3.5.0 ∫ cos6 (c+dx) ____ (a+b sin(c+dx))8 dx [468]

Integrand size = 21, antiderivative size = 407 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {5 a \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{9/2} d}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5} \]

5/8*a*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^(9/2)/d-1/7*cos(d*x+c)^5/b/d/(a+b*sin(d*x+c))

^7+1/168*a*(4*a^2-b^2)*cos(d*x+c)/b^5/(a^2-b^2)/d/(a+b*sin(d*x+c))^4+1/168*(4*a^4-9*a^2*b^2+12*b^4)*cos(d*x+c)

/b^5/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^3+1/336*a*(8*a^4-30*a^2*b^2+57*b^4)*cos(d*x+c)/b^5/(a^2-b^2)^3/d/(a+b*sin(

d*x+c))^2+1/336*(8*a^6-38*a^4*b^2+87*a^2*b^4+48*b^6)*cos(d*x+c)/b^5/(a^2-b^2)^4/d/(a+b*sin(d*x+c))+5/42*cos(d*

x+c)^3*(2*a+3*b*sin(d*x+c))/b^3/d/(a+b*sin(d*x+c))^6-1/42*cos(d*x+c)*(4*a^2+9*b^2+10*a*b*sin(d*x+c))/b^5/d/(a+

Rubi [A] (veriﬁed)

Time = 0.97 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2772, 2942, 2833, 12, 2739, 632, 210} \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {5 a \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{9/2}}-\frac {\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{42 b^5 d (a+b \sin (c+d x))^5}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^3}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^8,x]

(5*a*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(9/2)*d) - Cos[c + d*x]^5/(7*b*d*(a + b*

Sin[c + d*x])^7) + (a*(4*a^2 - b^2)*Cos[c + d*x])/(168*b^5*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^4) + ((4*a^4 - 9

*a^2*b^2 + 12*b^4)*Cos[c + d*x])/(168*b^5*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^3) + (a*(8*a^4 - 30*a^2*b^2 + 5

7*b^4)*Cos[c + d*x])/(336*b^5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + ((8*a^6 - 38*a^4*b^2 + 87*a^2*b^4 + 48

*b^6)*Cos[c + d*x])/(336*b^5*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (5*Cos[c + d*x]^3*(2*a + 3*b*Sin[c + d*x]

))/(42*b^3*d*(a + b*Sin[c + d*x])^6) - (Cos[c + d*x]*(4*a^2 + 9*b^2 + 10*a*b*Sin[c + d*x]))/(42*b^5*d*(a + b*S

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

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])

], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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]

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] &&

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C

os[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]

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]

