3.5.0 ∫ cos2 (c+dx) ____ (a+b sin(c+dx))8 dx [470]

Integrand size = 21, antiderivative size = 422 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {a \left (8 a^4+20 a^2 b^2+5 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{13/2} d}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))} \]

1/8*a*(8*a^4+20*a^2*b^2+5*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^2)^(13/2)/d-1/7*cos(d*x

+c)/b/d/(a+b*sin(d*x+c))^7+1/42*a*cos(d*x+c)/b/(a^2-b^2)/d/(a+b*sin(d*x+c))^6+1/210*(5*a^2+6*b^2)*cos(d*x+c)/b

/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^5+1/840*a*(20*a^2+79*b^2)*cos(d*x+c)/b/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^4+1/840*

(20*a^4+179*a^2*b^2+32*b^4)*cos(d*x+c)/b/(a^2-b^2)^4/d/(a+b*sin(d*x+c))^3+1/1680*a*(40*a^4+718*a^2*b^2+397*b^4

)*cos(d*x+c)/b/(a^2-b^2)^5/d/(a+b*sin(d*x+c))^2+1/1680*(40*a^6+1518*a^4*b^2+1779*a^2*b^4+128*b^6)*cos(d*x+c)/b

Rubi [A] (veriﬁed)

Time = 0.54 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2772, 2833, 12, 2739, 632, 210} \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^4}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^5}+\frac {a \cos (c+d x)}{42 b d \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}+\frac {a \left (8 a^4+20 a^2 b^2+5 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{13/2}}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b d \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^2}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^3}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b d \left (a^2-b^2\right )^6 (a+b \sin (c+d x))}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]

(a*(8*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(13/2)*d) - C

os[c + d*x]/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x])/(42*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) + ((

5*a^2 + 6*b^2)*Cos[c + d*x])/(210*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^5) + (a*(20*a^2 + 79*b^2)*Cos[c + d*x

])/(840*b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^4) + ((20*a^4 + 179*a^2*b^2 + 32*b^4)*Cos[c + d*x])/(840*b*(a^2

- b^2)^4*d*(a + b*Sin[c + d*x])^3) + (a*(40*a^4 + 718*a^2*b^2 + 397*b^4)*Cos[c + d*x])/(1680*b*(a^2 - b^2)^5*

d*(a + b*Sin[c + d*x])^2) + ((40*a^6 + 1518*a^4*b^2 + 1779*a^2*b^4 + 128*b^6)*Cos[c + d*x])/(1680*b*(a^2 - b^2

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}-\frac {\int \frac {\sin (c+d x)}{(a+b \sin (c+d x))^7} \, dx}{7 b} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\int \frac {6 b-5 a \sin (c+d x)}{(a+b \sin (c+d x))^6} \, dx}{42 b \left (a^2-b^2\right )} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}-\frac {\int \frac {-55 a b+4 \left (5 a^2+6 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^5} \, dx}{210 b \left (a^2-b^2\right )^2} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\int \frac {12 b \left (25 a^2+8 b^2\right )-3 a \left (20 a^2+79 b^2\right ) \sin (c+d x)}{(a+b \sin (c+d x))^4} \, dx}{840 b \left (a^2-b^2\right )^3} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}-\frac {\int \frac {-27 a b \left (40 a^2+37 b^2\right )+6 \left (20 a^4+179 a^2 b^2+32 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^3} \, dx}{2520 b \left (a^2-b^2\right )^4} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\int \frac {6 b \left (400 a^4+691 a^2 b^2+64 b^4\right )-3 a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{5040 b \left (a^2-b^2\right )^5} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}-\frac {\int -\frac {315 a b \left (8 a^4+20 a^2 b^2+5 b^4\right )}{a+b \sin (c+d x)} \, dx}{5040 b \left (a^2-b^2\right )^6} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}+\frac {\left (a \left (8 a^4+20 a^2 b^2+5 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^6} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}+\frac {\left (a \left (8 a^4+20 a^2 b^2+5 b^4\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 \left (a^2-b^2\right )^6 d} \\ & = -\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))}-\frac {\left (a \left (8 a^4+20 a^2 b^2+5 b^4\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 \left (a^2-b^2\right )^6 d} \\ & = \frac {a \left (8 a^4+20 a^2 b^2+5 b^4\right ) \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 )^{13/2} d}-\frac {\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x)}{42 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))^6}+\frac {\left (5 a^2+6 b^2\right ) \cos (c+d x)}{210 b \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^5}+\frac {a \left (20 a^2+79 b^2\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^4}+\frac {\left (20 a^4+179 a^2 b^2+32 b^4\right ) \cos (c+d x)}{840 b \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^3}+\frac {a \left (40 a^4+718 a^2 b^2+397 b^4\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^2}+\frac {\left (40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right ) \cos (c+d x)}{1680 b \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(1896\) vs. \(2(422)=844\).

