3.467 \(\int \frac{\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=491 \[ -\frac{a \left (-56 a^4 b^2+70 a^2 b^4+16 a^6-35 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 b^8 d \left (a^2-b^2\right )^{7/2}}+\frac{a \cos ^7(c+d x)}{6 b d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}-\frac{\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{24 b^5 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^3}+\frac{a \left (-22 a^2 b^2+8 a^4+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \left (a b \left (-22 a^2 b^2+8 a^4+19 b^4\right ) \sin (c+d x)+16 \left (a^2-b^2\right )^3\right )}{16 b^7 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{x}{b^8} \]

[Out]

x/b^8 - (a*(16*a^6 - 56*a^4*b^2 + 70*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*b^
8*(a^2 - b^2)^(7/2)*d) - Cos[c + d*x]^7/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]^7)/(6*b*(a^2 - b^2)*d
*(a + b*Sin[c + d*x])^6) - (a*(6*a^2 - 11*b^2)*Cos[c + d*x]^5)/(24*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4)
+ (a*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Cos[c + d*x]^3)/(16*b^5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + (Cos[c +
d*x]^5*(6*(a^2 - b^2) + 5*a*b*Sin[c + d*x]))/(30*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) - (Cos[c + d*x]^3*
(8*(a^2 - b^2)^2 + a*b*(6*a^2 - 11*b^2)*Sin[c + d*x]))/(24*b^5*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^3) + (Cos[
c + d*x]*(16*(a^2 - b^2)^3 + a*b*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Sin[c + d*x]))/(16*b^7*(a^2 - b^2)^3*d*(a + b*S
in[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.27382, antiderivative size = 491, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2693, 2864, 2863, 2735, 2660, 618, 204} \[ -\frac{a \left (-56 a^4 b^2+70 a^2 b^4+16 a^6-35 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 b^8 d \left (a^2-b^2\right )^{7/2}}+\frac{a \cos ^7(c+d x)}{6 b d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}-\frac{\cos ^3(c+d x) \left (a b \left (6 a^2-11 b^2\right ) \sin (c+d x)+8 \left (a^2-b^2\right )^2\right )}{24 b^5 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^3}+\frac{a \left (-22 a^2 b^2+8 a^4+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \left (a b \left (-22 a^2 b^2+8 a^4+19 b^4\right ) \sin (c+d x)+16 \left (a^2-b^2\right )^3\right )}{16 b^7 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{x}{b^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/b^8 - (a*(16*a^6 - 56*a^4*b^2 + 70*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*b^
8*(a^2 - b^2)^(7/2)*d) - Cos[c + d*x]^7/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]^7)/(6*b*(a^2 - b^2)*d
*(a + b*Sin[c + d*x])^6) - (a*(6*a^2 - 11*b^2)*Cos[c + d*x]^5)/(24*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4)
