3.1255 \(\int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=198 \[ -\frac{\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac{\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a b \cot (c+d x) \csc (c+d x)}{128 d} \]

[Out]

(3*a*b*ArcTanh[Cos[c + d*x]])/(128*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - ((2*a^2 + b^2)*Cot[c + d*x]^9)/(9
*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a*b*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a*b*Cot[c + d*x]*Csc[c + d*x
]^3)/(64*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a*b*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a*b*Cot[
c + d*x]^5*Csc[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.456852, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2911, 2611, 3768, 3770, 448} \[ -\frac{\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac{\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac{3 a b \cot (c+d x) \csc (c+d x)}{128 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]

[Out]

(3*a*b*ArcTanh[Cos[c + d*x]])/(128*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - ((2*a^2 + b^2)*Cot[c + d*x]^9)/(9
*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a*b*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a*b*Cot[c + d*x]*Csc[c + d*x
]^3)/(64*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a*b*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a*b*Cot[
c + d*x]^5*Csc[c + d*x]^5)/(5*d)

Rule 2911

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^2, x_Symbol] :> Dist[(2*a*b)/d, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e
+ f*x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[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 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx &=(2 a b) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\int \cot ^6(c+d x) \csc ^6(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-(a b) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right ) \left (a^2+\left (a^2+b^2\right ) x^2\right )}{x^{12}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac{1}{8} (3 a b) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^{12}}+\frac{2 a^2+b^2}{x^{10}}+\frac{a^2+b^2}{x^8}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac{\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac{1}{16} (a b) \int \csc ^5(c+d x) \, dx\\ &=-\frac{\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac{\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac{1}{64} (3 a b) \int \csc ^3(c+d x) \, dx\\ &=-\frac{\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac{\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a b \cot (c+d x) \csc (c+d x)}{128 d}+\frac{a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac{1}{128} (3 a b) \int \csc (c+d x) \, dx\\ &=\frac{3 a b \tanh ^{-1}(\cos (c+d x))}{128 d}-\frac{\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac{\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac{a^2 \cot ^{11}(c+d x)}{11 d}+\frac{3 a b \cot (c+d x) \csc (c+d x)}{128 d}+\frac{a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac{a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac{a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac{a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 1.71349, size = 250, normalized size = 1.26 \[ -\frac{\csc ^{11}(c+d x) \left (1478400 \left (8 a^2+b^2\right ) \cos (c+d x)+42240 \left (160 a^2-b^2\right ) \cos (3 (c+d x))+1943040 a^2 \cos (5 (c+d x))+140800 a^2 \cos (7 (c+d x))-28160 a^2 \cos (9 (c+d x))+2560 a^2 \cos (11 (c+d x))+5828130 a b \sin (2 (c+d x))+4790016 a b \sin (4 (c+d x))+2302839 a b \sin (6 (c+d x))+110880 a b \sin (8 (c+d x))-10395 a b \sin (10 (c+d x))-865920 b^2 \cos (5 (c+d x))-499840 b^2 \cos (7 (c+d x))-77440 b^2 \cos (9 (c+d x))+7040 b^2 \cos (11 (c+d x))\right )+5322240 a b \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-5322240 a b \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{227082240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]

[Out]

-(-5322240*a*b*Log[Cos[(c + d*x)/2]] + 5322240*a*b*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^11*(1478400*(8*a^2 + b
^2)*Cos[c + d*x] + 42240*(160*a^2 - b^2)*Cos[3*(c + d*x)] + 1943040*a^2*Cos[5*(c + d*x)] - 865920*b^2*Cos[5*(c
+ d*x)] + 140800*a^2*Cos[7*(c + d*x)] - 499840*b^2*Cos[7*(c + d*x)] - 28160*a^2*Cos[9*(c + d*x)] - 77440*b^2*
Cos[9*(c + d*x)] + 2560*a^2*Cos[11*(c + d*x)] + 7040*b^2*Cos[11*(c + d*x)] + 5828130*a*b*Sin[2*(c + d*x)] + 47
90016*a*b*Sin[4*(c + d*x)] + 2302839*a*b*Sin[6*(c + d*x)] + 110880*a*b*Sin[8*(c + d*x)] - 10395*a*b*Sin[10*(c
+ d*x)]))/(227082240*d)

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Maple [A]  time = 0.101, size = 303, normalized size = 1.5 \begin{align*} -{\frac{{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{11\,d \left ( \sin \left ( dx+c \right ) \right ) ^{11}}}-{\frac{4\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{8\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{5\,d \left ( \sin \left ( dx+c \right ) \right ) ^{10}}}-{\frac{3\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{40\,d \left ( \sin \left ( dx+c \right ) \right ) ^{8}}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80\,d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{320\,d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{640\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,ab \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{640\,d}}-{\frac{ab \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{128\,d}}-{\frac{3\,ab\cos \left ( dx+c \right ) }{128\,d}}-{\frac{3\,ab\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{128\,d}}-{\frac{{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9\,d \left ( \sin \left ( dx+c \right ) \right ) ^{9}}}-{\frac{2\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63\,d \left ( \sin \left ( dx+c \right ) \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x)

[Out]

-1/11/d*a^2/sin(d*x+c)^11*cos(d*x+c)^7-4/99/d*a^2/sin(d*x+c)^9*cos(d*x+c)^7-8/693/d*a^2/sin(d*x+c)^7*cos(d*x+c
)^7-1/5/d*a*b/sin(d*x+c)^10*cos(d*x+c)^7-3/40/d*a*b/sin(d*x+c)^8*cos(d*x+c)^7-1/80/d*a*b/sin(d*x+c)^6*cos(d*x+
c)^7+1/320/d*a*b/sin(d*x+c)^4*cos(d*x+c)^7-3/640/d*a*b/sin(d*x+c)^2*cos(d*x+c)^7-3/640*a*b*cos(d*x+c)^5/d-1/12
8*a*b*cos(d*x+c)^3/d-3/128*a*b*cos(d*x+c)/d-3/128/d*a*b*ln(csc(d*x+c)-cot(d*x+c))-1/9/d*b^2/sin(d*x+c)^9*cos(d
*x+c)^7-2/63/d*b^2/sin(d*x+c)^7*cos(d*x+c)^7

