\(\int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) [1331]

Optimal result

Integrand size = 29, antiderivative size = 476 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {2 b^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^9 d}+\frac {\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \text {arctanh}(\cos (c+d x))}{128 a^9 d}-\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d} \]

[Out]

2*b^3*(a^2-b^2)^(5/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^9/d+1/128*(5*a^8+40*a^6*b^2-240*a^4*b
^4+320*a^2*b^6-128*b^8)*arctanh(cos(d*x+c))/a^9/d-1/105*b*(15*a^6-161*a^4*b^2+245*a^2*b^4-105*b^6)*cot(d*x+c)/
a^8/d+1/128*(5*a^6-88*a^4*b^2+144*a^2*b^4-64*b^6)*cot(d*x+c)*csc(d*x+c)/a^7/d+1/105*b*(45*a^4-77*a^2*b^2+35*b^
4)*cot(d*x+c)*csc(d*x+c)^2/a^6/d-1/192*(59*a^4-104*a^2*b^2+48*b^4)*cot(d*x+c)*csc(d*x+c)^3/a^5/d-1/4*cot(d*x+c
)*csc(d*x+c)^4/b/d+1/140*(35*a^4-60*a^2*b^2+28*b^4)*cot(d*x+c)*csc(d*x+c)^4/a^4/b/d+1/5*a*cot(d*x+c)*csc(d*x+c
)^5/b^2/d-1/240*(48*a^4-85*a^2*b^2+40*b^4)*cot(d*x+c)*csc(d*x+c)^5/a^3/b^2/d+1/7*b*cot(d*x+c)*csc(d*x+c)^6/a^2
/d-1/8*cot(d*x+c)*csc(d*x+c)^7/a/d

Rubi [A] (verified)

Time = 1.48 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2975, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {2 b^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^9 d}+\frac {\left (35 a^4-60 a^2 b^2+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}-\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \text {arctanh}(\cos (c+d x))}{128 a^9 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d} \]

[In]

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

[Out]