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {5 \int \frac {\cos ^4(c+d x) \sin (c+d x)}{(a+b \sin (c+d x))^7} \, dx}{7 b} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {5 \int \frac {\cos ^2(c+d x) (-6 b-4 a \sin (c+d x))}{(a+b \sin (c+d x))^6} \, dx}{28 b^3} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}-\frac {\int \frac {20 a b+2 \left (4 a^2+9 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5} \, dx}{84 b^5} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}+\frac {\int \frac {-24 b \left (2 a^2-3 b^2\right )-6 a \left (4 a^2-b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4} \, dx}{336 b^5 \left (a^2-b^2\right )} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}-\frac {\int \frac {18 a b \left (4 a^2-11 b^2\right )+12 \left (4 a^4-9 a^2 b^2+12 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{1008 b^5 \left (a^2-b^2\right )^2} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}+\frac {\int \frac {-12 b \left (4 a^4-15 a^2 b^2-24 b^4\right )-6 a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2016 b^5 \left (a^2-b^2\right )^3} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}-\frac {\int -\frac {630 a b^5}{a+b \sin (c+d x)} \, dx}{2016 b^5 \left (a^2-b^2\right )^4} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}+\frac {(5 a) \int \frac {1}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^4} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}+\frac {(5 a) \text {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 )^4 d} \\ & = -\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5}-\frac {(5 a) \text {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 )^4 d} \\ & = \frac {5 a \arctan \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 )^{9/2} d}-\frac {\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^4}+\frac {\left (4 a^4-9 a^2 b^2+12 b^4\right ) \cos (c+d x)}{168 b^5 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^3}+\frac {a \left (8 a^4-30 a^2 b^2+57 b^4\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}+\frac {\left (8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right ) \cos (c+d x)}{336 b^5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac {\cos (c+d x) \left (4 a^2+9 b^2+10 a b \sin (c+d x)\right )}{42 b^5 d (a+b \sin (c+d x))^5} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 3.76 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\cos (c+d x) \left (\frac {6 \cos ^6(c+d x)}{(a+b \sin (c+d x))^7}+\frac {6 a (-1+\sin (c+d x))^2 (1+\sin (c+d x))^4}{(a-b) (a+b \sin (c+d x))^7}-\frac {5 a (-1+\sin (c+d x)) (1+\sin (c+d x))^4}{(a-b)^2 (a+b \sin (c+d x))^6}+\frac {3 a (1+\sin (c+d x))^4}{(a-b)^3 (a+b \sin (c+d x))^5}+\frac {3 a (1+\sin (c+d x))^3}{4 (-a+b)^3 (a+b) (a+b \sin (c+d x))^4}+\frac {7 a (1+\sin (c+d x))^2}{4 (-a+b)^3 (a+b)^2 (a+b \sin (c+d x))^3}+\frac {35 a (1+\sin (c+d x))}{8 (-a+b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\frac {105}{8} a \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {a-b} \sqrt {1-\sin (c+d x)}}{\sqrt {-a-b} \sqrt {1+\sin (c+d x)}}\right )}{(-a-b)^{9/2} (a-b)^{7/2} \sqrt {\cos ^2(c+d x)}}-\frac {1}{(a-b)^3 (a+b)^4 (a+b \sin (c+d x))}\right )\right )}{42 (a-b) d} \]

Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^8,x]

(Cos[c + d*x]*((6*Cos[c + d*x]^6)/(a + b*Sin[c + d*x])^7 + (6*a*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^4)/((

a - b)*(a + b*Sin[c + d*x])^7) - (5*a*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^4)/((a - b)^2*(a + b*Sin[c + d*x]

)^6) + (3*a*(1 + Sin[c + d*x])^4)/((a - b)^3*(a + b*Sin[c + d*x])^5) + (3*a*(1 + Sin[c + d*x])^3)/(4*(-a + b)^

3*(a + b)*(a + b*Sin[c + d*x])^4) + (7*a*(1 + Sin[c + d*x])^2)/(4*(-a + b)^3*(a + b)^2*(a + b*Sin[c + d*x])^3)

+ (35*a*(1 + Sin[c + d*x]))/(8*(-a + b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2) + (105*a*((2*ArcTanh[(Sqrt[a - b]

*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(9/2)*(a - b)^(7/2)*Sqrt[Cos[c + d*

x]^2]) - 1/((a - b)^3*(a + b)^4*(a + b*Sin[c + d*x]))))/8))/(42*(a - b)*d)

Maple [C] (warning: unable to verify)

Time = 12.77 (sec) , antiderivative size = 1388, normalized size of antiderivative = 3.41


method	result	size

risch	\(\text {Expression too large to display}\)	\(1388\)
derivativedivides	\(\text {Expression too large to display}\)	\(1389\)
default	\(\text {Expression too large to display}\)	\(1389\)

int(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

-1/168*I*(80192*I*a^6*b^7*exp(6*I*(d*x+c))+1344*I*b^7*a^6*exp(12*I*(d*x+c))-2016*I*b^9*a^4*exp(12*I*(d*x+c))-7

9128*a^5*b^8*exp(5*I*(d*x+c))-44660*a^3*b^10*exp(5*I*(d*x+c))-5635*a*b^12*exp(5*I*(d*x+c))-2688*a^11*b^2*exp(5

*I*(d*x+c))+8288*a^9*b^4*exp(5*I*(d*x+c))-11312*a^7*b^6*exp(5*I*(d*x+c))-19488*I*a^6*b^7*exp(4*I*(d*x+c))+336*