Time = 6.24 (sec) , antiderivative size = 1896, normalized size of antiderivative = 4.49 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\cos ^3(c+d x)}{3 (a-b) d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x) \left (-\frac {b (1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{5/2}}{7 (-a+b) (a+b) (a+b \sin (c+d x))^7}-\frac {-\frac {(3 a b+(7 a-b) b) (1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{5/2}}{6 (-a+b) (a+b) (a+b \sin (c+d x))^6}-\frac {-\frac {\left (2 a (10 a-b) b+b \left (42 a^2-16 a b+19 b^2\right )\right ) (1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{5/2}}{5 (-a+b) (a+b) (a+b \sin (c+d x))^5}-\frac {-\frac {\left (a b \left (62 a^2-18 a b+19 b^2\right )+b \left (210 a^3-142 a^2 b+213 a b^2-29 b^3\right )\right ) (1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{5/2}}{4 (-a+b) (a+b) (a+b \sin (c+d x))^4}-\frac {105 \left (8 a^4-8 a^3 b+12 a^2 b^2-4 a b^3+b^4\right ) \left (-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{5/2}}{3 (-a+b) (a+b \sin (c+d x))^3}-\frac {-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}+\frac {3 \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)^{3/2} \sqrt {a-b}}+\frac {\sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}{(-a-b) (a+b \sin (c+d x))}\right )}{2 (a+b)}}{3 (-a+b)}\right )}{4 (-a+b) (a+b)}}{5 (-a+b) (a+b)}}{6 (-a+b) (a+b)}}{7 (-a+b) (a+b)}\right )}{(a-b) d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}+\frac {4 b \left (\frac {\cos ^5(c+d x)}{5 (a-b) d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x) \left (-\frac {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}-\frac {-\frac {(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}-\frac {7 \left (6 a^2-2 a b+b^2\right ) \left (-\frac {(1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{7/2}}{5 (-a+b) (a+b \sin (c+d x))^5}-\frac {3 \left (-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{7/2}}{4 (-a+b) (a+b \sin (c+d x))^4}-\frac {-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}+\frac {5 \left (-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}+\frac {3 \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)^{3/2} \sqrt {a-b}}+\frac {\sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}{(-a-b) (a+b \sin (c+d x))}\right )}{2 (a+b)}\right )}{3 (a+b)}}{4 (-a+b)}\right )}{5 (-a+b)}\right )}{6 (-a+b) (a+b)}}{7 (-a+b) (a+b)}\right )}{(a-b) d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}+\frac {2 b \left (\frac {\cos ^7(c+d x)}{7 (a-b) d (a+b \sin (c+d x))^7}+\frac {a \cos (c+d x) \left (-\frac {(1-\sin (c+d x))^{5/2} (1+\sin (c+d x))^{9/2}}{7 (-a+b) (a+b \sin (c+d x))^7}-\frac {5 \left (-\frac {(1-\sin (c+d x))^{3/2} (1+\sin (c+d x))^{9/2}}{6 (-a+b) (a+b \sin (c+d x))^6}-\frac {-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{9/2}}{5 (-a+b) (a+b \sin (c+d x))^5}-\frac {-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{7/2}}{4 (a+b) (a+b \sin (c+d x))^4}+\frac {7 \left (-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}+\frac {5 \left (-\frac {\sqrt {1-\sin (c+d x)} (1+\sin (c+d x))^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}+\frac {3 \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)^{3/2} \sqrt {a-b}}+\frac {\sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}{(-a-b) (a+b \sin (c+d x))}\right )}{2 (a+b)}\right )}{3 (a+b)}\right )}{4 (a+b)}}{5 (-a+b)}}{2 (-a+b)}\right )}{7 (-a+b)}\right )}{(a-b) d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}\right )}{5 (a-b)}\right )}{3 (a-b)} \]

Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]

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

in[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-1/6*((3*a*b + (7*a - b)*b)*(1 - Sin[c + d*x]

)^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (-1/5*((2*a*(10*a - b)*b + b*(42

*a^2 - 16*a*b + 19*b^2))*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d

*x])^5) - (-1/4*((a*b*(62*a^2 - 18*a*b + 19*b^2) + b*(210*a^3 - 142*a^2*b + 213*a*b^2 - 29*b^3))*(1 - Sin[c +

d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^4) - (105*(8*a^4 - 8*a^3*b + 12*a

^2*b^2 - 4*a*b^3 + b^4)*(-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b*Sin[c + d*x])

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

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

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

(-1/6*((a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c +

d*x])^6) - (7*(6*a^2 - 2*a*b + b^2)*(-1/5*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a +

b*Sin[c + d*x])^5) - (3*(-1/4*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])

^4) - (-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3) + (5*(-1/2*(Sqr

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

^(5/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^7) - (5*(-1/6*((1 - Sin[c + d*x])^(3/2)*(1 + S

in[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^6) - (-1/5*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(9/2)

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

Sin[c + d*x])^4) + (7*(-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3)

+ (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTa

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

Maple [C] (warning: unable to verify)

Time = 14.28 (sec) , antiderivative size = 1703, normalized size of antiderivative = 4.04


method	result	size

risch	\(\text {Expression too large to display}\)	\(1703\)
derivativedivides	\(\text {Expression too large to display}\)	\(1872\)
default	\(\text {Expression too large to display}\)	\(1872\)

int(cos(d*x+c)^2/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

-1/840*I*(-128*I*b^13-1497888*a^5*b^8*exp(5*I*(d*x+c))-304640*a^3*b^10*exp(5*I*(d*x+c))-16975*a*b^12*exp(5*I*(

d*x+c))-27300*I*b^9*a^4*exp(12*I*(d*x+c))-26880*a^11*b^2*exp(5*I*(d*x+c))-756448*a^9*b^4*exp(5*I*(d*x+c))-1856

