3.1275 \(\int \frac{\cot ^6(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=600 \[ -\frac{3 b^2 \sqrt{a^2-b^2} \left (-23 a^2 b^2+4 a^4+24 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^{10} d}+\frac{\left (-889 a^4 b^2+3255 a^2 b^4+10 a^6-2520 b^6\right ) \cot (c+d x)}{70 a^9 d}-\frac{3 b \left (-100 a^4 b^2+280 a^2 b^4+5 a^6-192 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}-\frac{3 \left (-185 a^2 b^2+35 a^4+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}+\frac{\left (-81 a^2 b^2+16 a^4+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{\left (-973 a^2 b^2+205 a^4+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{3 b \left (-116 a^2 b^2+27 a^4+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}+\frac{\left (-65 a^2 b^2+12 a^4+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left (-35 a^2 b^2+7 a^4+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2} \]

[Out]

(-3*b^2*Sqrt[a^2 - b^2]*(4*a^4 - 23*a^2*b^2 + 24*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^10*
d) - (3*b*(5*a^6 - 100*a^4*b^2 + 280*a^2*b^4 - 192*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^10*d) + ((10*a^6 - 889*a^
4*b^2 + 3255*a^2*b^4 - 2520*b^6)*Cot[c + d*x])/(70*a^9*d) + (3*b*(27*a^4 - 116*a^2*b^2 + 96*b^4)*Cot[c + d*x]*
Csc[c + d*x])/(16*a^8*d) - ((205*a^4 - 973*a^2*b^2 + 840*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(70*a^7*d) + ((16*a
^4 - 81*a^2*b^2 + 72*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^6*b*d) - (3*(35*a^4 - 185*a^2*b^2 + 168*b^4)*Cot[c
+ d*x]*Csc[c + d*x]^4)/(70*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(5*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot
[c + d*x]*Csc[c + d*x]^4)/(10*b^2*d*(a + b*Sin[c + d*x])^2) + ((7*a^4 - 35*a^2*b^2 + 30*b^4)*Cot[c + d*x]*Csc[
c + d*x]^4)/(35*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (3*b*Cot[c + d*x]*Csc[c + d*x]^5)/(14*a^2*d*(a + b*Sin[c +
d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d*(a + b*Sin[c + d*x])^2) + ((12*a^4 - 65*a^2*b^2 + 60*b^4)*Cot
[c + d*x]*Csc[c + d*x]^4)/(10*a^4*b^2*d*(a + b*Sin[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 3.22025, antiderivative size = 600, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2896, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac{3 b^2 \sqrt{a^2-b^2} \left (-23 a^2 b^2+4 a^4+24 b^4\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^{10} d}+\frac{\left (-889 a^4 b^2+3255 a^2 b^4+10 a^6-2520 b^6\right ) \cot (c+d x)}{70 a^9 d}-\frac{3 b \left (-100 a^4 b^2+280 a^2 b^4+5 a^6-192 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}-\frac{3 \left (-185 a^2 b^2+35 a^4+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}+\frac{\left (-81 a^2 b^2+16 a^4+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{\left (-973 a^2 b^2+205 a^4+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{3 b \left (-116 a^2 b^2+27 a^4+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}+\frac{\left (-65 a^2 b^2+12 a^4+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left (-35 a^2 b^2+7 a^4+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cot[c + d*x]^6*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]

[Out]

(-3*b^2*Sqrt[a^2 - b^2]*(4*a^4 - 23*a^2*b^2 + 24*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^10*
