Integrand size = 21, antiderivative size = 141 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=a^3 x+\frac {a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {4 a^3 \csc ^7(c+d x)}{7 d} \] 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-4/7*a ^3*cot(d*x+c)^7/d+3*a^3*csc(d*x+c)/d-10/3*a^3*csc(d*x+c)^3/d+11/5*a^3*csc( d*x+c)^5/d-4/7*a^3*csc(d*x+c)^7/d
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.66 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.87 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=-\frac {3 a^3 \cot ^7(c+d x)}{7 d}+\frac {3 a^3 \csc (c+d x)}{d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {4 a^3 \csc ^7(c+d x)}{7 d}-\frac {a^3 \cot ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(c+d x)\right )}{7 d} \] Input:
Integrate[Cot[c + d*x]^8*(a + a*Sec[c + d*x])^3,x]
Output:
(-3*a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Csc[c + d*x])/d - (10*a^3*Csc[c + d *x]^3)/(3*d) + (11*a^3*Csc[c + d*x]^5)/(5*d) - (4*a^3*Csc[c + d*x]^7)/(7*d ) - (a^3*Cot[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2]) /(7*d)
Time = 0.43 (sec) , antiderivative size = 141, 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.
\(\displaystyle \int \cot ^8(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\cot \left (c+d x+\frac {\pi }{2}\right )^8}dx\) |
\(\Big \downarrow \) 4374 |
\(\displaystyle \int \left (a^3 \cot ^8(c+d x)+3 a^3 \cot ^7(c+d x) \csc (c+d x)+3 a^3 \cot ^6(c+d x) \csc ^2(c+d x)+a^3 \cot ^5(c+d x) \csc ^3(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 a^3 \cot ^7(c+d x)}{7 d}+\frac {a^3 \cot ^5(c+d x)}{5 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {4 a^3 \csc ^7(c+d x)}{7 d}+\frac {11 a^3 \csc ^5(c+d x)}{5 d}-\frac {10 a^3 \csc ^3(c+d x)}{3 d}+\frac {3 a^3 \csc (c+d x)}{d}+a^3 x\) |
Input:
Int[Cot[c + d*x]^8*(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) - (4*a^3*Cot[c + d*x]^7)/(7*d) + (3*a^3*Csc[c + d*x])/d - (10 *a^3*Csc[c + d*x]^3)/(3*d) + (11*a^3*Csc[c + d*x]^5)/(5*d) - (4*a^3*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 = 3.82 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86
method | result | size |
risch | \(a^{3} x +\frac {2 i a^{3} \left (315 \,{\mathrm e}^{7 i \left (d x +c \right )}-1155 \,{\mathrm e}^{6 i \left (d x +c \right )}+1715 \,{\mathrm e}^{5 i \left (d x +c \right )}-525 \,{\mathrm e}^{4 i \left (d x +c \right )}-1379 \,{\mathrm e}^{3 i \left (d x +c \right )}+1939 \,{\mathrm e}^{2 i \left (d x +c \right )}-1011 \,{\mathrm e}^{i \left (d x +c \right )}+221\right )}{105 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}\) | \(121\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{7 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{6}}{35 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{6}}{105 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{6}}{35 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(293\) |
default | \(\frac {a^{3} \left (-\frac {\cos \left (d x +c \right )^{6}}{7 \sin \left (d x +c \right )^{7}}-\frac {\cos \left (d x +c \right )^{6}}{35 \sin \left (d x +c \right )^{5}}+\frac {\cos \left (d x +c \right )^{6}}{105 \sin \left (d x +c \right )^{3}}-\frac {\cos \left (d x +c \right )^{6}}{35 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {3 a^{3} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}+3 a^{3} \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )+a^{3} \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )}{d}\) | \(293\) |
Input:
int(cot(d*x+c)^8*(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
a^3*x+2/105*I*a^3*(315*exp(7*I*(d*x+c))-1155*exp(6*I*(d*x+c))+1715*exp(5*I *(d*x+c))-525*exp(4*I*(d*x+c))-1379*exp(3*I*(d*x+c))+1939*exp(2*I*(d*x+c)) -1011*exp(I*(d*x+c))+221)/d/(exp(I*(d*x+c))-1)^7/(exp(I*(d*x+c))+1)