624*a^7*b^6*exp(5*I*(d*x+c))+2560*a^13*exp(7*I*(d*x+c))+16380*I*b^11*a^2*exp(2*I*(d*x+c))+508200*I*b^7*a^6*exp

(10*I*(d*x+c))-890190*I*b^9*a^4*exp(8*I*(d*x+c))-35385*I*b^11*a^2*exp(8*I*(d*x+c))+8960*I*b*a^12*exp(6*I*(d*x+

c))+2574992*I*b^7*a^6*exp(6*I*(d*x+c))-8960*I*b^13*exp(8*I*(d*x+c))-4480*I*b^13*exp(6*I*(d*x+c))-2688*I*b^13*e

xp(4*I*(d*x+c))+896*I*b^13*exp(2*I*(d*x+c))+3500*b^12*a*exp(11*I*(d*x+c))+124032*b^2*a^11*exp(7*I*(d*x+c))+356

580*a^3*b^10*exp(7*I*(d*x+c))+17920*a*b^12*exp(7*I*(d*x+c))-560*exp(I*(d*x+c))*b^6*a^7-20412*b^8*exp(I*(d*x+c)

)*a^5-22806*exp(I*(d*x+c))*b^10*a^3-1267*exp(I*(d*x+c))*b^12*a-308448*b^4*a^9*exp(9*I*(d*x+c))-1047424*b^6*a^7

*exp(9*I*(d*x+c))-899388*b^8*a^5*exp(9*I*(d*x+c))-212310*b^10*a^3*exp(9*I*(d*x+c))-9905*b^12*a*exp(9*I*(d*x+c)

)-22400*I*b^3*a^10*exp(4*I*(d*x+c))-585088*I*b^9*a^4*exp(4*I*(d*x+c))-245952*I*b^3*a^10*exp(8*I*(d*x+c))-17475

92*I*a^6*b^7*exp(8*I*(d*x+c))-10920*I*b^7*a^6*exp(12*I*(d*x+c))-40*I*a^6*b^7-1518*I*a^4*b^9-1779*I*a^2*b^11+15

5400*b^8*a^5*exp(11*I*(d*x+c))+51450*b^10*a^3*exp(11*I*(d*x+c))-1522416*I*a^8*b^5*exp(8*I*(d*x+c))-8960*I*b*a^

12*exp(8*I*(d*x+c))+501312*I*a^10*b^3*exp(6*I*(d*x+c))+1858416*I*a^8*b^5*exp(6*I*(d*x+c))-6825*I*b^11*a^2*exp(

12*I*(d*x+c))+178640*I*b^5*a^8*exp(10*I*(d*x+c))+265650*I*b^9*a^4*exp(10*I*(d*x+c))-840*b^8*a^5*exp(13*I*(d*x+

c))-2100*b^10*a^3*exp(13*I*(d*x+c))+59920*b^6*a^7*exp(11*I*(d*x+c))-525*b^12*a*exp(13*I*(d*x+c))+11200*a^9*b^4

*exp(3*I*(d*x+c))+368480*a^7*b^6*exp(3*I*(d*x+c))+470232*a^5*b^8*exp(3*I*(d*x+c))+133826*a^3*b^10*exp(3*I*(d*x

+c))+7252*a*b^12*exp(3*I*(d*x+c))+1167552*a^9*b^4*exp(7*I*(d*x+c))+2484400*a^7*b^6*exp(7*I*(d*x+c))+1792896*a^

5*b^8*exp(7*I*(d*x+c))+38500*I*b^11*a^2*exp(10*I*(d*x+c))+872340*I*a^4*b^9*exp(6*I*(d*x+c))+134400*I*a^2*b^11*

exp(6*I*(d*x+c))-688240*I*a^8*b^5*exp(4*I*(d*x+c))-1126440*I*a^6*b^7*exp(4*I*(d*x+c))+3360*I*a^8*b^5*exp(2*I*(

d*x+c))+116872*I*a^6*b^7*exp(2*I*(d*x+c))-52619*I*b^11*a^2*exp(4*I*(d*x+c))+132762*I*b^9*a^4*exp(2*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)^6/d/b^2-1/2/(-a^2+b^2)^(1/2)*a^5/(a+b)^6/(a-b)^6/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))-5/4/(-a^2+b^2)^(1/2)*a^3/(a+b)^6/(a-b)^6

/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))*b^2-5/16/(-a^2+b^2)^(1/2)*a/(a+b)^6/(a

-b)^6/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))*b^4+1/2/(-a^2+b^2)^(1/2)*a^5/(a+b

)^6/(a-b)^6/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))+5/4/(-a^2+b^2)^(1/2)*a^3/(a

+b)^6/(a-b)^6/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))*b^2+5/16/(-a^2+b^2)^(1/2)

*a/(a+b)^6/(a-b)^6/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))*b^4

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1444 vs. \(2 (401) = 802\).

