\(\int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx\) [270]

Optimal result

Integrand size = 19, antiderivative size = 111 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=a x-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d} \]

[Out]

a*x-1/7*cot(d*x+c)^7*(a+b*sec(d*x+c))/d+1/35*cot(d*x+c)^5*(7*a+6*b*sec(d*x+c))/d+1/35*cot(d*x+c)*(35*a+16*b*se
c(d*x+c))/d-1/105*cot(d*x+c)^3*(35*a+24*b*sec(d*x+c))/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3967, 8} \[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}+a x \]

[In]

Int[Cot[c + d*x]^8*(a + b*Sec[c + d*x]),x]

[Out]

a*x - (Cot[c + d*x]^7*(a + b*Sec[c + d*x]))/(7*d) + (Cot[c + d*x]^5*(7*a + 6*b*Sec[c + d*x]))/(35*d) + (Cot[c
+ d*x]*(35*a + 16*b*Sec[c + d*x]))/(35*d) - (Cot[c + d*x]^3*(35*a + 24*b*Sec[c + d*x]))/(105*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3967

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-(e*Cot[c
+ d*x])^(m + 1))*((a + b*Csc[c + d*x])/(d*e*(m + 1))), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)
*(a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {1}{7} \int \cot ^6(c+d x) (-7 a-6 b \sec (c+d x)) \, dx \\ & = -\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}+\frac {1}{35} \int \cot ^4(c+d x) (35 a+24 b \sec (c+d x)) \, dx \\ & = -\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac {1}{105} \int \cot ^2(c+d x) (-105 a-48 b \sec (c+d x)) \, dx \\ & = -\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d}+\frac {1}{105} \int 105 a \, dx \\ & = a x-\frac {\cot ^7(c+d x) (a+b \sec (c+d x))}{7 d}+\frac {\cot ^5(c+d x) (7 a+6 b \sec (c+d x))}{35 d}+\frac {\cot (c+d x) (35 a+16 b \sec (c+d x))}{35 d}-\frac {\cot ^3(c+d x) (35 a+24 b \sec (c+d x))}{105 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.83 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=\frac {b \csc (c+d x)}{d}-\frac {b \csc ^3(c+d x)}{d}+\frac {3 b \csc ^5(c+d x)}{5 d}-\frac {b \csc ^7(c+d x)}{7 d}-\frac {a \cot ^7(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},1,-\frac {5}{2},-\tan ^2(c+d x)\right )}{7 d} \]

[In]

Integrate[Cot[c + d*x]^8*(a + b*Sec[c + d*x]),x]

[Out]

(b*Csc[c + d*x])/d - (b*Csc[c + d*x]^3)/d + (3*b*Csc[c + d*x]^5)/(5*d) - (b*Csc[c + d*x]^7)/(7*d) - (a*Cot[c +
d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, -Tan[c + d*x]^2])/(7*d)

Maple [A] (verified)

Time = 1.69 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(\frac {a \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+b \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )}{d}\)	\(162\)
default	\(\frac {a \left (-\frac {\cot \left (d x +c \right )^{7}}{7}+\frac {\cot \left (d x +c \right )^{5}}{5}-\frac {\cot \left (d x +c \right )^{3}}{3}+\cot \left (d x +c \right )+d x +c \right )+b \left (-\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )^{7}}+\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{5}}-\frac {\cos \left (d x +c \right )^{8}}{35 \sin \left (d x +c \right )^{3}}+\frac {\cos \left (d x +c \right )^{8}}{7 \sin \left (d x +c \right )}+\frac {\left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}\right )}{d}\)	\(162\)
risch	\(a x +\frac {2 i \left (105 b \,{\mathrm e}^{13 i \left (d x +c \right )}+420 a \,{\mathrm e}^{12 i \left (d x +c \right )}-210 b \,{\mathrm e}^{11 i \left (d x +c \right )}-1260 a \,{\mathrm e}^{10 i \left (d x +c \right )}+903 b \,{\mathrm e}^{9 i \left (d x +c \right )}+3080 a \,{\mathrm e}^{8 i \left (d x +c \right )}-636 b \,{\mathrm e}^{7 i \left (d x +c \right )}-3080 a \,{\mathrm e}^{6 i \left (d x +c \right )}+903 b \,{\mathrm e}^{5 i \left (d x +c \right )}+2436 a \,{\mathrm e}^{4 i \left (d x +c \right )}-210 b \,{\mathrm e}^{3 i \left (d x +c \right )}-812 a \,{\mathrm e}^{2 i \left (d x +c \right )}+105 b \,{\mathrm e}^{i \left (d x +c \right )}+176 a \right )}{105 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\)	\(184\)