I*a^8*b^5*exp(2*I*(d*x+c))-1792*I*a^6*b^7*exp(2*I*(d*x+c))+4620*I*a^2*b^11*exp(2*I*(d*x+c))+1792*I*b^3*a^10*ex

p(8*I*(d*x+c))-48*I*b^13+512*a^13*exp(7*I*(d*x+c))+1820*b^12*a*exp(11*I*(d*x+c))-8960*I*b^5*a^8*exp(10*I*(d*x+

c))+13440*I*b^7*a^6*exp(10*I*(d*x+c))+13370*I*b^9*a^4*exp(10*I*(d*x+c))+9940*I*b^11*a^2*exp(10*I*(d*x+c))+8218

0*I*a^4*b^9*exp(6*I*(d*x+c))+26880*I*a^2*b^11*exp(6*I*(d*x+c))+8960*I*a^8*b^5*exp(4*I*(d*x+c))-2688*b^2*a^11*e

xp(9*I*(d*x+c))+2944*b^2*a^11*exp(7*I*(d*x+c))+52500*a^3*b^10*exp(7*I*(d*x+c))+6720*a*b^12*exp(7*I*(d*x+c))-11

2*exp(I*(d*x+c))*b^6*a^7+532*b^8*exp(I*(d*x+c))*a^5-1218*exp(I*(d*x+c))*b^10*a^3-567*exp(I*(d*x+c))*b^12*a+828

8*b^4*a^9*exp(9*I*(d*x+c))-6272*b^6*a^7*exp(9*I*(d*x+c))-42588*b^8*a^5*exp(9*I*(d*x+c))-27370*b^10*a^3*exp(9*I

*(d*x+c))-4445*b^12*a*exp(9*I*(d*x+c))-70210*I*b^9*a^4*exp(8*I*(d*x+c))-16485*I*b^11*a^2*exp(8*I*(d*x+c))-2240

*I*a^10*b^3*exp(4*I*(d*x+c))-12047*I*a^2*b^11*exp(4*I*(d*x+c))+5026*I*a^4*b^9*exp(2*I*(d*x+c))-55832*I*a^6*b^7

*exp(8*I*(d*x+c))+6720*b^8*a^5*exp(11*I*(d*x+c))+3010*b^10*a^3*exp(11*I*(d*x+c))-8*I*a^6*b^7+38*I*a^4*b^9-87*I

*a^2*b^11-1008*I*b^13*exp(4*I*(d*x+c))-1680*I*b^13*exp(8*I*(d*x+c))-336*I*b^13*exp(12*I*(d*x+c))-4480*b^6*a^7*

exp(11*I*(d*x+c))-105*b^12*a*exp(13*I*(d*x+c))+1120*b^4*a^9*exp(11*I*(d*x+c))+1120*a^9*b^4*exp(3*I*(d*x+c))-54

88*a^7*b^6*exp(3*I*(d*x+c))+14448*a^5*b^8*exp(3*I*(d*x+c))+17738*a^3*b^10*exp(3*I*(d*x+c))+2212*a*b^12*exp(3*I

*(d*x+c))-13248*a^9*b^4*exp(7*I*(d*x+c))+30736*a^7*b^6*exp(7*I*(d*x+c))+100016*a^5*b^8*exp(7*I*(d*x+c))-1792*I

*b*a^12*exp(8*I*(d*x+c))+1792*I*b*a^12*exp(6*I*(d*x+c))-1792*I*a^10*b^3*exp(6*I*(d*x+c))-9072*I*a^8*b^5*exp(6*

I*(d*x+c))-336*I*b^5*a^8*exp(12*I*(d*x+c))-21*I*b^11*a^2*exp(12*I*(d*x+c))+2240*I*b^3*a^10*exp(10*I*(d*x+c))-4

9252*I*a^4*b^9*exp(4*I*(d*x+c))+9072*I*a^8*b^5*exp(8*I*(d*x+c)))/(-I*b*exp(2*I*(d*x+c))+2*a*exp(I*(d*x+c))+I*b

)^7/(a^2-b^2)^4/d/b^6-5/16/(-a^2+b^2)^(1/2)*a/(a+b)^4/(a-b)^4/d*ln(exp(I*(d*x+c))+(I*a*(-a^2+b^2)^(1/2)-a^2+b^

2)/b/(-a^2+b^2)^(1/2))+5/16/(-a^2+b^2)^(1/2)*a/(a+b)^4/(a-b)^4/d*ln(exp(I*(d*x+c))+(I*a*(-a^2+b^2)^(1/2)+a^2-b

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (386) = 772\).

