3.469 \(\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

Optimal. Leaf size=411 \[ \frac{3 a \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{11/2}}-\frac{\left (-40 a^4 b^2-247 a^2 b^4+4 a^6-32 b^6\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))}-\frac{a \left (-36 a^2 b^2+4 a^4-73 b^4\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^2}-\frac{\left (-15 a^2 b^2+2 a^4-8 b^4\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^3}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

[Out]

(3*a*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(11/2)*d) - Cos[c + d*x]^3
/(7*b*d*(a + b*Sin[c + d*x])^7) - ((a^2 - 3*b^2)*Cos[c + d*x])/(140*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5)
- (a*(2*a^2 - 11*b^2)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4) - ((2*a^4 - 15*a^2*b^2 -
8*b^4)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^3) - (a*(4*a^4 - 36*a^2*b^2 - 73*b^4)*Cos[c
+ d*x])/(560*b^3*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^2) - ((4*a^6 - 40*a^4*b^2 - 247*a^2*b^4 - 32*b^6)*Cos[c
+ d*x])/(560*b^3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*(a + 3*b*Sin[c + d*x]))/(28*b^3*d*(a +
b*Sin[c + d*x])^6)

________________________________________________________________________________________

Rubi [A]  time = 0.790644, antiderivative size = 411, 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, 2863, 2754, 12, 2660, 618, 204} \[ \frac{3 a \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{11/2}}-\frac{\left (-40 a^4 b^2-247 a^2 b^4+4 a^6-32 b^6\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))}-\frac{a \left (-36 a^2 b^2+4 a^4-73 b^4\right ) \cos (c+d x)}{560 b^3 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^2}-\frac{\left (-15 a^2 b^2+2 a^4-8 b^4\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^3}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^4}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^5}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]

[Out]

(3*a*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(11/2)*d) - Cos[c + d*x]^3
/(7*b*d*(a + b*Sin[c + d*x])^7) - ((a^2 - 3*b^2)*Cos[c + d*x])/(140*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5)
- (a*(2*a^2 - 11*b^2)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4) - ((2*a^4 - 15*a^2*b^2 -
8*b^4)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^3) - (a*(4*a^4 - 36*a^2*b^2 - 73*b^4)*Cos[c
+ d*x])/(560*b^3*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^2) - ((4*a^6 - 40*a^4*b^2 - 247*a^2*b^4 - 32*b^6)*Cos[c
+ d*x])/(560*b^3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*(a + 3*b*Sin[c + d*x]))/(28*b^3*d*(a +
b*Sin[c + d*x])^6)

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 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 2754

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{3 \int \frac{\cos ^2(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^7} \, dx}{7 b}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\int \frac{-6 b-2 a \sin (c+d x)}{(a+b \sin (c+d x))^6} \, dx}{56 b^3}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac{\int \frac{20 a b+8 \left (a^2-3 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5} \, dx}{280 b^3 \left (a^2-b^2\right )}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\int \frac{-48 b \left (a^2+2 b^2\right )-12 a \left (2 a^2-11 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4} \, dx}{1120 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac{\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac{\int \frac{36 a b \left (2 a^2+19 b^2\right )+24 \left (a^2-8 b^2\right ) \left (2 a^2+b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{3360 b^3 \left (a^2-b^2\right )^3}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac{\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac{a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\int \frac{-24 b \left (2 a^4+87 a^2 b^2+16 b^4\right )-12 a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{6720 b^3 \left (a^2-b^2\right )^4}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac{\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac{a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac{\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac{\int \frac{1260 a b^3 \left (2 a^2+b^2\right )}{a+b \sin (c+d x)} \, dx}{6720 b^3 \left (a^2-b^2\right )^5}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac{\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac{a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac{\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac{\left (3 a \left (2 a^2+b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^5}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac{\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac{a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac{\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}+\frac{\left (3 a \left (2 a^2+b^2\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 \left (a^2-b^2\right )^5 d}\\ &=-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac{\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac{a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac{\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\left (3 a \left (2 a^2+b^2\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 \left (a^2-b^2\right )^5 d}\\ &=\frac{3 a \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{11/2} d}-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac{\left (a^2-3 b^2\right ) \cos (c+d x)}{140 b^3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^5}-\frac{a \left (2 a^2-11 b^2\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^4}-\frac{\left (2 a^4-15 a^2 b^2-8 b^4\right ) \cos (c+d x)}{280 b^3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^3}-\frac{a \left (4 a^4-36 a^2 b^2-73 b^4\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^2}-\frac{\left (4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right ) \cos (c+d x)}{560 b^3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}\\ \end{align*}