Time = 0.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.13 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {221 \, a^{3} \cos \left (d x + c\right )^{4} - 348 \, a^{3} \cos \left (d x + c\right )^{3} - 25 \, a^{3} \cos \left (d x + c\right )^{2} + 303 \, a^{3} \cos \left (d x + c\right ) - 136 \, a^{3} + 105 \, {\left (a^{3} d x \cos \left (d x + c\right )^{3} - 3 \, a^{3} d x \cos \left (d x + c\right )^{2} + 3 \, a^{3} d x \cos \left (d x + c\right ) - a^{3} d x\right )} \sin \left (d x + c\right )}{105 \, {\left (d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 3 \, d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^8*(a+a*sec(d*x+c))^3,x, algorithm="fricas")
Output:
1/105*(221*a^3*cos(d*x + c)^4 - 348*a^3*cos(d*x + c)^3 - 25*a^3*cos(d*x + c)^2 + 303*a^3*cos(d*x + c) - 136*a^3 + 105*(a^3*d*x*cos(d*x + c)^3 - 3*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)^3 - 3*d*cos(d*x + c)^2 + 3*d*cos(d*x + c) - d)*sin(d*x + c))
Timed out. \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**8*(a+a*sec(d*x+c))**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.08 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a^{3} + \frac {9 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} a^{3}}{\sin \left (d x + c\right )^{7}} - \frac {{\left (35 \, \sin \left (d x + c\right )^{4} - 42 \, \sin \left (d x + c\right )^{2} + 15\right )} a^{3}}{\sin \left (d x + c\right )^{7}} - \frac {45 \, a^{3}}{\tan \left (d x + c\right )^{7}}}{105 \, d} \] Input:
integrate(cot(d*x+c)^8*(a+a*sec(d*x+c))^3,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^3 + 9*(35*sin(d*x + c)^6 - 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 - 5)*a^3/sin(d*x + c)^7 - (35*sin(d*x + c)^4 - 42*sin(d*x + c)^2 + 15)*a^3/sin(d*x + c)^7 - 45*a^3/tan(d*x + c)^7)/d
Time = 0.20 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.68 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {1680 \, {\left (d x + c\right )} a^{3} - 105 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2730 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 560 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 126 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{1680 \, d} \] Input:
integrate(cot(d*x+c)^8*(a+a*sec(d*x+c))^3,x, algorithm="giac")
Output:
1/1680*(1680*(d*x + c)*a^3 - 105*a^3*tan(1/2*d*x + 1/2*c) + (2730*a^3*tan( 1/2*d*x + 1/2*c)^6 - 560*a^3*tan(1/2*d*x + 1/2*c)^4 + 126*a^3*tan(1/2*d*x + 1/2*c)^2 - 15*a^3)/tan(1/2*d*x + 1/2*c)^7)/d
Time = 12.56 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.65 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^3\,\left (126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-560\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+2730\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+1680\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (c+d\,x\right )-15\right )}{1680\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \] Input:
int(cot(c + d*x)^8*(a + a/cos(c + d*x))^3,x)
Output:
(a^3*(126*tan(c/2 + (d*x)/2)^2 - 560*tan(c/2 + (d*x)/2)^4 + 2730*tan(c/2 + (d*x)/2)^6 - 105*tan(c/2 + (d*x)/2)^8 + 1680*tan(c/2 + (d*x)/2)^7*(c + d* x) - 15))/(1680*d*tan(c/2 + (d*x)/2)^7)
Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.62 \[ \int \cot ^8(c+d x) (a+a \sec (c+d x))^3 \, dx=\frac {a^{3} \left (-105 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+1680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} d x +2730 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-560 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+126 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-15\right )}{1680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} d} \] Input:
int(cot(d*x+c)^8*(a+a*sec(d*x+c))^3,x)
Output:
(a**3*( - 105*tan((c + d*x)/2)**8 + 1680*tan((c + d*x)/2)**7*d*x + 2730*ta n((c + d*x)/2)**6 - 560*tan((c + d*x)/2)**4 + 126*tan((c + d*x)/2)**2 - 15 ))/(1680*tan((c + d*x)/2)**7*d)