Integrand size = 21, antiderivative size = 179 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-a^2 x-\frac {a^2 \cot (c+d x)}{d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {2 a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \csc (c+d x)}{d}+\frac {8 a^2 \csc ^3(c+d x)}{3 d}-\frac {12 a^2 \csc ^5(c+d x)}{5 d}+\frac {8 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d} \] 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-2/9*a^2*cot(d*x+c)^9/d-2*a^2*csc(d*x+c)/d+8/3*a^2*csc(d *x+c)^3/d-12/5*a^2*csc(d*x+c)^5/d+8/7*a^2*csc(d*x+c)^7/d-2/9*a^2*csc(d*x+c )^9/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.55 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.79 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 \cot ^9(c+d x)}{9 d}-\frac {2 a^2 \csc (c+d x)}{d}+\frac {8 a^2 \csc ^3(c+d x)}{3 d}-\frac {12 a^2 \csc ^5(c+d x)}{5 d}+\frac {8 a^2 \csc ^7(c+d x)}{7 d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}-\frac {a^2 \cot ^9(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},1,-\frac {7}{2},-\tan ^2(c+d x)\right )}{9 d} \] Input:
Integrate[Cot[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]
Output:
-1/9*(a^2*Cot[c + d*x]^9)/d - (2*a^2*Csc[c + d*x])/d + (8*a^2*Csc[c + d*x] ^3)/(3*d) - (12*a^2*Csc[c + d*x]^5)/(5*d) + (8*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*Csc[c + d*x]^9)/(9*d) - (a^2*Cot[c + d*x]^9*Hypergeometric2F1[-9/2 , 1, -7/2, -Tan[c + d*x]^2])/(9*d)
Time = 0.43 (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.
| \(\displaystyle \int \cot ^{10}(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}{\cot \left (c+d x+\frac {\pi }{2}\right )^{10}}dx\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \int \left (a^2 \cot ^{10}(c+d x)+2 a^2 \cot ^9(c+d x) \csc (c+d x)+a^2 \cot ^8(c+d x) \csc ^2(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^2 \cot ^9(c+d x)}{9 d}+\frac {a^2 \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-\frac {2 a^2 \csc ^9(c+d x)}{9 d}+\frac {8 a^2 \csc ^7(c+d x)}{7 d}-\frac {12 a^2 \csc ^5(c+d x)}{5 d}+\frac {8 a^2 \csc ^3(c+d x)}{3 d}-\frac {2 a^2 \csc (c+d x)}{d}-a^2 x\) |
Input:
Int[Cot[c + d*x]^10*(a + a*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) - (2*a^2*Cot[c + d*x]^9)/(9*d ) - (2*a^2*Csc[c + d*x])/d + (8*a^2*Csc[c + d*x]^3)/(3*d) - (12*a^2*Csc[c + d*x]^5)/(5*d) + (8*a^2*Csc[c + d*x]^7)/(7*d) - (2*a^2*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 = 3.89 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(-a^{2} x -\frac {4 i a^{2} \left (315 \,{\mathrm e}^{13 i \left (d x +c \right )}-315 \,{\mathrm e}^{12 i \left (d x +c \right )}-1470 \,{\mathrm e}^{11 i \left (d x +c \right )}+3360 \,{\mathrm e}^{10 i \left (d x +c \right )}+1113 \,{\mathrm e}^{9 i \left (d x +c \right )}-6447 \,{\mathrm e}^{8 i \left (d x +c \right )}+2028 \,{\mathrm e}^{7 i \left (d x +c \right )}+7008 \,{\mathrm e}^{6 i \left (d x +c \right )}-4867 \,{\mathrm e}^{5 i \left (d x +c \right )}-2321 \,{\mathrm e}^{4 i \left (d x +c \right )}+3314 \,{\mathrm e}^{3 i \left (d x +c \right )}-16 \,{\mathrm e}^{2 i \left (d x +c \right )}-881 \,{\mathrm e}^{i \left (d x +c \right )}+299\right )}{315 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{5}}\) | \(188\) |
| derivativedivides | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{9}}{9 \sin \left (d x +c \right )^{9}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{10}}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{9}}{9}+\frac {\cot \left (d x +c \right )^{7}}{7}-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(231\) |