(2*b^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^9*d) + ((5*a^8 + 40*a^6*b^2 - 24
0*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*ArcTanh[Cos[c + d*x]])/(128*a^9*d) - (b*(15*a^6 - 161*a^4*b^2 + 245*a^2*b^4
- 105*b^6)*Cot[c + d*x])/(105*a^8*d) + ((5*a^6 - 88*a^4*b^2 + 144*a^2*b^4 - 64*b^6)*Cot[c + d*x]*Csc[c + d*x]
)/(128*a^7*d) + (b*(45*a^4 - 77*a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(105*a^6*d) - ((59*a^4 - 104*a^
2*b^2 + 48*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(192*a^5*d) - (Cot[c + d*x]*Csc[c + d*x]^4)/(4*b*d) + ((35*a^4 -
60*a^2*b^2 + 28*b^4)*Cot[c + d*x]*Csc[c + d*x]^4)/(140*a^4*b*d) + (a*Cot[c + d*x]*Csc[c + d*x]^5)/(5*b^2*d) -
((48*a^4 - 85*a^2*b^2 + 40*b^4)*Cot[c + d*x]*Csc[c + d*x]^5)/(240*a^3*b^2*d) + (b*Cot[c + d*x]*Csc[c + d*x]^6)
/(7*a^2*d) - (Cot[c + d*x]*Csc[c + d*x]^7)/(8*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 ^4(c+d x)}{4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\int \frac {\csc ^7(c+d x) \left (28 \left (48 a^4-85 a^2 b^2+40 b^4\right )-4 a b \left (14 a^2-5 b^2\right ) \sin (c+d x)-40 \left (28 a^4-49 a^2 b^2+24 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1120 a^2 b^2} \\ & = -\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\int \frac {\csc ^6(c+d x) \left (-240 b \left (35 a^4-60 a^2 b^2+28 b^4\right )-20 a b^2 \left (7 a^2+8 b^2\right ) \sin (c+d x)+140 b \left (48 a^4-85 a^2 b^2+40 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6720 a^3 b^2} \\ & = -\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\int \frac {\csc ^5(c+d x) \left (700 b^2 \left (59 a^4-104 a^2 b^2+48 b^4\right )-20 a b^3 \left (95 a^2-56 b^2\right ) \sin (c+d x)-960 b^2 \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}{33600 a^4 b^2} \\ & = -\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\int \frac {\csc ^4(c+d x) \left (-3840 b^3 \left (45 a^4-77 a^2 b^2+35 b^4\right )-60 a b^2 \left (175 a^4-200 a^2 b^2+112 b^4\right ) \sin (c+d x)+2100 b^3 \left (59 a^4-104 a^2 b^2+48 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{134400 a^5 b^2} \\ & = \frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\int \frac {\csc ^3(c+d x) \left (-6300 b^2 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right )+60 a b^3 \left (435 a^4-1064 a^2 b^2+560 b^4\right ) \sin (c+d x)-7680 b^4 \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}{403200 a^6 b^2} \\ & = \frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\int \frac {\csc ^2(c+d x) \left (7680 b^3 \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right )-60 a b^2 \left (525 a^6+2280 a^4 b^2-4592 a^2 b^4+2240 b^6\right ) \sin (c+d x)-6300 b^3 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{806400 a^7 b^2} \\ & = -\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\int \frac {\csc (c+d x) \left (-6300 b^2 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right )-6300 a b^3 \left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{806400 a^8 b^2} \\ & = -\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\left (b^3 \left (a^2-b^2\right )^3\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^9}-\frac {\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \int \csc (c+d x) \, dx}{128 a^9} \\ & = \frac {\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \text {arctanh}(\cos (c+d x))}{128 a^9 d}-\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}+\frac {\left (2 b^3 \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^9 d} \\ & = \frac {\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \text {arctanh}(\cos (c+d x))}{128 a^9 d}-\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\left (4 b^3 \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^9 d} \\ & = \frac {2 b^3 \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^9 d}+\frac {\left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \text {arctanh}(\cos (c+d x))}{128 a^9 d}-\frac {b \left (15 a^6-161 a^4 b^2+245 a^2 b^4-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (5 a^6-88 a^4 b^2+144 a^2 b^4-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \left (45 a^4-77 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}-\frac {\left (59 a^4-104 a^2 b^2+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b 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^4 b d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\left (48 a^4-85 a^2 b^2+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.26 (sec) , antiderivative size = 593, normalized size of antiderivative = 1.25 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {1720320 b^3 \left (a^2-b^2\right )^{5/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+6720 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6720 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+a \csc ^8(c+d x) \left (-35 a \left (1765 a^6+680 a^4 b^2-1392 a^2 b^4+960 b^6\right ) \cos (c+d x)-35 \left (895 a^7-904 a^5 b^2+2736 a^3 b^4-1728 a b^6\right ) \cos (3 (c+d x))-13895 a^7 \cos (5 (c+d x))-17080 a^5 b^2 \cos (5 (c+d x))+62160 a^3 b^4 \cos (5 (c+d x))-33600 a b^6 \cos (5 (c+d x))-525 a^7 \cos (7 (c+d x))+9240 a^5 b^2 \cos (7 (c+d x))-15120 a^3 b^4 \cos (7 (c+d x))+6720 a b^6 \cos (7 (c+d x))+13440 a^6 b \sin (2 (c+d x))+88704 a^4 b^3 \sin (2 (c+d x))-174720 a^2 b^5 \sin (2 (c+d x))+94080 b^7 \sin (2 (c+d x))+13440 a^6 b \sin (4 (c+d x))-86912 a^4 b^3 \sin (4 (c+d x))+183680 a^2 b^5 \sin (4 (c+d x))-94080 b^7 \sin (4 (c+d x))+5760 a^6 b \sin (6 (c+d x))+42112 a^4 b^3 \sin (6 (c+d x))-85120 a^2 b^5 \sin (6 (c+d x))+40320 b^7 \sin (6 (c+d x))+960 a^6 b \sin (8 (c+d x))-10304 a^4 b^3 \sin (8 (c+d x))+15680 a^2 b^5 \sin (8 (c+d x))-6720 b^7 \sin (8 (c+d x))\right )}{860160 a^9 d} \]

[In]

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

[Out]