Time = 0.48 (sec) , antiderivative size = 2250, normalized size of antiderivative = 5.53 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[1/672*(2*(8*a^8*b - 46*a^6*b^3 + 125*a^4*b^5 - 39*a^2*b^7 - 48*b^9)*cos(d*x + c)^7 + 28*(7*a^8*b - 56*a^6*b^3

- 44*a^4*b^5 + 93*a^2*b^7)*cos(d*x + c)^5 + 70*(7*a^8*b + 83*a^6*b^3 - 43*a^4*b^5 - 47*a^2*b^7)*cos(d*x + c)^

3 - 105*(7*a^2*b^6*cos(d*x + c)^6 - a^8 - 21*a^6*b^2 - 35*a^4*b^4 - 7*a^2*b^6 - 7*(5*a^4*b^4 + 3*a^2*b^6)*cos(

d*x + c)^4 + 7*(3*a^6*b^2 + 10*a^4*b^4 + 3*a^2*b^6)*cos(d*x + c)^2 + (a*b^7*cos(d*x + c)^6 - 7*a^7*b - 35*a^5*

b^3 - 21*a^3*b^5 - a*b^7 - 3*(7*a^3*b^5 + a*b^7)*cos(d*x + c)^4 + (35*a^5*b^3 + 42*a^3*b^5 + 3*a*b^7)*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*(3*a^8*b + 7*a^6*b^3 - 7*a^4*b^5 - 3*a^2*b^7)*cos(d*x + c) - 14*((8*a^9 - 46*a^7*b^2 + 125*a^5

*b^4 - 54*a^3*b^6 - 33*a*b^8)*cos(d*x + c)^5 + 10*(a^9 - 11*a^7*b^2 - 25*a^5*b^4 + 31*a^3*b^6 + 4*a*b^8)*cos(d

*x + c)^3 + 15*(a^9 + 14*a^7*b^2 - 14*a^3*b^6 - a*b^8)*cos(d*x + c))*sin(d*x + c))/(7*(a^11*b^6 - 5*a^9*b^8 +

10*a^7*b^10 - 10*a^5*b^12 + 5*a^3*b^14 - a*b^16)*d*cos(d*x + c)^6 - 7*(5*a^13*b^4 - 22*a^11*b^6 + 35*a^9*b^8 -

20*a^7*b^10 - 5*a^5*b^12 + 10*a^3*b^14 - 3*a*b^16)*d*cos(d*x + c)^4 + 7*(3*a^15*b^2 - 5*a^13*b^4 - 17*a^11*b^

6 + 55*a^9*b^8 - 55*a^7*b^10 + 17*a^5*b^12 + 5*a^3*b^14 - 3*a*b^16)*d*cos(d*x + c)^2 - (a^17 + 16*a^15*b^2 - 6

0*a^13*b^4 + 32*a^11*b^6 + 110*a^9*b^8 - 176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d + ((a^10*b^7 - 5*a^8*b^9 + 1

0*a^6*b^11 - 10*a^4*b^13 + 5*a^2*b^15 - b^17)*d*cos(d*x + c)^6 - 3*(7*a^12*b^5 - 34*a^10*b^7 + 65*a^8*b^9 - 60

*a^6*b^11 + 25*a^4*b^13 - 2*a^2*b^15 - b^17)*d*cos(d*x + c)^4 + (35*a^14*b^3 - 133*a^12*b^5 + 143*a^10*b^7 + 5

5*a^8*b^9 - 215*a^6*b^11 + 145*a^4*b^13 - 27*a^2*b^15 - 3*b^17)*d*cos(d*x + c)^2 - (7*a^16*b - 84*a^12*b^5 + 1

76*a^10*b^7 - 110*a^8*b^9 - 32*a^6*b^11 + 60*a^4*b^13 - 16*a^2*b^15 - b^17)*d)*sin(d*x + c)), 1/336*((8*a^8*b