Mathematica [B]  time = 6.07593, size = 1167, normalized size = 2.84 \[ \frac{\cos ^5(c+d x)}{5 (a-b) d (a+b \sin (c+d x))^7}+\frac{a \left (-\frac{b (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{7 (b-a) (a+b) (a+b \sin (c+d x))^7}-\frac{-\frac{(a b+(7 a-b) b) (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{6 (b-a) (a+b) (a+b \sin (c+d x))^6}-\frac{7 \left (6 a^2-2 b a+b^2\right ) \left (-\frac{(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{7/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac{3 \left (-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (b-a) (a+b \sin (c+d x))^4}-\frac{\frac{5 \left (\frac{3 \left (\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b-a} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right )}{(-a-b)^{3/2} \sqrt{b-a}}\right )}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right )}{3 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}}{4 (b-a)}\right )}{5 (b-a)}\right )}{6 (b-a) (a+b)}}{7 (b-a) (a+b)}\right ) \cos (c+d x)}{(a-b) d \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}+\frac{2 b \left (\frac{\cos ^7(c+d x)}{7 (a-b) d (a+b \sin (c+d x))^7}+\frac{a \left (-\frac{(1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{9/2}}{7 (b-a) (a+b \sin (c+d x))^7}-\frac{5 \left (-\frac{(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{9/2}}{6 (b-a) (a+b \sin (c+d x))^6}-\frac{-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{9/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac{\frac{7 \left (\frac{5 \left (\frac{3 \left (\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b-a} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right )}{(-a-b)^{3/2} \sqrt{b-a}}\right )}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right )}{3 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}\right )}{4 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (a+b) (a+b \sin (c+d x))^4}}{5 (b-a)}}{2 (b-a)}\right )}{7 (b-a)}\right ) \cos (c+d x)}{(a-b) d \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}\right )}{5 (a-b)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]

[Out]

Cos[c + d*x]^5/(5*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-(b*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c
+ d*x])^(7/2))/(7*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-((a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(5/2)*
(1 + Sin[c + d*x])^(7/2))/(6*(-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (7*(6*a^2 - 2*a*b + b^2)*(-((1 - Sin[c
+ d*x])^(3/2)*(1 + Sin[c + d*x])^(7/2))/(5*(-a + b)*(a + b*Sin[c + d*x])^5) - (3*(-(Sqrt[1 - Sin[c + d*x]]*(1
+ Sin[c + d*x])^(7/2))/(4*(-a + b)*(a + b*Sin[c + d*x])^4) - (-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/
2))/(3*(a + b)*(a + b*Sin[c + d*x])^3) + (5*(-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/(2*(a + b)*(a
+ b*Sin[c + d*x])^2) + (3*((-2*ArcTan[(Sqrt[-a + b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x
]])])/((-a - b)^(3/2)*Sqrt[-a + b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Sin[c +
d*x]))))/(2*(a + b))))/(3*(a + b)))/(4*(-a + b))))/(5*(-a + b))))/(6*(-a + b)*(a + b)))/(7*(-a + b)*(a + b)))
)/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]) + (2*b*(Cos[c + d*x]^7/(7*(a - b)*d*(a + b*Sin[c +
d*x])^7) + (a*Cos[c + d*x]*(-((1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(9/2))/(7*(-a + b)*(a + b*Sin[c + d
*x])^7) - (5*(-((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(9/2))/(6*(-a + b)*(a + b*Sin[c + d*x])^6) - (-(Sq
rt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(9/2))/(5*(-a + b)*(a + b*Sin[c + d*x])^5) - (-(Sqrt[1 - Sin[c + d*x]]
*(1 + Sin[c + d*x])^(7/2))/(4*(a + b)*(a + b*Sin[c + d*x])^4) + (7*(-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x]
)^(5/2))/(3*(a + b)*(a + b*Sin[c + d*x])^3) + (5*(-(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/(2*(a + b
)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTan[(Sqrt[-a + b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c
+ d*x]])])/((-a - b)^(3/2)*Sqrt[-a + b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Si
n[c + d*x]))))/(2*(a + b))))/(3*(a + b))))/(4*(a + b)))/(5*(-a + b)))/(2*(-a + b))))/(7*(-a + b))))/((a - b)*d
*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])))/(5*(a - b))