+ (a*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Cos[c + d*x]^3)/(16*b^5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + (Cos[c +
d*x]^5*(6*(a^2 - b^2) + 5*a*b*Sin[c + d*x]))/(30*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) - (Cos[c + d*x]^3*
(8*(a^2 - b^2)^2 + a*b*(6*a^2 - 11*b^2)*Sin[c + d*x]))/(24*b^5*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^3) + (Cos[
c + d*x]*(16*(a^2 - b^2)^3 + a*b*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Sin[c + d*x]))/(16*b^7*(a^2 - b^2)^3*d*(a + b*S
in[c + d*x]))

Rule 2693

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(g^2*(p - 1))/(b*(m + 1)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Sin[e + f*x], x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a
^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && IntegersQ[2*m, 2*p]

Rule 2864

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((b*c - a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a
^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m + 1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*Sim
p[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 2863

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) -
a*d*p + b*d*(m + 1)*Sin[e + f*x]))/(b^2*f*(m + 1)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + 1)*(m + p +
1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^(m + 1)*Simp[b*d*(m + 1) + (b*c*(m + p + 1) - a*d*p)*Si
n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[p, 1] && N
eQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\int \frac{\cos ^6(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^7} \, dx}{b}\\ &=-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\int \frac{\cos ^6(c+d x) (6 b+a \sin (c+d x))}{(a+b \sin (c+d x))^6} \, dx}{6 b \left (a^2-b^2\right )}\\ &=-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\int \frac{\cos ^4(c+d x) \left (-5 a b-6 \left (a^2-b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^5} \, dx}{6 b^3 \left (a^2-b^2\right )}\\ &=-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}+\frac{\int \frac{\cos ^4(c+d x) \left (-4 b \left (a^2-6 b^2\right )-a \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^4} \, dx}{24 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}-\frac{\int \frac{\cos ^2(c+d x) \left (3 a b \left (6 a^2-11 b^2\right )+24 \left (a^2-b^2\right )^2 \sin (c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{24 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac{\int \frac{\cos ^2(c+d x) \left (6 b \left (2 a^4-5 a^2 b^2+8 b^4\right )+3 a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{48 b^5 \left (a^2-b^2\right )^3}\\ &=-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac{\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{16 b^7 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac{\int \frac{-3 a b \left (8 a^4-22 a^2 b^2+19 b^4\right )-48 \left (a^2-b^2\right )^3 \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{48 b^7 \left (a^2-b^2\right )^3}\\ &=\frac{x}{b^8}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac{\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{16 b^7 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac{\left (a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{16 b^8 \left (a^2-b^2\right )^3}\\ &=\frac{x}{b^8}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac{\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{16 b^7 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac{\left (a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{8 b^8 \left (a^2-b^2\right )^3 d}\\ &=\frac{x}{b^8}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac{\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{16 b^7 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}+\frac{\left (a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{4 b^8 \left (a^2-b^2\right )^3 d}\\ &=\frac{x}{b^8}-\frac{a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{8 b^8 \left (a^2-b^2\right )^{7/2} d}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{a \cos ^7(c+d x)}{6 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}-\frac{a \left (6 a^2-11 b^2\right ) \cos ^5(c+d x)}{24 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{a \left (8 a^4-22 a^2 b^2+19 b^4\right ) \cos ^3(c+d x)}{16 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac{\cos ^5(c+d x) \left (6 \left (a^2-b^2\right )+5 a b \sin (c+d x)\right )}{30 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{\cos ^3(c+d x) \left (8 \left (a^2-b^2\right )^2+a b \left (6 a^2-11 b^2\right ) \sin (c+d x)\right )}{24 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac{\cos (c+d x) \left (16 \left (a^2-b^2\right )^3+a b \left (8 a^4-22 a^2 b^2+19 b^4\right ) \sin (c+d x)\right )}{16 b^7 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [B]  time = 8.54895, size = 6586, normalized size = 13.41 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.227, size = 9454, normalized size = 19.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 10.6953, size = 8992, normalized size = 18.31 \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)^8/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

[1/3360*(23520*(a^9*b^6 - 4*a^7*b^8 + 6*a^5*b^10 - 4*a^3*b^12 + a*b^14)*d*x*cos(d*x + c)^6 + 2*(4356*a^8*b^7 -
16864*a^6*b^9 + 24001*a^4*b^11 - 14309*a^2*b^13 + 2816*b^15)*cos(d*x + c)^7 - 23520*(5*a^11*b^4 - 17*a^9*b^6
+ 18*a^7*b^8 - 2*a^5*b^10 - 7*a^3*b^12 + 3*a*b^14)*d*x*cos(d*x + c)^4 - 28*(2754*a^10*b^5 - 9717*a^8*b^7 + 115
28*a^6*b^9 - 3782*a^4*b^11 - 1247*a^2*b^13 + 464*b^15)*cos(d*x + c)^5 + 23520*(3*a^13*b^2 - 2*a^11*b^4 - 19*a^
9*b^6 + 36*a^7*b^8 - 19*a^5*b^10 - 2*a^3*b^12 + 3*a*b^14)*d*x*cos(d*x + c)^2 + 70*(856*a^12*b^3 - 1090*a^10*b^
