Integrand size = 21, antiderivative size = 215 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {x}{a^3}-\frac {\cot (c+d x)}{a^3 d}+\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {\cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\cot ^9(c+d x)}{9 a^3 d}+\frac {4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac {3 \csc (c+d x)}{a^3 d}-\frac {16 \csc ^3(c+d x)}{3 a^3 d}+\frac {34 \csc ^5(c+d x)}{5 a^3 d}-\frac {36 \csc ^7(c+d x)}{7 a^3 d}+\frac {19 \csc ^9(c+d x)}{9 a^3 d}-\frac {4 \csc ^{11}(c+d x)}{11 a^3 d} \] Output:
-x/a^3-cot(d*x+c)/a^3/d+1/3*cot(d*x+c)^3/a^3/d-1/5*cot(d*x+c)^5/a^3/d+1/7* cot(d*x+c)^7/a^3/d-1/9*cot(d*x+c)^9/a^3/d+4/11*cot(d*x+c)^11/a^3/d+3*csc(d *x+c)/a^3/d-16/3*csc(d*x+c)^3/a^3/d+34/5*csc(d*x+c)^5/a^3/d-36/7*csc(d*x+c )^7/a^3/d+19/9*csc(d*x+c)^9/a^3/d-4/11*csc(d*x+c)^11/a^3/d
Time = 3.82 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.83 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (231 (-25+28 \cos (c+d x)) \cot ^2\left (\frac {c}{2}\right ) \csc ^4\left (\frac {1}{2} (c+d x)\right )+561145 \sec ^2\left (\frac {1}{2} (c+d x)\right )-184650 \sec ^4\left (\frac {1}{2} (c+d x)\right )+41320 \sec ^6\left (\frac {1}{2} (c+d x)\right )-5425 \sec ^8\left (\frac {1}{2} (c+d x)\right )+315 \sec ^{10}\left (\frac {1}{2} (c+d x)\right )-1736335 \csc \left (\frac {c}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+561145 \csc \left (\frac {c}{2}\right ) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )-184650 \csc \left (\frac {c}{2}\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+41320 \csc \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )-5425 \csc \left (\frac {c}{2}\right ) \sec ^9\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+315 \csc \left (\frac {c}{2}\right ) \sec ^{11}\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+6468 \csc ^3\left (\frac {c}{2}\right ) \csc ^3\left (\frac {1}{2} (c+d x)\right ) \sin (c) \sin \left (\frac {d x}{2}\right )+231 \cot \left (\frac {c}{2}\right ) \left (3840 d x-\csc \left (\frac {c}{2}\right ) \csc \left (\frac {1}{2} (c+d x)\right ) \left (743+3 \csc ^4\left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (\frac {d x}{2}\right )\right )\right ) \tan \left (\frac {c}{2}\right )}{110880 a^3 d (1+\sec (c+d x))^3} \] Input:
Integrate[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]
Output:
-1/110880*(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(231*(-25 + 28*Cos[c + d*x])* Cot[c/2]^2*Csc[(c + d*x)/2]^4 + 561145*Sec[(c + d*x)/2]^2 - 184650*Sec[(c + d*x)/2]^4 + 41320*Sec[(c + d*x)/2]^6 - 5425*Sec[(c + d*x)/2]^8 + 315*Sec [(c + d*x)/2]^10 - 1736335*Csc[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + 561145 *Csc[c/2]*Sec[(c + d*x)/2]^3*Sin[(d*x)/2] - 184650*Csc[c/2]*Sec[(c + d*x)/ 2]^5*Sin[(d*x)/2] + 41320*Csc[c/2]*Sec[(c + d*x)/2]^7*Sin[(d*x)/2] - 5425* Csc[c/2]*Sec[(c + d*x)/2]^9*Sin[(d*x)/2] + 315*Csc[c/2]*Sec[(c + d*x)/2]^1 1*Sin[(d*x)/2] + 6468*Csc[c/2]^3*Csc[(c + d*x)/2]^3*Sin[c]*Sin[(d*x)/2] + 231*Cot[c/2]*(3840*d*x - Csc[c/2]*Csc[(c + d*x)/2]*(743 + 3*Csc[(c + d*x)/ 2]^4)*Sin[(d*x)/2]))*Tan[c/2])/(a^3*d*(1 + Sec[c + d*x])^3)
Time = 0.63 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4376, 25, 3042, 4374, 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.
