\(\int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [1330]

Optimal result

Integrand size = 29, antiderivative size = 417 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}-\frac {b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d} \]

[Out]

-2*b^2*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^8/d-1/16*b*(5*a^6-30*a^4*b^2+40*a^2*
b^4-16*b^6)*arctanh(cos(d*x+c))/a^8/d+1/105*(15*a^6-161*a^4*b^2+245*a^2*b^4-105*b^6)*cot(d*x+c)/a^7/d+1/16*b*(
11*a^4-18*a^2*b^2+8*b^4)*cot(d*x+c)*csc(d*x+c)/a^6/d-1/105*(45*a^4-77*a^2*b^2+35*b^4)*cot(d*x+c)*csc(d*x+c)^2/
a^5/d-1/3*cot(d*x+c)*csc(d*x+c)^3/b/d+1/24*(8*a^4-13*a^2*b^2+6*b^4)*cot(d*x+c)*csc(d*x+c)^3/a^4/b/d+1/4*a*cot(
d*x+c)*csc(d*x+c)^4/b^2/d-1/140*(35*a^4-60*a^2*b^2+28*b^4)*cot(d*x+c)*csc(d*x+c)^4/a^3/b^2/d+1/6*b*cot(d*x+c)*
csc(d*x+c)^5/a^2/d-1/7*cot(d*x+c)*csc(d*x+c)^6/a/d

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2975, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {2 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}-\frac {b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d} \]

[In]

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

[Out]

(-2*b^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^8*d) - (b*(5*a^6 - 30*a^4*b^2 +
40*a^2*b^4 - 16*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^8*d) + ((15*a^6 - 161*a^4*b^2 + 245*a^2*b^4 - 105*b^6)*Cot[
c + d*x])/(105*a^7*d) + (b*(11*a^4 - 18*a^2*b^2 + 8*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^6*d) - ((45*a^4 - 77
*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^3)/(3*b*d) + ((8*a^4
- 13*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(24*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^4)/(4*b^2*d) -
((35*a^4 - 60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(140*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^5)
/(6*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^6)/(7*a*d)

Rule 210

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])

Rule 632

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 2739

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 2975

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*S
in[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[Cos
[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 3080

