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

Optimal. Leaf size=91 \[ \frac{\cot ^9(c+d x)}{9 a d}+\frac{3 \cot ^7(c+d x)}{7 a d}+\frac{3 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^9(c+d x)}{9 a d} \]

[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)

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Rubi [A]  time = 0.150765, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3872, 2839, 2606, 30, 2607, 270} \[ \frac{\cot ^9(c+d x)}{9 a d}+\frac{3 \cot ^7(c+d x)}{7 a d}+\frac{3 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^9(c+d x)}{9 a d} \]

Antiderivative was successfully verified.

[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 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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]

Rule 2839

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 2606

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 30

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

Rule 2607

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 270

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]

Rubi steps

\begin{align*} \int \frac{\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx &=-\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{\operatorname{Subst}\left (\int x^8 \, dx,x,\csc (c+d x)\right )}{a d}-\frac{\operatorname{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{\operatorname{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]  time = 0.979606, size = 200, normalized size = 2.2 \[ -\frac{\csc (c) (-85750 \sin (c+d x)-17150 \sin (2 (c+d x))+51450 \sin (3 (c+d x))+17150 \sin (4 (c+d x))-17150 \sin (5 (c+d x))-7350 \sin (6 (c+d x))+2450 \sin (7 (c+d x))+1225 \sin (8 (c+d x))-28672 \sin (c+2 d x)+86016 \sin (2 c+3 d x)+28672 \sin (3 c+4 d x)-28672 \sin (4 c+5 d x)-12288 \sin (5 c+6 d x)+4096 \sin (6 c+7 d x)+2048 \sin (7 c+8 d x)+645120 \sin (c)-143360 \sin (d x)) \csc ^7(c+d x) \sec (c+d x)}{5160960 a d (\sec (c+d x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[c]*Csc[c + d*x]^7*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)] + 245
0*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]))/(516
0960*a*d*(1 + Sec[c + d*x]))

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Maple [A]  time = 0.065, size = 114, normalized size = 1.3 \begin{align*}{\frac{1}{256\,da} \left ( -{\frac{1}{9} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9}}-{\frac{6}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{14}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{14}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{14}{3} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-14\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{-1}-{\frac{6}{5} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{1}{7} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256/d/a*(-1/9*tan(1/2*d*x+1/2*c)^9-6/7*tan(1/2*d*x+1/2*c)^7-14/5*tan(1/2*d*x+1/2*c)^5-14/3*tan(1/2*d*x+1/2*c
)^3-14/3/tan(1/2*d*x+1/2*c)^3-14/tan(1/2*d*x+1/2*c)-6/5/tan(1/2*d*x+1/2*c)^5-1/7/tan(1/2*d*x+1/2*c)^7)

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Maxima [B]  time = 1.00339, size = 238, normalized size = 2.62 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.79155, size = 466, normalized size = 5.12 \begin{align*} -\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.2839, size = 178, normalized size = 1.96 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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