5 - 3477*a^8*b^7 + 7907*a^6*b^9 - 4423*a^4*b^11 + 67*a^2*b^13 + 160*b^15)*cos(d*x + c)^3 - 3360*(a^15 + 17*a^1
3*b^2 - 43*a^11*b^4 - 11*a^9*b^6 + 99*a^7*b^8 - 77*a^5*b^10 + 7*a^3*b^12 + 7*a*b^14)*d*x + 105*(16*a^14 + 280*
a^12*b^2 - 546*a^10*b^4 - 413*a^8*b^6 + 1323*a^6*b^8 - 735*a^4*b^10 - 245*a^2*b^12 - 7*(16*a^8*b^6 - 56*a^6*b^
8 + 70*a^4*b^10 - 35*a^2*b^12)*cos(d*x + c)^6 + 7*(80*a^10*b^4 - 232*a^8*b^6 + 182*a^6*b^8 + 35*a^4*b^10 - 105
*a^2*b^12)*cos(d*x + c)^4 - 7*(48*a^12*b^2 - 8*a^10*b^4 - 302*a^8*b^6 + 427*a^6*b^8 - 140*a^4*b^10 - 105*a^2*b
^12)*cos(d*x + c)^2 + (112*a^13*b + 168*a^11*b^3 - 1134*a^9*b^5 + 1045*a^7*b^7 + 189*a^5*b^9 - 665*a^3*b^11 -
35*a*b^13 - (16*a^7*b^7 - 56*a^5*b^9 + 70*a^3*b^11 - 35*a*b^13)*cos(d*x + c)^6 + 3*(112*a^9*b^5 - 376*a^7*b^7
+ 434*a^5*b^9 - 175*a^3*b^11 - 35*a*b^13)*cos(d*x + c)^4 - (560*a^11*b^3 - 1288*a^9*b^5 + 146*a^7*b^7 + 1547*a
^5*b^9 - 1260*a^3*b^11 - 105*a*b^13)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*
x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))
/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 420*(8*a^14*b + 112*a^12*b^3 - 322*a^10*b^5 + 63*a^8
*b^7 + 479*a^6*b^9 - 379*a^4*b^11 + 31*a^2*b^13 + 8*b^15)*cos(d*x + c) + 14*(240*(a^8*b^7 - 4*a^6*b^9 + 6*a^4*
b^11 - 4*a^2*b^13 + b^15)*d*x*cos(d*x + c)^6 - 720*(7*a^10*b^5 - 27*a^8*b^7 + 38*a^6*b^9 - 22*a^4*b^11 + 3*a^2
*b^13 + b^15)*d*x*cos(d*x + c)^4 - (2676*a^9*b^6 - 10264*a^7*b^8 + 14371*a^5*b^10 - 8204*a^3*b^12 + 1421*a*b^1
4)*cos(d*x + c)^5 + 240*(35*a^12*b^3 - 98*a^10*b^5 + 45*a^8*b^7 + 100*a^6*b^9 - 115*a^4*b^11 + 30*a^2*b^13 + 3
*b^15)*d*x*cos(d*x + c)^2 + 10*(638*a^11*b^4 - 1925*a^9*b^6 + 1427*a^7*b^8 + 861*a^5*b^10 - 1253*a^3*b^12 + 25
2*a*b^14)*cos(d*x + c)^3 - 240*(7*a^14*b + 7*a^12*b^3 - 77*a^10*b^5 + 99*a^8*b^7 - 11*a^6*b^9 - 43*a^4*b^11 +
17*a^2*b^13 + b^15)*d*x - 15*(104*a^13*b^2 + 26*a^11*b^4 - 897*a^9*b^6 + 1306*a^7*b^8 - 308*a^5*b^10 - 308*a^3
*b^12 + 77*a*b^14)*cos(d*x + c))*sin(d*x + c))/(7*(a^9*b^14 - 4*a^7*b^16 + 6*a^5*b^18 - 4*a^3*b^20 + a*b^22)*d
*cos(d*x + c)^6 - 7*(5*a^11*b^12 - 17*a^9*b^14 + 18*a^7*b^16 - 2*a^5*b^18 - 7*a^3*b^20 + 3*a*b^22)*d*cos(d*x +
c)^4 + 7*(3*a^13*b^10 - 2*a^11*b^12 - 19*a^9*b^14 + 36*a^7*b^16 - 19*a^5*b^18 - 2*a^3*b^20 + 3*a*b^22)*d*cos(
d*x + c)^2 - (a^15*b^8 + 17*a^13*b^10 - 43*a^11*b^12 - 11*a^9*b^14 + 99*a^7*b^16 - 77*a^5*b^18 + 7*a^3*b^20 +
7*a*b^22)*d + ((a^8*b^15 - 4*a^6*b^17 + 6*a^4*b^19 - 4*a^2*b^21 + b^23)*d*cos(d*x + c)^6 - 3*(7*a^10*b^13 - 27
*a^8*b^15 + 38*a^6*b^17 - 22*a^4*b^19 + 3*a^2*b^21 + b^23)*d*cos(d*x + c)^4 + (35*a^12*b^11 - 98*a^10*b^13 + 4
5*a^8*b^15 + 100*a^6*b^17 - 115*a^4*b^19 + 30*a^2*b^21 + 3*b^23)*d*cos(d*x + c)^2 - (7*a^14*b^9 + 7*a^12*b^11
- 77*a^10*b^13 + 99*a^8*b^15 - 11*a^6*b^17 - 43*a^4*b^19 + 17*a^2*b^21 + b^23)*d)*sin(d*x + c)), 1/1680*(11760