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 3134

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] + D
ist[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] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}+\frac {\int \frac {\csc ^6(c+d x) \left (18 \left (35 a^4-60 a^2 b^2+28 b^4\right )-6 a b \left (7 a^2-2 b^2\right ) \sin (c+d x)-84 \left (6 a^4-10 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{504 a^2 b^2} \\ & = -\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}+\frac {\int \frac {\csc ^5(c+d x) \left (-420 b \left (8 a^4-13 a^2 b^2+6 b^4\right )-12 a b^2 \left (10 a^2+7 b^2\right ) \sin (c+d x)+72 b \left (35 a^4-60 a^2 b^2+28 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2520 a^3 b^2} \\ & = -\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}+\frac {\int \frac {\csc ^4(c+d x) \left (288 b^2 \left (45 a^4-77 a^2 b^2+35 b^4\right )-36 a b^3 \left (25 a^2-14 b^2\right ) \sin (c+d x)-1260 b^2 \left (8 a^4-13 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{10080 a^4 b^2} \\ & = -\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}+\frac {\int \frac {\csc ^3(c+d x) \left (-3780 b^3 \left (11 a^4-18 a^2 b^2+8 b^4\right )-36 a b^2 \left (120 a^4-133 a^2 b^2+70 b^4\right ) \sin (c+d x)+576 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{30240 a^5 b^2} \\ & = \frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}+\frac {\int \frac {\csc ^2(c+d x) \left (-576 b^2 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right )+36 a b^3 \left (285 a^4-574 a^2 b^2+280 b^4\right ) \sin (c+d x)-3780 b^4 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60480 a^6 b^2} \\ & = \frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}+\frac {\int \frac {\csc (c+d x) \left (3780 b^3 \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )-3780 a b^4 \left (11 a^4-18 a^2 b^2+8 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60480 a^7 b^2} \\ & = \frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\left (b^2 \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^8}+\frac {\left (b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right )\right ) \int \csc (c+d x) \, dx}{16 a^8} \\ & = -\frac {b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}-\frac {\left (2 b^2 \left (a^2-b^2\right )^3\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 )}{a^8 d} \\ & = -\frac {b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d}+\frac {\left (4 b^2 \left (a^2-b^2\right )^3\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 )}{a^8 d} \\ & = -\frac {2 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}-\frac {b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {\left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^7 d}+\frac {b \left (11 a^4-18 a^2 b^2+8 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}-\frac {\left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{3 b d}+\frac {\left (8 a^4-13 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b d}+\frac {a \cot (c+d x) \csc ^4(c+d x)}{4 b^2 d}-\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^5(c+d x)}{6 a^2 d}-\frac {\cot (c+d x) \csc ^6(c+d x)}{7 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.06 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-107520 b^2 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+3360 \left (-5 a^6 b+30 a^4 b^3-40 a^2 b^5+16 b^7\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3360 b \left (5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 a \cot (c+d x) \csc ^6(c+d x) \left (1200 a^6+8176 a^4 b^2-16240 a^2 b^4+8400 b^6+8 \left (225 a^6-1519 a^4 b^2+3115 a^2 b^4-1575 b^6\right ) \cos (2 (c+d x))+16 \left (45 a^6+329 a^4 b^2-665 a^2 b^4+315 b^6\right ) \cos (4 (c+d x))+120 a^6 \cos (6 (c+d x))-1288 a^4 b^2 \cos (6 (c+d x))+1960 a^2 b^4 \cos (6 (c+d x))-840 b^6 \cos (6 (c+d x))-5110 a^5 b \sin (c+d x)+13860 a^3 b^3 \sin (c+d x)-8400 a b^5 \sin (c+d x)+2135 a^5 b \sin (3 (c+d x))-7770 a^3 b^3 \sin (3 (c+d x))+4200 a b^5 \sin (3 (c+d x))-1155 a^5 b \sin (5 (c+d x))+1890 a^3 b^3 \sin (5 (c+d x))-840 a b^5 \sin (5 (c+d x))\right )}{53760 a^8 d} \]

[In]

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

[Out]

(-107520*b^2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 3360*(-5*a^6*b + 30*a^4*b^3
- 40*a^2*b^5 + 16*b^7)*Log[Cos[(c + d*x)/2]] + 3360*b*(5*a^6 - 30*a^4*b^2 + 40*a^2*b^4 - 16*b^6)*Log[Sin[(c +
d*x)/2]] - 2*a*Cot[c + d*x]*Csc[c + d*x]^6*(1200*a^6 + 8176*a^4*b^2 - 16240*a^2*b^4 + 8400*b^6 + 8*(225*a^6 -
1519*a^4*b^2 + 3115*a^2*b^4 - 1575*b^6)*Cos[2*(c + d*x)] + 16*(45*a^6 + 329*a^4*b^2 - 665*a^2*b^4 + 315*b^6)*C
os[4*(c + d*x)] + 120*a^6*Cos[6*(c + d*x)] - 1288*a^4*b^2*Cos[6*(c + d*x)] + 1960*a^2*b^4*Cos[6*(c + d*x)] - 8
40*b^6*Cos[6*(c + d*x)] - 5110*a^5*b*Sin[c + d*x] + 13860*a^3*b^3*Sin[c + d*x] - 8400*a*b^5*Sin[c + d*x] + 213
5*a^5*b*Sin[3*(c + d*x)] - 7770*a^3*b^3*Sin[3*(c + d*x)] + 4200*a*b^5*Sin[3*(c + d*x)] - 1155*a^5*b*Sin[5*(c +
d*x)] + 1890*a^3*b^3*Sin[5*(c + d*x)] - 840*a*b^5*Sin[5*(c + d*x)]))/(53760*a^8*d)