Time = 0.54 (sec) , antiderivative size = 2972, normalized size of antiderivative = 7.04 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]

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

[1/3360*(2*(40*a^8*b^5 + 1478*a^6*b^7 + 261*a^4*b^9 - 1651*a^2*b^11 - 128*b^13)*cos(d*x + c)^7 - 28*(60*a^10*b

^3 + 1837*a^8*b^5 + 176*a^6*b^7 - 1680*a^4*b^9 - 361*a^2*b^11 - 32*b^13)*cos(d*x + c)^5 + 70*(40*a^12*b + 900*

a^10*b^3 + 1111*a^8*b^5 - 501*a^6*b^7 - 1395*a^4*b^9 - 139*a^2*b^11 - 16*b^13)*cos(d*x + c)^3 + 105*(8*a^12 +

188*a^10*b^2 + 705*a^8*b^4 + 861*a^6*b^6 + 315*a^4*b^8 + 35*a^2*b^10 - 7*(8*a^6*b^6 + 20*a^4*b^8 + 5*a^2*b^10)

*cos(d*x + c)^6 + 7*(40*a^8*b^4 + 124*a^6*b^6 + 85*a^4*b^8 + 15*a^2*b^10)*cos(d*x + c)^4 - 7*(24*a^10*b^2 + 14

0*a^8*b^4 + 239*a^6*b^6 + 110*a^4*b^8 + 15*a^2*b^10)*cos(d*x + c)^2 + (56*a^11*b + 420*a^9*b^3 + 903*a^7*b^5 +

603*a^5*b^7 + 125*a^3*b^9 + 5*a*b^11 - (8*a^5*b^7 + 20*a^3*b^9 + 5*a*b^11)*cos(d*x + c)^6 + 3*(56*a^7*b^5 + 1

48*a^5*b^7 + 55*a^3*b^9 + 5*a*b^11)*cos(d*x + c)^4 - (280*a^9*b^3 + 1036*a^7*b^5 + 1039*a^5*b^7 + 270*a^3*b^9

+ 15*a*b^11)*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*(24*a^12*b + 116*a^10*b^3 + 99*a^8*b^5 - 129*a^6*b^7 - 95*a^4*b^9 - 15*a^

2*b^11)*cos(d*x + c) - 14*((40*a^9*b^4 + 1358*a^7*b^6 + 81*a^5*b^8 - 1426*a^3*b^10 - 53*a*b^12)*cos(d*x + c)^5

- 10*(20*a^11*b^2 + 535*a^9*b^4 + 147*a^7*b^6 - 407*a^5*b^8 - 283*a^3*b^10 - 12*a*b^12)*cos(d*x + c)^3 + 15*(

8*a^13 + 132*a^11*b^2 + 285*a^9*b^4 - 42*a^7*b^6 - 288*a^5*b^8 - 90*a^3*b^10 - 5*a*b^12)*cos(d*x + c))*sin(d*x

+ c))/(7*(a^15*b^6 - 7*a^13*b^8 + 21*a^11*b^10 - 35*a^9*b^12 + 35*a^7*b^14 - 21*a^5*b^16 + 7*a^3*b^18 - a*b^2

0)*d*cos(d*x + c)^6 - 7*(5*a^17*b^4 - 32*a^15*b^6 + 84*a^13*b^8 - 112*a^11*b^10 + 70*a^9*b^12 - 28*a^5*b^16 +

16*a^3*b^18 - 3*a*b^20)*d*cos(d*x + c)^4 + 7*(3*a^19*b^2 - 11*a^17*b^4 - 4*a^15*b^6 + 84*a^13*b^8 - 182*a^11*b

^10 + 182*a^9*b^12 - 84*a^7*b^14 + 4*a^5*b^16 + 11*a^3*b^18 - 3*a*b^20)*d*cos(d*x + c)^2 - (a^21 + 14*a^19*b^2

- 91*a^17*b^4 + 168*a^15*b^6 - 14*a^13*b^8 - 364*a^11*b^10 + 546*a^9*b^12 - 344*a^7*b^14 + 77*a^5*b^16 + 14*a

^3*b^18 - 7*a*b^20)*d + ((a^14*b^7 - 7*a^12*b^9 + 21*a^10*b^11 - 35*a^8*b^13 + 35*a^6*b^15 - 21*a^4*b^17 + 7*a

^2*b^19 - b^21)*d*cos(d*x + c)^6 - 3*(7*a^16*b^5 - 48*a^14*b^7 + 140*a^12*b^9 - 224*a^10*b^11 + 210*a^8*b^13 -

112*a^6*b^15 + 28*a^4*b^17 - b^21)*d*cos(d*x + c)^4 + (35*a^18*b^3 - 203*a^16*b^5 + 444*a^14*b^7 - 364*a^12*b

^9 - 182*a^10*b^11 + 630*a^8*b^13 - 532*a^6*b^15 + 196*a^4*b^17 - 21*a^2*b^19 - 3*b^21)*d*cos(d*x + c)^2 - (7*

a^20*b - 14*a^18*b^3 - 77*a^16*b^5 + 344*a^14*b^7 - 546*a^12*b^9 + 364*a^10*b^11 + 14*a^8*b^13 - 168*a^6*b^15