(1720320*b^3*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 6720*(5*a^8 + 40*a^6*b^2 - 2
40*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*Log[Cos[(c + d*x)/2]] - 6720*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b
^6 - 128*b^8)*Log[Sin[(c + d*x)/2]] + a*Csc[c + d*x]^8*(-35*a*(1765*a^6 + 680*a^4*b^2 - 1392*a^2*b^4 + 960*b^6
)*Cos[c + d*x] - 35*(895*a^7 - 904*a^5*b^2 + 2736*a^3*b^4 - 1728*a*b^6)*Cos[3*(c + d*x)] - 13895*a^7*Cos[5*(c
+ d*x)] - 17080*a^5*b^2*Cos[5*(c + d*x)] + 62160*a^3*b^4*Cos[5*(c + d*x)] - 33600*a*b^6*Cos[5*(c + d*x)] - 525
*a^7*Cos[7*(c + d*x)] + 9240*a^5*b^2*Cos[7*(c + d*x)] - 15120*a^3*b^4*Cos[7*(c + d*x)] + 6720*a*b^6*Cos[7*(c +
d*x)] + 13440*a^6*b*Sin[2*(c + d*x)] + 88704*a^4*b^3*Sin[2*(c + d*x)] - 174720*a^2*b^5*Sin[2*(c + d*x)] + 940
80*b^7*Sin[2*(c + d*x)] + 13440*a^6*b*Sin[4*(c + d*x)] - 86912*a^4*b^3*Sin[4*(c + d*x)] + 183680*a^2*b^5*Sin[4
*(c + d*x)] - 94080*b^7*Sin[4*(c + d*x)] + 5760*a^6*b*Sin[6*(c + d*x)] + 42112*a^4*b^3*Sin[6*(c + d*x)] - 8512
0*a^2*b^5*Sin[6*(c + d*x)] + 40320*b^7*Sin[6*(c + d*x)] + 960*a^6*b*Sin[8*(c + d*x)] - 10304*a^4*b^3*Sin[8*(c
+ d*x)] + 15680*a^2*b^5*Sin[8*(c + d*x)] - 6720*b^7*Sin[8*(c + d*x)]))/(860160*a^9*d)