*(a^9*b^6 - 4*a^7*b^8 + 6*a^5*b^10 - 4*a^3*b^12 + a*b^14)*d*x*cos(d*x + c)^6 + (4356*a^8*b^7 - 16864*a^6*b^9 +
24001*a^4*b^11 - 14309*a^2*b^13 + 2816*b^15)*cos(d*x + c)^7 - 11760*(5*a^11*b^4 - 17*a^9*b^6 + 18*a^7*b^8 - 2
*a^5*b^10 - 7*a^3*b^12 + 3*a*b^14)*d*x*cos(d*x + c)^4 - 14*(2754*a^10*b^5 - 9717*a^8*b^7 + 11528*a^6*b^9 - 378
2*a^4*b^11 - 1247*a^2*b^13 + 464*b^15)*cos(d*x + c)^5 + 11760*(3*a^13*b^2 - 2*a^11*b^4 - 19*a^9*b^6 + 36*a^7*b
^8 - 19*a^5*b^10 - 2*a^3*b^12 + 3*a*b^14)*d*x*cos(d*x + c)^2 + 35*(856*a^12*b^3 - 1090*a^10*b^5 - 3477*a^8*b^7
+ 7907*a^6*b^9 - 4423*a^4*b^11 + 67*a^2*b^13 + 160*b^15)*cos(d*x + c)^3 - 1680*(a^15 + 17*a^13*b^2 - 43*a^11*
b^4 - 11*a^9*b^6 + 99*a^7*b^8 - 77*a^5*b^10 + 7*a^3*b^12 + 7*a*b^14)*d*x - 105*(16*a^14 + 280*a^12*b^2 - 546*a
^10*b^4 - 413*a^8*b^6 + 1323*a^6*b^8 - 735*a^4*b^10 - 245*a^2*b^12 - 7*(16*a^8*b^6 - 56*a^6*b^8 + 70*a^4*b^10
- 35*a^2*b^12)*cos(d*x + c)^6 + 7*(80*a^10*b^4 - 232*a^8*b^6 + 182*a^6*b^8 + 35*a^4*b^10 - 105*a^2*b^12)*cos(d
*x + c)^4 - 7*(48*a^12*b^2 - 8*a^10*b^4 - 302*a^8*b^6 + 427*a^6*b^8 - 140*a^4*b^10 - 105*a^2*b^12)*cos(d*x + c
)^2 + (112*a^13*b + 168*a^11*b^3 - 1134*a^9*b^5 + 1045*a^7*b^7 + 189*a^5*b^9 - 665*a^3*b^11 - 35*a*b^13 - (16*
a^7*b^7 - 56*a^5*b^9 + 70*a^3*b^11 - 35*a*b^13)*cos(d*x + c)^6 + 3*(112*a^9*b^5 - 376*a^7*b^7 + 434*a^5*b^9 -
175*a^3*b^11 - 35*a*b^13)*cos(d*x + c)^4 - (560*a^11*b^3 - 1288*a^9*b^5 + 146*a^7*b^7 + 1547*a^5*b^9 - 1260*a^
3*b^11 - 105*a*b^13)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^
2)*cos(d*x + c))) - 210*(8*a^14*b + 112*a^12*b^3 - 322*a^10*b^5 + 63*a^8*b^7 + 479*a^6*b^9 - 379*a^4*b^11 + 31
*a^2*b^13 + 8*b^15)*cos(d*x + c) + 7*(240*(a^8*b^7 - 4*a^6*b^9 + 6*a^4*b^11 - 4*a^2*b^13 + b^15)*d*x*cos(d*x +
c)^6 - 720*(7*a^10*b^5 - 27*a^8*b^7 + 38*a^6*b^9 - 22*a^4*b^11 + 3*a^2*b^13 + b^15)*d*x*cos(d*x + c)^4 - (267
6*a^9*b^6 - 10264*a^7*b^8 + 14371*a^5*b^10 - 8204*a^3*b^12 + 1421*a*b^14)*cos(d*x + c)^5 + 240*(35*a^12*b^3 -
98*a^10*b^5 + 45*a^8*b^7 + 100*a^6*b^9 - 115*a^4*b^11 + 30*a^2*b^13 + 3*b^15)*d*x*cos(d*x + c)^2 + 10*(638*a^1
1*b^4 - 1925*a^9*b^6 + 1427*a^7*b^8 + 861*a^5*b^10 - 1253*a^3*b^12 + 252*a*b^14)*cos(d*x + c)^3 - 240*(7*a^14*
b + 7*a^12*b^3 - 77*a^10*b^5 + 99*a^8*b^7 - 11*a^6*b^9 - 43*a^4*b^11 + 17*a^2*b^13 + b^15)*d*x - 15*(104*a^13*
b^2 + 26*a^11*b^4 - 897*a^9*b^6 + 1306*a^7*b^8 - 308*a^5*b^10 - 308*a^3*b^12 + 77*a*b^14)*cos(d*x + c))*sin(d*
x + c))/(7*(a^9*b^14 - 4*a^7*b^16 + 6*a^5*b^18 - 4*a^3*b^20 + a*b^22)*d*cos(d*x + c)^6 - 7*(5*a^11*b^12 - 17*a
^9*b^14 + 18*a^7*b^16 - 2*a^5*b^18 - 7*a^3*b^20 + 3*a*b^22)*d*cos(d*x + c)^4 + 7*(3*a^13*b^10 - 2*a^11*b^12 -
19*a^9*b^14 + 36*a^7*b^16 - 19*a^5*b^18 - 2*a^3*b^20 + 3*a*b^22)*d*cos(d*x + c)^2 - (a^15*b^8 + 17*a^13*b^10 -
43*a^11*b^12 - 11*a^9*b^14 + 99*a^7*b^16 - 77*a^5*b^18 + 7*a^3*b^20 + 7*a*b^22)*d + ((a^8*b^15 - 4*a^6*b^17 +