| \(\displaystyle \int \frac {\cot ^6(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cot \left (c+d x+\frac {\pi }{2}\right )^6 \left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {\int -\cot ^{12}(c+d x) (a-a \sec (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \cot ^{12}(c+d x) (a-a \sec (c+d x))^3dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\cot \left (c+d x+\frac {\pi }{2}\right )^{12}}dx}{a^6}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle -\frac {\int \left (a^3 \cot ^{12}(c+d x)-3 a^3 \csc (c+d x) \cot ^{11}(c+d x)+3 a^3 \csc ^2(c+d x) \cot ^{10}(c+d x)-a^3 \csc ^3(c+d x) \cot ^9(c+d x)\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {4 a^3 \cot ^{11}(c+d x)}{11 d}+\frac {a^3 \cot ^9(c+d x)}{9 d}-\frac {a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}+\frac {4 a^3 \csc ^{11}(c+d x)}{11 d}-\frac {19 a^3 \csc ^9(c+d x)}{9 d}+\frac {36 a^3 \csc ^7(c+d x)}{7 d}-\frac {34 a^3 \csc ^5(c+d x)}{5 d}+\frac {16 a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc (c+d x)}{d}+a^3 x}{a^6}\) |
Input:
Int[Cot[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]
Output:
-((a^3*x + (a^3*Cot[c + d*x])/d - (a^3*Cot[c + d*x]^3)/(3*d) + (a^3*Cot[c + d*x]^5)/(5*d) - (a^3*Cot[c + d*x]^7)/(7*d) + (a^3*Cot[c + d*x]^9)/(9*d) - (4*a^3*Cot[c + d*x]^11)/(11*d) - (3*a^3*Csc[c + d*x])/d + (16*a^3*Csc[c + d*x]^3)/(3*d) - (34*a^3*Csc[c + d*x]^5)/(5*d) + (36*a^3*Csc[c + d*x]^7)/ (7*d) - (19*a^3*Csc[c + d*x]^9)/(9*d) + (4*a^3*Csc[c + d*x]^11)/(11*d))/a^ 6)
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Time = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.64
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}+\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+382 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-512 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {10}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {46}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{256 d \,a^{3}}\) | \(137\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{11}+\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {46 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+382 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-512 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {10}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {46}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{256 d \,a^{3}}\) | \(137\) |
| risch | \(-\frac {x}{a^{3}}+\frac {2 i \left (10395 \,{\mathrm e}^{15 i \left (d x +c \right )}+31185 \,{\mathrm e}^{14 i \left (d x +c \right )}+1155 \,{\mathrm e}^{13 i \left (d x +c \right )}-148995 \,{\mathrm e}^{12 i \left (d x +c \right )}-190113 \,{\mathrm e}^{11 i \left (d x +c \right )}+117117 \,{\mathrm e}^{10 i \left (d x +c \right )}+434775 \,{\mathrm e}^{9 i \left (d x +c \right )}+138105 \,{\mathrm e}^{8 i \left (d x +c \right )}-385055 \,{\mathrm e}^{7 i \left (d x +c \right )}-374781 \,{\mathrm e}^{6 i \left (d x +c \right )}+63289 \,{\mathrm e}^{5 i \left (d x +c \right )}+223655 \,{\mathrm e}^{4 i \left (d x +c \right )}+75685 \,{\mathrm e}^{3 i \left (d x +c \right )}-43345 \,{\mathrm e}^{2 i \left (d x +c \right )}-34323 \,{\mathrm e}^{i \left (d x +c \right )}-7453\right )}{3465 d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{11} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\) | \(210\) |
Input:
int(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/256/d/a^3*(-1/11*tan(1/2*d*x+1/2*c)^11+10/9*tan(1/2*d*x+1/2*c)^9-46/7*ta n(1/2*d*x+1/2*c)^7+26*tan(1/2*d*x+1/2*c)^5-256/3*tan(1/2*d*x+1/2*c)^3+382* tan(1/2*d*x+1/2*c)-512*arctan(tan(1/2*d*x+1/2*c))-1/5/tan(1/2*d*x+1/2*c)^5 +10/3/tan(1/2*d*x+1/2*c)^3-46/tan(1/2*d*x+1/2*c))
Time = 0.09 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.31 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {7453 \, \cos \left (d x + c\right )^{8} + 11964 \, \cos \left (d x + c\right )^{7} - 11866 \, \cos \left (d x + c\right )^{6} - 30542 \, \cos \left (d x + c\right )^{5} + 90 \, \cos \left (d x + c\right )^{4} + 26438 \, \cos \left (d x + c\right )^{3} + 8539 \, \cos \left (d x + c\right )^{2} + 3465 \, {\left (d x \cos \left (d x + c\right )^{7} + 3 \, d x \cos \left (d x + c\right )^{6} + d x \cos \left (d x + c\right )^{5} - 5 \, d x \cos \left (d x + c\right )^{4} - 5 \, d x \cos \left (d x + c\right )^{3} + d x \cos \left (d x + c\right )^{2} + 3 \, d x \cos \left (d x + c\right ) + d x\right )} \sin \left (d x + c\right ) - 7671 \, \cos \left (d x + c\right ) - 3712}{3465 \, {\left (a^{3} d \cos \left (d x + c\right )^{7} + 3 \, a^{3} d \cos \left (d x + c\right )^{6} + a^{3} d \cos \left (d x + c\right )^{5} - 5 \, a^{3} d \cos \left (d x + c\right )^{4} - 5 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
-1/3465*(7453*cos(d*x + c)^8 + 11964*cos(d*x + c)^7 - 11866*cos(d*x + c)^6 - 30542*cos(d*x + c)^5 + 90*cos(d*x + c)^4 + 26438*cos(d*x + c)^3 + 8539* cos(d*x + c)^2 + 3465*(d*x*cos(d*x + c)^7 + 3*d*x*cos(d*x + c)^6 + d*x*cos (d*x + c)^5 - 5*d*x*cos(d*x + c)^4 - 5*d*x*cos(d*x + c)^3 + d*x*cos(d*x + c)^2 + 3*d*x*cos(d*x + c) + d*x)*sin(d*x + c) - 7671*cos(d*x + c) - 3712)/ ((a^3*d*cos(d*x + c)^7 + 3*a^3*d*cos(d*x + c)^6 + a^3*d*cos(d*x + c)^5 - 5 *a^3*d*cos(d*x + c)^4 - 5*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2 + 3* a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))