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 728, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{7}}{8}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6} b}{7}-\frac {2 a^{7} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b^{2}}{3}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6} b -\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b^{3}}{5}+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{7}-6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b^{2}+4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{4}-6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6} b +\frac {56 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b^{3}}{3}-\frac {32 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{5}}{3}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{7}+30 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b^{2}-64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{4}+32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{6}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6} b -176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{3}+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{5}-128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{7}}{256 a^{8}}-\frac {1}{2048 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {-4 a^{2}+4 b^{2}}{1536 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {4 a^{4}-24 a^{2} b^{2}+16 b^{4}}{1024 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {4 a^{6}+60 a^{4} b^{2}-128 a^{2} b^{4}+64 b^{6}}{512 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{8}-80 a^{6} b^{2}+480 a^{4} b^{4}-640 a^{2} b^{6}+256 b^{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 a^{9}}+\frac {b}{896 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{640 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \left (9 a^{4}-28 a^{2} b^{2}+16 b^{4}\right )}{384 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{6}-88 a^{4} b^{2}+144 a^{2} b^{4}-64 b^{6}\right )}{128 a^{8} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{3} \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^{9} \sqrt {a^{2}-b^{2}}}}{d}\)	\(728\)
default	\(\frac {\frac {\frac {\left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{7}}{8}-\frac {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6} b}{7}-\frac {2 a^{7} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b^{2}}{3}+2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6} b -\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b^{3}}{5}+\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{7}-6 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b^{2}+4 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{4}-6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6} b +\frac {56 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4} b^{3}}{3}-\frac {32 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b^{5}}{3}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{7}+30 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b^{2}-64 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3} b^{4}+32 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{6}+10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{6} b -176 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4} b^{3}+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{5}-128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{7}}{256 a^{8}}-\frac {1}{2048 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}-\frac {-4 a^{2}+4 b^{2}}{1536 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {4 a^{4}-24 a^{2} b^{2}+16 b^{4}}{1024 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {4 a^{6}+60 a^{4} b^{2}-128 a^{2} b^{4}+64 b^{6}}{512 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{8}-80 a^{6} b^{2}+480 a^{4} b^{4}-640 a^{2} b^{6}+256 b^{8}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 a^{9}}+\frac {b}{896 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {b \left (5 a^{2}-4 b^{2}\right )}{640 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {b \left (9 a^{4}-28 a^{2} b^{2}+16 b^{4}\right )}{384 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b \left (5 a^{6}-88 a^{4} b^{2}+144 a^{2} b^{4}-64 b^{6}\right )}{128 a^{8} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 b^{3} \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^{9} \sqrt {a^{2}-b^{2}}}}{d}\)	\(728\)
risch	\(\text {Expression too large to display}\)	\(1576\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^9/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/256/a^8*(1/8*tan(1/2*d*x+1/2*c)^8*a^7-2/7*tan(1/2*d*x+1/2*c)^7*a^6*b-2/3*a^7*tan(1/2*d*x+1/2*c)^6+2/3*t
an(1/2*d*x+1/2*c)^6*a^5*b^2+2*tan(1/2*d*x+1/2*c)^5*a^6*b-8/5*tan(1/2*d*x+1/2*c)^5*a^4*b^3+tan(1/2*d*x+1/2*c)^4
*a^7-6*tan(1/2*d*x+1/2*c)^4*a^5*b^2+4*tan(1/2*d*x+1/2*c)^4*a^3*b^4-6*tan(1/2*d*x+1/2*c)^3*a^6*b+56/3*tan(1/2*d
*x+1/2*c)^3*a^4*b^3-32/3*tan(1/2*d*x+1/2*c)^3*a^2*b^5+2*tan(1/2*d*x+1/2*c)^2*a^7+30*tan(1/2*d*x+1/2*c)^2*a^5*b
^2-64*tan(1/2*d*x+1/2*c)^2*a^3*b^4+32*tan(1/2*d*x+1/2*c)^2*a*b^6+10*tan(1/2*d*x+1/2*c)*a^6*b-176*tan(1/2*d*x+1
/2*c)*a^4*b^3+288*tan(1/2*d*x+1/2*c)*a^2*b^5-128*tan(1/2*d*x+1/2*c)*b^7)-1/2048/a/tan(1/2*d*x+1/2*c)^8-1/1536*
(-4*a^2+4*b^2)/a^3/tan(1/2*d*x+1/2*c)^6-1/1024*(4*a^4-24*a^2*b^2+16*b^4)/a^5/tan(1/2*d*x+1/2*c)^4-1/512*(4*a^6
+60*a^4*b^2-128*a^2*b^4+64*b^6)/a^7/tan(1/2*d*x+1/2*c)^2+1/256/a^9*(-10*a^8-80*a^6*b^2+480*a^4*b^4-640*a^2*b^6
+256*b^8)*ln(tan(1/2*d*x+1/2*c))+1/896/a^2*b/tan(1/2*d*x+1/2*c)^7-1/640/a^4*b*(5*a^2-4*b^2)/tan(1/2*d*x+1/2*c)
^5+1/384/a^6*b*(9*a^4-28*a^2*b^2+16*b^4)/tan(1/2*d*x+1/2*c)^3-1/128*b*(5*a^6-88*a^4*b^2+144*a^2*b^4-64*b^6)/a^
8/tan(1/2*d*x+1/2*c)+2*b^3*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/a^9/(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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 999 vs. \(2 (449) = 898\).

Time = 1.50 (sec) , antiderivative size = 2082, normalized size of antiderivative = 4.37 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