+ 91*a^4*b^17 - 14*a^2*b^19 - b^21)*d)*sin(d*x + c)), 1/1680*((40*a^8*b^5 + 1478*a^6*b^7 + 261*a^4*b^9 - 1651*

a^2*b^11 - 128*b^13)*cos(d*x + c)^7 - 14*(60*a^10*b^3 + 1837*a^8*b^5 + 176*a^6*b^7 - 1680*a^4*b^9 - 361*a^2*b^

11 - 32*b^13)*cos(d*x + c)^5 + 35*(40*a^12*b + 900*a^10*b^3 + 1111*a^8*b^5 - 501*a^6*b^7 - 1395*a^4*b^9 - 139*

a^2*b^11 - 16*b^13)*cos(d*x + c)^3 + 105*(8*a^12 + 188*a^10*b^2 + 705*a^8*b^4 + 861*a^6*b^6 + 315*a^4*b^8 + 35

*a^2*b^10 - 7*(8*a^6*b^6 + 20*a^4*b^8 + 5*a^2*b^10)*cos(d*x + c)^6 + 7*(40*a^8*b^4 + 124*a^6*b^6 + 85*a^4*b^8

+ 15*a^2*b^10)*cos(d*x + c)^4 - 7*(24*a^10*b^2 + 140*a^8*b^4 + 239*a^6*b^6 + 110*a^4*b^8 + 15*a^2*b^10)*cos(d*

x + c)^2 + (56*a^11*b + 420*a^9*b^3 + 903*a^7*b^5 + 603*a^5*b^7 + 125*a^3*b^9 + 5*a*b^11 - (8*a^5*b^7 + 20*a^3

*b^9 + 5*a*b^11)*cos(d*x + c)^6 + 3*(56*a^7*b^5 + 148*a^5*b^7 + 55*a^3*b^9 + 5*a*b^11)*cos(d*x + c)^4 - (280*a

^9*b^3 + 1036*a^7*b^5 + 1039*a^5*b^7 + 270*a^3*b^9 + 15*a*b^11)*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*(24*a^12*b + 116*a^10*b^3 + 99*a^8*b^5 - 12

9*a^6*b^7 - 95*a^4*b^9 - 15*a^2*b^11)*cos(d*x + c) - 7*((40*a^9*b^4 + 1358*a^7*b^6 + 81*a^5*b^8 - 1426*a^3*b^1

0 - 53*a*b^12)*cos(d*x + c)^5 - 10*(20*a^11*b^2 + 535*a^9*b^4 + 147*a^7*b^6 - 407*a^5*b^8 - 283*a^3*b^10 - 12*

a*b^12)*cos(d*x + c)^3 + 15*(8*a^13 + 132*a^11*b^2 + 285*a^9*b^4 - 42*a^7*b^6 - 288*a^5*b^8 - 90*a^3*b^10 - 5*

a*b^12)*cos(d*x + c))*sin(d*x + c))/(7*(a^15*b^6 - 7*a^13*b^8 + 21*a^11*b^10 - 35*a^9*b^12 + 35*a^7*b^14 - 21*

a^5*b^16 + 7*a^3*b^18 - a*b^20)*d*cos(d*x + c)^6 - 7*(5*a^17*b^4 - 32*a^15*b^6 + 84*a^13*b^8 - 112*a^11*b^10 +

70*a^9*b^12 - 28*a^5*b^16 + 16*a^3*b^18 - 3*a*b^20)*d*cos(d*x + c)^4 + 7*(3*a^19*b^2 - 11*a^17*b^4 - 4*a^15*b

^6 + 84*a^13*b^8 - 182*a^11*b^10 + 182*a^9*b^12 - 84*a^7*b^14 + 4*a^5*b^16 + 11*a^3*b^18 - 3*a*b^20)*d*cos(d*x

+ c)^2 - (a^21 + 14*a^19*b^2 - 91*a^17*b^4 + 168*a^15*b^6 - 14*a^13*b^8 - 364*a^11*b^10 + 546*a^9*b^12 - 344*

a^7*b^14 + 77*a^5*b^16 + 14*a^3*b^18 - 7*a*b^20)*d + ((a^14*b^7 - 7*a^12*b^9 + 21*a^10*b^11 - 35*a^8*b^13 + 35

*a^6*b^15 - 21*a^4*b^17 + 7*a^2*b^19 - b^21)*d*cos(d*x + c)^6 - 3*(7*a^16*b^5 - 48*a^14*b^7 + 140*a^12*b^9 - 2

24*a^10*b^11 + 210*a^8*b^13 - 112*a^6*b^15 + 28*a^4*b^17 - b^21)*d*cos(d*x + c)^4 + (35*a^18*b^3 - 203*a^16*b^

5 + 444*a^14*b^7 - 364*a^12*b^9 - 182*a^10*b^11 + 630*a^8*b^13 - 532*a^6*b^15 + 196*a^4*b^17 - 21*a^2*b^19 - 3

*b^21)*d*cos(d*x + c)^2 - (7*a^20*b - 14*a^18*b^3 - 77*a^16*b^5 + 344*a^14*b^7 - 546*a^12*b^9 + 364*a^10*b^11

+ 14*a^8*b^13 - 168*a^6*b^15 + 91*a^4*b^17 - 14*a^2*b^19 - b^21)*d)*sin(d*x + c))]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \]

integrate(cos(d*x+c)**2/(a+b*sin(d*x+c))**8,x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Exception raised: ValueError} \]

integrate(cos(d*x+c)^2/(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. 2207 vs. \(2 (401) = 802\).