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6}}{7}-\frac {b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{3}-a^{6} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{4} b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+3 b \,a^{5} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{3} b^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{6} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {28 a^{4} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {16 a^{2} b^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 a^{5} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 a^{3} b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a \,b^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6}+88 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{2}-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{4}+64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{6}}{128 a^{7}}-\frac {2 b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{8} \sqrt {a^{2}-b^{2}}}-\frac {1}{896 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {-5 a^{2}+4 b^{2}}{640 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {9 a^{4}-28 a^{2} b^{2}+16 b^{4}}{384 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{6}+88 a^{4} b^{2}-144 a^{2} b^{4}+64 b^{6}}{128 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {b \left (3 a^{2}-2 b^{2}\right )}{128 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b \left (15 a^{4}-32 a^{2} b^{2}+16 b^{4}\right )}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (5 a^{6}-30 a^{4} b^{2}+40 a^{2} b^{4}-16 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{8}}}{d}\)	\(604\)
default	\(\frac {\frac {\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6}}{7}-\frac {b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{3}-a^{6} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{4} b^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+3 b \,a^{5} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a^{3} b^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 a^{6} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {28 a^{4} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {16 a^{2} b^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-15 a^{5} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+32 a^{3} b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a \,b^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6}+88 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{2}-144 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{4}+64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{6}}{128 a^{7}}-\frac {2 b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{8} \sqrt {a^{2}-b^{2}}}-\frac {1}{896 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {-5 a^{2}+4 b^{2}}{640 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {9 a^{4}-28 a^{2} b^{2}+16 b^{4}}{384 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-5 a^{6}+88 a^{4} b^{2}-144 a^{2} b^{4}+64 b^{6}}{128 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {b \left (3 a^{2}-2 b^{2}\right )}{128 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {b \left (15 a^{4}-32 a^{2} b^{2}+16 b^{4}\right )}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (5 a^{6}-30 a^{4} b^{2}+40 a^{2} b^{4}-16 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 a^{8}}}{d}\)	\(604\)
risch	\(\text {Expression too large to display}\)	\(1188\)

[In]

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

[Out]

1/d*(1/128/a^7*(1/7*tan(1/2*d*x+1/2*c)^7*a^6-1/3*b*tan(1/2*d*x+1/2*c)^6*a^5-a^6*tan(1/2*d*x+1/2*c)^5+4/5*a^4*b
^2*tan(1/2*d*x+1/2*c)^5+3*b*a^5*tan(1/2*d*x+1/2*c)^4-2*a^3*b^3*tan(1/2*d*x+1/2*c)^4+3*a^6*tan(1/2*d*x+1/2*c)^3
-28/3*a^4*b^2*tan(1/2*d*x+1/2*c)^3+16/3*a^2*b^4*tan(1/2*d*x+1/2*c)^3-15*a^5*b*tan(1/2*d*x+1/2*c)^2+32*a^3*b^3*
tan(1/2*d*x+1/2*c)^2-16*a*b^5*tan(1/2*d*x+1/2*c)^2-5*tan(1/2*d*x+1/2*c)*a^6+88*tan(1/2*d*x+1/2*c)*a^4*b^2-144*
tan(1/2*d*x+1/2*c)*a^2*b^4+64*tan(1/2*d*x+1/2*c)*b^6)-2*b^2*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/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))-1/896/a/tan(1/2*d*x+1/2*c)^7-1/640*(-5*a^2+4*b^2)/a^3
/tan(1/2*d*x+1/2*c)^5-1/384/a^5*(9*a^4-28*a^2*b^2+16*b^4)/tan(1/2*d*x+1/2*c)^3-1/128*(-5*a^6+88*a^4*b^2-144*a^
2*b^4+64*b^6)/a^7/tan(1/2*d*x+1/2*c)+1/384/a^2*b/tan(1/2*d*x+1/2*c)^6-1/128/a^4*b*(3*a^2-2*b^2)/tan(1/2*d*x+1/
2*c)^4+1/128/a^6*b*(15*a^4-32*a^2*b^2+16*b^4)/tan(1/2*d*x+1/2*c)^2+1/16/a^8*b*(5*a^6-30*a^4*b^2+40*a^2*b^4-16*
b^6)*ln(tan(1/2*d*x+1/2*c)))