[-1/26880*(210*(5*a^8 - 88*a^6*b^2 + 144*a^4*b^4 - 64*a^2*b^6)*cos(d*x + c)^7 + 70*(73*a^8 + 584*a^6*b^2 - 120
0*a^4*b^4 + 576*a^2*b^6)*cos(d*x + c)^5 - 70*(55*a^8 + 440*a^6*b^2 - 1104*a^4*b^4 + 576*a^2*b^6)*cos(d*x + c)^
3 - 13440*((a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^8 + a^4*b^3 - 2*a^2*b^5 + b^7 - 4*(a^4*b^3 - 2*a^2*b^5 + b
^7)*cos(d*x + c)^6 + 6*(a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^4 - 4*(a^4*b^3 - 2*a^2*b^5 + b^7)*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)*s
in(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)) + 210*(
5*a^8 + 40*a^6*b^2 - 112*a^4*b^4 + 64*a^2*b^6)*cos(d*x + c) - 105*((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2
*b^6 - 128*b^8)*cos(d*x + c)^8 + 5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8 - 4*(5*a^8 + 40*a^6*
b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^6 + 6*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6
- 128*b^8)*cos(d*x + c)^4 - 4*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^2)*log(1
/2*cos(d*x + c) + 1/2) + 105*((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^8 + 5*a^
8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8 - 4*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128
*b^8)*cos(d*x + c)^6 + 6*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^4 - 4*(5*a^8
+ 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 256*((15*a^
7*b - 161*a^5*b^3 + 245*a^3*b^5 - 105*a*b^7)*cos(d*x + c)^7 + 7*(58*a^5*b^3 - 100*a^3*b^5 + 45*a*b^7)*cos(d*x
+ c)^5 - 35*(10*a^5*b^3 - 19*a^3*b^5 + 9*a*b^7)*cos(d*x + c)^3 + 105*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c
))*sin(d*x + c))/(a^9*d*cos(d*x + c)^8 - 4*a^9*d*cos(d*x + c)^6 + 6*a^9*d*cos(d*x + c)^4 - 4*a^9*d*cos(d*x + c
)^2 + a^9*d), -1/26880*(210*(5*a^8 - 88*a^6*b^2 + 144*a^4*b^4 - 64*a^2*b^6)*cos(d*x + c)^7 + 70*(73*a^8 + 584*
a^6*b^2 - 1200*a^4*b^4 + 576*a^2*b^6)*cos(d*x + c)^5 - 70*(55*a^8 + 440*a^6*b^2 - 1104*a^4*b^4 + 576*a^2*b^6)*
cos(d*x + c)^3 + 26880*((a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^8 + a^4*b^3 - 2*a^2*b^5 + b^7 - 4*(a^4*b^3 -
2*a^2*b^5 + b^7)*cos(d*x + c)^6 + 6*(a^4*b^3 - 2*a^2*b^5 + b^7)*cos(d*x + c)^4 - 4*(a^4*b^3 - 2*a^2*b^5 + b^7)
*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))) + 210*(5*a^8 + 4
0*a^6*b^2 - 112*a^4*b^4 + 64*a^2*b^6)*cos(d*x + c) - 105*((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 12
8*b^8)*cos(d*x + c)^8 + 5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8 - 4*(5*a^8 + 40*a^6*b^2 - 240
*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^6 + 6*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8
)*cos(d*x + c)^4 - 4*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^2)*log(1/2*cos(d*
x + c) + 1/2) + 105*((5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^8 + 5*a^8 + 40*a^
6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8 - 4*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos
(d*x + c)^6 + 6*(5*a^8 + 40*a^6*b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^4 - 4*(5*a^8 + 40*a^6*
b^2 - 240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2) - 256*((15*a^7*b - 161
*a^5*b^3 + 245*a^3*b^5 - 105*a*b^7)*cos(d*x + c)^7 + 7*(58*a^5*b^3 - 100*a^3*b^5 + 45*a*b^7)*cos(d*x + c)^5 -
35*(10*a^5*b^3 - 19*a^3*b^5 + 9*a*b^7)*cos(d*x + c)^3 + 105*(a^5*b^3 - 2*a^3*b^5 + a*b^7)*cos(d*x + c))*sin(d*
x + c))/(a^9*d*cos(d*x + c)^8 - 4*a^9*d*cos(d*x + c)^6 + 6*a^9*d*cos(d*x + c)^4 - 4*a^9*d*cos(d*x + c)^2 + a^9
*d)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**9/(a+b*sin(d*x+c)),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 948 vs. \(2 (449) = 898\).

Time = 0.49 (sec) , antiderivative size = 948, normalized size of antiderivative = 1.99 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^9/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

