3.718 \(\int \frac{\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=134 \[ \frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d} \]

[Out]

(5*ArcTanh[Cos[c + d*x]])/(128*a*d) + Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(128*a*d) - (5*Co
t[c + d*x]*Csc[c + d*x]^3)/(64*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) - (Cot[c + d*x]^5*Csc[c + d*x
]^3)/(8*a*d)

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Rubi [A]  time = 0.218482, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2839, 2611, 3768, 3770, 2607, 30} \[ \frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*ArcTanh[Cos[c + d*x]])/(128*a*d) + Cot[c + d*x]^7/(7*a*d) + (5*Cot[c + d*x]*Csc[c + d*x])/(128*a*d) - (5*Co
t[c + d*x]*Csc[c + d*x]^3)/(64*a*d) + (5*Cot[c + d*x]^3*Csc[c + d*x]^3)/(48*a*d) - (Cot[c + d*x]^5*Csc[c + d*x
]^3)/(8*a*d)

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 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + n - 1), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 30

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

Rubi steps

\begin{align*} \int \frac{\cot ^8(c+d x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a}+\frac{\int \cot ^6(c+d x) \csc ^3(c+d x) \, dx}{a}\\ &=-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx}{8 a}-\frac{\operatorname{Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}+\frac{5 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{16 a}\\ &=\frac{\cot ^7(c+d x)}{7 a d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \int \csc ^3(c+d x) \, dx}{64 a}\\ &=\frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}-\frac{5 \int \csc (c+d x) \, dx}{128 a}\\ &=\frac{5 \tanh ^{-1}(\cos (c+d x))}{128 a d}+\frac{\cot ^7(c+d x)}{7 a d}+\frac{5 \cot (c+d x) \csc (c+d x)}{128 a d}-\frac{5 \cot (c+d x) \csc ^3(c+d x)}{64 a d}+\frac{5 \cot ^3(c+d x) \csc ^3(c+d x)}{48 a d}-\frac{\cot ^5(c+d x) \csc ^3(c+d x)}{8 a d}\\ \end{align*}

Mathematica [B]  time = 0.997881, size = 291, normalized size = 2.17 \[ \frac{\csc ^8(c+d x) \left (5376 \sin (2 (c+d x))+5376 \sin (4 (c+d x))+2304 \sin (6 (c+d x))+384 \sin (8 (c+d x))-24710 \cos (c+d x)-12530 \cos (3 (c+d x))-5558 \cos (5 (c+d x))-210 \cos (7 (c+d x))-3675 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-5880 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2940 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-840 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+105 \cos (8 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3675 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+5880 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2940 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+840 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-105 \cos (8 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{344064 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Csc[c + d*x]^8*(-24710*Cos[c + d*x] - 12530*Cos[3*(c + d*x)] - 5558*Cos[5*(c + d*x)] - 210*Cos[7*(c + d*x)] +
3675*Log[Cos[(c + d*x)/2]] - 5880*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 2940*Cos[4*(c + d*x)]*Log[Cos[(c +
d*x)/2]] - 840*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 105*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 3675*Log
[Sin[(c + d*x)/2]] + 5880*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 2940*Cos[4*(c + d*x)]*Log[Sin[(c + d*x)/2]]
+ 840*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 105*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 5376*Sin[2*(c + d
*x)] + 5376*Sin[4*(c + d*x)] + 2304*Sin[6*(c + d*x)] + 384*Sin[8*(c + d*x)]))/(344064*a*d)

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Maple [B]  time = 0.174, size = 322, normalized size = 2.4 \begin{align*}{\frac{1}{2048\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{8}}-{\frac{1}{896\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}}+{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{1}{256\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4}}-{\frac{3}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}+{\frac{5}{128\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{1}{896\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}-{\frac{5}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{2048\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-8}}-{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}}-{\frac{1}{256\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-4}}-{\frac{5}{128\,da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+{\frac{1}{384\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-6}}+{\frac{3}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{1}{128\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x)

[Out]

1/2048/d/a*tan(1/2*d*x+1/2*c)^8-1/896/d/a*tan(1/2*d*x+1/2*c)^7-1/384/d/a*tan(1/2*d*x+1/2*c)^6+1/128/d/a*tan(1/
2*d*x+1/2*c)^5+1/256/d/a*tan(1/2*d*x+1/2*c)^4-3/128/d/a*tan(1/2*d*x+1/2*c)^3+1/128/d/a*tan(1/2*d*x+1/2*c)^2+5/
128/d/a*tan(1/2*d*x+1/2*c)+1/896/d/a/tan(1/2*d*x+1/2*c)^7-5/128/d/a/tan(1/2*d*x+1/2*c)-1/2048/d/a/tan(1/2*d*x+
1/2*c)^8-1/128/d/a/tan(1/2*d*x+1/2*c)^5-1/256/d/a/tan(1/2*d*x+1/2*c)^4-5/128/d/a*ln(tan(1/2*d*x+1/2*c))+1/384/
d/a/tan(1/2*d*x+1/2*c)^6+3/128/d/a/tan(1/2*d*x+1/2*c)^3-1/128/d/a/tan(1/2*d*x+1/2*c)^2