- 46*a^6*b^3 + 125*a^4*b^5 - 39*a^2*b^7 - 48*b^9)*cos(d*x + c)^7 + 14*(7*a^8*b - 56*a^6*b^3 - 44*a^4*b^5 + 93*

a^2*b^7)*cos(d*x + c)^5 + 35*(7*a^8*b + 83*a^6*b^3 - 43*a^4*b^5 - 47*a^2*b^7)*cos(d*x + c)^3 - 105*(7*a^2*b^6*

cos(d*x + c)^6 - a^8 - 21*a^6*b^2 - 35*a^4*b^4 - 7*a^2*b^6 - 7*(5*a^4*b^4 + 3*a^2*b^6)*cos(d*x + c)^4 + 7*(3*a

^6*b^2 + 10*a^4*b^4 + 3*a^2*b^6)*cos(d*x + c)^2 + (a*b^7*cos(d*x + c)^6 - 7*a^7*b - 35*a^5*b^3 - 21*a^3*b^5 -

a*b^7 - 3*(7*a^3*b^5 + a*b^7)*cos(d*x + c)^4 + (35*a^5*b^3 + 42*a^3*b^5 + 3*a*b^7)*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*(3*a^8*b + 7*a^6*b^3 - 7

*a^4*b^5 - 3*a^2*b^7)*cos(d*x + c) - 7*((8*a^9 - 46*a^7*b^2 + 125*a^5*b^4 - 54*a^3*b^6 - 33*a*b^8)*cos(d*x + c

)^5 + 10*(a^9 - 11*a^7*b^2 - 25*a^5*b^4 + 31*a^3*b^6 + 4*a*b^8)*cos(d*x + c)^3 + 15*(a^9 + 14*a^7*b^2 - 14*a^3

*b^6 - a*b^8)*cos(d*x + c))*sin(d*x + c))/(7*(a^11*b^6 - 5*a^9*b^8 + 10*a^7*b^10 - 10*a^5*b^12 + 5*a^3*b^14 -

a*b^16)*d*cos(d*x + c)^6 - 7*(5*a^13*b^4 - 22*a^11*b^6 + 35*a^9*b^8 - 20*a^7*b^10 - 5*a^5*b^12 + 10*a^3*b^14 -

3*a*b^16)*d*cos(d*x + c)^4 + 7*(3*a^15*b^2 - 5*a^13*b^4 - 17*a^11*b^6 + 55*a^9*b^8 - 55*a^7*b^10 + 17*a^5*b^1

2 + 5*a^3*b^14 - 3*a*b^16)*d*cos(d*x + c)^2 - (a^17 + 16*a^15*b^2 - 60*a^13*b^4 + 32*a^11*b^6 + 110*a^9*b^8 -

176*a^7*b^10 + 84*a^5*b^12 - 7*a*b^16)*d + ((a^10*b^7 - 5*a^8*b^9 + 10*a^6*b^11 - 10*a^4*b^13 + 5*a^2*b^15 - b

^17)*d*cos(d*x + c)^6 - 3*(7*a^12*b^5 - 34*a^10*b^7 + 65*a^8*b^9 - 60*a^6*b^11 + 25*a^4*b^13 - 2*a^2*b^15 - b^

17)*d*cos(d*x + c)^4 + (35*a^14*b^3 - 133*a^12*b^5 + 143*a^10*b^7 + 55*a^8*b^9 - 215*a^6*b^11 + 145*a^4*b^13 -

27*a^2*b^15 - 3*b^17)*d*cos(d*x + c)^2 - (7*a^16*b - 84*a^12*b^5 + 176*a^10*b^7 - 110*a^8*b^9 - 32*a^6*b^11 +

60*a^4*b^13 - 16*a^2*b^15 - b^17)*d)*sin(d*x + c))]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \]

integrate(cos(d*x+c)**6/(a+b*sin(d*x+c))**8,x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Exception raised: ValueError} \]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x, algorithm="maxima")

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1650 vs. \(2 (386) = 772\).

