3.389 \(\int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=176 \[ -\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{11 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

(-11*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (11*a^
2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(6*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)

________________________________________________________________________________________

Rubi [A]  time = 0.32037, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2611, 3768, 3770, 2607, 14} \[ -\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{11 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

(-11*a^2*ArcTanh[Cos[c + d*x]])/(128*d) - (2*a^2*Cot[c + d*x]^5)/(5*d) - (2*a^2*Cot[c + d*x]^7)/(7*d) - (11*a^
2*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (7*a^2*Cot[c + d*x]*Csc[c + d*x]^3)/(64*d) - (a^2*Cot[c + d*x]^3*Csc[c
+ d*x]^3)/(6*d) + (a^2*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) - (a^2*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d)

Rule 2873

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \cot ^4(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^4(c+d x) \csc ^3(c+d x)+2 a^2 \cot ^4(c+d x) \csc ^4(c+d x)+a^2 \cot ^4(c+d x) \csc ^5(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{1}{8} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx-\frac{1}{2} a^2 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{16} a^2 \int \csc ^5(c+d x) \, dx+\frac{1}{8} a^2 \int \csc ^3(c+d x) \, dx+\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot (c+d x) \csc (c+d x)}{16 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{64} \left (3 a^2\right ) \int \csc ^3(c+d x) \, dx+\frac{1}{16} a^2 \int \csc (c+d x) \, dx\\ &=-\frac{a^2 \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}+\frac{1}{128} \left (3 a^2\right ) \int \csc (c+d x) \, dx\\ &=-\frac{11 a^2 \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{11 a^2 \cot (c+d x) \csc (c+d x)}{128 d}+\frac{7 a^2 \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a^2 \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac{a^2 \cot (c+d x) \csc ^5(c+d x)}{16 d}-\frac{a^2 \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.911599, size = 291, normalized size = 1.65 \[ -\frac{a^2 \csc ^8(c+d x) \left (86016 \sin (2 (c+d x))+64512 \sin (4 (c+d x))+12288 \sin (6 (c+d x))-1536 \sin (8 (c+d x))+158270 \cos (c+d x)+77210 \cos (3 (c+d x))-18130 \cos (5 (c+d x))-2310 \cos (7 (c+d x))-40425 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-64680 \cos (2 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+32340 \cos (4 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-9240 \cos (6 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+1155 \cos (8 (c+d x)) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+40425 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+64680 \cos (2 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-32340 \cos (4 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+9240 \cos (6 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-1155 \cos (8 (c+d x)) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )\right )}{1720320 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Csc[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]

[Out]

-(a^2*Csc[c + d*x]^8*(158270*Cos[c + d*x] + 77210*Cos[3*(c + d*x)] - 18130*Cos[5*(c + d*x)] - 2310*Cos[7*(c +
d*x)] + 40425*Log[Cos[(c + d*x)/2]] - 64680*Cos[2*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 32340*Cos[4*(c + d*x)]*Lo
g[Cos[(c + d*x)/2]] - 9240*Cos[6*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 1155*Cos[8*(c + d*x)]*Log[Cos[(c + d*x)/2]
] - 40425*Log[Sin[(c + d*x)/2]] + 64680*Cos[2*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 32340*Cos[4*(c + d*x)]*Log[Si
n[(c + d*x)/2]] + 9240*Cos[6*(c + d*x)]*Log[Sin[(c + d*x)/2]] - 1155*Cos[8*(c + d*x)]*Log[Sin[(c + d*x)/2]] +
86016*Sin[2*(c + d*x)] + 64512*Sin[4*(c + d*x)] + 12288*Sin[6*(c + d*x)] - 1536*Sin[8*(c + d*x)]))/(1720320*d)

________________________________________________________________________________________

Maple [A]  time = 0.086, size = 200, normalized size = 1.1 \begin{align*} -{\frac{11\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{48\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}-{\frac{11\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{192\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{11\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{384\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{11\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{384\,d}}+{\frac{11\,{a}^{2}\cos \left ( dx+c \right ) }{128\,d}}+{\frac{11\,{a}^{2}\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{4\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35\,d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x)

[Out]

-11/48/d*a^2/sin(d*x+c)^6*cos(d*x+c)^5-11/192/d*a^2/sin(d*x+c)^4*cos(d*x+c)^5+11/384/d*a^2/sin(d*x+c)^2*cos(d*
x+c)^5+11/384*a^2*cos(d*x+c)^3/d+11/128*a^2*cos(d*x+c)/d+11/128/d*a^2*ln(csc(d*x+c)-cot(d*x+c))-2/7/d*a^2/sin(
d*x+c)^7*cos(d*x+c)^5-4/35/d*a^2/sin(d*x+c)^5*cos(d*x+c)^5-1/8/d*a^2/sin(d*x+c)^8*cos(d*x+c)^5