1/215040*((105*a^7*tan(1/2*d*x + 1/2*c)^8 - 240*a^6*b*tan(1/2*d*x + 1/2*c)^7 - 560*a^7*tan(1/2*d*x + 1/2*c)^6
+ 560*a^5*b^2*tan(1/2*d*x + 1/2*c)^6 + 1680*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 1344*a^4*b^3*tan(1/2*d*x + 1/2*c)^5
+ 840*a^7*tan(1/2*d*x + 1/2*c)^4 - 5040*a^5*b^2*tan(1/2*d*x + 1/2*c)^4 + 3360*a^3*b^4*tan(1/2*d*x + 1/2*c)^4
- 5040*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 15680*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 8960*a^2*b^5*tan(1/2*d*x + 1/2*c)
^3 + 1680*a^7*tan(1/2*d*x + 1/2*c)^2 + 25200*a^5*b^2*tan(1/2*d*x + 1/2*c)^2 - 53760*a^3*b^4*tan(1/2*d*x + 1/2*
c)^2 + 26880*a*b^6*tan(1/2*d*x + 1/2*c)^2 + 8400*a^6*b*tan(1/2*d*x + 1/2*c) - 147840*a^4*b^3*tan(1/2*d*x + 1/2
*c) + 241920*a^2*b^5*tan(1/2*d*x + 1/2*c) - 107520*b^7*tan(1/2*d*x + 1/2*c))/a^8 - 1680*(5*a^8 + 40*a^6*b^2 -
240*a^4*b^4 + 320*a^2*b^6 - 128*b^8)*log(abs(tan(1/2*d*x + 1/2*c)))/a^9 + 430080*(a^6*b^3 - 3*a^4*b^5 + 3*a^2*
b^7 - b^9)*(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)))/(s
qrt(a^2 - b^2)*a^9) + (22830*a^8*tan(1/2*d*x + 1/2*c)^8 + 182640*a^6*b^2*tan(1/2*d*x + 1/2*c)^8 - 1095840*a^4*
b^4*tan(1/2*d*x + 1/2*c)^8 + 1461120*a^2*b^6*tan(1/2*d*x + 1/2*c)^8 - 584448*b^8*tan(1/2*d*x + 1/2*c)^8 - 8400
*a^7*b*tan(1/2*d*x + 1/2*c)^7 + 147840*a^5*b^3*tan(1/2*d*x + 1/2*c)^7 - 241920*a^3*b^5*tan(1/2*d*x + 1/2*c)^7
+ 107520*a*b^7*tan(1/2*d*x + 1/2*c)^7 - 1680*a^8*tan(1/2*d*x + 1/2*c)^6 - 25200*a^6*b^2*tan(1/2*d*x + 1/2*c)^6
+ 53760*a^4*b^4*tan(1/2*d*x + 1/2*c)^6 - 26880*a^2*b^6*tan(1/2*d*x + 1/2*c)^6 + 5040*a^7*b*tan(1/2*d*x + 1/2*
c)^5 - 15680*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 + 8960*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 - 840*a^8*tan(1/2*d*x + 1/2*
c)^4 + 5040*a^6*b^2*tan(1/2*d*x + 1/2*c)^4 - 3360*a^4*b^4*tan(1/2*d*x + 1/2*c)^4 - 1680*a^7*b*tan(1/2*d*x + 1/
2*c)^3 + 1344*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 + 560*a^8*tan(1/2*d*x + 1/2*c)^2 - 560*a^6*b^2*tan(1/2*d*x + 1/2*
c)^2 + 240*a^7*b*tan(1/2*d*x + 1/2*c) - 105*a^8)/(a^9*tan(1/2*d*x + 1/2*c)^8))/d

Mupad [B] (verification not implemented)

Time = 13.03 (sec) , antiderivative size = 1861, normalized size of antiderivative = 3.91 \[ \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^9*(a + b*sin(c + d*x))),x)

[Out]