| default | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{9}}{9 \sin \left (d x +c \right )^{9}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )^{9}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{10}}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{10}}{63 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{10}}{9 \sin \left (d x +c \right )}-\frac {\left (\frac {128}{35}+\cos \left (d x +c \right )^{8}+\frac {8 \cos \left (d x +c \right )^{6}}{7}+\frac {48 \cos \left (d x +c \right )^{4}}{35}+\frac {64 \cos \left (d x +c \right )^{2}}{35}\right ) \sin \left (d x +c \right )}{9}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{9}}{9}+\frac {\cot \left (d x +c \right )^{7}}{7}-\frac {\cot \left (d x +c \right )^{5}}{5}+\frac {\cot \left (d x +c \right )^{3}}{3}-\cot \left (d x +c \right )-d x -c \right )}{d}\) | \(231\) |
Input:
int(cot(d*x+c)^10*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-a^2*x-4/315*I*a^2*(315*exp(13*I*(d*x+c))-315*exp(12*I*(d*x+c))-1470*exp(1 1*I*(d*x+c))+3360*exp(10*I*(d*x+c))+1113*exp(9*I*(d*x+c))-6447*exp(8*I*(d* x+c))+2028*exp(7*I*(d*x+c))+7008*exp(6*I*(d*x+c))-4867*exp(5*I*(d*x+c))-23 21*exp(4*I*(d*x+c))+3314*exp(3*I*(d*x+c))-16*exp(2*I*(d*x+c))-881*exp(I*(d *x+c))+299)/d/(exp(I*(d*x+c))-1)^9/(exp(I*(d*x+c))+1)^5
Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.53 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {598 \, a^{2} \cos \left (d x + c\right )^{7} - 566 \, a^{2} \cos \left (d x + c\right )^{6} - 1212 \, a^{2} \cos \left (d x + c\right )^{5} + 1310 \, a^{2} \cos \left (d x + c\right )^{4} + 860 \, a^{2} \cos \left (d x + c\right )^{3} - 1014 \, a^{2} \cos \left (d x + c\right )^{2} - 197 \, a^{2} \cos \left (d x + c\right ) + 256 \, a^{2} + 315 \, {\left (a^{2} d x \cos \left (d x + c\right )^{6} - 2 \, a^{2} d x \cos \left (d x + c\right )^{5} - a^{2} d x \cos \left (d x + c\right )^{4} + 4 \, a^{2} d x \cos \left (d x + c\right )^{3} - a^{2} d x \cos \left (d x + c\right )^{2} - 2 \, a^{2} d x \cos \left (d x + c\right ) + a^{2} d x\right )} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{6} - 2 \, d \cos \left (d x + c\right )^{5} - d \cos \left (d x + c\right )^{4} + 4 \, d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Output:
-1/315*(598*a^2*cos(d*x + c)^7 - 566*a^2*cos(d*x + c)^6 - 1212*a^2*cos(d*x + c)^5 + 1310*a^2*cos(d*x + c)^4 + 860*a^2*cos(d*x + c)^3 - 1014*a^2*cos( d*x + c)^2 - 197*a^2*cos(d*x + c) + 256*a^2 + 315*(a^2*d*x*cos(d*x + c)^6 - 2*a^2*d*x*cos(d*x + c)^5 - a^2*d*x*cos(d*x + c)^4 + 4*a^2*d*x*cos(d*x + c)^3 - a^2*d*x*cos(d*x + c)^2 - 2*a^2*d*x*cos(d*x + c) + a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^6 - 2*d*cos(d*x + c)^5 - d*cos(d*x + c)^4 + 4*d*cos (d*x + c)^3 - d*cos(d*x + c)^2 - 2*d*cos(d*x + c) + d)*sin(d*x + c))
Timed out. \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**10*(a+a*sec(d*x+c))**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.77 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {{\left (315 \, d x + 315 \, c + \frac {315 \, \tan \left (d x + c\right )^{8} - 105 \, \tan \left (d x + c\right )^{6} + 63 \, \tan \left (d x + c\right )^{4} - 45 \, \tan \left (d x + c\right )^{2} + 35}{\tan \left (d x + c\right )^{9}}\right )} a^{2} + \frac {2 \, {\left (315 \, \sin \left (d x + c\right )^{8} - 420 \, \sin \left (d x + c\right )^{6} + 378 \, \sin \left (d x + c\right )^{4} - 180 \, \sin \left (d x + c\right )^{2} + 35\right )} a^{2}}{\sin \left (d x + c\right )^{9}} + \frac {35 \, a^{2}}{\tan \left (d x + c\right )^{9}}}{315 \, d} \] Input:
integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^2,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^2 + 2*(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^2/sin(d*x + c)^9 + 35*a^2/tan(d*x + c)^9)/d
Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.81 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {63 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 945 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40320 \, {\left (d x + c\right )} a^{2} + 11655 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {51345 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 9765 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 2331 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 405 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 35 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{40320 \, d} \] Input:
integrate(cot(d*x+c)^10*(a+a*sec(d*x+c))^2,x, algorithm="giac")
Output:
1/40320*(63*a^2*tan(1/2*d*x + 1/2*c)^5 - 945*a^2*tan(1/2*d*x + 1/2*c)^3 - 40320*(d*x + c)*a^2 + 11655*a^2*tan(1/2*d*x + 1/2*c) - (51345*a^2*tan(1/2* d*x + 1/2*c)^8 - 9765*a^2*tan(1/2*d*x + 1/2*c)^6 + 2331*a^2*tan(1/2*d*x + 1/2*c)^4 - 405*a^2*tan(1/2*d*x + 1/2*c)^2 + 35*a^2)/tan(1/2*d*x + 1/2*c)^9 )/d
Time = 14.66 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.28 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2\,\left (35\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-63\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+945\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-11655\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+51345\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-9765\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2331\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-405\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+40320\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (c+d\,x\right )\right )}{40320\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9} \] Input:
int(cot(c + d*x)^10*(a + a/cos(c + d*x))^2,x)
Output:
-(a^2*(35*cos(c/2 + (d*x)/2)^14 - 63*sin(c/2 + (d*x)/2)^14 + 945*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^12 - 11655*cos(c/2 + (d*x)/2)^4*sin(c/2 + ( d*x)/2)^10 + 51345*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^8 - 9765*cos(c/ 2 + (d*x)/2)^8*sin(c/2 + (d*x)/2)^6 + 2331*cos(c/2 + (d*x)/2)^10*sin(c/2 + (d*x)/2)^4 - 405*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^2 + 40320*cos(c /2 + (d*x)/2)^5*sin(c/2 + (d*x)/2)^9*(c + d*x)))/(40320*d*cos(c/2 + (d*x)/ 2)^5*sin(c/2 + (d*x)/2)^9)
Time = 0.17 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.71 \[ \int \cot ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^{2} \left (63 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}-945 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+11655 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-40320 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} d x -51345 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+9765 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-2331 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+405 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-35\right )}{40320 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} d} \] Input:
int(cot(d*x+c)^10*(a+a*sec(d*x+c))^2,x)
Output:
(a**2*(63*tan((c + d*x)/2)**14 - 945*tan((c + d*x)/2)**12 + 11655*tan((c + d*x)/2)**10 - 40320*tan((c + d*x)/2)**9*d*x - 51345*tan((c + d*x)/2)**8 + 9765*tan((c + d*x)/2)**6 - 2331*tan((c + d*x)/2)**4 + 405*tan((c + d*x)/2 )**2 - 35))/(40320*tan((c + d*x)/2)**9*d)