b^ 5)*cos(d*x + c))/((a + b)^3*(a - b)^3*d*(cos(d*x + c) + 1)^2*(cos(d*x + c) - 1)^2)
Time = 12.07 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.24 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx=\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (8\,a^2+21\,a\,b+15\,b^2\right )}{d\,\left (8\,a^3+24\,a^2\,b+24\,a\,b^2+8\,b^3\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4\,d\,\left (16\,a-16\,b\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {16\,b}{{\left (16\,a-16\,b\right )}^2}-\frac {3}{16\,a-16\,b}\right )}{d}-\frac {\frac {a^2-2\,a\,b+b^2}{4\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-3\,a^3+2\,a^2\,b+5\,a\,b^2-4\,b^3\right )}{{\left (a+b\right )}^2}}{d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (16\,a^2-32\,a\,b+16\,b^2\right )}-\frac {b^6\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a\,d\,{\left (a^2-b^2\right )}^3} \] Input:
int(cot(c + d*x)^5/(a + b/cos(c + d*x)),x)
Output:
(log(tan(c/2 + (d*x)/2))*(21*a*b + 8*a^2 + 15*b^2))/(d*(24*a*b^2 + 24*a^2* b + 8*a^3 + 8*b^3)) - tan(c/2 + (d*x)/2)^4/(4*d*(16*a - 16*b)) - log(tan(c /2 + (d*x)/2)^2 + 1)/(a*d) - (tan(c/2 + (d*x)/2)^2*((16*b)/(16*a - 16*b)^2 - 3/(16*a - 16*b)))/d - ((a^2 - 2*a*b + b^2)/(4*(a + b)) + (tan(c/2 + (d* x)/2)^2*(5*a*b^2 + 2*a^2*b - 3*a^3 - 4*b^3))/(a + b)^2)/(d*tan(c/2 + (d*x) /2)^4*(16*a^2 - 32*a*b + 16*b^2)) - (b^6*log(a + b - a*tan(c/2 + (d*x)/2)^ 2 + b*tan(c/2 + (d*x)/2)^2))/(a*d*(a^2 - b^2)^3)
Time = 0.21 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.42 \[ \int \frac {\cot ^5(c+d x)}{a+b \sec (c+d x)} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^5/(a+b*sec(d*x+c)),x)
Output:
( - 20*cos(c + d*x)*sin(c + d*x)**2*a**5*b + 56*cos(c + d*x)*sin(c + d*x)* *2*a**3*b**3 - 36*cos(c + d*x)*sin(c + d*x)**2*a*b**5 + 8*cos(c + d*x)*a** 5*b - 16*cos(c + d*x)*a**3*b**3 + 8*cos(c + d*x)*a*b**5 - 32*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a**6 + 96*log(tan((c + d*x)/2)**2 + 1)*sin (c + d*x)**4*a**4*b**2 - 96*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*a **2*b**4 + 32*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)**4*b**6 - 32*log(t an((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*sin(c + d*x)**4*b**6 + 32*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**6 - 12*log(tan((c + d*x)/2) )*sin(c + d*x)**4*a**5*b - 96*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**4*b **2 + 40*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**3*b**3 + 96*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**2*b**4 - 60*log(tan((c + d*x)/2))*sin(c + d*x )**4*a*b**5 - 13*sin(c + d*x)**4*a**6 + 34*sin(c + d*x)**4*a**4*b**2 - 21* sin(c + d*x)**4*a**2*b**4 + 32*sin(c + d*x)**2*a**6 - 80*sin(c + d*x)**2*a **4*b**2 + 48*sin(c + d*x)**2*a**2*b**4 - 8*a**6 + 16*a**4*b**2 - 8*a**2*b **4)/(32*sin(c + d*x)**4*a*d*(a**6 - 3*a**4*b**2 + 3*a**2*b**4 - b**6))