Time = 0.52 (sec) , antiderivative size = 2207, normalized size of antiderivative = 5.23 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]

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

1/840*(105*(8*a^5 + 20*a^3*b^2 + 5*a*b^4)*(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^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b^10 + b^12)*s

qrt(a^2 - b^2)) - (840*a^18*tan(1/2*d*x + 1/2*c)^13 - 12180*a^16*b^2*tan(1/2*d*x + 1/2*c)^13 + 24675*a^14*b^4*

tan(1/2*d*x + 1/2*c)^13 - 33600*a^12*b^6*tan(1/2*d*x + 1/2*c)^13 + 25200*a^10*b^8*tan(1/2*d*x + 1/2*c)^13 - 10

080*a^8*b^10*tan(1/2*d*x + 1/2*c)^13 + 1680*a^6*b^12*tan(1/2*d*x + 1/2*c)^13 - 840*a^17*b*tan(1/2*d*x + 1/2*c)

^12 - 87780*a^15*b^3*tan(1/2*d*x + 1/2*c)^12 + 144375*a^13*b^5*tan(1/2*d*x + 1/2*c)^12 - 201600*a^11*b^7*tan(1

/2*d*x + 1/2*c)^12 + 151200*a^9*b^9*tan(1/2*d*x + 1/2*c)^12 - 60480*a^7*b^11*tan(1/2*d*x + 1/2*c)^12 + 10080*a

^5*b^13*tan(1/2*d*x + 1/2*c)^12 + 3360*a^18*tan(1/2*d*x + 1/2*c)^11 - 94080*a^16*b^2*tan(1/2*d*x + 1/2*c)^11 -

220500*a^14*b^4*tan(1/2*d*x + 1/2*c)^11 + 287350*a^12*b^6*tan(1/2*d*x + 1/2*c)^11 - 537600*a^10*b^8*tan(1/2*d

*x + 1/2*c)^11 + 450240*a^8*b^10*tan(1/2*d*x + 1/2*c)^11 - 192640*a^6*b^12*tan(1/2*d*x + 1/2*c)^11 + 33600*a^4

*b^14*tan(1/2*d*x + 1/2*c)^11 - 13440*a^17*b*tan(1/2*d*x + 1/2*c)^10 - 554400*a^15*b^3*tan(1/2*d*x + 1/2*c)^10

- 165900*a^13*b^5*tan(1/2*d*x + 1/2*c)^10 - 66850*a^11*b^7*tan(1/2*d*x + 1/2*c)^10 - 621600*a^9*b^9*tan(1/2*d

*x + 1/2*c)^10 + 719040*a^7*b^11*tan(1/2*d*x + 1/2*c)^10 - 355040*a^5*b^13*tan(1/2*d*x + 1/2*c)^10 + 67200*a^3

*b^15*tan(1/2*d*x + 1/2*c)^10 + 4200*a^18*tan(1/2*d*x + 1/2*c)^9 - 304500*a^16*b^2*tan(1/2*d*x + 1/2*c)^9 - 14

18025*a^14*b^4*tan(1/2*d*x + 1/2*c)^9 + 147070*a^12*b^6*tan(1/2*d*x + 1/2*c)^9 - 1316700*a^10*b^8*tan(1/2*d*x

+ 1/2*c)^9 + 242592*a^8*b^10*tan(1/2*d*x + 1/2*c)^9 + 439376*a^6*b^12*tan(1/2*d*x + 1/2*c)^9 - 352128*a^4*b^14

*tan(1/2*d*x + 1/2*c)^9 + 80640*a^2*b^16*tan(1/2*d*x + 1/2*c)^9 - 49000*a^17*b*tan(1/2*d*x + 1/2*c)^8 - 135730

0*a^15*b^3*tan(1/2*d*x + 1/2*c)^8 - 1726305*a^13*b^5*tan(1/2*d*x + 1/2*c)^8 - 346570*a^11*b^7*tan(1/2*d*x + 1/

2*c)^8 - 1972600*a^9*b^9*tan(1/2*d*x + 1/2*c)^8 + 1360128*a^7*b^11*tan(1/2*d*x + 1/2*c)^8 - 298816*a^5*b^13*ta

n(1/2*d*x + 1/2*c)^8 - 122752*a^3*b^15*tan(1/2*d*x + 1/2*c)^8 + 53760*a*b^17*tan(1/2*d*x + 1/2*c)^8 - 509600*a

^16*b^2*tan(1/2*d*x + 1/2*c)^7 - 2685200*a^14*b^4*tan(1/2*d*x + 1/2*c)^7 - 900900*a^12*b^6*tan(1/2*d*x + 1/2*c

)^7 - 2070320*a^10*b^8*tan(1/2*d*x + 1/2*c)^7 - 278096*a^8*b^10*tan(1/2*d*x + 1/2*c)^7 + 952320*a^6*b^12*tan(1