d) - (3*b*(5*a^6 - 100*a^4*b^2 + 280*a^2*b^4 - 192*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^10*d) + ((10*a^6 - 889*a^
4*b^2 + 3255*a^2*b^4 - 2520*b^6)*Cot[c + d*x])/(70*a^9*d) + (3*b*(27*a^4 - 116*a^2*b^2 + 96*b^4)*Cot[c + d*x]*
Csc[c + d*x])/(16*a^8*d) - ((205*a^4 - 973*a^2*b^2 + 840*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(70*a^7*d) + ((16*a
^4 - 81*a^2*b^2 + 72*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(8*a^6*b*d) - (3*(35*a^4 - 185*a^2*b^2 + 168*b^4)*Cot[c
+ d*x]*Csc[c + d*x]^4)/(70*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(5*b*d*(a + b*Sin[c + d*x])^2) + (a*Cot
[c + d*x]*Csc[c + d*x]^4)/(10*b^2*d*(a + b*Sin[c + d*x])^2) + ((7*a^4 - 35*a^2*b^2 + 30*b^4)*Cot[c + d*x]*Csc[
c + d*x]^4)/(35*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (3*b*Cot[c + d*x]*Csc[c + d*x]^5)/(14*a^2*d*(a + b*Sin[c +
d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d*(a + b*Sin[c + d*x])^2) + ((12*a^4 - 65*a^2*b^2 + 60*b^4)*Cot
[c + d*x]*Csc[c + d*x]^4)/(10*a^4*b^2*d*(a + b*Sin[c + d*x]))

Rule 2896

Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] +
(Dist[1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*
x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m +
n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e +
f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(
n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(d*
Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*d^2*f*(n + 1)*(n + 2)), x] - Simp[(a*(n + 5)*Cos[e +
f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d^3*f*(m + n + 5)*(m + n + 6)), x] + Simp[(Co
s[e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1))/(b*d^4*f*(m + n + 6)), x]) /; FreeQ[{a, b, d
, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0]
&& NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, 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{\cot ^6(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^6(c+d x) \left (90 \left (7 a^4-30 a^2 b^2+24 b^4\right )-18 a b \left (7 a^2-5 b^2\right ) \sin (c+d x)-126 \left (4 a^4-18 a^2 b^2+15 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{1260 a^2 b^2}\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\int \frac{\csc ^6(c+d x) \left (180 \left (21 a^6-121 a^4 b^2+184 a^2 b^4-84 b^6\right )-36 a b \left (14 a^4-29 a^2 b^2+15 b^4\right ) \sin (c+d x)-432 \left (7 a^6-42 a^4 b^2+65 a^2 b^4-30 b^6\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2520 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^6(c+d x) \left (540 \left (a^2-b^2\right )^2 \left (35 a^4-185 a^2 b^2+168 b^4\right )-180 a b \left (7 a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-1260 \left (a^2-b^2\right )^2 \left (12 a^4-65 a^2 b^2+60 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2520 a^4 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^5(c+d x) \left (-6300 b \left (a^2-b^2\right )^2 \left (16 a^4-81 a^2 b^2+72 b^4\right )+180 a b^2 \left (55 a^2-84 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+2160 b \left (a^2-b^2\right )^2 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12600 a^5 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^4(c+d x) \left (2160 b^2 \left (a^2-b^2\right )^2 \left (205 a^4-973 a^2 