Integrand size = 21, antiderivative size = 179 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-a^3 x-\frac {a^3 \cot (c+d x)}{d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {4 a^3 \cot ^9(c+d x)}{9 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d} \] Output:
-a^3*x-a^3*cot(d*x+c)/d+1/3*a^3*cot(d*x+c)^3/d-1/5*a^3*cot(d*x+c)^5/d+1/7* a^3*cot(d*x+c)^7/d-4/9*a^3*cot(d*x+c)^9/d-3*a^3*csc(d*x+c)/d+13/3*a^3*csc( d*x+c)^3/d-21/5*a^3*csc(d*x+c)^5/d+15/7*a^3*csc(d*x+c)^7/d-4/9*a^3*csc(d*x +c)^9/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.60 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.79 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3 \cot ^9(c+d x)}{3 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d}-\frac {a^3 \cot ^9(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},-\tan ^2(c+d x)\right )}{9 d} \] Input:
Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]
Output:
-1/3*(a^3*Cot[c + d*x]^9)/d - (3*a^3*Csc[c + d*x])/d + (13*a^3*Csc[c + d*x ]^3)/(3*d) - (21*a^3*Csc[c + d*x]^5)/(5*d) + (15*a^3*Csc[c + d*x]^7)/(7*d) - (4*a^3*Csc[c + d*x]^9)/(9*d) - (a^3*Cot[c + d*x]^9*Hypergeometric2F1[-9 /2, 1, -7/2, -Tan[c + d*x]^2])/(9*d)
Time = 0.47 (sec) , antiderivative size = 179, 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.
| \(\displaystyle \int \cot ^{10}(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\cot \left (c+d x+\frac {\pi }{2}\right )^{10}}dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \int \left (a^3 \cot ^{10}(c+d x)+3 a^3 \cot ^9(c+d x) \csc (c+d x)+3 a^3 \cot ^8(c+d x) \csc ^2(c+d x)+a^3 \cot ^7(c+d x) \csc ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 a^3 \cot ^9(c+d x)}{9 d}+\frac {a^3 \cot ^7(c+d x)}{7 d}-\frac {a^3 \cot ^5(c+d x)}{5 d}+\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \csc ^9(c+d x)}{9 d}+\frac {15 a^3 \csc ^7(c+d x)}{7 d}-\frac {21 a^3 \csc ^5(c+d x)}{5 d}+\frac {13 a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc (c+d x)}{d}-a^3 x\) |
Input:
Int[Cot[c + d*x]^10*(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) - (4*a^3*Cot[c + d*x]^9)/(9*d ) - (3*a^3*Csc[c + d*x])/d + (13*a^3*Csc[c + d*x]^3)/(3*d) - (21*a^3*Csc[c + d*x]^5)/(5*d) + (15*a^3*Csc[c + d*x]^7)/(7*d) - (4*a^3*Csc[c + d*x]^9)/ (9*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]
Result contains complex when optimal does not.
Time = 13.46 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(-a^{3} x -\frac {2 i a^{3} \left (945 \,{\mathrm e}^{11 i \left (d x +c \right )}-3150 \,{\mathrm e}^{10 i \left (d x +c \right )}+2625 \,{\mathrm e}^{9 i \left (d x +c \right )}+6300 \,{\mathrm e}^{8 i \left (d x +c \right )}-13482 \,{\mathrm e}^{7 i \left (d x +c \right )}+5292 \,{\mathrm e}^{6 i \left (d x +c \right )}+10566 \,{\mathrm e}^{5 i \left (d x +c \right )}-11736 \,{\mathrm e}^{4 i \left (d x +c \right )}+1289 \,{\mathrm e}^{3 i \left (d x +c \right )}+4866 \,{\mathrm e}^{2 i \left (d x +c \right )}-3063 \,{\mathrm e}^{i \left (d x +c \right )}+668\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(166\) |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{8}}{9 \sin \left (d x +c \right )^{9}}-\frac {\cos \left (d x +c \right )^{8}}{63 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{315 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{315 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{63 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {a^{3} \cos \left (d x +c \right )^{9}}{3 \sin \left (d x +c \right )^{9}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{10}}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{9}}{9}+\frac {\cot \left (d x +c \right )^{7}}{7}-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(364\) |
| default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{8}}{9 \sin \left (d x +c \right )^{9}}-\frac {\cos \left (d x +c \right )^{8}}{63 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{315 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{315 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{63 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {a^{3} \cos \left (d x +c \right )^{9}}{3 \sin \left (d x +c \right )^{9}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{10}}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{9}}{9}+\frac {\cot \left (d x +c \right )^{7}}{7}-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(364\) |
Input:
int(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-a^3*x-2/315*I*a^3*(945*exp(11*I*(d*x+c))-3150*exp(10*I*(d*x+c))+2625*exp( 9*I*(d*x+c))+6300*exp(8*I*(d*x+c))-13482*exp(7*I*(d*x+c))+5292*exp(6*I*(d* x+c))+10566*exp(5*I*(d*x+c))-11736*exp(4*I*(d*x+c))+1289*exp(3*I*(d*x+c))+ 4866*exp(2*I*(d*x+c))-3063*exp(I*(d*x+c))+668)/d/(exp(I*(d*x+c))-1)^9/(exp (I*(d*x+c))+1)^3
