Integrand size = 21, antiderivative size = 153 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=a^2 x+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {b^2 \cot ^7(c+d x)}{7 d}+\frac {2 a b \csc (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^7(c+d x)}{7 d} \] Output:
a^2*x+a^2*cot(d*x+c)/d-1/3*a^2*cot(d*x+c)^3/d+1/5*a^2*cot(d*x+c)^5/d-1/7*a ^2*cot(d*x+c)^7/d-1/7*b^2*cot(d*x+c)^7/d+2*a*b*csc(d*x+c)/d-2*a*b*csc(d*x+ c)^3/d+6/5*a*b*csc(d*x+c)^5/d-2/7*a*b*csc(d*x+c)^7/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.38 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.61 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {b \left (-5 b \cot ^7(c+d x)+2 a \csc (c+d x) \left (35-35 \csc ^2(c+d x)+21 \csc ^4(c+d x)-5 \csc ^6(c+d x)\right )\right )-5 a^2 \cot ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(c+d x)\right )}{35 d} \] Input:
Integrate[Cot[c + d*x]^8*(a + b*Sec[c + d*x])^2,x]
Output:
(b*(-5*b*Cot[c + d*x]^7 + 2*a*Csc[c + d*x]*(35 - 35*Csc[c + d*x]^2 + 21*Cs c[c + d*x]^4 - 5*Csc[c + d*x]^6)) - 5*a^2*Cot[c + d*x]^7*Hypergeometric2F1 [-7/2, 1, -5/2, -Tan[c + d*x]^2])/(35*d)
Time = 0.38 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {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.
| \(\displaystyle \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\cot \left (c+d x+\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \int \left (a^2 \cot ^8(c+d x)+2 a b \cot ^7(c+d x) \csc (c+d x)+b^2 \cot ^6(c+d x) \csc ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x-\frac {2 a b \csc ^7(c+d x)}{7 d}+\frac {6 a b \csc ^5(c+d x)}{5 d}-\frac {2 a b \csc ^3(c+d x)}{d}+\frac {2 a b \csc (c+d x)}{d}-\frac {b^2 \cot ^7(c+d x)}{7 d}\) |
Input:
Int[Cot[c + d*x]^8*(a + b*Sec[c + d*x])^2,x]
Output:
a^2*x + (a^2*Cot[c + d*x])/d - (a^2*Cot[c + d*x]^3)/(3*d) + (a^2*Cot[c + d *x]^5)/(5*d) - (a^2*Cot[c + d*x]^7)/(7*d) - (b^2*Cot[c + d*x]^7)/(7*d) + ( 2*a*b*Csc[c + d*x])/d - (2*a*b*Csc[c + d*x]^3)/d + (6*a*b*Csc[c + d*x]^5)/ (5*d) - (2*a*b*Csc[c + d*x]^7)/(7*d)
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]
Time = 2.50 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )-\frac {b^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(187\) |
| default | \(\frac {a^{2} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )-\frac {b^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(187\) |
| risch | \(a^{2} x +\frac {2 i \left (210 a b \,{\mathrm e}^{13 i \left (d x +c \right )}+420 a^{2} {\mathrm e}^{12 i \left (d x +c \right )}+105 b^{2} {\mathrm e}^{12 i \left (d x +c \right )}-420 a b \,{\mathrm e}^{11 i \left (d x +c \right )}-1260 a^{2} {\mathrm e}^{10 i \left (d x +c \right )}+1806 a b \,{\mathrm e}^{9 i \left (d x +c \right )}+3080 a^{2} {\mathrm e}^{8 i \left (d x +c \right )}+525 b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-1272 a b \,{\mathrm e}^{7 i \left (d x +c \right )}-3080 a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+1806 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+2436 a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+315 b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-420 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-812 a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+210 a b \,{\mathrm e}^{i \left (d x +c \right )}+176 a^{2}+15 b^{2}\right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\) | \(254\) |
Input:
int(cot(d*x+c)^8*(a+b*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-1/7*cot(d*x+c)^7+1/5*cot(d*x+c)^5-1/3*cot(d*x+c)^3+cot(d*x+c)+d *x+c)+2*a*b*(-1/7/sin(d*x+c)^7*cos(d*x+c)^8+1/35/sin(d*x+c)^5*cos(d*x+c)^8 -1/35/sin(d*x+c)^3*cos(d*x+c)^8+1/7/sin(d*x+c)*cos(d*x+c)^8+1/7*(16/5+cos( d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))-1/7*b^2/sin(d*x+c) ^7*cos(d*x+c)^7)
Time = 0.12 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.35 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {210 \, a b \cos \left (d x + c\right )^{6} + {\left (176 \, a^{2} + 15 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 406 \, a^{2} \cos \left (d x + c\right )^{5} - 420 \, a b \cos \left (d x + c\right )^{4} + 350 \, a^{2} \cos \left (d x + c\right )^{3} + 336 \, a b \cos \left (d x + c\right )^{2} - 105 \, a^{2} \cos \left (d x + c\right ) - 96 \, a b + 105 \, {\left (a^{2} d x \cos \left (d x + c\right )^{6} - 3 \, a^{2} d x \cos \left (d x + c\right )^{4} + 3 \, a^{2} d x \cos \left (d x + c\right )^{2} - a^{2} d x\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8*(a+b*sec(d*x+c))^2,x, algorithm="fricas")
Output:
1/105*(210*a*b*cos(d*x + c)^6 + (176*a^2 + 15*b^2)*cos(d*x + c)^7 - 406*a^ 2*cos(d*x + c)^5 - 420*a*b*cos(d*x + c)^4 + 350*a^2*cos(d*x + c)^3 + 336*a *b*cos(d*x + c)^2 - 105*a^2*cos(d*x + c) - 96*a*b + 105*(a^2*d*x*cos(d*x + c)^6 - 3*a^2*d*x*cos(d*x + c)^4 + 3*a^2*d*x*cos(d*x + c)^2 - a^2*d*x)*sin (d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))
Timed out. \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*(a+b*sec(d*x+c))**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.76 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{2} + \frac {6 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a b}{\sin \left (d x + c\right )^{7}} - \frac {15 \, b^{2}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \] Input:
integrate(cot(d*x+c)^8*(a+b*sec(d*x+c))^2,x, algorithm="maxima")
Output:
1/105*((105*d*x + 105*c + (105*tan(d*x + c)^6 - 35*tan(d*x + c)^4 + 21*tan (d*x + c)^2 - 15)/tan(d*x + c)^7)*a^2 + 6*(35*sin(d*x + c)^6 - 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 - 5)*a*b/sin(d*x + c)^7 - 15*b^2/tan(d*x + c)^7 )/d
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (141) = 282\).
Time = 0.20 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.39 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 294 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1295 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1470 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 315 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13440 \, {\left (d x + c\right )} a^{2} - 9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7350 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 525 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 7350 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 525 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1295 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1470 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 315 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 294 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 105 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} - 30 \, a b - 15 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \] Input:
integrate(cot(d*x+c)^8*(a+b*sec(d*x+c))^2,x, algorithm="giac")
Output:
1/13440*(15*a^2*tan(1/2*d*x + 1/2*c)^7 - 30*a*b*tan(1/2*d*x + 1/2*c)^7 + 1 5*b^2*tan(1/2*d*x + 1/2*c)^7 - 189*a^2*tan(1/2*d*x + 1/2*c)^5 + 294*a*b*ta n(1/2*d*x + 1/2*c)^5 - 105*b^2*tan(1/2*d*x + 1/2*c)^5 + 1295*a^2*tan(1/2*d *x + 1/2*c)^3 - 1470*a*b*tan(1/2*d*x + 1/2*c)^3 + 315*b^2*tan(1/2*d*x + 1/ 2*c)^3 + 13440*(d*x + c)*a^2 - 9765*a^2*tan(1/2*d*x + 1/2*c) + 7350*a*b*ta n(1/2*d*x + 1/2*c) - 525*b^2*tan(1/2*d*x + 1/2*c) + (9765*a^2*tan(1/2*d*x + 1/2*c)^6 + 7350*a*b*tan(1/2*d*x + 1/2*c)^6 + 525*b^2*tan(1/2*d*x + 1/2*c )^6 - 1295*a^2*tan(1/2*d*x + 1/2*c)^4 - 1470*a*b*tan(1/2*d*x + 1/2*c)^4 - 315*b^2*tan(1/2*d*x + 1/2*c)^4 + 189*a^2*tan(1/2*d*x + 1/2*c)^2 + 294*a*b* tan(1/2*d*x + 1/2*c)^2 + 105*b^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2 - 30*a*b - 15*b^2)/tan(1/2*d*x + 1/2*c)^7)/d
Time = 10.93 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.69 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=a^2\,x+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\left (a-b\right )}^2}{896\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {3\,a^2}{32}-\frac {5\,a\,b}{48}+\frac {b^2}{48}+\frac {{\left (a-b\right )}^2}{384}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {a^2}{80}-\frac {3\,a\,b}{160}+\frac {b^2}{160}+\frac {{\left (a-b\right )}^2}{640}\right )}{d}-\frac {\frac {2\,a\,b}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {37\,a^2}{3}+14\,a\,b+3\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (93\,a^2+70\,a\,b+5\,b^2\right )+\frac {a^2}{7}+\frac {b^2}{7}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {9\,a^2}{5}+\frac {14\,a\,b}{5}+b^2\right )}{128\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {23\,a^2}{32}-\frac {17\,a\,b}{32}+\frac {b^2}{32}+\frac {{\left (a-b\right )}^2}{128}\right )}{d} \] Input:
int(cot(c + d*x)^8*(a + b/cos(c + d*x))^2,x)
Output:
a^2*x + (tan(c/2 + (d*x)/2)^7*(a - b)^2)/(896*d) + (tan(c/2 + (d*x)/2)^3*( (3*a^2)/32 - (5*a*b)/48 + b^2/48 + (a - b)^2/384))/d - (tan(c/2 + (d*x)/2) ^5*(a^2/80 - (3*a*b)/160 + b^2/160 + (a - b)^2/640))/d - ((2*a*b)/7 + tan( c/2 + (d*x)/2)^4*(14*a*b + (37*a^2)/3 + 3*b^2) - tan(c/2 + (d*x)/2)^6*(70* a*b + 93*a^2 + 5*b^2) + a^2/7 + b^2/7 - tan(c/2 + (d*x)/2)^2*((14*a*b)/5 + (9*a^2)/5 + b^2))/(128*d*tan(c/2 + (d*x)/2)^7) - (tan(c/2 + (d*x)/2)*((23 *a^2)/32 - (17*a*b)/32 + b^2/32 + (a - b)^2/128))/d
Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.34 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x))^2 \, dx=\frac {176 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}-122 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}-45 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+66 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}+45 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-15 \cos \left (d x +c \right ) a^{2}-15 \cos \left (d x +c \right ) b^{2}+105 \sin \left (d x +c \right )^{7} a^{2} d x +210 \sin \left (d x +c \right )^{6} a b -210 \sin \left (d x +c \right )^{4} a b +126 \sin \left (d x +c \right )^{2} a b -30 a b}{105 \sin \left (d x +c \right )^{7} d} \] Input:
int(cot(d*x+c)^8*(a+b*sec(d*x+c))^2,x)
Output:
(176*cos(c + d*x)*sin(c + d*x)**6*a**2 + 15*cos(c + d*x)*sin(c + d*x)**6*b **2 - 122*cos(c + d*x)*sin(c + d*x)**4*a**2 - 45*cos(c + d*x)*sin(c + d*x) **4*b**2 + 66*cos(c + d*x)*sin(c + d*x)**2*a**2 + 45*cos(c + d*x)*sin(c + d*x)**2*b**2 - 15*cos(c + d*x)*a**2 - 15*cos(c + d*x)*b**2 + 105*sin(c + d *x)**7*a**2*d*x + 210*sin(c + d*x)**6*a*b - 210*sin(c + d*x)**4*a*b + 126* sin(c + d*x)**2*a*b - 30*a*b)/(105*sin(c + d*x)**7*d)