b^2+840 b^4\right )-540 a b^3 \left (125 a^2-168 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-18900 b^2 \left (a^2-b^2\right )^2 \left (16 a^4-81 a^2 b^2+72 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{50400 a^6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (205 a^4-973 a^2 b^2+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^3(c+d x) \left (-56700 b^3 \left (a^2-b^2\right )^2 \left (27 a^4-116 a^2 b^2+96 b^4\right )-540 a b^2 \left (a^2-b^2\right )^2 \left (40 a^4-721 a^2 b^2+840 b^4\right ) \sin (c+d x)+4320 b^3 \left (a^2-b^2\right )^2 \left (205 a^4-973 a^2 b^2+840 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{151200 a^7 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{3 b \left (27 a^4-116 a^2 b^2+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}-\frac{\left (205 a^4-973 a^2 b^2+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (-4320 b^2 \left (a^2-b^2\right )^2 \left (10 a^6-889 a^4 b^2+3255 a^2 b^4-2520 b^6\right )+540 a b^3 \left (a^2-b^2\right )^2 \left (445 a^4-3388 a^2 b^2+3360 b^4\right ) \sin (c+d x)-56700 b^4 \left (a^2-b^2\right )^2 \left (27 a^4-116 a^2 b^2+96 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{302400 a^8 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (10 a^6-889 a^4 b^2+3255 a^2 b^4-2520 b^6\right ) \cot (c+d x)}{70 a^9 d}+\frac{3 b \left (27 a^4-116 a^2 b^2+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}-\frac{\left (205 a^4-973 a^2 b^2+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (56700 b^3 \left (a^2-b^2\right )^2 \left (5 a^6-100 a^4 b^2+280 a^2 b^4-192 b^6\right )-56700 a b^4 \left (a^2-b^2\right )^2 \left (27 a^4-116 a^2 b^2+96 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{302400 a^9 b^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (10 a^6-889 a^4 b^2+3255 a^2 b^4-2520 b^6\right ) \cot (c+d x)}{70 a^9 d}+\frac{3 b \left (27 a^4-116 a^2 b^2+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}-\frac{\left (205 a^4-973 a^2 b^2+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}-\frac{\left (3 b^2 \left (a^2-b^2\right ) \left (4 a^4-23 a^2 b^2+24 b^4\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{2 a^{10}}+\frac{\left (3 b \left (5 a^6-100 a^4 b^2+280 a^2 b^4-192 b^6\right )\right ) \int \csc (c+d x) \, dx}{16 a^{10}}\\ &=-\frac{3 b \left (5 a^6-100 a^4 b^2+280 a^2 b^4-192 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}+\frac{\left (10 a^6-889 a^4 b^2+3255 a^2 b^4-2520 b^6\right ) \cot (c+d x)}{70 a^9 d}+\frac{3 b \left (27 a^4-116 a^2 b^2+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}-\frac{\left (205 a^4-973 a^2 b^2+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}-\frac{\left (3 b^2 \left (a^2-b^2\right ) \left (4 a^4-23 a^2 b^2+24 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^{10} d}\\ &=-\frac{3 b \left (5 a^6-100 a^4 b^2+280 a^2 b^4-192 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}+\frac{\left (10 a^6-889 a^4 b^2+3255 a^2 b^4-2520 b^6\right ) \cot (c+d x)}{70 a^9 d}+\frac{3 b \left (27 a^4-116 a^2 b^2+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}-\frac{\left (205 a^4-973 a^2 b^2+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}+\frac{\left (6 b^2 \left (a^2-b^2\right ) \left (4 a^4-23 a^2 b^2+24 