6*a^4*b^19 - 4*a^2*b^21 + b^23)*d*cos(d*x + c)^6 - 3*(7*a^10*b^13 - 27*a^8*b^15 + 38*a^6*b^17 - 22*a^4*b^19 +
3*a^2*b^21 + b^23)*d*cos(d*x + c)^4 + (35*a^12*b^11 - 98*a^10*b^13 + 45*a^8*b^15 + 100*a^6*b^17 - 115*a^4*b^1
9 + 30*a^2*b^21 + 3*b^23)*d*cos(d*x + c)^2 - (7*a^14*b^9 + 7*a^12*b^11 - 77*a^10*b^13 + 99*a^8*b^15 - 11*a^6*b
^17 - 43*a^4*b^19 + 17*a^2*b^21 + b^23)*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)**8/(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [B]  time = 1.5888, size = 3140, normalized size = 6.4 \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)^8/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

-1/840*(105*(16*a^7 - 56*a^5*b^2 + 70*a^3*b^4 - 35*a*b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a
*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^6*b^8 - 3*a^4*b^10 + 3*a^2*b^12 - b^14)*sqrt(a^2 - b^2)) - (8
40*a^18*b*tan(1/2*d*x + 1/2*c)^13 - 2310*a^16*b^3*tan(1/2*d*x + 1/2*c)^13 + 1995*a^14*b^5*tan(1/2*d*x + 1/2*c)
^13 - 1680*a^12*b^7*tan(1/2*d*x + 1/2*c)^13 + 5040*a^10*b^9*tan(1/2*d*x + 1/2*c)^13 - 5040*a^8*b^11*tan(1/2*d*
x + 1/2*c)^13 + 1680*a^6*b^13*tan(1/2*d*x + 1/2*c)^13 + 1680*a^19*tan(1/2*d*x + 1/2*c)^12 + 5880*a^17*b^2*tan(
1/2*d*x + 1/2*c)^12 - 24990*a^15*b^4*tan(1/2*d*x + 1/2*c)^12 + 24255*a^13*b^6*tan(1/2*d*x + 1/2*c)^12 - 10080*
a^11*b^8*tan(1/2*d*x + 1/2*c)^12 + 30240*a^9*b^10*tan(1/2*d*x + 1/2*c)^12 - 30240*a^7*b^12*tan(1/2*d*x + 1/2*c
)^12 + 10080*a^5*b^14*tan(1/2*d*x + 1/2*c)^12 + 26880*a^18*b*tan(1/2*d*x + 1/2*c)^11 - 19320*a^16*b^3*tan(1/2*
d*x + 1/2*c)^11 - 87640*a^14*b^5*tan(1/2*d*x + 1/2*c)^11 + 118790*a^12*b^7*tan(1/2*d*x + 1/2*c)^11 - 26880*a^1
0*b^9*tan(1/2*d*x + 1/2*c)^11 + 94080*a^8*b^11*tan(1/2*d*x + 1/2*c)^11 - 98560*a^6*b^13*tan(1/2*d*x + 1/2*c)^1
1 + 33600*a^4*b^15*tan(1/2*d*x + 1/2*c)^11 + 10080*a^19*tan(1/2*d*x + 1/2*c)^10 + 144480*a^17*b^2*tan(1/2*d*x
+ 1/2*c)^10 - 299880*a^15*b^4*tan(1/2*d*x + 1/2*c)^10 - 15680*a^13*b^6*tan(1/2*d*x + 1/2*c)^10 + 276430*a^11*b
^8*tan(1/2*d*x + 1/2*c)^10 + 36960*a^9*b^10*tan(1/2*d*x + 1/2*c)^10 + 97440*a^7*b^12*tan(1/2*d*x + 1/2*c)^10 -
166880*a^5*b^14*tan(1/2*d*x + 1/2*c)^10 + 67200*a^3*b^16*tan(1/2*d*x + 1/2*c)^10 + 121800*a^18*b*tan(1/2*d*x
+ 1/2*c)^9 + 238770*a^16*b^3*tan(1/2*d*x + 1/2*c)^9 - 1067605*a^14*b^5*tan(1/2*d*x + 1/2*c)^9 + 656390*a^12*b^
7*tan(1/2*d*x + 1/2*c)^9 + 345156*a^10*b^9*tan(1/2*d*x + 1/2*c)^9 + 214032*a^8*b^11*tan(1/2*d*x + 1/2*c)^9 - 8
7472*a^6*b^13*tan(1/2*d*x + 1/2*c)^9 - 126336*a^4*b^15*tan(1/2*d*x + 1/2*c)^9 + 80640*a^2*b^17*tan(1/2*d*x + 1
/2*c)^9 + 25200*a^19*tan(1/2*d*x + 1/2*c)^8 + 514360*a^17*b^2*tan(1/2*d*x + 1/2*c)^8 - 490350*a^15*b^4*tan(1/2
*d*x + 1/2*c)^8 - 1389885*a^13*b^6*tan(1/2*d*x + 1/2*c)^8 + 1764630*a^11*b^8*tan(1/2*d*x + 1/2*c)^8 + 201544*a
^9*b^10*tan(1/2*d*x + 1/2*c)^8 + 305088*a^7*b^12*tan(1/2*d*x + 1/2*c)^8 - 336448*a^5*b^14*tan(1/2*d*x + 1/2*c)
