\(\int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx\) [72]

Optimal result

Integrand size = 21, antiderivative size = 91 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]

[Out]

1/3*cot(d*x+c)^3/a/d+3/5*cot(d*x+c)^5/a/d+3/7*cot(d*x+c)^7/a/d+1/9*cot(d*x+c)^9/a/d-1/9*csc(d*x+c)^9/a/d

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2918, 2686, 30, 2687, 276} \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cot ^9(c+d x)}{9 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^9(c+d x)}{9 a d} \]

[In]

Int[Csc[c + d*x]^8/(a + a*Sec[c + d*x]),x]

[Out]

Cot[c + d*x]^3/(3*a*d) + (3*Cot[c + d*x]^5)/(5*a*d) + (3*Cot[c + d*x]^7)/(7*a*d) + Cot[c + d*x]^9/(9*a*d) - Cs
c[c + d*x]^9/(9*a*d)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2918

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_
.)*(x_)]), x_Symbol] :> Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[g^2/(b*d),
Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2
- b^2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cot (c+d x) \csc ^7(c+d x)}{-a-a \cos (c+d x)} \, dx \\ & = -\frac {\int \cot ^2(c+d x) \csc ^8(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^9(c+d x) \, dx}{a} \\ & = -\frac {\text {Subst}\left (\int x^8 \, dx,x,\csc (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^3 \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = -\frac {\csc ^9(c+d x)}{9 a d}-\frac {\text {Subst}\left (\int \left (x^2+3 x^4+3 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a d} \\ & = \frac {\cot ^3(c+d x)}{3 a d}+\frac {3 \cot ^5(c+d x)}{5 a d}+\frac {3 \cot ^7(c+d x)}{7 a d}+\frac {\cot ^9(c+d x)}{9 a d}-\frac {\csc ^9(c+d x)}{9 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(91)=182\).

Time = 2.07 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.20 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\csc (c) \csc ^7(c+d x) \sec (c+d x) (645120 \sin (c)-143360 \sin (d x)-85750 \sin (c+d x)-17150 \sin (2 (c+d x))+51450 \sin (3 (c+d x))+17150 \sin (4 (c+d x))-17150 \sin (5 (c+d x))-7350 \sin (6 (c+d x))+2450 \sin (7 (c+d x))+1225 \sin (8 (c+d x))-28672 \sin (c+2 d x)+86016 \sin (2 c+3 d x)+28672 \sin (3 c+4 d x)-28672 \sin (4 c+5 d x)-12288 \sin (5 c+6 d x)+4096 \sin (6 c+7 d x)+2048 \sin (7 c+8 d x))}{5160960 a d (1+\sec (c+d x))} \]

[In]

Integrate[Csc[c + d*x]^8/(a + a*Sec[c + d*x]),x]

[Out]

-1/5160960*(Csc[c]*Csc[c + d*x]^7*Sec[c + d*x]*(645120*Sin[c] - 143360*Sin[d*x] - 85750*Sin[c + d*x] - 17150*S
in[2*(c + d*x)] + 51450*Sin[3*(c + d*x)] + 17150*Sin[4*(c + d*x)] - 17150*Sin[5*(c + d*x)] - 7350*Sin[6*(c + d
*x)] + 2450*Sin[7*(c + d*x)] + 1225*Sin[8*(c + d*x)] - 28672*Sin[c + 2*d*x] + 86016*Sin[2*c + 3*d*x] + 28672*S
in[3*c + 4*d*x] - 28672*Sin[4*c + 5*d*x] - 12288*Sin[5*c + 6*d*x] + 4096*Sin[6*c + 7*d*x] + 2048*Sin[7*c + 8*d
*x]))/(a*d*(1 + Sec[c + d*x]))

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.23

method	result	size
parallelrisch	\(\frac {-35 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-270 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-45 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-882 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-378 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-1470 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-1470 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-4410 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{80640 d a}\)	\(112\)
derivativedivides	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\)	\(114\)
default	\(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{9}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}-\frac {14}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {6}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {1}{7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {14}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{256 d a}\)	\(114\)
risch	\(\frac {32 i \left (315 \,{\mathrm e}^{8 i \left (d x +c \right )}+70 \,{\mathrm e}^{7 i \left (d x +c \right )}+14 \,{\mathrm e}^{6 i \left (d x +c \right )}-42 \,{\mathrm e}^{5 i \left (d x +c \right )}-14 \,{\mathrm e}^{4 i \left (d x +c \right )}+14 \,{\mathrm e}^{3 i \left (d x +c \right )}+6 \,{\mathrm e}^{2 i \left (d x +c \right )}-2 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{315 a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{9} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{7}}\)	\(126\)
norman	\(\frac {-\frac {1}{1792 a d}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{640 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{896 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{2304 a d}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{640 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{384 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{128 d a}-\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{384 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}\)	\(155\)

[In]

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

[Out]

1/80640*(-35*tan(1/2*d*x+1/2*c)^9-270*tan(1/2*d*x+1/2*c)^7-45*cot(1/2*d*x+1/2*c)^7-882*tan(1/2*d*x+1/2*c)^5-37
8*cot(1/2*d*x+1/2*c)^5-1470*tan(1/2*d*x+1/2*c)^3-1470*cot(1/2*d*x+1/2*c)^3-4410*cot(1/2*d*x+1/2*c))/d/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (81) = 162\).

Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.95 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {16 \, \cos \left (d x + c\right )^{8} + 16 \, \cos \left (d x + c\right )^{7} - 56 \, \cos \left (d x + c\right )^{6} - 56 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{4} + 70 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )^{2} - 35 \, \cos \left (d x + c\right ) - 35}{315 \, {\left (a d \cos \left (d x + c\right )^{7} + a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{5} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{3} + 3 \, a d \cos \left (d x + c\right )^{2} - a d \cos \left (d x + c\right ) - a d\right )} \sin \left (d x + c\right )} \]

[In]

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

[Out]

-1/315*(16*cos(d*x + c)^8 + 16*cos(d*x + c)^7 - 56*cos(d*x + c)^6 - 56*cos(d*x + c)^5 + 70*cos(d*x + c)^4 + 70
*cos(d*x + c)^3 - 35*cos(d*x + c)^2 - 35*cos(d*x + c) - 35)/((a*d*cos(d*x + c)^7 + a*d*cos(d*x + c)^6 - 3*a*d*
cos(d*x + c)^5 - 3*a*d*cos(d*x + c)^4 + 3*a*d*cos(d*x + c)^3 + 3*a*d*cos(d*x + c)^2 - a*d*cos(d*x + c) - a*d)*
sin(d*x + c))

Sympy [F]

\[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\csc ^{8}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate(csc(d*x+c)**8/(a+a*sec(d*x+c)),x)

[Out]

Integral(csc(c + d*x)**8/(sec(c + d*x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (81) = 162\).

Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.93 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {\frac {1470 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {882 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {270 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {35 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a} + \frac {3 \, {\left (\frac {126 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {490 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1470 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 15\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{7}}{a \sin \left (d x + c\right )^{7}}}{80640 \, d} \]

[In]

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

[Out]

-1/80640*((1470*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 882*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 270*sin(d*x +
c)^7/(cos(d*x + c) + 1)^7 + 35*sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/a + 3*(126*sin(d*x + c)^2/(cos(d*x + c) +
1)^2 + 490*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1470*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 15)*(cos(d*x + c)
+ 1)^7/(a*sin(d*x + c)^7))/d

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.45 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {3 \, {\left (1470 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 490 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 126 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7}} + \frac {35 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 270 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 882 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1470 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{9}}}{80640 \, d} \]

[In]

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

[Out]

-1/80640*(3*(1470*tan(1/2*d*x + 1/2*c)^6 + 490*tan(1/2*d*x + 1/2*c)^4 + 126*tan(1/2*d*x + 1/2*c)^2 + 15)/(a*ta
n(1/2*d*x + 1/2*c)^7) + (35*a^8*tan(1/2*d*x + 1/2*c)^9 + 270*a^8*tan(1/2*d*x + 1/2*c)^7 + 882*a^8*tan(1/2*d*x
+ 1/2*c)^5 + 1470*a^8*tan(1/2*d*x + 1/2*c)^3)/a^9)/d

Mupad [B] (verification not implemented)

Time = 13.93 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.21 \[ \int \frac {\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {45\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+378\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1470\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4410\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1470\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+882\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+270\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+35\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}}{80640\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7} \]

[In]

int(1/(sin(c + d*x)^8*(a + a/cos(c + d*x))),x)

[Out]

-(45*cos(c/2 + (d*x)/2)^16 + 35*sin(c/2 + (d*x)/2)^16 + 270*cos(c/2 + (d*x)/2)^2*sin(c/2 + (d*x)/2)^14 + 882*c
os(c/2 + (d*x)/2)^4*sin(c/2 + (d*x)/2)^12 + 1470*cos(c/2 + (d*x)/2)^6*sin(c/2 + (d*x)/2)^10 + 4410*cos(c/2 + (
d*x)/2)^10*sin(c/2 + (d*x)/2)^6 + 1470*cos(c/2 + (d*x)/2)^12*sin(c/2 + (d*x)/2)^4 + 378*cos(c/2 + (d*x)/2)^14*
sin(c/2 + (d*x)/2)^2)/(80640*a*d*cos(c/2 + (d*x)/2)^9*sin(c/2 + (d*x)/2)^7)