[In]

int(cot(d*x+c)^8*(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(a*(-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)+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)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.61 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=\frac {176 \, a \cos \left (d x + c\right )^{7} + 105 \, b \cos \left (d x + c\right )^{6} - 406 \, a \cos \left (d x + c\right )^{5} - 210 \, b \cos \left (d x + c\right )^{4} + 350 \, a \cos \left (d x + c\right )^{3} + 168 \, b \cos \left (d x + c\right )^{2} - 105 \, a \cos \left (d x + c\right ) + 105 \, {\left (a d x \cos \left (d x + c\right )^{6} - 3 \, a d x \cos \left (d x + c\right )^{4} + 3 \, a d x \cos \left (d x + c\right )^{2} - a d x\right )} \sin \left (d x + c\right ) - 48 \, b}{105 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cot(d*x+c)^8*(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

1/105*(176*a*cos(d*x + c)^7 + 105*b*cos(d*x + c)^6 - 406*a*cos(d*x + c)^5 - 210*b*cos(d*x + c)^4 + 350*a*cos(d
*x + c)^3 + 168*b*cos(d*x + c)^2 - 105*a*cos(d*x + c) + 105*(a*d*x*cos(d*x + c)^6 - 3*a*d*x*cos(d*x + c)^4 + 3
*a*d*x*cos(d*x + c)^2 - a*d*x)*sin(d*x + c) - 48*b)/((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))

Sympy [F]

\[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{8}{\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)**8*(a+b*sec(d*x+c)),x)

[Out]

Integral((a + b*sec(c + d*x))*cot(c + d*x)**8, x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.90 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=\frac {{\left (105 \, d x + 105 \, c + \frac {105 \, \tan \left (d x + c\right )^{6} - 35 \, \tan \left (d x + c\right )^{4} + 21 \, \tan \left (d x + c\right )^{2} - 15}{\tan \left (d x + c\right )^{7}}\right )} a + \frac {3 \, {\left (35 \, \sin \left (d x + c\right )^{6} - 35 \, \sin \left (d x + c\right )^{4} + 21 \, \sin \left (d x + c\right )^{2} - 5\right )} b}{\sin \left (d x + c\right )^{7}}}{105 \, d} \]

[In]

integrate(cot(d*x+c)^8*(a+b*sec(d*x+c)),x, algorithm="maxima")

[Out]

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*(35*sin(d*x + c)^6 - 35*sin(d*x + c)^4 + 21*sin(d*x + c)^2 - 5)*b/sin(d*x + c)^7)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 225 vs. \(2 (103) = 206\).

Time = 0.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.03 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=\frac {15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 189 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 147 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1295 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 735 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13440 \, {\left (d x + c\right )} a - 9765 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3675 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {9765 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3675 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1295 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 735 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 189 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 147 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a - 15 \, b}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}}}{13440 \, d} \]

[In]

integrate(cot(d*x+c)^8*(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

1/13440*(15*a*tan(1/2*d*x + 1/2*c)^7 - 15*b*tan(1/2*d*x + 1/2*c)^7 - 189*a*tan(1/2*d*x + 1/2*c)^5 + 147*b*tan(
1/2*d*x + 1/2*c)^5 + 1295*a*tan(1/2*d*x + 1/2*c)^3 - 735*b*tan(1/2*d*x + 1/2*c)^3 + 13440*(d*x + c)*a - 9765*a
*tan(1/2*d*x + 1/2*c) + 3675*b*tan(1/2*d*x + 1/2*c) + (9765*a*tan(1/2*d*x + 1/2*c)^6 + 3675*b*tan(1/2*d*x + 1/
2*c)^6 - 1295*a*tan(1/2*d*x + 1/2*c)^4 - 735*b*tan(1/2*d*x + 1/2*c)^4 + 189*a*tan(1/2*d*x + 1/2*c)^2 + 147*b*t
an(1/2*d*x + 1/2*c)^2 - 15*a - 15*b)/tan(1/2*d*x + 1/2*c)^7)/d

Mupad [B] (verification not implemented)

Time = 14.66 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.57 \[ \int \cot ^8(c+d x) (a+b \sec (c+d x)) \, dx=a\,x+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {37\,a}{384}-\frac {7\,b}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {9\,a}{640}-\frac {7\,b}{640}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {a}{896}-\frac {b}{896}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\left (-93\,a-35\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {37\,a}{3}+7\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {9\,a}{5}-\frac {7\,b}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{7}+\frac {b}{7}\right )}{128\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {93\,a}{128}-\frac {35\,b}{128}\right )}{d} \]

[In]

int(cot(c + d*x)^8*(a + b/cos(c + d*x)),x)

[Out]

a*x + (tan(c/2 + (d*x)/2)^3*((37*a)/384 - (7*b)/128))/d - (tan(c/2 + (d*x)/2)^5*((9*a)/640 - (7*b)/640))/d + (
tan(c/2 + (d*x)/2)^7*(a/896 - b/896))/d - (cot(c/2 + (d*x)/2)^7*(a/7 + b/7 - tan(c/2 + (d*x)/2)^2*((9*a)/5 + (
7*b)/5) + tan(c/2 + (d*x)/2)^4*((37*a)/3 + 7*b) - tan(c/2 + (d*x)/2)^6*(93*a + 35*b)))/(128*d) - (tan(c/2 + (d
*x)/2)*((93*a)/128 - (35*b)/128))/d