Fricas [A] (verification not implemented)

none

Time = 0.94 (sec) , antiderivative size = 1645, normalized size of antiderivative = 3.94 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/3360*(32*(15*a^7 - 161*a^5*b^2 + 245*a^3*b^4 - 105*a*b^6)*cos(d*x + c)^7 + 224*(58*a^5*b^2 - 100*a^3*b^4 +
45*a*b^6)*cos(d*x + c)^5 - 1120*(10*a^5*b^2 - 19*a^3*b^4 + 9*a*b^6)*cos(d*x + c)^3 + 1680*((a^4*b^2 - 2*a^2*b^
4 + b^6)*cos(d*x + c)^6 - a^4*b^2 + 2*a^2*b^4 - b^6 - 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^4 + 3*(a^4*b^
2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2)*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*si
n(d*x + c) - a^2 - b^2))*sin(d*x + c) + 105*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7 - (5*a^6*b - 30*a^4*b^
3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^6 + 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^4 - 3*(5
*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 105*(5*a
^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7 - (5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^6 + 3*(5*a
^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d
*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3360*(a^5*b^2 - 2*a^3*b^4 + a*b^6)*cos(d*x + c) - 70*(3
*(11*a^6*b - 18*a^4*b^3 + 8*a^2*b^5)*cos(d*x + c)^5 - 8*(5*a^6*b - 12*a^4*b^3 + 6*a^2*b^5)*cos(d*x + c)^3 + 3*
(5*a^6*b - 14*a^4*b^3 + 8*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^8*d*cos(d*x + c)^6 - 3*a^8*d*cos(d*x + c)^4
+ 3*a^8*d*cos(d*x + c)^2 - a^8*d)*sin(d*x + c)), 1/3360*(32*(15*a^7 - 161*a^5*b^2 + 245*a^3*b^4 - 105*a*b^6)*
cos(d*x + c)^7 + 224*(58*a^5*b^2 - 100*a^3*b^4 + 45*a*b^6)*cos(d*x + c)^5 - 1120*(10*a^5*b^2 - 19*a^3*b^4 + 9*
a*b^6)*cos(d*x + c)^3 + 3360*((a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^6 - a^4*b^2 + 2*a^2*b^4 - b^6 - 3*(a^4*
b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^4 + 3*(a^4*b^2 - 2*a^2*b^4 + b^6)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-
(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c)))*sin(d*x + c) + 105*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 1
6*b^7 - (5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^6 + 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 1
6*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) +
1/2)*sin(d*x + c) - 105*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7 - (5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*
b^7)*cos(d*x + c)^6 + 3*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 30*a^4*b^3
+ 40*a^2*b^5 - 16*b^7)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3360*(a^5*b^2 - 2*a^3*b^4 +
a*b^6)*cos(d*x + c) - 70*(3*(11*a^6*b - 18*a^4*b^3 + 8*a^2*b^5)*cos(d*x + c)^5 - 8*(5*a^6*b - 12*a^4*b^3 + 6*
a^2*b^5)*cos(d*x + c)^3 + 3*(5*a^6*b - 14*a^4*b^3 + 8*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/((a^8*d*cos(d*x + c
)^6 - 3*a^8*d*cos(d*x + c)^4 + 3*a^8*d*cos(d*x + c)^2 - a^8*d)*sin(d*x + c))]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