________________________________________________________________________________________

Maple [B]  time = 0.208, size = 9171, normalized size = 22.3 \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)^4/(a+b*sin(d*x+c))^8,x)

[Out]

result too large to display

________________________________________________________________________________________

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)^4/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 6.72923, size = 6137, normalized size = 14.93 \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)^4/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

[-1/1120*(2*(4*a^8*b^3 - 44*a^6*b^5 - 207*a^4*b^7 + 215*a^2*b^9 + 32*b^11)*cos(d*x + c)^7 - 28*(6*a^10*b - 65*
a^8*b^3 - 224*a^6*b^5 + 222*a^4*b^7 + 53*a^2*b^9 + 8*b^11)*cos(d*x + c)^5 - 70*(14*a^10*b + 173*a^8*b^3 - 3*a^
6*b^5 - 137*a^4*b^7 - 47*a^2*b^9)*cos(d*x + c)^3 + 105*(2*a^10 + 43*a^8*b^2 + 91*a^6*b^4 + 49*a^4*b^6 + 7*a^2*
b^8 - 7*(2*a^4*b^6 + a^2*b^8)*cos(d*x + c)^6 + 7*(10*a^6*b^4 + 11*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^4 - 7*(6*a
^8*b^2 + 23*a^6*b^4 + 16*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^2 + (14*a^9*b + 77*a^7*b^3 + 77*a^5*b^5 + 23*a^3*b^
7 + a*b^9 - (2*a^3*b^7 + a*b^9)*cos(d*x + c)^6 + 3*(14*a^5*b^5 + 9*a^3*b^7 + a*b^9)*cos(d*x + c)^4 - (70*a^7*b
^3 + 119*a^5*b^5 + 48*a^3*b^7 + 3*a*b^9)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*co
s(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*(6*a^10*b + 17*a^8*b^3 - 7*a^6*b^5 - 13*a^4*
b^7 - 3*a^2*b^9)*cos(d*x + c) - 14*((4*a^9*b^2 - 44*a^7*b^4 - 177*a^5*b^6 + 200*a^3*b^8 + 17*a*b^10)*cos(d*x +
c)^5 - 10*(2*a^11 - 21*a^9*b^2 - 61*a^7*b^4 + 37*a^5*b^6 + 39*a^3*b^8 + 4*a*b^10)*cos(d*x + c)^3 - 15*(2*a^11
+ 29*a^9*b^2 + 14*a^7*b^4 - 28*a^5*b^6 - 16*a^3*b^8 - a*b^10)*cos(d*x + c))*sin(d*x + c))/(7*(a^13*b^6 - 6*a^
11*b^8 + 15*a^9*b^10 - 20*a^7*b^12 + 15*a^5*b^14 - 6*a^3*b^16 + a*b^18)*d*cos(d*x + c)^6 - 7*(5*a^15*b^4 - 27*
a^13*b^6 + 57*a^11*b^8 - 55*a^9*b^10 + 15*a^7*b^12 + 15*a^5*b^14 - 13*a^3*b^16 + 3*a*b^18)*d*cos(d*x + c)^4 +
7*(3*a^17*b^2 - 8*a^15*b^4 - 12*a^13*b^6 + 72*a^11*b^8 - 110*a^9*b^10 + 72*a^7*b^12 - 12*a^5*b^14 - 8*a^3*b^16
+ 3*a*b^18)*d*cos(d*x + c)^2 - (a^19 + 15*a^17*b^2 - 76*a^15*b^4 + 92*a^13*b^6 + 78*a^11*b^8 - 286*a^9*b^10 +
260*a^7*b^12 - 84*a^5*b^14 - 7*a^3*b^16 + 7*a*b^18)*d + ((a^12*b^7 - 6*a^10*b^9 + 15*a^8*b^11 - 20*a^6*b^13 +
15*a^4*b^15 - 6*a^2*b^17 + b^19)*d*cos(d*x + c)^6 - 3*(7*a^14*b^5 - 41*a^12*b^7 + 99*a^10*b^9 - 125*a^8*b^11