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^{10} d}\\ &=-\frac{3 b^2 \sqrt{a^2-b^2} \left (4 a^4-23 a^2 b^2+24 b^4\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^{10} d}-\frac{3 b \left (5 a^6-100 a^4 b^2+280 a^2 b^4-192 b^6\right ) \tanh ^{-1}(\cos (c+d x))}{16 a^{10} d}+\frac{\left (10 a^6-889 a^4 b^2+3255 a^2 b^4-2520 b^6\right ) \cot (c+d x)}{70 a^9 d}+\frac{3 b \left (27 a^4-116 a^2 b^2+96 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^8 d}-\frac{\left (205 a^4-973 a^2 b^2+840 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{70 a^7 d}+\frac{\left (16 a^4-81 a^2 b^2+72 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{8 a^6 b d}-\frac{3 \left (35 a^4-185 a^2 b^2+168 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{70 a^5 b^2 d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{5 b d (a+b \sin (c+d x))^2}+\frac{a \cot (c+d x) \csc ^4(c+d x)}{10 b^2 d (a+b \sin (c+d x))^2}+\frac{\left (7 a^4-35 a^2 b^2+30 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{35 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac{3 b \cot (c+d x) \csc ^5(c+d x)}{14 a^2 d (a+b \sin (c+d x))^2}-\frac{\cot (c+d x) \csc ^6(c+d x)}{7 a d (a+b \sin (c+d x))^2}+\frac{\left (12 a^4-65 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{10 a^4 b^2 d (a+b \sin (c+d x))}\\ \end{align*}

Mathematica [A]  time = 2.96027, size = 728, normalized size = 1.21 \[ \frac{\frac{215040 b^2 \left (27 a^4 b^2-47 a^2 b^4-4 a^6+24 b^6\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{\sqrt{a^2-b^2}}+13440 b \left (-100 a^4 b^2+280 a^2 b^4+5 a^6-192 b^6\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+13440 b \left (100 a^4 b^2-280 a^2 b^4-5 a^6+192 b^6\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\frac{a \csc ^9(c+d x) \left (194334 a^5 b^3 \sin (2 (c+d x))-190582 a^5 b^3 \sin (4 (c+d x))+77462 a^5 b^3 \sin (6 (c+d x))-11389 a^5 b^3 \sin (8 (c+d x))-592200 a^3 b^5 \sin (2 (c+d x))+585480 a^3 b^5 \sin (4 (c+d x))-246120 a^3 b^5 \sin (6 (c+d x))+39900 a^3 b^5 \sin (8 (c+d x))+22948 a^6 b^2 \cos (5 (c+d x))-5884 a^6 b^2 \cos (7 (c+d x))-40 a^6 b^2 \cos (9 (c+d x))-18144 a^4 b^4 \cos (5 (c+d x))-5964 a^4 b^4 \cos (7 (c+d x))+3556 a^4 b^4 \cos (9 (c+d x))-193200 a^2 b^6 \cos (5 (c+d x))+77700 a^2 b^6 \cos (7 (c+d x))-13020 a^2 b^6 \cos (9 (c+d x))+28 \left (795 a^6 b^2-1218 a^4 b^4-4110 a^2 b^6+200 a^8+5040 b^8\right ) \cos (c+d x)+28 \left (-1403 a^6 b^2+1952 a^4 b^4+8700 a^2 b^6+120 a^8-10080 b^8\right ) \cos (3 (c+d x))-9660 a^7 b \sin (2 (c+d x))+6160 a^7 b \sin (4 (c+d x))-3660 a^7 b \sin (6 (c+d x))+160 a^7 b \sin (8 (c+d x))+1120 a^8 \cos (5 (c+d x))+160 a^8 \cos (7 (c+d x))+423360 a b^7 \sin (2 (c+d x))-423360 a b^7 \sin (4 (c+d x))+181440 a b^7 \sin (6 (c+d x))-30240 a b^7 \sin (8 (c+d x))+201600 b^8 \cos (5 (c+d x))-70560 b^8 \cos (7 (c+d x))+10080 b^8 \cos (9 (c+d x))\right )}{(a \csc (c+d x)+b)^2}}{71680 a^{10} d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cot[c + d*x]^6*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^3,x]

[Out]

((215040*b^2*(-4*a^6 + 27*a^4*b^2 - 47*a^2*b^4 + 24*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqr
t[a^2 - b^2] + 13440*b*(-5*a^6 + 100*a^4*b^2 - 280*a^2*b^4 + 192*b^6)*Log[Cos[(c + d*x)/2]] + 13440*b*(5*a^6 -