________________________________________________________________________________________

Maxima [A]  time = 1.12378, size = 315, normalized size = 1.79 \begin{align*} \frac{105 \, a^{2}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{7} - 11 \, \cos \left (d x + c\right )^{5} - 11 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )\right )}}{\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} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 280 \, a^{2}{\left (\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac{1536 \,{\left (7 \, \tan \left (d x + c\right )^{2} + 5\right )} a^{2}}{\tan \left (d x + c\right )^{7}}}{26880 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

1/26880*(105*a^2*(2*(3*cos(d*x + c)^7 - 11*cos(d*x + c)^5 - 11*cos(d*x + c)^3 + 3*cos(d*x + c))/(cos(d*x + c)^
8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c)
- 1)) + 280*a^2*(2*(3*cos(d*x + c)^5 + 8*cos(d*x + c)^3 - 3*cos(d*x + c))/(cos(d*x + c)^6 - 3*cos(d*x + c)^4
+ 3*cos(d*x + c)^2 - 1) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 1536*(7*tan(d*x + c)^2 + 5)*a^2
/tan(d*x + c)^7)/d

________________________________________________________________________________________

Fricas [A]  time = 1.64732, size = 709, normalized size = 4.03 \begin{align*} \frac{2310 \, a^{2} \cos \left (d x + c\right )^{7} + 490 \, a^{2} \cos \left (d x + c\right )^{5} - 8470 \, a^{2} \cos \left (d x + c\right )^{3} + 2310 \, a^{2} \cos \left (d x + c\right ) - 1155 \,{\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 1155 \,{\left (a^{2} \cos \left (d x + c\right )^{8} - 4 \, a^{2} \cos \left (d x + c\right )^{6} + 6 \, a^{2} \cos \left (d x + c\right )^{4} - 4 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 1536 \,{\left (2 \, a^{2} \cos \left (d x + c\right )^{7} - 7 \, a^{2} \cos \left (d x + c\right )^{5}\right )} \sin \left (d x + c\right )}{26880 \,{\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/26880*(2310*a^2*cos(d*x + c)^7 + 490*a^2*cos(d*x + c)^5 - 8470*a^2*cos(d*x + c)^3 + 2310*a^2*cos(d*x + c) -
1155*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x + c)^2 + a^2)*log(1/2*c
os(d*x + c) + 1/2) + 1155*(a^2*cos(d*x + c)^8 - 4*a^2*cos(d*x + c)^6 + 6*a^2*cos(d*x + c)^4 - 4*a^2*cos(d*x +
c)^2 + a^2)*log(-1/2*cos(d*x + c) + 1/2) + 1536*(2*a^2*cos(d*x + c)^7 - 7*a^2*cos(d*x + c)^5)*sin(d*x + c))/(d
*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)

________________________________________________________________________________________

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)**4*csc(d*x+c)**9*(a+a*sin(d*x+c))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.495, size = 396, normalized size = 2.25 \begin{align*} \frac{105 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 480 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 672 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 3360 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1680 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 18480 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) + 10080 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{50226 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 10080 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1680 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 3360 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2520 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 672 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 560 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 480 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 105 \, a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8}}}{215040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*csc(d*x+c)^9*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/215040*(105*a^2*tan(1/2*d*x + 1/2*c)^8 + 480*a^2*tan(1/2*d*x + 1/2*c)^7 + 560*a^2*tan(1/2*d*x + 1/2*c)^6 - 6
72*a^2*tan(1/2*d*x + 1/2*c)^5 - 2520*a^2*tan(1/2*d*x + 1/2*c)^4 - 3360*a^2*tan(1/2*d*x + 1/2*c)^3 - 1680*a^2*t
an(1/2*d*x + 1/2*c)^2 + 18480*a^2*log(abs(tan(1/2*d*x + 1/2*c))) + 10080*a^2*tan(1/2*d*x + 1/2*c) - (50226*a^2
*tan(1/2*d*x + 1/2*c)^8 + 10080*a^2*tan(1/2*d*x + 1/2*c)^7 - 1680*a^2*tan(1/2*d*x + 1/2*c)^6 - 3360*a^2*tan(1/
2*d*x + 1/2*c)^5 - 2520*a^2*tan(1/2*d*x + 1/2*c)^4 - 672*a^2*tan(1/2*d*x + 1/2*c)^3 + 560*a^2*tan(1/2*d*x + 1/
2*c)^2 + 480*a^2*tan(1/2*d*x + 1/2*c) + 105*a^2)/tan(1/2*d*x + 1/2*c)^8)/d