+ 85*a^6*b^13 - 27*a^4*b^15 + a^2*b^17 + b^19)*d*cos(d*x + c)^4 + (35*a^16*b^3 - 168*a^14*b^5 + 276*a^12*b^7 -
88*a^10*b^9 - 270*a^8*b^11 + 360*a^6*b^13 - 172*a^4*b^15 + 24*a^2*b^17 + 3*b^19)*d*cos(d*x + c)^2 - (7*a^18*b
- 7*a^16*b^3 - 84*a^14*b^5 + 260*a^12*b^7 - 286*a^10*b^9 + 78*a^8*b^11 + 92*a^6*b^13 - 76*a^4*b^15 + 15*a^2*b
^17 + b^19)*d)*sin(d*x + c)), -1/560*((4*a^8*b^3 - 44*a^6*b^5 - 207*a^4*b^7 + 215*a^2*b^9 + 32*b^11)*cos(d*x +
c)^7 - 14*(6*a^10*b - 65*a^8*b^3 - 224*a^6*b^5 + 222*a^4*b^7 + 53*a^2*b^9 + 8*b^11)*cos(d*x + c)^5 - 35*(14*a
^10*b + 173*a^8*b^3 - 3*a^6*b^5 - 137*a^4*b^7 - 47*a^2*b^9)*cos(d*x + c)^3 - 105*(2*a^10 + 43*a^8*b^2 + 91*a^6
*b^4 + 49*a^4*b^6 + 7*a^2*b^8 - 7*(2*a^4*b^6 + a^2*b^8)*cos(d*x + c)^6 + 7*(10*a^6*b^4 + 11*a^4*b^6 + 3*a^2*b^
8)*cos(d*x + c)^4 - 7*(6*a^8*b^2 + 23*a^6*b^4 + 16*a^4*b^6 + 3*a^2*b^8)*cos(d*x + c)^2 + (14*a^9*b + 77*a^7*b^
3 + 77*a^5*b^5 + 23*a^3*b^7 + a*b^9 - (2*a^3*b^7 + a*b^9)*cos(d*x + c)^6 + 3*(14*a^5*b^5 + 9*a^3*b^7 + a*b^9)*
cos(d*x + c)^4 - (70*a^7*b^3 + 119*a^5*b^5 + 48*a^3*b^7 + 3*a*b^9)*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*(6*a^10*b + 17*a^8*b^3 - 7*a^6*b^5 - 13*
a^4*b^7 - 3*a^2*b^9)*cos(d*x + c) - 7*((4*a^9*b^2 - 44*a^7*b^4 - 177*a^5*b^6 + 200*a^3*b^8 + 17*a*b^10)*cos(d*
x + c)^5 - 10*(2*a^11 - 21*a^9*b^2 - 61*a^7*b^4 + 37*a^5*b^6 + 39*a^3*b^8 + 4*a*b^10)*cos(d*x + c)^3 - 15*(2*a
^11 + 29*a^9*b^2 + 14*a^7*b^4 - 28*a^5*b^6 - 16*a^3*b^8 - a*b^10)*cos(d*x + c))*sin(d*x + c))/(7*(a^13*b^6 - 6
*a^11*b^8 + 15*a^9*b^10 - 20*a^7*b^12 + 15*a^5*b^14 - 6*a^3*b^16 + a*b^18)*d*cos(d*x + c)^6 - 7*(5*a^15*b^4 -
27*a^13*b^6 + 57*a^11*b^8 - 55*a^9*b^10 + 15*a^7*b^12 + 15*a^5*b^14 - 13*a^3*b^16 + 3*a*b^18)*d*cos(d*x + c)^4
+ 7*(3*a^17*b^2 - 8*a^15*b^4 - 12*a^13*b^6 + 72*a^11*b^8 - 110*a^9*b^10 + 72*a^7*b^12 - 12*a^5*b^14 - 8*a^3*b
^16 + 3*a*b^18)*d*cos(d*x + c)^2 - (a^19 + 15*a^17*b^2 - 76*a^15*b^4 + 92*a^13*b^6 + 78*a^11*b^8 - 286*a^9*b^1
0 + 260*a^7*b^12 - 84*a^5*b^14 - 7*a^3*b^16 + 7*a*b^18)*d + ((a^12*b^7 - 6*a^10*b^9 + 15*a^8*b^11 - 20*a^6*b^1
3 + 15*a^4*b^15 - 6*a^2*b^17 + b^19)*d*cos(d*x + c)^6 - 3*(7*a^14*b^5 - 41*a^12*b^7 + 99*a^10*b^9 - 125*a^8*b^
11 + 85*a^6*b^13 - 27*a^4*b^15 + a^2*b^17 + b^19)*d*cos(d*x + c)^4 + (35*a^16*b^3 - 168*a^14*b^5 + 276*a^12*b^
7 - 88*a^10*b^9 - 270*a^8*b^11 + 360*a^6*b^13 - 172*a^4*b^15 + 24*a^2*b^17 + 3*b^19)*d*cos(d*x + c)^2 - (7*a^1
8*b - 7*a^16*b^3 - 84*a^14*b^5 + 260*a^12*b^7 - 286*a^10*b^9 + 78*a^8*b^11 + 92*a^6*b^13 - 76*a^4*b^15 + 15*a^
2*b^17 + b^19)*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)**4/(a+b*sin(d*x+c))**8,x)