Time = 0.08 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.31 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {668 \, a^{3} \cos \left (d x + c\right )^{6} - 1059 \, a^{3} \cos \left (d x + c\right )^{5} - 573 \, a^{3} \cos \left (d x + c\right )^{4} + 1813 \, a^{3} \cos \left (d x + c\right )^{3} - 393 \, a^{3} \cos \left (d x + c\right )^{2} - 789 \, a^{3} \cos \left (d x + c\right ) + 368 \, a^{3} + 315 \, {\left (a^{3} d x \cos \left (d x + c\right )^{5} - 3 \, a^{3} d x \cos \left (d x + c\right )^{4} + 2 \, a^{3} d x \cos \left (d x + c\right )^{3} + 2 \, a^{3} d x \cos \left (d x + c\right )^{2} - 3 \, a^{3} d x \cos \left (d x + c\right ) + a^{3} d x\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} - 3 \, d \cos \left (d x + c\right )^{4} + 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - 3 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
-1/315*(668*a^3*cos(d*x + c)^6 - 1059*a^3*cos(d*x + c)^5 - 573*a^3*cos(d*x + c)^4 + 1813*a^3*cos(d*x + c)^3 - 393*a^3*cos(d*x + c)^2 - 789*a^3*cos(d *x + c) + 368*a^3 + 315*(a^3*d*x*cos(d*x + c)^5 - 3*a^3*d*x*cos(d*x + c)^4 + 2*a^3*d*x*cos(d*x + c)^3 + 2*a^3*d*x*cos(d*x + c)^2 - 3*a^3*d*x*cos(d*x + c) + a^3*d*x)*sin(d*x + c))/((d*cos(d*x + c)^5 - 3*d*cos(d*x + c)^4 + 2 *d*cos(d*x + c)^3 + 2*d*cos(d*x + c)^2 - 3*d*cos(d*x + c) + d)*sin(d*x + c ))
Timed out. \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**10*(a+a*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.02 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {{\left (315 \, d x + 315 \, c + \frac {315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a^{3} + \frac {3 \, {\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a^{3}}{\sin \left (d x + c\right )^{9}} - \frac {{\left (105 \, \sin \left (d x + c\right )^{6} - 189 \, \sin \left (d x + c\right )^{4} + 135 \, \sin \left (d x + c\right )^{2} - 35\right )} a^{3}}{\sin \left (d x + c\right )^{9}} + \frac {105 \, a^{3}}{\tan \left (d x + c\right )^{9}}}{315 \, d} \] Input:
integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="maxima")
Output:
-1/315*((315*d*x + 315*c + (315*tan(d*x + c)^8 - 105*tan(d*x + c)^6 + 63*t an(d*x + c)^4 - 45*tan(d*x + c)^2 + 35)/tan(d*x + c)^9)*a^3 + 3*(315*sin(d *x + c)^8 - 420*sin(d*x + c)^6 + 378*sin(d*x + c)^4 - 180*sin(d*x + c)^2 + 35)*a^3/sin(d*x + c)^9 - (105*sin(d*x + c)^6 - 189*sin(d*x + c)^4 + 135*s in(d*x + c)^2 - 35)*a^3/sin(d*x + c)^9 + 105*a^3/tan(d*x + c)^9)/d
Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.72 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20160 \, {\left (d x + c\right )} a^{3} - 2520 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {31185 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 6720 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1827 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 360 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{20160 \, d} \] Input:
integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
-1/20160*(105*a^3*tan(1/2*d*x + 1/2*c)^3 + 20160*(d*x + c)*a^3 - 2520*a^3* tan(1/2*d*x + 1/2*c) + (31185*a^3*tan(1/2*d*x + 1/2*c)^8 - 6720*a^3*tan(1/ 2*d*x + 1/2*c)^6 + 1827*a^3*tan(1/2*d*x + 1/2*c)^4 - 360*a^3*tan(1/2*d*x + 1/2*c)^2 + 35*a^3)/tan(1/2*d*x + 1/2*c)^9)/d
Time = 12.92 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.15 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {a^3\,\left (35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+105\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-2520\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+31185\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-6720\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1827\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+20160\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )\right )}{20160\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \] Input:
int(cot(c + d*x)^10*(a + a/cos(c + d*x))^3,x)
Output:
-(a^3*(35*cos(c/2 + (d*x)/2)^12 + 105*sin(c/2 + (d*x)/2)^12 - 2520*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^10 + 31185*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^8 - 6720*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^6 + 1827*cos(c/ 2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^4 - 360*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^2 + 20160*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^9*(c + d*x)))/( 20160*d*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^9)
Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.64 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (-105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+2520 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-20160 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} d x -31185 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+6720 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-1827 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+360 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-35\right )}{20160 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} d} \] Input:
int(cot(d*x+c)^10*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*( - 105*tan((c + d*x)/2)**12 + 2520*tan((c + d*x)/2)**10 - 20160*tan ((c + d*x)/2)**9*d*x - 31185*tan((c + d*x)/2)**8 + 6720*tan((c + d*x)/2)** 6 - 1827*tan((c + d*x)/2)**4 + 360*tan((c + d*x)/2)**2 - 35))/(20160*tan(( c + d*x)/2)**9*d)