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?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.86 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/13440*((15*a^6*tan(1/2*d*x + 1/2*c)^7 - 35*a^5*b*tan(1/2*d*x + 1/2*c)^6 - 105*a^6*tan(1/2*d*x + 1/2*c)^5 + 8
4*a^4*b^2*tan(1/2*d*x + 1/2*c)^5 + 315*a^5*b*tan(1/2*d*x + 1/2*c)^4 - 210*a^3*b^3*tan(1/2*d*x + 1/2*c)^4 + 315
*a^6*tan(1/2*d*x + 1/2*c)^3 - 980*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 + 560*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 - 1575*a
^5*b*tan(1/2*d*x + 1/2*c)^2 + 3360*a^3*b^3*tan(1/2*d*x + 1/2*c)^2 - 1680*a*b^5*tan(1/2*d*x + 1/2*c)^2 - 525*a^
6*tan(1/2*d*x + 1/2*c) + 9240*a^4*b^2*tan(1/2*d*x + 1/2*c) - 15120*a^2*b^4*tan(1/2*d*x + 1/2*c) + 6720*b^6*tan
(1/2*d*x + 1/2*c))/a^7 + 840*(5*a^6*b - 30*a^4*b^3 + 40*a^2*b^5 - 16*b^7)*log(abs(tan(1/2*d*x + 1/2*c)))/a^8 -
26880*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - 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^8) - (10890*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 65340*a^4*b^3*
tan(1/2*d*x + 1/2*c)^7 + 87120*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 34848*b^7*tan(1/2*d*x + 1/2*c)^7 - 525*a^7*tan
(1/2*d*x + 1/2*c)^6 + 9240*a^5*b^2*tan(1/2*d*x + 1/2*c)^6 - 15120*a^3*b^4*tan(1/2*d*x + 1/2*c)^6 + 6720*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 + 3360*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 1680*a^2*b^
5*tan(1/2*d*x + 1/2*c)^5 + 315*a^7*tan(1/2*d*x + 1/2*c)^4 - 980*a^5*b^2*tan(1/2*d*x + 1/2*c)^4 + 560*a^3*b^4*t
an(1/2*d*x + 1/2*c)^4 + 315*a^6*b*tan(1/2*d*x + 1/2*c)^3 - 210*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 105*a^7*tan(1/
2*d*x + 1/2*c)^2 + 84*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) + 15*a^7)/(a^8*tan(1/2*d*
x + 1/2*c)^7))/d

Mupad [B] (verification not implemented)