Time = 0.56 (sec) , antiderivative size = 1650, normalized size of antiderivative = 4.05 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]

integrate(cos(d*x+c)^6/(a+b*sin(d*x+c))^8,x, algorithm="giac")

1/168*(105*(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/

((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) - (231*a^14*tan(1/2*d*x + 1/2*c)^13 - 1344*a

^12*b^2*tan(1/2*d*x + 1/2*c)^13 + 2016*a^10*b^4*tan(1/2*d*x + 1/2*c)^13 - 1344*a^8*b^6*tan(1/2*d*x + 1/2*c)^13

+ 336*a^6*b^8*tan(1/2*d*x + 1/2*c)^13 + 651*a^13*b*tan(1/2*d*x + 1/2*c)^12 - 8064*a^11*b^3*tan(1/2*d*x + 1/2*

c)^12 + 12096*a^9*b^5*tan(1/2*d*x + 1/2*c)^12 - 8064*a^7*b^7*tan(1/2*d*x + 1/2*c)^12 + 2016*a^5*b^9*tan(1/2*d*

x + 1/2*c)^12 + 196*a^14*tan(1/2*d*x + 1/2*c)^11 - 4354*a^12*b^2*tan(1/2*d*x + 1/2*c)^11 - 21504*a^10*b^4*tan(

1/2*d*x + 1/2*c)^11 + 36736*a^8*b^6*tan(1/2*d*x + 1/2*c)^11 - 25984*a^6*b^8*tan(1/2*d*x + 1/2*c)^11 + 6720*a^4

*b^10*tan(1/2*d*x + 1/2*c)^11 + 140*a^13*b*tan(1/2*d*x + 1/2*c)^10 - 40250*a^11*b^3*tan(1/2*d*x + 1/2*c)^10 -

6720*a^9*b^5*tan(1/2*d*x + 1/2*c)^10 + 49280*a^7*b^7*tan(1/2*d*x + 1/2*c)^10 - 45920*a^5*b^9*tan(1/2*d*x + 1/2

*c)^10 + 13440*a^3*b^11*tan(1/2*d*x + 1/2*c)^10 + 595*a^14*tan(1/2*d*x + 1/2*c)^9 - 20650*a^12*b^2*tan(1/2*d*x

+ 1/2*c)^9 - 103740*a^10*b^4*tan(1/2*d*x + 1/2*c)^9 + 70336*a^8*b^6*tan(1/2*d*x + 1/2*c)^9 + 2576*a^6*b^8*tan

(1/2*d*x + 1/2*c)^9 - 40320*a^4*b^10*tan(1/2*d*x + 1/2*c)^9 + 16128*a^2*b^12*tan(1/2*d*x + 1/2*c)^9 - 3045*a^1

3*b*tan(1/2*d*x + 1/2*c)^8 - 100450*a^11*b^3*tan(1/2*d*x + 1/2*c)^8 - 92120*a^9*b^5*tan(1/2*d*x + 1/2*c)^8 + 1

29024*a^7*b^7*tan(1/2*d*x + 1/2*c)^8 - 74816*a^5*b^9*tan(1/2*d*x + 1/2*c)^8 - 4480*a^3*b^11*tan(1/2*d*x + 1/2*

c)^8 + 10752*a*b^13*tan(1/2*d*x + 1/2*c)^8 - 39060*a^12*b^2*tan(1/2*d*x + 1/2*c)^7 - 188720*a^10*b^4*tan(1/2*d

*x + 1/2*c)^7 + 58352*a^8*b^6*tan(1/2*d*x + 1/2*c)^7 + 39936*a^6*b^8*tan(1/2*d*x + 1/2*c)^7 - 73216*a^4*b^10*t

an(1/2*d*x + 1/2*c)^7 + 19456*a^2*b^12*tan(1/2*d*x + 1/2*c)^7 + 3072*b^14*tan(1/2*d*x + 1/2*c)^7 - 6720*a^13*b

*tan(1/2*d*x + 1/2*c)^6 - 122500*a^11*b^3*tan(1/2*d*x + 1/2*c)^6 - 109760*a^9*b^5*tan(1/2*d*x + 1/2*c)^6 + 127

344*a^7*b^7*tan(1/2*d*x + 1/2*c)^6 - 74816*a^5*b^9*tan(1/2*d*x + 1/2*c)^6 - 4480*a^3*b^11*tan(1/2*d*x + 1/2*c)

