Integrand size = 21, antiderivative size = 139 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=a^2 x+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{d}+\frac {6 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d} \] 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-2/7*a ^2*cot(d*x+c)^7/d+2*a^2*csc(d*x+c)/d-2*a^2*csc(d*x+c)^3/d+6/5*a^2*csc(d*x+ c)^5/d-2/7*a^2*csc(d*x+c)^7/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.49 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.87 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 \cot ^7(c+d x)}{7 d}+\frac {2 a^2 \csc (c+d x)}{d}-\frac {2 a^2 \csc ^3(c+d x)}{d}+\frac {6 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}-\frac {a^2 \cot ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(c+d x)\right )}{7 d} \] Input:
Integrate[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^2,x]
Output:
-1/7*(a^2*Cot[c + d*x]^7)/d + (2*a^2*Csc[c + d*x])/d - (2*a^2*Csc[c + d*x] ^3)/d + (6*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*Csc[c + d*x]^7)/(7*d) - (a^2 *Cot[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2])/(7*d)
Time = 0.39 (sec) , antiderivative size = 139, 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.
\(\displaystyle \int \cot ^8(c+d x) (a \sec (c+d x)+a)^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2}{\cot \left (c+d x+\frac {\pi }{2}\right )^8}dx\) |
\(\Big \downarrow \) 4374 |
\(\displaystyle \int \left (a^2 \cot ^8(c+d x)+2 a^2 \cot ^7(c+d x) \csc (c+d x)+a^2 \cot ^6(c+d x) \csc ^2(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {2 a^2 \cot ^7(c+d x)}{7 d}+\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {2 a^2 \csc ^7(c+d x)}{7 d}+\frac {6 a^2 \csc ^5(c+d x)}{5 d}-\frac {2 a^2 \csc ^3(c+d x)}{d}+\frac {2 a^2 \csc (c+d x)}{d}+a^2 x\) |
Input:
Int[Cot[c + d*x]^8*(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) - (2*a^2*Cot[c + d*x]^7)/(7*d) + (2*a^2*Csc[c + d*x])/d - (2* a^2*Csc[c + d*x]^3)/d + (6*a^2*Csc[c + d*x]^5)/(5*d) - (2*a^2*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]
Result contains complex when optimal does not.
Time = 2.02 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.03
method | result | size |
risch | \(a^{2} x +\frac {2 i a^{2} \left (210 \,{\mathrm e}^{9 i \left (d x +c \right )}-315 \,{\mathrm e}^{8 i \left (d x +c \right )}-420 \,{\mathrm e}^{7 i \left (d x +c \right )}+1470 \,{\mathrm e}^{6 i \left (d x +c \right )}-504 \,{\mathrm e}^{5 i \left (d x +c \right )}-1204 \,{\mathrm e}^{4 i \left (d x +c \right )}+1108 \,{\mathrm e}^{3 i \left (d x +c \right )}+258 \,{\mathrm e}^{2 i \left (d x +c \right )}-554 \,{\mathrm e}^{i \left (d x +c \right )}+191\right )}{105 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{3}}\) | \(143\) |
derivativedivides | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(188\) |
default | \(\frac {-\frac {a^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}+2 a^{2} \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )+a^{2} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(188\) |
Input:
int(cot(d*x+c)^8*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
a^2*x+2/105*I*a^2*(210*exp(9*I*(d*x+c))-315*exp(8*I*(d*x+c))-420*exp(7*I*( d*x+c))+1470*exp(6*I*(d*x+c))-504*exp(5*I*(d*x+c))-1204*exp(4*I*(d*x+c))+1 108*exp(3*I*(d*x+c))+258*exp(2*I*(d*x+c))-554*exp(I*(d*x+c))+191)/d/(exp(I *(d*x+c))-1)^7/(exp(I*(d*x+c))+1)^3
Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.24 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {191 \, a^{2} \cos \left (d x + c\right )^{5} - 172 \, a^{2} \cos \left (d x + c\right )^{4} - 253 \, a^{2} \cos \left (d x + c\right )^{3} + 258 \, a^{2} \cos \left (d x + c\right )^{2} + 87 \, a^{2} \cos \left (d x + c\right ) - 96 \, a^{2} + 105 \, {\left (a^{2} d x \cos \left (d x + c\right )^{4} - 2 \, a^{2} d x \cos \left (d x + c\right )^{3} + 2 \, a^{2} d x \cos \left (d x + c\right ) - a^{2} d x\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8*(a+a*sec(d*x+c))^2,x, algorithm="fricas")
Output:
1/105*(191*a^2*cos(d*x + c)^5 - 172*a^2*cos(d*x + c)^4 - 253*a^2*cos(d*x + c)^3 + 258*a^2*cos(d*x + c)^2 + 87*a^2*cos(d*x + c) - 96*a^2 + 105*(a^2*d *x*cos(d*x + c)^4 - 2*a^2*d*x*cos(d*x + c)^3 + 2*a^2*d*x*cos(d*x + c) - a^ 2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^3 + 2*d*cos(d*x + c) - d)*sin(d*x + c))
Timed out. \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*(a+a*sec(d*x+c))**2,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{2} + \frac {6 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a^{2}}{\sin \left (d x + c\right )^{7}} - \frac {15 \, a^{2}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \] Input:
integrate(cot(d*x+c)^8*(a+a*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^2/sin(d*x + c)^7 - 15*a^2/tan(d*x + c)^7 )/d
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.81 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3360 \, {\left (d x + c\right )} a^{2} - 735 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {4410 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 770 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 147 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{3360 \, d} \] Input:
integrate(cot(d*x+c)^8*(a+a*sec(d*x+c))^2,x, algorithm="giac")
Output:
1/3360*(35*a^2*tan(1/2*d*x + 1/2*c)^3 + 3360*(d*x + c)*a^2 - 735*a^2*tan(1 /2*d*x + 1/2*c) + (4410*a^2*tan(1/2*d*x + 1/2*c)^6 - 770*a^2*tan(1/2*d*x + 1/2*c)^4 + 147*a^2*tan(1/2*d*x + 1/2*c)^2 - 15*a^2)/tan(1/2*d*x + 1/2*c)^ 7)/d
Time = 13.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.31 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^2\,\left (35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-735\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-770\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+147\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3360\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )\right )}{3360\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int(cot(c + d*x)^8*(a + a/cos(c + d*x))^2,x)
Output:
(a^2*(35*sin(c/2 + (d*x)/2)^10 - 15*cos(c/2 + (d*x)/2)^10 - 735*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^8 + 4410*cos(c/2 + (d*x)/2)^4*sin(c/2 + (d*x )/2)^6 - 770*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^4 + 147*cos(c/2 + (d* x)/2)^8*sin(c/2 + (d*x)/2)^2 + 3360*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2 )^7*(c + d*x)))/(3360*d*cos(c/2 + (d*x)/2)^3*sin(c/2 + (d*x)/2)^7)
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^2 \, dx=\frac {a^{2} \left (35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-735 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+3360 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} d x +4410 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-770 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+147 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-15\right )}{3360 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} d} \] Input:
int(cot(d*x+c)^8*(a+a*sec(d*x+c))^2,x)
Output:
(a**2*(35*tan((c + d*x)/2)**10 - 735*tan((c + d*x)/2)**8 + 3360*tan((c + d *x)/2)**7*d*x + 4410*tan((c + d*x)/2)**6 - 770*tan((c + d*x)/2)**4 + 147*t an((c + d*x)/2)**2 - 15))/(3360*tan((c + d*x)/2)**7*d)