∫ cos8 (c+dx)__ (a+b sin(c+dx))8 dx [467]

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

x/b^8-1/8*a*(16*a^6-56*a^4*b^2+70*a^2*b^4-35*b^6)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/b^8/(a^2-b^

2)^(7/2)/d-1/7*cos(d*x+c)^7/b/d/(a+b*sin(d*x+c))^7+1/6*a*cos(d*x+c)^7/b/(a^2-b^2)/d/(a+b*sin(d*x+c))^6-1/24*a*

(6*a^2-11*b^2)*cos(d*x+c)^5/b^3/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^4+1/16*a*(8*a^4-22*a^2*b^2+19*b^4)*cos(d*x+c)^3

/b^5/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^2+1/30*cos(d*x+c)^5*(6*a^2-6*b^2+5*a*b*sin(d*x+c))/b^3/(a^2-b^2)/d/(a+b*si

n(d*x+c))^5-1/24*cos(d*x+c)^3*(8*(a^2-b^2)^2+a*b*(6*a^2-11*b^2)*sin(d*x+c))/b^5/(a^2-b^2)^2/d/(a+b*sin(d*x+c))

^3+1/16*cos(d*x+c)*(16*(a^2-b^2)^3+a*b*(8*a^4-22*a^2*b^2+19*b^4)*sin(d*x+c))/b^7/(a^2-b^2)^3/d/(a+b*sin(d*x+c)

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Rubi [A]
time = 0.84, antiderivative size = 491, normalized size of antiderivative = 1.00, 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 = {2772, 2943, 2942, 2814, 2739, 632, 210} \begin {gather*} \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 ^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 {\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 (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 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {a \left (8 a^4-22 a^2 b^2+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 {a \left (16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right ) \text {ArcTan}\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 {\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {x}{b^8} \end {gather*}

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

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

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_)])/((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

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

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)*S

imp[(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]

\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 ) \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 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 ) \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 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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(6570\) vs. \(2(491)=982\).
time = 8.36, size = 6570, normalized size = 13.38 \begin {gather*} \text {Result too large to show} \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1651\) vs. \(2(469)=938\).
time = 3.68, size = 1652, normalized size = 3.36


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1652\)
default	\(\text {Expression too large to display}\)	\(1652\)
risch	\(\text {Expression too large to display}\)	\(1998\)

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

1/d*(-2/b^8*((-1/16*b^2*(8*a^12-22*a^10*b^2+19*a^8*b^4-16*a^6*b^6+48*a^4*b^8-48*a^2*b^10+16*b^12)/a/(a^6-3*a^4

*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^13-1/16*b*(16*a^14+56*a^12*b^2-238*a^10*b^4+231*a^8*b^6-96*a^6*b^8+288*

a^4*b^10-288*a^2*b^12+96*b^14)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/a^2*tan(1/2*d*x+1/2*c)^12-1/24/a^3*b^2*(384*a^14-

276*a^12*b^2-1252*a^10*b^4+1697*a^8*b^6-384*a^6*b^8+1344*a^4*b^10-1408*a^2*b^12+480*b^14)/(a^6-3*a^4*b^2+3*a^2

*b^4-b^6)*tan(1/2*d*x+1/2*c)^11-1/24/a^4*b*(144*a^16+2064*a^14*b^2-4284*a^12*b^4-224*a^10*b^6+3949*a^8*b^8+528

*a^6*b^10+1392*a^4*b^12-2384*a^2*b^14+960*b^16)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^10-1/240/a^5*

b^2*(17400*a^16+34110*a^14*b^2-152515*a^12*b^4+93770*a^10*b^6+49308*a^8*b^8+30576*a^6*b^10-12496*a^4*b^12-1804

8*a^2*b^14+11520*b^16)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^9-1/240/a^6*b*(3600*a^18+73480*a^16*b^

2-70050*a^14*b^4-198555*a^12*b^6+252090*a^10*b^8+28792*a^8*b^10+43584*a^6*b^12-48064*a^4*b^14+3968*a^2*b^16+76

80*b^18)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^8-1/420/a^7*b^2*(58800*a^18+186200*a^16*b^2-565950*a

^14*b^4+95655*a^12*b^6+444108*a^10*b^8+51212*a^8*b^10-11904*a^6*b^12-64640*a^4*b^14+27904*a^2*b^16+3840*b^18)/

(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^7-1/60/a^6*b*(1200*a^18+26080*a^16*b^2-19080*a^14*b^4-80730*a

^12*b^6+87285*a^10*b^8+21208*a^8*b^10+5316*a^6*b^12-12016*a^4*b^14+992*a^2*b^16+1920*b^18)/(a^6-3*a^4*b^2+3*a^

2*b^4-b^6)*tan(1/2*d*x+1/2*c)^6-1/240/a^5*b^2*(33000*a^16+91890*a^14*b^2-318025*a^12*b^4+107140*a^10*b^6+21184