^6 + 10752*a*b^13*tan(1/2*d*x + 1/2*c)^6 - 595*a^14*tan(1/2*d*x + 1/2*c)^5 - 37940*a^12*b^2*tan(1/2*d*x + 1/2*

c)^5 - 140280*a^10*b^4*tan(1/2*d*x + 1/2*c)^5 + 65296*a^8*b^6*tan(1/2*d*x + 1/2*c)^5 + 2576*a^6*b^8*tan(1/2*d*

x + 1/2*c)^5 - 40320*a^4*b^10*tan(1/2*d*x + 1/2*c)^5 + 16128*a^2*b^12*tan(1/2*d*x + 1/2*c)^5 - 5999*a^13*b*tan

(1/2*d*x + 1/2*c)^4 - 70084*a^11*b^3*tan(1/2*d*x + 1/2*c)^4 - 16800*a^9*b^5*tan(1/2*d*x + 1/2*c)^4 + 50288*a^7

*b^7*tan(1/2*d*x + 1/2*c)^4 - 45920*a^5*b^9*tan(1/2*d*x + 1/2*c)^4 + 13440*a^3*b^11*tan(1/2*d*x + 1/2*c)^4 - 1

96*a^14*tan(1/2*d*x + 1/2*c)^3 - 19082*a^12*b^2*tan(1/2*d*x + 1/2*c)^3 - 29232*a^10*b^4*tan(1/2*d*x + 1/2*c)^3

+ 37744*a^8*b^6*tan(1/2*d*x + 1/2*c)^3 - 25984*a^6*b^8*tan(1/2*d*x + 1/2*c)^3 + 6720*a^4*b^10*tan(1/2*d*x + 1

/2*c)^3 - 2604*a^13*b*tan(1/2*d*x + 1/2*c)^2 - 13090*a^11*b^3*tan(1/2*d*x + 1/2*c)^2 + 13888*a^9*b^5*tan(1/2*d

*x + 1/2*c)^2 - 8400*a^7*b^7*tan(1/2*d*x + 1/2*c)^2 + 2016*a^5*b^9*tan(1/2*d*x + 1/2*c)^2 - 231*a^14*tan(1/2*d

*x + 1/2*c) - 2562*a^12*b^2*tan(1/2*d*x + 1/2*c) + 2548*a^10*b^4*tan(1/2*d*x + 1/2*c) - 1456*a^8*b^6*tan(1/2*d

*x + 1/2*c) + 336*a^6*b^8*tan(1/2*d*x + 1/2*c) - 279*a^13*b + 326*a^11*b^3 - 200*a^9*b^5 + 48*a^7*b^7)/((a^15

- 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^7))

Mupad [B] (veriﬁcation not implemented)

Time = 12.13 (sec) , antiderivative size = 1868, normalized size of antiderivative = 4.59 \[ \int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]

int(cos(c + d*x)^6/(a + b*sin(c + d*x))^8,x)

((279*a^6*b - 48*b^7 + 200*a^2*b^5 - 326*a^4*b^3)/(168*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan

(c/2 + (d*x)/2)*(33*a^8 - 48*b^8 + 208*a^2*b^6 - 364*a^4*b^4 + 366*a^6*b^2))/(24*a*(a^8 + b^8 - 4*a^2*b^6 + 6*

a^4*b^4 - 4*a^6*b^2)) - (tan(c/2 + (d*x)/2)^9*(85*a^12 + 2304*b^12 - 5760*a^2*b^10 + 368*a^4*b^8 + 10048*a^6*b

^6 - 14820*a^8*b^4 - 2950*a^10*b^2))/(24*a^5*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*

x)/2)^5*(85*a^12 - 2304*b^12 + 5760*a^2*b^10 - 368*a^4*b^8 - 9328*a^6*b^6 + 20040*a^8*b^4 + 5420*a^10*b^2))/(2

4*a^5*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (tan(c/2 + (d*x)/2)^11*(14*a^10 + 480*b^10 - 1856*a^2

*b^8 + 2624*a^4*b^6 - 1536*a^6*b^4 - 311*a^8*b^2))/(12*a^3*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) +

(tan(c/2 + (d*x)/2)^3*(14*a^10 - 480*b^10 + 1856*a^2*b^8 - 2696*a^4*b^6 + 2088*a^6*b^4 + 1363*a^8*b^2))/(12*a^