100*a^4*b^2 + 280*a^2*b^4 - 192*b^6)*Log[Sin[(c + d*x)/2]] - (a*Csc[c + d*x]^9*(28*(200*a^8 + 795*a^6*b^2 - 1
218*a^4*b^4 - 4110*a^2*b^6 + 5040*b^8)*Cos[c + d*x] + 28*(120*a^8 - 1403*a^6*b^2 + 1952*a^4*b^4 + 8700*a^2*b^6
- 10080*b^8)*Cos[3*(c + d*x)] + 1120*a^8*Cos[5*(c + d*x)] + 22948*a^6*b^2*Cos[5*(c + d*x)] - 18144*a^4*b^4*Co
s[5*(c + d*x)] - 193200*a^2*b^6*Cos[5*(c + d*x)] + 201600*b^8*Cos[5*(c + d*x)] + 160*a^8*Cos[7*(c + d*x)] - 58
84*a^6*b^2*Cos[7*(c + d*x)] - 5964*a^4*b^4*Cos[7*(c + d*x)] + 77700*a^2*b^6*Cos[7*(c + d*x)] - 70560*b^8*Cos[7
*(c + d*x)] - 40*a^6*b^2*Cos[9*(c + d*x)] + 3556*a^4*b^4*Cos[9*(c + d*x)] - 13020*a^2*b^6*Cos[9*(c + d*x)] + 1
0080*b^8*Cos[9*(c + d*x)] - 9660*a^7*b*Sin[2*(c + d*x)] + 194334*a^5*b^3*Sin[2*(c + d*x)] - 592200*a^3*b^5*Sin
[2*(c + d*x)] + 423360*a*b^7*Sin[2*(c + d*x)] + 6160*a^7*b*Sin[4*(c + d*x)] - 190582*a^5*b^3*Sin[4*(c + d*x)]
+ 585480*a^3*b^5*Sin[4*(c + d*x)] - 423360*a*b^7*Sin[4*(c + d*x)] - 3660*a^7*b*Sin[6*(c + d*x)] + 77462*a^5*b^
3*Sin[6*(c + d*x)] - 246120*a^3*b^5*Sin[6*(c + d*x)] + 181440*a*b^7*Sin[6*(c + d*x)] + 160*a^7*b*Sin[8*(c + d*
x)] - 11389*a^5*b^3*Sin[8*(c + d*x)] + 39900*a^3*b^5*Sin[8*(c + d*x)] - 30240*a*b^7*Sin[8*(c + d*x)]))/(b + a*
Csc[c + d*x])^2)/(71680*a^10*d)

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Maple [B]  time = 0.252, size = 1576, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*csc(d*x+c)^8/(a+b*sin(d*x+c))^3,x)

[Out]

14/d/a^9*b^6*tan(1/2*d*x+1/2*c)+5/32/d/a^6*b^3/tan(1/2*d*x+1/2*c)^4+21/8/d/a^8*b^5/tan(1/2*d*x+1/2*c)^2-36/d/a
^10*b^7*ln(tan(1/2*d*x+1/2*c))-17/d*b^7/a^8/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2-3/80/d/a^5/tan
(1/2*d*x+1/2*c)^5*b^2-5/8/d/a^7/tan(1/2*d*x+1/2*c)^3*b^4-14/d/a^9/tan(1/2*d*x+1/2*c)*b^6+1/128/d/a^4*b/tan(1/2
*d*x+1/2*c)^6-1/128/d/a^4*b*tan(1/2*d*x+1/2*c)^6+3/80/d/a^5*tan(1/2*d*x+1/2*c)^5*b^2-5/32/d/a^6*tan(1/2*d*x+1/
2*c)^4*b^3+5/8/d/a^7*tan(1/2*d*x+1/2*c)^3*b^4-21/8/d/a^8*tan(1/2*d*x+1/2*c)^2*b^5-12/d/a^4/(a^2-b^2)^(1/2)*arc
tan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))*b^2-5/128/d/a^3*tan(1/2*d*x+1/2*c)+5/128/d/a^3/tan(1/2*d
*x+1/2*c)+105/2/d/a^8*b^5*ln(tan(1/2*d*x+1/2*c))+25/d/a^6/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*
b^5+7/16/d/a^5/tan(1/2*d*x+1/2*c)^3*b^2+9/128/d/a^4*tan(1/2*d*x+1/2*c)^4*b-7/16/d/a^5*tan(1/2*d*x+1/2*c)^3*b^2
+5/2/d/a^6*tan(1/2*d*x+1/2*c)^2*b^3-135/8/d/a^7*b^4*tan(1/2*d*x+1/2*c)+135/8/d/a^7/tan(1/2*d*x+1/2*c)*b^4-9/12
8/d/a^4*b/tan(1/2*d*x+1/2*c)^4-5/2/d/a^6*b^3/tan(1/2*d*x+1/2*c)^2+1/896/d/a^3*tan(1/2*d*x+1/2*c)^7-1/896/d/a^3
/tan(1/2*d*x+1/2*c)^7+1/128/d/a^3/tan(1/2*d*x+1/2*c)^5-1/128/d/a^3*tan(1/2*d*x+1/2*c)^5+15/16/d/a^4*b*ln(tan(1
/2*d*x+1/2*c))-18/d*b^8/a^9/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3-34/d*b^9/
a^10/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2-50/d*b^8/a^9/(tan(1/2*d*x+1/2*c)
^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)+72/d*b^8/a^10/(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))-8/d/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*b^3-33/8/d/a^5/tan(1/
2*d*x+1/2*c)*b^2+45/128/d/a^4*b/tan(1/2*d*x+1/2*c)^2-75/4/d/a^6*b^3*ln(tan(1/2*d*x+1/2*c))+73/d/a^7/(tan(1/2*d
*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)*b^6-141/d/a^8/(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))*b^6+27/d/a^7/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2