\[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\cot ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \] Input:
integrate(cot(d*x+c)**6/(a+a*sec(d*x+c))**3,x)
Output:
Integral(cot(c + d*x)**6/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3
Time = 0.12 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.01 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\frac {5 \, {\left (\frac {264726 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {59136 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {18018 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {4554 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {770 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {63 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}\right )}}{a^{3}} - \frac {1774080 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {231 \, {\left (\frac {50 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {690 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a^{3} \sin \left (d x + c\right )^{5}}}{887040 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
1/887040*(5*(264726*sin(d*x + c)/(cos(d*x + c) + 1) - 59136*sin(d*x + c)^3 /(cos(d*x + c) + 1)^3 + 18018*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 4554*s in(d*x + c)^7/(cos(d*x + c) + 1)^7 + 770*sin(d*x + c)^9/(cos(d*x + c) + 1) ^9 - 63*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)/a^3 - 1774080*arctan(sin(d* x + c)/(cos(d*x + c) + 1))/a^3 + 231*(50*sin(d*x + c)^2/(cos(d*x + c) + 1) ^2 - 690*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 3)*(cos(d*x + c) + 1)^5/(a^ 3*sin(d*x + c)^5))/d
Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.74 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {887040 \, {\left (d x + c\right )}}{a^{3}} + \frac {231 \, {\left (690 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 50 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {5 \, {\left (63 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 770 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 18018 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 59136 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 264726 \, a^{30} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{33}}}{887040 \, d} \] Input:
integrate(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
-1/887040*(887040*(d*x + c)/a^3 + 231*(690*tan(1/2*d*x + 1/2*c)^4 - 50*tan (1/2*d*x + 1/2*c)^2 + 3)/(a^3*tan(1/2*d*x + 1/2*c)^5) + 5*(63*a^30*tan(1/2 *d*x + 1/2*c)^11 - 770*a^30*tan(1/2*d*x + 1/2*c)^9 + 4554*a^30*tan(1/2*d*x + 1/2*c)^7 - 18018*a^30*tan(1/2*d*x + 1/2*c)^5 + 59136*a^30*tan(1/2*d*x + 1/2*c)^3 - 264726*a^30*tan(1/2*d*x + 1/2*c))/a^33)/d
Time = 14.44 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.18 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {693\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+315\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-3850\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+22770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-90090\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+295680\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-1323630\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+159390\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-11550\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+887040\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (c+d\,x\right )}{887040\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \] Input:
int(cot(c + d*x)^6/(a + a/cos(c + d*x))^3,x)
Output:
-(693*cos(c/2 + (d*x)/2)^16 + 315*sin(c/2 + (d*x)/2)^16 - 3850*cos(c/2 + ( d*x)/2)^2*sin(c/2 + (d*x)/2)^14 + 22770*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d* x)/2)^12 - 90090*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 295680*cos(c /2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^8 - 1323630*cos(c/2 + (d*x)/2)^10*sin(c /2 + (d*x)/2)^6 + 159390*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 - 1155 0*cos(c/2 + (d*x)/2)^14*sin(c/2 + (d*x)/2)^2 + 887040*cos(c/2 + (d*x)/2)^1 1*sin(c/2 + (d*x)/2)^5*(c + d*x))/(887040*a^3*d*cos(c/2 + (d*x)/2)^11*sin( c/2 + (d*x)/2)^5)
Time = 18.64 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.65 \[ \int \frac {\cot ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {-315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+3850 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}-22770 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+90090 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-295680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+1323630 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-887040 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} d x -159390 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+11550 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-693}{887040 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} a^{3} d} \] Input:
int(cot(d*x+c)^6/(a+a*sec(d*x+c))^3,x)
Output:
( - 315*tan((c + d*x)/2)**16 + 3850*tan((c + d*x)/2)**14 - 22770*tan((c + d*x)/2)**12 + 90090*tan((c + d*x)/2)**10 - 295680*tan((c + d*x)/2)**8 + 13 23630*tan((c + d*x)/2)**6 - 887040*tan((c + d*x)/2)**5*d*x - 159390*tan((c + d*x)/2)**4 + 11550*tan((c + d*x)/2)**2 - 693)/(887040*tan((c + d*x)/2)* *5*a**3*d)