^8 + 27776*a^3*b^16*tan(1/2*d*x + 1/2*c)^8 + 53760*a*b^18*tan(1/2*d*x + 1/2*c)^8 + 235200*a^18*b*tan(1/2*d*x +
1/2*c)^7 + 744800*a^16*b^3*tan(1/2*d*x + 1/2*c)^7 - 2263800*a^14*b^5*tan(1/2*d*x + 1/2*c)^7 + 382620*a^12*b^7
*tan(1/2*d*x + 1/2*c)^7 + 1776432*a^10*b^9*tan(1/2*d*x + 1/2*c)^7 + 204848*a^8*b^11*tan(1/2*d*x + 1/2*c)^7 - 4
7616*a^6*b^13*tan(1/2*d*x + 1/2*c)^7 - 258560*a^4*b^15*tan(1/2*d*x + 1/2*c)^7 + 111616*a^2*b^17*tan(1/2*d*x +
1/2*c)^7 + 15360*b^19*tan(1/2*d*x + 1/2*c)^7 + 33600*a^19*tan(1/2*d*x + 1/2*c)^6 + 730240*a^17*b^2*tan(1/2*d*x
+ 1/2*c)^6 - 534240*a^15*b^4*tan(1/2*d*x + 1/2*c)^6 - 2260440*a^13*b^6*tan(1/2*d*x + 1/2*c)^6 + 2443980*a^11*
b^8*tan(1/2*d*x + 1/2*c)^6 + 593824*a^9*b^10*tan(1/2*d*x + 1/2*c)^6 + 148848*a^7*b^12*tan(1/2*d*x + 1/2*c)^6 -
336448*a^5*b^14*tan(1/2*d*x + 1/2*c)^6 + 27776*a^3*b^16*tan(1/2*d*x + 1/2*c)^6 + 53760*a*b^18*tan(1/2*d*x + 1
/2*c)^6 + 231000*a^18*b*tan(1/2*d*x + 1/2*c)^5 + 643230*a^16*b^3*tan(1/2*d*x + 1/2*c)^5 - 2226175*a^14*b^5*tan
(1/2*d*x + 1/2*c)^5 + 749980*a^12*b^7*tan(1/2*d*x + 1/2*c)^5 + 1482936*a^10*b^9*tan(1/2*d*x + 1/2*c)^5 - 72128
*a^8*b^11*tan(1/2*d*x + 1/2*c)^5 - 87472*a^6*b^13*tan(1/2*d*x + 1/2*c)^5 - 126336*a^4*b^15*tan(1/2*d*x + 1/2*c
)^5 + 80640*a^2*b^17*tan(1/2*d*x + 1/2*c)^5 + 25200*a^19*tan(1/2*d*x + 1/2*c)^4 + 461160*a^17*b^2*tan(1/2*d*x
+ 1/2*c)^4 - 667674*a^15*b^4*tan(1/2*d*x + 1/2*c)^4 - 857003*a^13*b^6*tan(1/2*d*x + 1/2*c)^4 + 1686188*a^11*b^
8*tan(1/2*d*x + 1/2*c)^4 - 290976*a^9*b^10*tan(1/2*d*x + 1/2*c)^4 + 118160*a^7*b^12*tan(1/2*d*x + 1/2*c)^4 - 1
66880*a^5*b^14*tan(1/2*d*x + 1/2*c)^4 + 67200*a^3*b^16*tan(1/2*d*x + 1/2*c)^4 + 114240*a^18*b*tan(1/2*d*x + 1/
2*c)^3 + 89880*a^16*b^3*tan(1/2*d*x + 1/2*c)^3 - 881776*a^14*b^5*tan(1/2*d*x + 1/2*c)^3 + 996478*a^12*b^7*tan(
1/2*d*x + 1/2*c)^3 - 212688*a^10*b^9*tan(1/2*d*x + 1/2*c)^3 + 108976*a^8*b^11*tan(1/2*d*x + 1/2*c)^3 - 98560*a
^6*b^13*tan(1/2*d*x + 1/2*c)^3 + 33600*a^4*b^15*tan(1/2*d*x + 1/2*c)^3 + 10080*a^19*tan(1/2*d*x + 1/2*c)^2 + 1
01920*a^17*b^2*tan(1/2*d*x + 1/2*c)^2 - 344568*a^15*b^4*tan(1/2*d*x + 1/2*c)^2 + 331128*a^13*b^6*tan(1/2*d*x +
1/2*c)^2 - 79226*a^11*b^8*tan(1/2*d*x + 1/2*c)^2 + 44800*a^9*b^10*tan(1/2*d*x + 1/2*c)^2 - 33264*a^7*b^12*tan
(1/2*d*x + 1/2*c)^2 + 10080*a^5*b^14*tan(1/2*d*x + 1/2*c)^2 + 22680*a^18*b*tan(1/2*d*x + 1/2*c) - 64330*a^16*b
^3*tan(1/2*d*x + 1/2*c) + 58569*a^14*b^5*tan(1/2*d*x + 1/2*c) - 14322*a^12*b^7*tan(1/2*d*x + 1/2*c) + 8372*a^1
0*b^9*tan(1/2*d*x + 1/2*c) - 5824*a^8*b^11*tan(1/2*d*x + 1/2*c) + 1680*a^6*b^13*tan(1/2*d*x + 1/2*c) + 1680*a^
19 - 4760*a^17*b^2 + 4326*a^15*b^4 - 1143*a^13*b^6 + 958*a^11*b^8 - 776*a^9*b^10 + 240*a^7*b^12)/((a^13*b^7 -
3*a^11*b^9 + 3*a^9*b^11 - a^7*b^13)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^7) - 840*(d*x +
c)/b^8)/d