/2*d*x + 1/2*c)^7 - 538112*a^4*b^14*tan(1/2*d*x + 1/2*c)^7 + 68608*a^2*b^16*tan(1/2*d*x + 1/2*c)^7 + 15360*b^1

8*tan(1/2*d*x + 1/2*c)^7 - 78400*a^17*b*tan(1/2*d*x + 1/2*c)^6 - 1607200*a^15*b^3*tan(1/2*d*x + 1/2*c)^6 - 232

6800*a^13*b^5*tan(1/2*d*x + 1/2*c)^6 - 823060*a^11*b^7*tan(1/2*d*x + 1/2*c)^6 - 2094400*a^9*b^9*tan(1/2*d*x +

1/2*c)^6 + 1351728*a^7*b^11*tan(1/2*d*x + 1/2*c)^6 - 298816*a^5*b^13*tan(1/2*d*x + 1/2*c)^6 - 122752*a^3*b^15*

tan(1/2*d*x + 1/2*c)^6 + 53760*a*b^17*tan(1/2*d*x + 1/2*c)^6 - 4200*a^18*tan(1/2*d*x + 1/2*c)^5 - 459900*a^16*

b^2*tan(1/2*d*x + 1/2*c)^5 - 2100175*a^14*b^4*tan(1/2*d*x + 1/2*c)^5 - 647780*a^12*b^6*tan(1/2*d*x + 1/2*c)^5

- 1643880*a^10*b^8*tan(1/2*d*x + 1/2*c)^5 + 228592*a^8*b^10*tan(1/2*d*x + 1/2*c)^5 + 439376*a^6*b^12*tan(1/2*d

*x + 1/2*c)^5 - 352128*a^4*b^14*tan(1/2*d*x + 1/2*c)^5 + 80640*a^2*b^16*tan(1/2*d*x + 1/2*c)^5 - 63000*a^17*b*

tan(1/2*d*x + 1/2*c)^4 - 918540*a^15*b^3*tan(1/2*d*x + 1/2*c)^4 - 858683*a^13*b^5*tan(1/2*d*x + 1/2*c)^4 - 434

644*a^11*b^7*tan(1/2*d*x + 1/2*c)^4 - 634368*a^9*b^9*tan(1/2*d*x + 1/2*c)^4 + 719600*a^7*b^11*tan(1/2*d*x + 1/

2*c)^4 - 355040*a^5*b^13*tan(1/2*d*x + 1/2*c)^4 + 67200*a^3*b^15*tan(1/2*d*x + 1/2*c)^4 - 3360*a^18*tan(1/2*d*

x + 1/2*c)^3 - 211680*a^16*b^2*tan(1/2*d*x + 1/2*c)^3 - 575260*a^14*b^4*tan(1/2*d*x + 1/2*c)^3 + 43918*a^12*b^

6*tan(1/2*d*x + 1/2*c)^3 - 534576*a^10*b^8*tan(1/2*d*x + 1/2*c)^3 + 449008*a^8*b^10*tan(1/2*d*x + 1/2*c)^3 - 1

92640*a^6*b^12*tan(1/2*d*x + 1/2*c)^3 + 33600*a^4*b^14*tan(1/2*d*x + 1/2*c)^3 - 24640*a^17*b*tan(1/2*d*x + 1/2

*c)^2 - 199360*a^15*b^3*tan(1/2*d*x + 1/2*c)^2 + 44604*a^13*b^5*tan(1/2*d*x + 1/2*c)^2 - 186410*a^11*b^7*tan(1

/2*d*x + 1/2*c)^2 + 144928*a^9*b^9*tan(1/2*d*x + 1/2*c)^2 - 59472*a^7*b^11*tan(1/2*d*x + 1/2*c)^2 + 10080*a^5*

b^13*tan(1/2*d*x + 1/2*c)^2 - 840*a^18*tan(1/2*d*x + 1/2*c) - 38780*a^16*b^2*tan(1/2*d*x + 1/2*c) + 12565*a^14

*b^4*tan(1/2*d*x + 1/2*c) - 35322*a^12*b^6*tan(1/2*d*x + 1/2*c) + 25844*a^10*b^8*tan(1/2*d*x + 1/2*c) - 10192*

a^8*b^10*tan(1/2*d*x + 1/2*c) + 1680*a^6*b^12*tan(1/2*d*x + 1/2*c) - 3640*a^17*b + 2660*a^15*b^3 - 4923*a^13*b

^5 + 3646*a^11*b^7 - 1448*a^9*b^9 + 240*a^7*b^11)/((a^19 - 6*a^17*b^2 + 15*a^15*b^4 - 20*a^13*b^6 + 15*a^11*b^

8 - 6*a^9*b^10 + a^7*b^12)*(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^7))/d

Mupad [B] (veriﬁcation not implemented)

Time = 9.78 (sec) , antiderivative size = 2440, normalized size of antiderivative = 5.78 \[ \int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]

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

((3640*a^10*b - 240*b^11 + 1448*a^2*b^9 - 3646*a^4*b^7 + 4923*a^6*b^5 - 2660*a^8*b^3)/(840*(a^12 + b^12 - 6*a^

2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^6*(2800*a^16*b - 1920*b^17