[Out]

Timed out

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Giac [B]  time = 1.54474, size = 2608, normalized size = 6.35 \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)^4/(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/280*(105*(2*a^3 + a*b^2)*(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^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*sqrt(a^2 - b^2)) - (350*a^16*ta
n(1/2*d*x + 1/2*c)^13 - 2905*a^14*b^2*tan(1/2*d*x + 1/2*c)^13 + 5600*a^12*b^4*tan(1/2*d*x + 1/2*c)^13 - 5600*a
^10*b^6*tan(1/2*d*x + 1/2*c)^13 + 2800*a^8*b^8*tan(1/2*d*x + 1/2*c)^13 - 560*a^6*b^10*tan(1/2*d*x + 1/2*c)^13
+ 630*a^15*b*tan(1/2*d*x + 1/2*c)^12 - 18165*a^13*b^3*tan(1/2*d*x + 1/2*c)^12 + 33600*a^11*b^5*tan(1/2*d*x + 1
/2*c)^12 - 33600*a^9*b^7*tan(1/2*d*x + 1/2*c)^12 + 16800*a^7*b^9*tan(1/2*d*x + 1/2*c)^12 - 3360*a^5*b^11*tan(1
/2*d*x + 1/2*c)^12 + 840*a^16*tan(1/2*d*x + 1/2*c)^11 - 15680*a^14*b^2*tan(1/2*d*x + 1/2*c)^11 - 41090*a^12*b^
4*tan(1/2*d*x + 1/2*c)^11 + 89600*a^10*b^6*tan(1/2*d*x + 1/2*c)^11 - 100800*a^8*b^8*tan(1/2*d*x + 1/2*c)^11 +
53760*a^6*b^10*tan(1/2*d*x + 1/2*c)^11 - 11200*a^4*b^12*tan(1/2*d*x + 1/2*c)^11 - 840*a^15*b*tan(1/2*d*x + 1/2
*c)^10 - 102760*a^13*b^3*tan(1/2*d*x + 1/2*c)^10 + 11270*a^11*b^5*tan(1/2*d*x + 1/2*c)^10 + 78400*a^9*b^7*tan(
1/2*d*x + 1/2*c)^10 - 151200*a^7*b^9*tan(1/2*d*x + 1/2*c)^10 + 97440*a^5*b^11*tan(1/2*d*x + 1/2*c)^10 - 22400*
a^3*b^13*tan(1/2*d*x + 1/2*c)^10 + 630*a^16*tan(1/2*d*x + 1/2*c)^9 - 51905*a^14*b^2*tan(1/2*d*x + 1/2*c)^9 - 2
49410*a^12*b^4*tan(1/2*d*x + 1/2*c)^9 + 202244*a^10*b^6*tan(1/2*d*x + 1/2*c)^9 - 129360*a^8*b^8*tan(1/2*d*x +
1/2*c)^9 - 62832*a^6*b^10*tan(1/2*d*x + 1/2*c)^9 + 92288*a^4*b^12*tan(1/2*d*x + 1/2*c)^9 - 26880*a^2*b^14*tan(
1/2*d*x + 1/2*c)^9 - 8330*a^15*b*tan(1/2*d*x + 1/2*c)^8 - 248745*a^13*b^3*tan(1/2*d*x + 1/2*c)^8 - 190610*a^11
*b^5*tan(1/2*d*x + 1/2*c)^8 + 253736*a^9*b^7*tan(1/2*d*x + 1/2*c)^8 - 338240*a^7*b^9*tan(1/2*d*x + 1/2*c)^8 +
120512*a^5*b^11*tan(1/2*d*x + 1/2*c)^8 + 24192*a^3*b^13*tan(1/2*d*x + 1/2*c)^8 - 17920*a*b^15*tan(1/2*d*x + 1/