3*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (tan(c/2 + (d*x)/2)^13*(11*a^8 + 16*b^8 - 64*a^2*b^6 + 96

*a^4*b^4 - 64*a^6*b^2))/(8*a*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2)^6*(240*a^1

2*b - 384*b^13 + 160*a^2*b^11 + 2672*a^4*b^9 - 4548*a^6*b^7 + 3920*a^8*b^5 + 4375*a^10*b^3))/(6*a^6*(a^8 + b^8

- 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2)^8*(435*a^12*b - 1536*b^13 + 640*a^2*b^11 + 10688*

a^4*b^9 - 18432*a^6*b^7 + 13160*a^8*b^5 + 14350*a^10*b^3))/(24*a^6*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*

b^2)) - (5*tan(c/2 + (d*x)/2)^10*(2*a^10*b + 192*b^11 - 656*a^2*b^9 + 704*a^4*b^7 - 96*a^6*b^5 - 575*a^8*b^3))

/(12*a^4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2)^4*(857*a^10*b - 1920*b^11 + 65

60*a^2*b^9 - 7184*a^4*b^7 + 2400*a^6*b^5 + 10012*a^8*b^3))/(24*a^4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*

b^2)) - (tan(c/2 + (d*x)/2)^12*(31*a^8*b + 96*b^9 - 384*a^2*b^7 + 576*a^4*b^5 - 384*a^6*b^3))/(8*a^2*(a^8 + b^

8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (tan(c/2 + (d*x)/2)^2*(186*a^8*b - 144*b^9 + 600*a^2*b^7 - 992*a^4*b

^5 + 935*a^6*b^3))/(12*a^2*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (b*tan(c/2 + (d*x)/2)^7*(35*a^6

+ 16*b^6 + 168*a^2*b^4 + 210*a^4*b^2)*(279*a^6*b - 48*b^7 + 200*a^2*b^5 - 326*a^4*b^3))/(42*a^7*(a^8 + b^8 - 4

*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)))/(d*(tan(c/2 + (d*x)/2)^5*(210*a^6*b + 672*a^2*b^5 + 1120*a^4*b^3) + tan(c/

2 + (d*x)/2)^9*(210*a^6*b + 672*a^2*b^5 + 1120*a^4*b^3) + a^7*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^3*(84

*a^6*b + 280*a^4*b^3) + tan(c/2 + (d*x)/2)^11*(84*a^6*b + 280*a^4*b^3) + tan(c/2 + (d*x)/2)^6*(448*a*b^6 + 35*

a^7 + 1680*a^3*b^4 + 840*a^5*b^2) + tan(c/2 + (d*x)/2)^8*(448*a*b^6 + 35*a^7 + 1680*a^3*b^4 + 840*a^5*b^2) + t

an(c/2 + (d*x)/2)^7*(280*a^6*b + 128*b^7 + 1344*a^2*b^5 + 1680*a^4*b^3) + a^7 + tan(c/2 + (d*x)/2)^4*(21*a^7 +

560*a^3*b^4 + 420*a^5*b^2) + tan(c/2 + (d*x)/2)^10*(21*a^7 + 560*a^3*b^4 + 420*a^5*b^2) + tan(c/2 + (d*x)/2)^

2*(7*a^7 + 84*a^5*b^2) + tan(c/2 + (d*x)/2)^12*(7*a^7 + 84*a^5*b^2) + 14*a^6*b*tan(c/2 + (d*x)/2) + 14*a^6*b*t

an(c/2 + (d*x)/2)^13)) + (5*a*atan((8*((5*a^2*tan(c/2 + (d*x)/2))/(8*(a + b)^(9/2)*(a - b)^(9/2)) + (5*a*(16*a

^8*b + 16*b^9 - 64*a^2*b^7 + 96*a^4*b^5 - 64*a^6*b^3))/(128*(a + b)^(9/2)*(a - b)^(9/2)*(a^8 + b^8 - 4*a^2*b^6

+ 6*a^4*b^4 - 4*a^6*b^2)))*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2))/(5*a)))/(8*d*(a + b)^(9/2)*(a - b

\(\int \frac {\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [468]

Optimal result