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Maxima [B]  time = 1.00927, size = 478, normalized size = 3.57 \begin{align*} \frac{\frac{\frac{1680 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{336 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{1008 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{336 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{112 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{48 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{21 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}{a} - \frac{1680 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{{\left (\frac{48 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{112 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{336 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{168 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1008 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{336 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{1680 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 21\right )}{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}{a \sin \left (d x + c\right )^{8}}}{43008 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/43008*((1680*sin(d*x + c)/(cos(d*x + c) + 1) + 336*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 1008*sin(d*x + c)^3
/(cos(d*x + c) + 1)^3 + 168*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 336*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 11
2*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 48*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 21*sin(d*x + c)^8/(cos(d*x +
c) + 1)^8)/a - 1680*log(sin(d*x + c)/(cos(d*x + c) + 1))/a + (48*sin(d*x + c)/(cos(d*x + c) + 1) + 112*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 - 336*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 168*sin(d*x + c)^4/(cos(d*x + c) + 1)
^4 + 1008*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 336*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1680*sin(d*x + c)^7/
(cos(d*x + c) + 1)^7 - 21)*(cos(d*x + c) + 1)^8/(a*sin(d*x + c)^8))/d

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Fricas [A]  time = 1.16482, size = 602, normalized size = 4.49 \begin{align*} \frac{768 \, \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )^{7} - 1022 \, \cos \left (d x + c\right )^{5} + 770 \, \cos \left (d x + c\right )^{3} + 105 \,{\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 105 \,{\left (\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 210 \, \cos \left (d x + c\right )}{5376 \,{\left (a d \cos \left (d x + c\right )^{8} - 4 \, a d \cos \left (d x + c\right )^{6} + 6 \, a d \cos \left (d x + c\right )^{4} - 4 \, a d \cos \left (d x + c\right )^{2} + a d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/5376*(768*cos(d*x + c)^7*sin(d*x + c) - 210*cos(d*x + c)^7 - 1022*cos(d*x + c)^5 + 770*cos(d*x + c)^3 + 105*
(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(1/2*cos(d*x + c) + 1/2) - 10
5*(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1)*log(-1/2*cos(d*x + c) + 1/2) -
210*cos(d*x + c))/(a*d*cos(d*x + c)^8 - 4*a*d*cos(d*x + c)^6 + 6*a*d*cos(d*x + c)^4 - 4*a*d*cos(d*x + c)^2 +
a*d)

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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(cos(d*x+c)**8*csc(d*x+c)**9/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.43603, size = 370, normalized size = 2.76 \begin{align*} -\frac{\frac{1680 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{21 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 48 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 112 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 336 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 168 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1008 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 336 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1680 \, a^{7} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{8}} - \frac{4566 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1680 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 336 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 1008 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 168 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 336 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 112 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 48 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 21}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}}}{43008 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^9/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/43008*(1680*log(abs(tan(1/2*d*x + 1/2*c)))/a - (21*a^7*tan(1/2*d*x + 1/2*c)^8 - 48*a^7*tan(1/2*d*x + 1/2*c)
^7 - 112*a^7*tan(1/2*d*x + 1/2*c)^6 + 336*a^7*tan(1/2*d*x + 1/2*c)^5 + 168*a^7*tan(1/2*d*x + 1/2*c)^4 - 1008*a
^7*tan(1/2*d*x + 1/2*c)^3 + 336*a^7*tan(1/2*d*x + 1/2*c)^2 + 1680*a^7*tan(1/2*d*x + 1/2*c))/a^8 - (4566*tan(1/
2*d*x + 1/2*c)^8 - 1680*tan(1/2*d*x + 1/2*c)^7 - 336*tan(1/2*d*x + 1/2*c)^6 + 1008*tan(1/2*d*x + 1/2*c)^5 - 16
8*tan(1/2*d*x + 1/2*c)^4 - 336*tan(1/2*d*x + 1/2*c)^3 + 112*tan(1/2*d*x + 1/2*c)^2 + 48*tan(1/2*d*x + 1/2*c) -
21)/(a*tan(1/2*d*x + 1/2*c)^8))/d