Time = 12.46 (sec) , antiderivative size = 1513, normalized size of antiderivative = 3.63 \[ \int \frac {\cot ^6(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(tan(c/2 + (d*x)/2)*(b^2/(32*a^3) - 5/(128*a) + (2*b*(b/(64*a^2) + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/a + (2
*b*(b/(64*a^2) - (2*b*(b^2/(32*a^3) - 9/(128*a) + (2*b*(b/(64*a^2) + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/a))/
a + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/a))/d + tan(c/2 + (d*x)/2)^7/(896*a*d) + (tan(c/2 + (d*x)/2)^4*(b/(25
6*a^2) + (b*(5/(128*a) - b^2/(32*a^3)))/(2*a)))/d - (tan(c/2 + (d*x)/2)^2*(b/(128*a^2) - (b*(b^2/(32*a^3) - 9/
(128*a) + (2*b*(b/(64*a^2) + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/a))/a + (b*(5/(128*a) - b^2/(32*a^3)))/a))/d
- (tan(c/2 + (d*x)/2)^5*(1/(128*a) - b^2/(160*a^3)))/d - (tan(c/2 + (d*x)/2)^3*(b^2/(96*a^3) - 3/(128*a) + (2
*b*(b/(64*a^2) + (2*b*(5/(128*a) - b^2/(32*a^3)))/a))/(3*a)))/d + (tan(c/2 + (d*x)/2)^2*(a^6 - (4*a^4*b^2)/5)
- tan(c/2 + (d*x)/2)^3*(3*a^5*b - 2*a^3*b^3) - a^6/7 - tan(c/2 + (d*x)/2)^4*(3*a^6 + (16*a^2*b^4)/3 - (28*a^4*
b^2)/3) + tan(c/2 + (d*x)/2)^5*(16*a*b^5 + 15*a^5*b - 32*a^3*b^3) + tan(c/2 + (d*x)/2)^6*(5*a^6 - 64*b^6 + 144
*a^2*b^4 - 88*a^4*b^2) + (a^5*b*tan(c/2 + (d*x)/2))/3)/(128*a^7*d*tan(c/2 + (d*x)/2)^7) - (b*tan(c/2 + (d*x)/2
)^6)/(384*a^2*d) + (log(tan(c/2 + (d*x)/2))*(5*a^6*b - 16*b^7 + 40*a^2*b^5 - 30*a^4*b^3))/(16*a^8*d) - (b^2*at
an(((b^2*(-(a + b)^5*(a - b)^5)^(1/2)*((32*a^8*b^8 - 88*a^10*b^6 + 78*a^12*b^4 - 21*a^14*b^2)/(8*a^14) + (tan(
c/2 + (d*x)/2)*(5*a^14*b + 64*a^6*b^9 - 192*a^8*b^7 + 196*a^10*b^5 - 72*a^12*b^3))/(8*a^13) + (b^2*(2*a^2*b -
(tan(c/2 + (d*x)/2)*(48*a^16 - 64*a^14*b^2))/(8*a^13))*(-(a + b)^5*(a - b)^5)^(1/2))/a^8)*1i)/a^8 + (b^2*(-(a
+ b)^5*(a - b)^5)^(1/2)*((32*a^8*b^8 - 88*a^10*b^6 + 78*a^12*b^4 - 21*a^14*b^2)/(8*a^14) + (tan(c/2 + (d*x)/2)
*(5*a^14*b + 64*a^6*b^9 - 192*a^8*b^7 + 196*a^10*b^5 - 72*a^12*b^3))/(8*a^13) - (b^2*(2*a^2*b - (tan(c/2 + (d*
x)/2)*(48*a^16 - 64*a^14*b^2))/(8*a^13))*(-(a + b)^5*(a - b)^5)^(1/2))/a^8)*1i)/a^8)/((16*b^15 - 88*a^2*b^13 +
198*a^4*b^11 - 231*a^6*b^9 + 145*a^8*b^7 - 45*a^10*b^5 + 5*a^12*b^3)/(4*a^14) + (tan(c/2 + (d*x)/2)*(16*b^14
- 84*a^2*b^12 + 178*a^4*b^10 - 190*a^6*b^8 + 102*a^8*b^6 - 22*a^10*b^4))/(4*a^13) + (b^2*(-(a + b)^5*(a - b)^5
)^(1/2)*((32*a^8*b^8 - 88*a^10*b^6 + 78*a^12*b^4 - 21*a^14*b^2)/(8*a^14) + (tan(c/2 + (d*x)/2)*(5*a^14*b + 64*
a^6*b^9 - 192*a^8*b^7 + 196*a^10*b^5 - 72*a^12*b^3))/(8*a^13) + (b^2*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^16 -
64*a^14*b^2))/(8*a^13))*(-(a + b)^5*(a - b)^5)^(1/2))/a^8))/a^8 - (b^2*(-(a + b)^5*(a - b)^5)^(1/2)*((32*a^8*
b^8 - 88*a^10*b^6 + 78*a^12*b^4 - 21*a^14*b^2)/(8*a^14) + (tan(c/2 + (d*x)/2)*(5*a^14*b + 64*a^6*b^9 - 192*a^8
*b^7 + 196*a^10*b^5 - 72*a^12*b^3))/(8*a^13) - (b^2*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^16 - 64*a^14*b^2))/(8
*a^13))*(-(a + b)^5*(a - b)^5)^(1/2))/a^8))/a^8))*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(a^8*d)