2*c)^8 - 96040*a^14*b^2*tan(1/2*d*x + 1/2*c)^7 - 452340*a^12*b^4*tan(1/2*d*x + 1/2*c)^7 + 164528*a^10*b^6*tan(
1/2*d*x + 1/2*c)^7 - 99344*a^8*b^8*tan(1/2*d*x + 1/2*c)^7 - 177664*a^6*b^10*tan(1/2*d*x + 1/2*c)^7 + 153088*a^
4*b^12*tan(1/2*d*x + 1/2*c)^7 - 27648*a^2*b^14*tan(1/2*d*x + 1/2*c)^7 - 5120*b^16*tan(1/2*d*x + 1/2*c)^7 - 156
80*a^15*b*tan(1/2*d*x + 1/2*c)^6 - 296520*a^13*b^3*tan(1/2*d*x + 1/2*c)^6 - 247940*a^11*b^5*tan(1/2*d*x + 1/2*
c)^6 + 232736*a^9*b^7*tan(1/2*d*x + 1/2*c)^6 - 339920*a^7*b^9*tan(1/2*d*x + 1/2*c)^6 + 120512*a^5*b^11*tan(1/2
*d*x + 1/2*c)^6 + 24192*a^3*b^13*tan(1/2*d*x + 1/2*c)^6 - 17920*a*b^15*tan(1/2*d*x + 1/2*c)^6 - 630*a^16*tan(1
/2*d*x + 1/2*c)^5 - 92155*a^14*b^2*tan(1/2*d*x + 1/2*c)^5 - 333060*a^12*b^4*tan(1/2*d*x + 1/2*c)^5 + 151144*a^
10*b^6*tan(1/2*d*x + 1/2*c)^5 - 133280*a^8*b^8*tan(1/2*d*x + 1/2*c)^5 - 62832*a^6*b^10*tan(1/2*d*x + 1/2*c)^5
+ 92288*a^4*b^12*tan(1/2*d*x + 1/2*c)^5 - 26880*a^2*b^14*tan(1/2*d*x + 1/2*c)^5 - 13566*a^15*b*tan(1/2*d*x + 1
/2*c)^4 - 166775*a^13*b^3*tan(1/2*d*x + 1/2*c)^4 - 41412*a^11*b^5*tan(1/2*d*x + 1/2*c)^4 + 72128*a^9*b^7*tan(1
/2*d*x + 1/2*c)^4 - 150640*a^7*b^9*tan(1/2*d*x + 1/2*c)^4 + 97440*a^5*b^11*tan(1/2*d*x + 1/2*c)^4 - 22400*a^3*
b^13*tan(1/2*d*x + 1/2*c)^4 - 840*a^16*tan(1/2*d*x + 1/2*c)^3 - 41944*a^14*b^2*tan(1/2*d*x + 1/2*c)^3 - 76650*
a^12*b^4*tan(1/2*d*x + 1/2*c)^3 + 87472*a^10*b^6*tan(1/2*d*x + 1/2*c)^3 - 100688*a^8*b^8*tan(1/2*d*x + 1/2*c)^
3 + 53760*a^6*b^10*tan(1/2*d*x + 1/2*c)^3 - 11200*a^4*b^12*tan(1/2*d*x + 1/2*c)^3 - 5432*a^15*b*tan(1/2*d*x +
1/2*c)^2 - 33264*a^13*b^3*tan(1/2*d*x + 1/2*c)^2 + 34846*a^11*b^5*tan(1/2*d*x + 1/2*c)^2 - 34272*a^9*b^7*tan(1
/2*d*x + 1/2*c)^2 + 16912*a^7*b^9*tan(1/2*d*x + 1/2*c)^2 - 3360*a^5*b^11*tan(1/2*d*x + 1/2*c)^2 - 350*a^16*tan
(1/2*d*x + 1/2*c) - 6699*a^14*b^2*tan(1/2*d*x + 1/2*c) + 6790*a^12*b^4*tan(1/2*d*x + 1/2*c) - 6188*a^10*b^6*ta
n(1/2*d*x + 1/2*c) + 2912*a^8*b^8*tan(1/2*d*x + 1/2*c) - 560*a^6*b^10*tan(1/2*d*x + 1/2*c) - 686*a^15*b + 885*
a^13*b^3 - 842*a^11*b^5 + 408*a^9*b^7 - 80*a^7*b^9)/((a^17 - 5*a^15*b^2 + 10*a^13*b^4 - 10*a^11*b^6 + 5*a^9*b^
8 - a^7*b^10)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^7))/d