8*a^8*b^8-10304*a^6*b^10-12496*a^4*b^12-18048*a^2*b^14+11520*b^16)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1

/2*c)^5-1/240/a^4*b*(3600*a^16+65880*a^14*b^2-95382*a^12*b^4-122429*a^10*b^6+240884*a^8*b^8-41568*a^6*b^10+168

80*a^4*b^12-23840*a^2*b^14+9600*b^16)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^4-1/120/a^3*b^2*(8160*a

^14+6420*a^12*b^2-62984*a^10*b^4+71177*a^8*b^6-15192*a^6*b^8+7784*a^4*b^10-7040*a^2*b^12+2400*b^14)/(a^6-3*a^4

*b^2+3*a^2*b^4-b^6)*tan(1/2*d*x+1/2*c)^3-1/120*b*(720*a^14+7280*a^12*b^2-24612*a^10*b^4+23652*a^8*b^6-5659*a^6

*b^8+3200*a^4*b^10-2376*a^2*b^12+720*b^14)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/a^2*tan(1/2*d*x+1/2*c)^2-1/240*b^2*(3

240*a^12-9190*a^10*b^2+8367*a^8*b^4-2046*a^6*b^6+1196*a^4*b^8-832*a^2*b^10+240*b^12)/a/(a^6-3*a^4*b^2+3*a^2*b^

4-b^6)*tan(1/2*d*x+1/2*c)-1/1680*(1680*a^12-4760*a^10*b^2+4326*a^8*b^4-1143*a^6*b^6+958*a^4*b^8-776*a^2*b^10+2

40*b^12)*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6))/(a*tan(1/2*d*x+1/2*c)^2+2*b*tan(1/2*d*x+1/2*c)+a)^7+1/16*a*(16*a^6-5

6*a^4*b^2+70*a^2*b^4-35*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+

2*b)/(a^2-b^2)^(1/2)))+2/b^8*arctan(tan(1/2*d*x+1/2*c)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

integrate(cos(d*x+c)^8/(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?`

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1818 vs. \(2 (469) = 938\).
time = 0.73, size = 3721, normalized size = 7.58 \begin {gather*} \text {Too large to display} \end {gather*}

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

[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^...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2326 vs. \(2 (469) = 938\).
time = 4.26, size = 2326, normalized size = 4.74 \begin {gather*} \text {Too large to display} \end {gather*}

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

-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 +

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Mupad [B]
time = 32.22, size = 2500, normalized size = 5.09 \begin {gather*} \text {Too large to display} \end {gather*}

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

(2*atan((((((((32*a^2*b^35 - 192*a^4*b^33 + 480*a^6*b^31 - 640*a^8*b^29 + 480*a^10*b^27 - 192*a^12*b^25 + 32*a

^14*b^23)*1i)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) + (tan(c

/2 + (d*x)/2)*(768*a*b^37 - 5120*a^3*b^35 + 14592*a^5*b^33 - 23040*a^7*b^31 + 21760*a^9*b^29 - 12288*a^11*b^27

+ 3840*a^13*b^25 - 512*a^15*b^23)*1i)/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^1

0*b^23 + a^12*b^21)))*1i)/b^8 + ((32*a*b^28 - 154*a^3*b^26 + 322*a^5*b^24 - 378*a^7*b^22 + 262*a^9*b^20 - 100*

a^11*b^18 + 16*a^13*b^16)*1i)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^1

2*b^20) + (tan(c/2 + (d*x)/2)*(1120*a^2*b^28 - 5600*a^4*b^26 + 11872*a^6*b^24 - 13728*a^8*b^22 + 9152*a^10*b^2

0 - 3328*a^12*b^18 + 512*a^14*b^16)*1i)/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^

10*b^23 + a^12*b^21)))/b^8 + (32*a^2*b^19 - 192*a^4*b^17 + 480*a^6*b^15 - 640*a^8*b^13 + 480*a^10*b^11 - 192*a

^12*b^9 + 32*a^14*b^7)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20)

+ (tan(c/2 + (d*x)/2)*(512*a*b^21 - 4553*a^3*b^19 + 14116*a^5*b^17 - 22900*a^7*b^15 + 21760*a^9*b^13 - 12288*

a^11*b^11 + 3840*a^13*b^9 - 512*a^15*b^7))/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6

*a^10*b^23 + a^12*b^21)))/b^8 + ((32*a^2*b^19 - 192*a^4*b^17 + 480*a^6*b^15 - 640*a^8*b^13 + 480*a^10*b^11 - 1

92*a^12*b^9 + 32*a^14*b^7)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b

^20) - (((32*a*b^28 - 154*a^3*b^26 + 322*a^5*b^24 - 378*a^7*b^22 + 262*a^9*b^20 - 100*a^11*b^18 + 16*a^13*b^16

)*1i)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) - ((((32*a^2*b^3

5 - 192*a^4*b^33 + 480*a^6*b^31 - 640*a^8*b^29 + 480*a^10*b^27 - 192*a^12*b^25 + 32*a^14*b^23)*1i)/(b^32 - 6*a

