3.711 \(\int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=137 \[ -\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}-\frac{15 \cot (c+d x)}{8 a d}+\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{15 x}{8 a} \]

[Out]

(-15*x)/(8*a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*
a*d) - (15*Cot[c + d*x])/(8*a*d) + (5*Cos[c + d*x]^2*Cot[c + d*x])/(8*a*d) + (Cos[c + d*x]^4*Cot[c + d*x])/(4*
a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.173902, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2839, 2591, 288, 321, 203, 2592, 302, 206} \[ -\frac{\cos ^5(c+d x)}{5 a d}-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos (c+d x)}{a d}-\frac{15 \cot (c+d x)}{8 a d}+\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{15 x}{8 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*x)/(8*a) + ArcTanh[Cos[c + d*x]]/(a*d) - Cos[c + d*x]/(a*d) - Cos[c + d*x]^3/(3*a*d) - Cos[c + d*x]^5/(5*
a*d) - (15*Cot[c + d*x])/(8*a*d) + (5*Cos[c + d*x]^2*Cot[c + d*x])/(8*a*d) + (Cos[c + d*x]^4*Cot[c + d*x])/(4*
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 2591

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[(b*ff)/f, Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, (b*Tan[e + f*x])/f
f], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x) \cot ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\int \cos ^5(c+d x) \cot (c+d x) \, dx}{a}+\frac{\int \cos ^4(c+d x) \cot ^2(c+d x) \, dx}{a}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a d}-\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^3} \, dx,x,\cot (c+d x)\right )}{a d}\\ &=\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac{\operatorname{Subst}\left (\int \left (-1-x^2-x^4+\frac{1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{a d}-\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{4 a d}\\ &=-\frac{\cos (c+d x)}{a d}-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{a d}-\frac{15 \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 a d}\\ &=\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cos (c+d x)}{a d}-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos ^5(c+d x)}{5 a d}-\frac{15 \cot (c+d x)}{8 a d}+\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 a d}\\ &=-\frac{15 x}{8 a}+\frac{\tanh ^{-1}(\cos (c+d x))}{a d}-\frac{\cos (c+d x)}{a d}-\frac{\cos ^3(c+d x)}{3 a d}-\frac{\cos ^5(c+d x)}{5 a d}-\frac{15 \cot (c+d x)}{8 a d}+\frac{5 \cos ^2(c+d x) \cot (c+d x)}{8 a d}+\frac{\cos ^4(c+d x) \cot (c+d x)}{4 a d}\\ \end{align*}

Mathematica [A]  time = 0.705235, size = 146, normalized size = 1.07 \[ -\frac{\csc \left (\frac{1}{2} (c+d x)\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \left (1800 c \sin (c+d x)+1800 d x \sin (c+d x)+590 \sin (2 (c+d x))+64 \sin (4 (c+d x))+6 \sin (6 (c+d x))+1200 \cos (c+d x)-225 \cos (3 (c+d x))-15 \cos (5 (c+d x))+960 \sin (c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-960 \sin (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{1920 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(1200*Cos[c + d*x] - 225*Cos[3*(c + d*x)] - 15*Cos[5*(c + d*x)] + 1800*c*S
in[c + d*x] + 1800*d*x*Sin[c + d*x] - 960*Log[Cos[(c + d*x)/2]]*Sin[c + d*x] + 960*Log[Sin[(c + d*x)/2]]*Sin[c
+ d*x] + 590*Sin[2*(c + d*x)] + 64*Sin[4*(c + d*x)] + 6*Sin[6*(c + d*x)]))/(1920*a*d)

________________________________________________________________________________________

Maple [B]  time = 0.135, size = 367, normalized size = 2.7 \begin{align*}{\frac{1}{2\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{9}{4\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{9} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-6\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{5}}}+{\frac{5}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-12\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}}{da \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{5}}}-{\frac{56}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{4} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{5}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{28}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{9}{4\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{46}{15\,da} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-5}}-{\frac{15}{4\,da}\arctan \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{1}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{da}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x)

[Out]