+ 4384*a^2*b^15 + 10672*a^4*b^13 - 48276*a^6*b^11 + 74800*a^8*b^9 + 29395*a^10*b^7 + 83100*a^12*b^5 + 57400*a^

14*b^3))/(30*a^6*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 +

(d*x)/2)^8*(7000*a^16*b - 7680*b^17 + 17536*a^2*b^15 + 42688*a^4*b^13 - 194304*a^6*b^11 + 281800*a^8*b^9 + 495

10*a^10*b^7 + 246615*a^12*b^5 + 193900*a^14*b^3))/(120*a^6*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6

+ 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^10*(192*a^14*b - 960*b^15 + 5072*a^2*b^13 - 10272*a^4*b^11

+ 8880*a^6*b^9 + 955*a^8*b^7 + 2370*a^10*b^5 + 7920*a^12*b^3))/(12*a^4*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8

- 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^4*(9000*a^14*b - 9600*b^15 + 50720*a^2*b^13 - 1

02800*a^4*b^11 + 90624*a^6*b^9 + 62092*a^8*b^7 + 122669*a^10*b^5 + 131220*a^12*b^3))/(120*a^4*(a^12 + b^12 - 6

*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^12*(8*a^12*b - 96*b^13 +

576*a^2*b^11 - 1440*a^4*b^9 + 1920*a^6*b^7 - 1375*a^8*b^5 + 836*a^10*b^3))/(8*a^2*(a^12 + b^12 - 6*a^2*b^10 +

15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^2*(1760*a^12*b - 720*b^13 + 4248*a^

2*b^11 - 10352*a^4*b^9 + 13315*a^6*b^7 - 3186*a^8*b^5 + 14240*a^10*b^3))/(60*a^2*(a^12 + b^12 - 6*a^2*b^10 + 1

5*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)*(120*a^12 - 240*b^12 + 1456*a^2*b^10

- 3692*a^4*b^8 + 5046*a^6*b^6 - 1795*a^8*b^4 + 5540*a^10*b^2))/(120*a*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 -

20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) - (tan(c/2 + (d*x)/2)^9*(600*a^16 + 11520*b^16 - 50304*a^2*b^14 + 6276

8*a^4*b^12 + 34656*a^6*b^10 - 188100*a^8*b^8 + 21010*a^10*b^6 - 202575*a^12*b^4 - 43500*a^14*b^2))/(120*a^5*(a

^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^5*(600*a^1

6 - 11520*b^16 + 50304*a^2*b^14 - 62768*a^4*b^12 - 32656*a^6*b^10 + 234840*a^8*b^8 + 92540*a^10*b^6 + 300025*a

^12*b^4 + 65700*a^14*b^2))/(120*a^5*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*

b^2)) - (tan(c/2 + (d*x)/2)^11*(48*a^14 + 480*b^14 - 2752*a^2*b^12 + 6432*a^4*b^10 - 7680*a^6*b^8 + 4105*a^8*b

^6 - 3150*a^10*b^4 - 1344*a^12*b^2))/(12*a^3*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4

- 6*a^10*b^2)) + (tan(c/2 + (d*x)/2)^3*(240*a^14 - 2400*b^14 + 13760*a^2*b^12 - 32072*a^4*b^10 + 38184*a^6*b^8

- 3137*a^8*b^6 + 41090*a^10*b^4 + 15120*a^12*b^2))/(60*a^3*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^

6 + 15*a^8*b^4 - 6*a^10*b^2)) - (tan(c/2 + (d*x)/2)^13*(8*a^12 + 16*b^12 - 96*a^2*b^10 + 240*a^4*b^8 - 320*a^6

*b^6 + 235*a^8*b^4 - 116*a^10*b^2))/(8*a*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*

a^10*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)*(3640*a^10*b - 240*b^11 + 1

448*a^2*b^9 - 3646*a^4*b^7 + 4923*a^6*b^5 - 2660*a^8*b^3))/(210*a^7*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 2

0*a^6*b^6 + 15*a^8*b^4 - 6*a^10*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)

+ tan(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*tan(c/2 + (d*x)/2)^13)) + (a*atan((8*((a^2*tan(c/2 + (d*x)/2)*(8*a^4 + 5*b^4 + 20*a^2*b^2))/(8*(a + b)^(13/2

)*(a - b)^(13/2)) + (a*(8*a^4 + 5*b^4 + 20*a^2*b^2)*(16*a^12*b + 16*b^13 - 96*a^2*b^11 + 240*a^4*b^9 - 320*a^6

*b^7 + 240*a^8*b^5 - 96*a^10*b^3))/(128*(a + b)^(13/2)*(a - b)^(13/2)*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 -

20*a^6*b^6 + 15*a^8*b^4 - 6*a^10*b^2)))*(a^12 + b^12 - 6*a^2*b^10 + 15*a^4*b^8 - 20*a^6*b^6 + 15*a^8*b^4 - 6*

a^10*b^2))/(5*a*b^4 + 8*a^5 + 20*a^3*b^2))*(8*a^4 + 5*b^4 + 20*a^2*b^2))/(8*d*(a + b)^(13/2)*(a - b)^(13/2))

\(\int \frac {\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [470]

Optimal result