*d*x+1/2*c)^3*b^6+33/d/a^8/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*b^7-3/128/
d/a^3/tan(1/2*d*x+1/2*c)^3-9/d/a^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^3*b^
4+3/128/d/a^3*tan(1/2*d*x+1/2*c)^3-8/d/a^4/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2
*c)^2*b^3-45/128/d/a^4*tan(1/2*d*x+1/2*c)^2*b+33/8/d/a^5*b^2*tan(1/2*d*x+1/2*c)+9/d/a^6/(tan(1/2*d*x+1/2*c)^2*
a+2*tan(1/2*d*x+1/2*c)*b+a)^2*tan(1/2*d*x+1/2*c)^2*b^5-23/d/a^5/(tan(1/2*d*x+1/2*c)^2*a+2*tan(1/2*d*x+1/2*c)*b
+a)^2*tan(1/2*d*x+1/2*c)*b^4+81/d/a^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))
*b^4

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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*csc(d*x+c)^8/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 6.90324, size = 8743, normalized size = 14.57 \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*csc(d*x+c)^8/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/1120*(16*(10*a^7*b^2 - 889*a^5*b^4 + 3255*a^3*b^6 - 2520*a*b^8)*cos(d*x + c)^9 - 4*(40*a^9 - 1381*a^7*b^2 -
9492*a^5*b^4 + 48720*a^3*b^6 - 40320*a*b^8)*cos(d*x + c)^7 - 28*(563*a^7*b^2 + 1068*a^5*b^4 - 9720*a^3*b^6 +
8640*a*b^8)*cos(d*x + c)^5 + 140*(105*a^7*b^2 + 20*a^5*b^4 - 1200*a^3*b^6 + 1152*a*b^8)*cos(d*x + c)^3 + 840*(
2*(4*a^5*b^3 - 23*a^3*b^5 + 24*a*b^7)*cos(d*x + c)^8 + 8*a^5*b^3 - 46*a^3*b^5 + 48*a*b^7 - 8*(4*a^5*b^3 - 23*a
^3*b^5 + 24*a*b^7)*cos(d*x + c)^6 + 12*(4*a^5*b^3 - 23*a^3*b^5 + 24*a*b^7)*cos(d*x + c)^4 - 8*(4*a^5*b^3 - 23*
a^3*b^5 + 24*a*b^7)*cos(d*x + c)^2 + ((4*a^4*b^4 - 23*a^2*b^6 + 24*b^8)*cos(d*x + c)^8 + 4*a^6*b^2 - 19*a^4*b^
4 + a^2*b^6 + 24*b^8 - (4*a^6*b^2 - 7*a^4*b^4 - 68*a^2*b^6 + 96*b^8)*cos(d*x + c)^6 + 3*(4*a^6*b^2 - 15*a^4*b^
4 - 22*a^2*b^6 + 48*b^8)*cos(d*x + c)^4 - (12*a^6*b^2 - 53*a^4*b^4 - 20*a^2*b^6 + 96*b^8)*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*(11*a^7*b^2 - 8*a^5*b^4 - 92*a^3*b^6 + 96*a*b^8)*cos(d*x + c) - 105*(10*a^7*b^2 - 200*a^5*b^4 + 560*a^3*b^
6 - 384*a*b^8 + 2*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^8 - 8*(5*a^7*b^2 - 100*a^5*
b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^6 + 12*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x
+ c)^4 - 8*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^2 + (5*a^8*b - 95*a^6*b^3 + 180*a
^4*b^5 + 88*a^2*b^7 - 192*b^9 + (5*a^6*b^3 - 100*a^4*b^5 + 280*a^2*b^7 - 192*b^9)*cos(d*x + c)^8 - (5*a^8*b -
80*a^6*b^3 - 120*a^4*b^5 + 928*a^2*b^7 - 768*b^9)*cos(d*x + c)^6 + 3*(5*a^8*b - 90*a^6*b^3 + 80*a^4*b^5 + 368*
a^2*b^7 - 384*b^9)*cos(d*x + c)^4 - (15*a^8*b - 280*a^6*b^3 + 440*a^4*b^5 + 544*a^2*b^7 - 768*b^9)*cos(d*x + c
)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 105*(10*a^7*b^2 - 200*a^5*b^4 + 560*a^3*b^6 - 384*a*b^8 + 2*(
5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^8 - 8*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 -
192*a*b^8)*cos(d*x + c)^6 + 12*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^4 - 8*(5*a^7*
b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^2 + (5*a^8*b - 95*a^6*b^3 + 180*a^4*b^5 + 88*a^2*b^7
- 192*b^9 + (5*a^6*b^3 - 100*a^4*b^5 + 280*a^2*b^7 - 192*b^9)*cos(d*x + c)^8 - (5*a^8*b - 80*a^6*b^3 - 120*a^