^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) + (tan(c/2 + (d*x)/2)*(768*a*b^37

- 5120*a^3*b^35 + 14592*a^5*b^33 - 23040*a^7*b^31 + 21760*a^9*b^29 - 12288*a^11*b^27 + 3840*a^13*b^25 - 512*a

^15*b^23)*1i)/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*b^23 + a^12*b^21)))*1i)

/b^8 + (tan(c/2 + (d*x)/2)*(1120*a^2*b^28 - 5600*a^4*b^26 + 11872*a^6*b^24 - 13728*a^8*b^22 + 9152*a^10*b^20 -

3328*a^12*b^18 + 512*a^14*b^16)*1i)/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*

b^23 + a^12*b^21)))/b^8 + (tan(c/2 + (d*x)/2)*(512*a*b^21 - 4553*a^3*b^19 + 14116*a^5*b^17 - 22900*a^7*b^15 +

21760*a^9*b^13 - 12288*a^11*b^11 + 3840*a^13*b^9 - 512*a^15*b^7))/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6

*b^27 + 15*a^8*b^25 - 6*a^10*b^23 + a^12*b^21)))/b^8)/((32*a^13 - (665*a^3*b^10)/4 + 525*a^5*b^8 - 721*a^7*b^6

+ 524*a^9*b^4 - 200*a^11*b^2)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^

12*b^20) - (((32*a^2*b^19 - 192*a^4*b^17 + 480*a^6*b^15 - 640*a^8*b^13 + 480*a^10*b^11 - 192*a^12*b^9 + 32*a^1

4*b^7)*1i)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) - ((((32*a*

b^28 - 154*a^3*b^26 + 322*a^5*b^24 - 378*a^7*b^22 + 262*a^9*b^20 - 100*a^11*b^18 + 16*a^13*b^16)*1i)/(b^32 - 6

*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) - ((((32*a^2*b^35 - 192*a^4*b^3

3 + 480*a^6*b^31 - 640*a^8*b^29 + 480*a^10*b^27 - 192*a^12*b^25 + 32*a^14*b^23)*1i)/(b^32 - 6*a^2*b^30 + 15*a^

4*b^28 - 20*a^6*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) + (tan(c/2 + (d*x)/2)*(768*a*b^37 - 5120*a^3*b^3

5 + 14592*a^5*b^33 - 23040*a^7*b^31 + 21760*a^9*b^29 - 12288*a^11*b^27 + 3840*a^13*b^25 - 512*a^15*b^23)*1i)/(

8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*b^23 + a^12*b^21)))*1i)/b^8 + (tan(c/2

+ (d*x)/2)*(1120*a^2*b^28 - 5600*a^4*b^26 + 11872*a^6*b^24 - 13728*a^8*b^22 + 9152*a^10*b^20 - 3328*a^12*b^18

+ 512*a^14*b^16)*1i)/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*b^23 + a^12*b^2

1)))*1i)/b^8 + (tan(c/2 + (d*x)/2)*(512*a*b^21 - 4553*a^3*b^19 + 14116*a^5*b^17 - 22900*a^7*b^15 + 21760*a^9*b

^13 - 12288*a^11*b^11 + 3840*a^13*b^9 - 512*a^15*b^7)*1i)/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b^29 - 20*a^6*b^27 +

15*a^8*b^25 - 6*a^10*b^23 + a^12*b^21)))/b^8 + (((((((32*a^2*b^35 - 192*a^4*b^33 + 480*a^6*b^31 - 640*a^8*b^29

+ 480*a^10*b^27 - 192*a^12*b^25 + 32*a^14*b^23)*1i)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6*b^26 + 15*a^8*b

^24 - 6*a^10*b^22 + a^12*b^20) + (tan(c/2 + (d*x)/2)*(768*a*b^37 - 5120*a^3*b^35 + 14592*a^5*b^33 - 23040*a^7*

b^31 + 21760*a^9*b^29 - 12288*a^11*b^27 + 3840*a^13*b^25 - 512*a^15*b^23)*1i)/(8*(b^33 - 6*a^2*b^31 + 15*a^4*b

^29 - 20*a^6*b^27 + 15*a^8*b^25 - 6*a^10*b^23 + a^12*b^21)))*1i)/b^8 + ((32*a*b^28 - 154*a^3*b^26 + 322*a^5*b^

24 - 378*a^7*b^22 + 262*a^9*b^20 - 100*a^11*b^18 + 16*a^13*b^16)*1i)/(b^32 - 6*a^2*b^30 + 15*a^4*b^28 - 20*a^6

*b^26 + 15*a^8*b^24 - 6*a^10*b^22 + a^12*b^20) + (tan(c/2 + (d*x)/2)*(1120*a^2*b^28 - 5600*a^4*b^26 + 11872*a^

6*b^24 - 13728*a^8*b^22 + 9152*a^10*b^20 - 3328...

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3.5.67 \(\int \frac {\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [467]