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Maxima [A]  time = 1.00085, size = 265, normalized size = 1.34 \begin{align*} -\frac{693 \, a b{\left (\frac{2 \,{\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac{14080 \,{\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} b^{2}}{\tan \left (d x + c\right )^{9}} + \frac{1280 \,{\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{887040 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/887040*(693*a*b*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos
(d*x + c))/(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)
- 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 14080*(9*tan(d*x + c)^2 + 7)*b^2/tan(d*x + c)^9 + 12
80*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^2/tan(d*x + c)^11)/d

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Fricas [B]  time = 2.02084, size = 990, normalized size = 5. \begin{align*} \frac{2560 \,{\left (4 \, a^{2} + 11 \, b^{2}\right )} \cos \left (d x + c\right )^{11} - 14080 \,{\left (4 \, a^{2} + 11 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 126720 \,{\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} + 10395 \,{\left (a b \cos \left (d x + c\right )^{10} - 5 \, a b \cos \left (d x + c\right )^{8} + 10 \, a b \cos \left (d x + c\right )^{6} - 10 \, a b \cos \left (d x + c\right )^{4} + 5 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 10395 \,{\left (a b \cos \left (d x + c\right )^{10} - 5 \, a b \cos \left (d x + c\right )^{8} + 10 \, a b \cos \left (d x + c\right )^{6} - 10 \, a b \cos \left (d x + c\right )^{4} + 5 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 1386 \,{\left (15 \, a b \cos \left (d x + c\right )^{9} - 70 \, a b \cos \left (d x + c\right )^{7} - 128 \, a b \cos \left (d x + c\right )^{5} + 70 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \,{\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/887040*(2560*(4*a^2 + 11*b^2)*cos(d*x + c)^11 - 14080*(4*a^2 + 11*b^2)*cos(d*x + c)^9 + 126720*(a^2 + b^2)*c
os(d*x + c)^7 + 10395*(a*b*cos(d*x + c)^10 - 5*a*b*cos(d*x + c)^8 + 10*a*b*cos(d*x + c)^6 - 10*a*b*cos(d*x + c
)^4 + 5*a*b*cos(d*x + c)^2 - a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 10395*(a*b*cos(d*x + c)^10 - 5*a*
b*cos(d*x + c)^8 + 10*a*b*cos(d*x + c)^6 - 10*a*b*cos(d*x + c)^4 + 5*a*b*cos(d*x + c)^2 - a*b)*log(-1/2*cos(d*
x + c) + 1/2)*sin(d*x + c) - 1386*(15*a*b*cos(d*x + c)^9 - 70*a*b*cos(d*x + c)^7 - 128*a*b*cos(d*x + c)^5 + 70
*a*b*cos(d*x + c)^3 - 15*a*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d
*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - 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(cos(d*x+c)**6*csc(d*x+c)**12*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.28743, size = 678, normalized size = 3.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/7096320*(315*a^2*tan(1/2*d*x + 1/2*c)^11 + 1386*a*b*tan(1/2*d*x + 1/2*c)^10 - 385*a^2*tan(1/2*d*x + 1/2*c)^9
+ 1540*b^2*tan(1/2*d*x + 1/2*c)^9 - 3465*a*b*tan(1/2*d*x + 1/2*c)^8 - 2475*a^2*tan(1/2*d*x + 1/2*c)^7 - 5940*
b^2*tan(1/2*d*x + 1/2*c)^7 - 6930*a*b*tan(1/2*d*x + 1/2*c)^6 + 3465*a^2*tan(1/2*d*x + 1/2*c)^5 + 27720*a*b*tan
(1/2*d*x + 1/2*c)^4 + 11550*a^2*tan(1/2*d*x + 1/2*c)^3 + 36960*b^2*tan(1/2*d*x + 1/2*c)^3 + 13860*a*b*tan(1/2*
d*x + 1/2*c)^2 - 166320*a*b*log(abs(tan(1/2*d*x + 1/2*c))) - 34650*a^2*tan(1/2*d*x + 1/2*c) - 83160*b^2*tan(1/
2*d*x + 1/2*c) + (502266*a*b*tan(1/2*d*x + 1/2*c)^11 + 34650*a^2*tan(1/2*d*x + 1/2*c)^10 + 83160*b^2*tan(1/2*d
*x + 1/2*c)^10 - 13860*a*b*tan(1/2*d*x + 1/2*c)^9 - 11550*a^2*tan(1/2*d*x + 1/2*c)^8 - 36960*b^2*tan(1/2*d*x +
1/2*c)^8 - 27720*a*b*tan(1/2*d*x + 1/2*c)^7 - 3465*a^2*tan(1/2*d*x + 1/2*c)^6 + 6930*a*b*tan(1/2*d*x + 1/2*c)
^5 + 2475*a^2*tan(1/2*d*x + 1/2*c)^4 + 5940*b^2*tan(1/2*d*x + 1/2*c)^4 + 3465*a*b*tan(1/2*d*x + 1/2*c)^3 + 385
*a^2*tan(1/2*d*x + 1/2*c)^2 - 1540*b^2*tan(1/2*d*x + 1/2*c)^2 - 1386*a*b*tan(1/2*d*x + 1/2*c) - 315*a^2)/tan(1
/2*d*x + 1/2*c)^11)/d