4*b^5 + 928*a^2*b^7 - 768*b^9)*cos(d*x + c)^6 + 3*(5*a^8*b - 90*a^6*b^3 + 80*a^4*b^5 + 368*a^2*b^7 - 384*b^9)*
cos(d*x + c)^4 - (15*a^8*b - 280*a^6*b^3 + 440*a^4*b^5 + 544*a^2*b^7 - 768*b^9)*cos(d*x + c)^2)*sin(d*x + c))*
log(-1/2*cos(d*x + c) + 1/2) - 2*((160*a^8*b - 11389*a^6*b^3 + 39900*a^4*b^5 - 30240*a^2*b^7)*cos(d*x + c)^7 -
7*(165*a^8*b - 5207*a^6*b^3 + 17340*a^4*b^5 - 12960*a^2*b^7)*cos(d*x + c)^5 + 35*(40*a^8*b - 1097*a^6*b^3 + 3
516*a^4*b^5 - 2592*a^2*b^7)*cos(d*x + c)^3 - 105*(5*a^8*b - 127*a^6*b^3 + 396*a^4*b^5 - 288*a^2*b^7)*cos(d*x +
c))*sin(d*x + c))/(2*a^11*b*d*cos(d*x + c)^8 - 8*a^11*b*d*cos(d*x + c)^6 + 12*a^11*b*d*cos(d*x + c)^4 - 8*a^1
1*b*d*cos(d*x + c)^2 + 2*a^11*b*d + (a^10*b^2*d*cos(d*x + c)^8 - (a^12 + 4*a^10*b^2)*d*cos(d*x + c)^6 + 3*(a^1
2 + 2*a^10*b^2)*d*cos(d*x + c)^4 - (3*a^12 + 4*a^10*b^2)*d*cos(d*x + c)^2 + (a^12 + a^10*b^2)*d)*sin(d*x + c))
, 1/1120*(16*(10*a^7*b^2 - 889*a^5*b^4 + 3255*a^3*b^6 - 2520*a*b^8)*cos(d*x + c)^9 - 4*(40*a^9 - 1381*a^7*b^2
- 9492*a^5*b^4 + 48720*a^3*b^6 - 40320*a*b^8)*cos(d*x + c)^7 - 28*(563*a^7*b^2 + 1068*a^5*b^4 - 9720*a^3*b^6 +
8640*a*b^8)*cos(d*x + c)^5 + 140*(105*a^7*b^2 + 20*a^5*b^4 - 1200*a^3*b^6 + 1152*a*b^8)*cos(d*x + c)^3 + 1680
*(2*(4*a^5*b^3 - 23*a^3*b^5 + 24*a*b^7)*cos(d*x + c)^8 + 8*a^5*b^3 - 46*a^3*b^5 + 48*a*b^7 - 8*(4*a^5*b^3 - 23
*a^3*b^5 + 24*a*b^7)*cos(d*x + c)^6 + 12*(4*a^5*b^3 - 23*a^3*b^5 + 24*a*b^7)*cos(d*x + c)^4 - 8*(4*a^5*b^3 - 2
3*a^3*b^5 + 24*a*b^7)*cos(d*x + c)^2 + ((4*a^4*b^4 - 23*a^2*b^6 + 24*b^8)*cos(d*x + c)^8 + 4*a^6*b^2 - 19*a^4*
b^4 + a^2*b^6 + 24*b^8 - (4*a^6*b^2 - 7*a^4*b^4 - 68*a^2*b^6 + 96*b^8)*cos(d*x + c)^6 + 3*(4*a^6*b^2 - 15*a^4*
b^4 - 22*a^2*b^6 + 48*b^8)*cos(d*x + c)^4 - (12*a^6*b^2 - 53*a^4*b^4 - 20*a^2*b^6 + 96*b^8)*cos(d*x + c)^2)*si
n(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 420*(11*a^7*b^2 - 8
*a^5*b^4 - 92*a^3*b^6 + 96*a*b^8)*cos(d*x + c) - 105*(10*a^7*b^2 - 200*a^5*b^4 + 560*a^3*b^6 - 384*a*b^8 + 2*(
5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^8 - 8*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 -
192*a*b^8)*cos(d*x + c)^6 + 12*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^4 - 8*(5*a^7*
b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^2 + (5*a^8*b - 95*a^6*b^3 + 180*a^4*b^5 + 88*a^2*b^7
- 192*b^9 + (5*a^6*b^3 - 100*a^4*b^5 + 280*a^2*b^7 - 192*b^9)*cos(d*x + c)^8 - (5*a^8*b - 80*a^6*b^3 - 120*a^
4*b^5 + 928*a^2*b^7 - 768*b^9)*cos(d*x + c)^6 + 3*(5*a^8*b - 90*a^6*b^3 + 80*a^4*b^5 + 368*a^2*b^7 - 384*b^9)*
cos(d*x + c)^4 - (15*a^8*b - 280*a^6*b^3 + 440*a^4*b^5 + 544*a^2*b^7 - 768*b^9)*cos(d*x + c)^2)*sin(d*x + c))*
log(1/2*cos(d*x + c) + 1/2) + 105*(10*a^7*b^2 - 200*a^5*b^4 + 560*a^3*b^6 - 384*a*b^8 + 2*(5*a^7*b^2 - 100*a^5
*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^8 - 8*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x
+ c)^6 + 12*(5*a^7*b^2 - 100*a^5*b^4 + 280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^4 - 8*(5*a^7*b^2 - 100*a^5*b^4 +
280*a^3*b^6 - 192*a*b^8)*cos(d*x + c)^2 + (5*a^8*b - 95*a^6*b^3 + 180*a^4*b^5 + 88*a^2*b^7 - 192*b^9 + (5*a^6
*b^3 - 100*a^4*b^5 + 280*a^2*b^7 - 192*b^9)*cos(d*x + c)^8 - (5*a^8*b - 80*a^6*b^3 - 120*a^4*b^5 + 928*a^2*b^7
- 768*b^9)*cos(d*x + c)^6 + 3*(5*a^8*b - 90*a^6*b^3 + 80*a^4*b^5 + 368*a^2*b^7 - 384*b^9)*cos(d*x + c)^4 - (1