1/2/d/a*tan(1/2*d*x+1/2*c)+9/4/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^9-6/d/a/(1+tan(1/2*d*x+1/2*c)
^2)^5*tan(1/2*d*x+1/2*c)^8+5/2/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^7-12/d/a/(1+tan(1/2*d*x+1/2*c
)^2)^5*tan(1/2*d*x+1/2*c)^6-56/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^4-5/2/d/a/(1+tan(1/2*d*x+1/
2*c)^2)^5*tan(1/2*d*x+1/2*c)^3-28/3/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)^2-9/4/d/a/(1+tan(1/2*d*x
+1/2*c)^2)^5*tan(1/2*d*x+1/2*c)-46/15/d/a/(1+tan(1/2*d*x+1/2*c)^2)^5-15/4/a/d*arctan(tan(1/2*d*x+1/2*c))-1/2/d
/a/tan(1/2*d*x+1/2*c)-1/d/a*ln(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [B]  time = 1.57044, size = 512, normalized size = 3.74 \begin{align*} -\frac{\frac{\frac{184 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{285 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{560 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{450 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{1120 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{300 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{720 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{360 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac{105 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 30}{\frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{5 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{10 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{10 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac{5 \, a \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac{a \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}} + \frac{225 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{30 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/60*((184*sin(d*x + c)/(cos(d*x + c) + 1) + 285*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 560*sin(d*x + c)^3/(co
s(d*x + c) + 1)^3 + 450*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 1120*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 300*s
in(d*x + c)^6/(cos(d*x + c) + 1)^6 + 720*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 360*sin(d*x + c)^9/(cos(d*x + c
) + 1)^9 - 105*sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 30)/(a*sin(d*x + c)/(cos(d*x + c) + 1) + 5*a*sin(d*x +
c)^3/(cos(d*x + c) + 1)^3 + 10*a*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 10*a*sin(d*x + c)^7/(cos(d*x + c) + 1)^
7 + 5*a*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 + a*sin(d*x + c)^11/(cos(d*x + c) + 1)^11) + 225*arctan(sin(d*x +
c)/(cos(d*x + c) + 1))/a + 60*log(sin(d*x + c)/(cos(d*x + c) + 1))/a - 30*sin(d*x + c)/(a*(cos(d*x + c) + 1)))
/d

________________________________________________________________________________________

Fricas [A]  time = 1.151, size = 354, normalized size = 2.58 \begin{align*} \frac{30 \, \cos \left (d x + c\right )^{5} + 75 \, \cos \left (d x + c\right )^{3} -{\left (24 \, \cos \left (d x + c\right )^{5} + 40 \, \cos \left (d x + c\right )^{3} + 225 \, d x + 120 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 60 \, \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 60 \, \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 225 \, \cos \left (d x + c\right )}{120 \, a d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/120*(30*cos(d*x + c)^5 + 75*cos(d*x + c)^3 - (24*cos(d*x + c)^5 + 40*cos(d*x + c)^3 + 225*d*x + 120*cos(d*x
+ c))*sin(d*x + c) + 60*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 60*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c
) - 225*cos(d*x + c))/(a*d*sin(d*x + c))

________________________________________________________________________________________

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)**2/(a+a*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.21551, size = 269, normalized size = 1.96 \begin{align*} -\frac{\frac{225 \,{\left (d x + c\right )}}{a} + \frac{120 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a} - \frac{60 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{60 \,{\left (2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}}{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} - \frac{2 \,{\left (135 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 360 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 150 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 720 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1120 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 150 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 560 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 135 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 184\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )}^{5} a}}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^8*csc(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/120*(225*(d*x + c)/a + 120*log(abs(tan(1/2*d*x + 1/2*c)))/a - 60*tan(1/2*d*x + 1/2*c)/a - 60*(2*tan(1/2*d*x
+ 1/2*c) - 1)/(a*tan(1/2*d*x + 1/2*c)) - 2*(135*tan(1/2*d*x + 1/2*c)^9 - 360*tan(1/2*d*x + 1/2*c)^8 + 150*tan
(1/2*d*x + 1/2*c)^7 - 720*tan(1/2*d*x + 1/2*c)^6 - 1120*tan(1/2*d*x + 1/2*c)^4 - 150*tan(1/2*d*x + 1/2*c)^3 -
560*tan(1/2*d*x + 1/2*c)^2 - 135*tan(1/2*d*x + 1/2*c) - 184)/((tan(1/2*d*x + 1/2*c)^2 + 1)^5*a))/d