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

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

[Out]

(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
in[c + d*x])^5)

________________________________________________________________________________________

Rubi [A]  time = 0.79152, antiderivative size = 407, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2693, 2863, 2754, 12, 2660, 618, 204} \[ \frac{5 a \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{8 d \left (a^2-b^2\right )^{9/2}}-\frac{\cos (c+d x) \left (4 a^2+10 a b \sin (c+d x)+9 b^2\right )}{42 b^5 d (a+b \sin (c+d x))^5}+\frac{\left (-38 a^4 b^2+87 a^2 b^4+8 a^6+48 b^6\right ) \cos (c+d x)}{336 b^5 d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac{a \left (-30 a^2 b^2+8 a^4+57 b^4\right ) \cos (c+d x)}{336 b^5 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac{\left (-9 a^2 b^2+4 a^4+12 b^4\right ) \cos (c+d x)}{168 b^5 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^3}+\frac{a \left (4 a^2-b^2\right ) \cos (c+d x)}{168 b^5 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^4}+\frac{5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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
in[c + d*x])^5)

Rule 2693

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

Rule 2863

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

Rule 2754

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

Rule 12

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

Rule 2660

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

Rule 618

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

Rule 204

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

Rubi steps

\begin{align*} \int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=-\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) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{8 \left (a^2-b^2\right )^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) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{4 \left (a^2-b^2\right )^4 d}\\ &=\frac{5 a \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{8 \left (a^2-b^2\right )^{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]  time = 6.04288, size = 552, normalized size = 1.36 \[ \frac{\cos ^7(c+d x)}{7 d (a-b) (a+b \sin (c+d x))^7}+\frac{a \cos (c+d x) \left (-\frac{(1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{9/2}}{7 (b-a) (a+b \sin (c+d x))^7}-\frac{5 \left (-\frac{(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{9/2}}{6 (b-a) (a+b \sin (c+d x))^6}-\frac{-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{9/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac{\frac{7 \left (\frac{5 \left (\frac{3 \left (\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{b-a} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right )}{(-a-b)^{3/2} \sqrt{b-a}}\right )}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right )}{3 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}\right )}{4 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (a+b) (a+b \sin (c+d x))^4}}{5 (b-a)}}{2 (b-a)}\right )}{7 (b-a)}\right )}{d (a-b) \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.211, size = 6933, normalized size = 17. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 6.25552, size = 5125, normalized size = 12.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[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)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.55771, size = 2228, normalized size = 5.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))
/d