5*a^8*b - 280*a^6*b^3 + 440*a^4*b^5 + 544*a^2*b^7 - 768*b^9)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x +
c) + 1/2) - 2*((160*a^8*b - 11389*a^6*b^3 + 39900*a^4*b^5 - 30240*a^2*b^7)*cos(d*x + c)^7 - 7*(165*a^8*b - 520
7*a^6*b^3 + 17340*a^4*b^5 - 12960*a^2*b^7)*cos(d*x + c)^5 + 35*(40*a^8*b - 1097*a^6*b^3 + 3516*a^4*b^5 - 2592*
a^2*b^7)*cos(d*x + c)^3 - 105*(5*a^8*b - 127*a^6*b^3 + 396*a^4*b^5 - 288*a^2*b^7)*cos(d*x + c))*sin(d*x + c))/
(2*a^11*b*d*cos(d*x + c)^8 - 8*a^11*b*d*cos(d*x + c)^6 + 12*a^11*b*d*cos(d*x + c)^4 - 8*a^11*b*d*cos(d*x + c)^
2 + 2*a^11*b*d + (a^10*b^2*d*cos(d*x + c)^8 - (a^12 + 4*a^10*b^2)*d*cos(d*x + c)^6 + 3*(a^12 + 2*a^10*b^2)*d*c
os(d*x + c)^4 - (3*a^12 + 4*a^10*b^2)*d*cos(d*x + c)^2 + (a^12 + a^10*b^2)*d)*sin(d*x + c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**8/(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.32264, size = 1374, normalized size = 2.29 \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*csc(d*x+c)^8/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/4480*(840*(5*a^6*b - 100*a^4*b^3 + 280*a^2*b^5 - 192*b^7)*log(abs(tan(1/2*d*x + 1/2*c)))/a^10 - 13440*(4*a^6
*b^2 - 27*a^4*b^4 + 47*a^2*b^6 - 24*b^8)*(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)))/(sqrt(a^2 - b^2)*a^10) - 4480*(9*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 27*a^3*b^6*tan(1
/2*d*x + 1/2*c)^3 + 18*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 8*a^6*b^3*tan(1/2*d*x + 1/2*c)^2 - 9*a^4*b^5*tan(1/2*d*x
+ 1/2*c)^2 - 33*a^2*b^7*tan(1/2*d*x + 1/2*c)^2 + 34*b^9*tan(1/2*d*x + 1/2*c)^2 + 23*a^5*b^4*tan(1/2*d*x + 1/2
*c) - 73*a^3*b^6*tan(1/2*d*x + 1/2*c) + 50*a*b^8*tan(1/2*d*x + 1/2*c) + 8*a^6*b^3 - 25*a^4*b^5 + 17*a^2*b^7)/(
(a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^10) - (10890*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 2178
00*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 609840*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 418176*b^7*tan(1/2*d*x + 1/2*c)^7
- 175*a^7*tan(1/2*d*x + 1/2*c)^6 + 18480*a^5*b^2*tan(1/2*d*x + 1/2*c)^6 - 75600*a^3*b^4*tan(1/2*d*x + 1/2*c)^6
+ 62720*a*b^6*tan(1/2*d*x + 1/2*c)^6 - 1575*a^6*b*tan(1/2*d*x + 1/2*c)^5 + 11200*a^4*b^3*tan(1/2*d*x + 1/2*c)
^5 - 11760*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 105*a^7*tan(1/2*d*x + 1/2*c)^4 - 1960*a^5*b^2*tan(1/2*d*x + 1/2*c)
^4 + 2800*a^3*b^4*tan(1/2*d*x + 1/2*c)^4 + 315*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 700*a^4*b^3*tan(1/2*d*x + 1/2*c)
^3 - 35*a^7*tan(1/2*d*x + 1/2*c)^2 + 168*a^5*b^2*tan(1/2*d*x + 1/2*c)^2 - 35*a^6*b*tan(1/2*d*x + 1/2*c) + 5*a^
7)/(a^10*tan(1/2*d*x + 1/2*c)^7) + (5*a^18*tan(1/2*d*x + 1/2*c)^7 - 35*a^17*b*tan(1/2*d*x + 1/2*c)^6 - 35*a^18
*tan(1/2*d*x + 1/2*c)^5 + 168*a^16*b^2*tan(1/2*d*x + 1/2*c)^5 + 315*a^17*b*tan(1/2*d*x + 1/2*c)^4 - 700*a^15*b
^3*tan(1/2*d*x + 1/2*c)^4 + 105*a^18*tan(1/2*d*x + 1/2*c)^3 - 1960*a^16*b^2*tan(1/2*d*x + 1/2*c)^3 + 2800*a^14
*b^4*tan(1/2*d*x + 1/2*c)^3 - 1575*a^17*b*tan(1/2*d*x + 1/2*c)^2 + 11200*a^15*b^3*tan(1/2*d*x + 1/2*c)^2 - 117
60*a^13*b^5*tan(1/2*d*x + 1/2*c)^2 - 175*a^18*tan(1/2*d*x + 1/2*c) + 18480*a^16*b^2*tan(1/2*d*x + 1/2*c) - 756
00*a^14*b^4*tan(1/2*d*x + 1/2*c) + 62720*a^12*b^6*tan(1/2*d*x + 1/2*c))/a^21)/d