tan(c/2 + (d*x)/2)^8/(2048*a*d) + (tan(c/2 + (d*x)/2)^5*(b/(640*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/(5*a)))
/d - (tan(c/2 + (d*x)/2)^3*(b/(384*a^2) - (2*b*(b^2/(64*a^3) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) -
b^2/(64*a^3)))/a))/a))/(3*a) + (2*b*(1/(64*a) - b^2/(64*a^3)))/(3*a)))/d + (tan(c/2 + (d*x)/2)^2*(1/(128*a) +
b^2/(128*a^3) + (b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a + (b*(b/(128*a^2) - (2*b*(b^2/(64*a^3
) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a))/a + (2*b*(1/(64*a) - b^2/(64*a^3)))
/a))/a))/d + (tan(c/2 + (d*x)/2)*(b/(128*a^2) - (2*b*(b^2/(64*a^3) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(6
4*a) - b^2/(64*a^3)))/a))/a))/a - (2*b*(1/(64*a) + b^2/(64*a^3) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(64
*a^3)))/a))/a + (2*b*(b/(128*a^2) - (2*b*(b^2/(64*a^3) - 1/(64*a) + (2*b*(b/(128*a^2) + (2*b*(1/(64*a) - b^2/(
64*a^3)))/a))/a))/a + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/a))/a + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/d - (tan
(c/2 + (d*x)/2)^6*(1/(384*a) - b^2/(384*a^3)))/d - (tan(c/2 + (d*x)/2)^4*(b^2/(256*a^3) - 1/(256*a) + (b*(b/(1
28*a^2) + (2*b*(1/(64*a) - b^2/(64*a^3)))/a))/(2*a)))/d - (log(tan(c/2 + (d*x)/2))*(5*a^8 - 128*b^8 + 320*a^2*
b^6 - 240*a^4*b^4 + 40*a^6*b^2))/(128*a^9*d) - (cot(c/2 + (d*x)/2)^8*(tan(c/2 + (d*x)/2)^3*(2*a^6*b - (8*a^4*b
^3)/5) - tan(c/2 + (d*x)/2)^5*(6*a^6*b + (32*a^2*b^5)/3 - (56*a^4*b^3)/3) + tan(c/2 + (d*x)/2)^6*(32*a*b^6 + 2
*a^7 - 64*a^3*b^4 + 30*a^5*b^2) + tan(c/2 + (d*x)/2)^7*(10*a^6*b - 128*b^7 + 288*a^2*b^5 - 176*a^4*b^3) + tan(
c/2 + (d*x)/2)^4*(a^7 + 4*a^3*b^4 - 6*a^5*b^2) + a^7/8 - tan(c/2 + (d*x)/2)^2*((2*a^7)/3 - (2*a^5*b^2)/3) - (2
*a^6*b*tan(c/2 + (d*x)/2))/7))/(256*a^8*d) - (b*tan(c/2 + (d*x)/2)^7)/(896*a^2*d) + (b^3*atan(((b^3*(-(a + b)^
5*(a - b)^5)^(1/2)*((tan(c/2 + (d*x)/2)*(5*a^17 + 512*a^7*b^10 - 1536*a^9*b^8 + 1568*a^11*b^6 - 576*a^13*b^4 +
30*a^15*b^2))/(64*a^15) - (5*a^17*b - 256*a^9*b^9 + 704*a^11*b^7 - 624*a^13*b^5 + 168*a^15*b^3)/(64*a^16) + (
b^3*(2*a^2*b - (tan(c/2 + (d*x)/2)*(384*a^18 - 512*a^16*b^2))/(64*a^15))*(-(a + b)^5*(a - b)^5)^(1/2))/a^9)*1i
)/a^9 - (b^3*(-(a + b)^5*(a - b)^5)^(1/2)*((5*a^17*b - 256*a^9*b^9 + 704*a^11*b^7 - 624*a^13*b^5 + 168*a^15*b^
3)/(64*a^16) - (tan(c/2 + (d*x)/2)*(5*a^17 + 512*a^7*b^10 - 1536*a^9*b^8 + 1568*a^11*b^6 - 576*a^13*b^4 + 30*a
^15*b^2))/(64*a^15) + (b^3*(2*a^2*b - (tan(c/2 + (d*x)/2)*(384*a^18 - 512*a^16*b^2))/(64*a^15))*(-(a + b)^5*(a
- b)^5)^(1/2))/a^9)*1i)/a^9)/((128*b^17 - 704*a^2*b^15 + 1584*a^4*b^13 - 1848*a^6*b^11 + 1155*a^8*b^9 - 345*a
^10*b^7 + 25*a^12*b^5 + 5*a^14*b^3)/(32*a^16) + (tan(c/2 + (d*x)/2)*(128*b^16 - 672*a^2*b^14 + 1424*a^4*b^12 -
1530*a^6*b^10 + 846*a^8*b^8 - 206*a^10*b^6 + 10*a^12*b^4))/(32*a^15) + (b^3*(-(a + b)^5*(a - b)^5)^(1/2)*((ta
n(c/2 + (d*x)/2)*(5*a^17 + 512*a^7*b^10 - 1536*a^9*b^8 + 1568*a^11*b^6 - 576*a^13*b^4 + 30*a^15*b^2))/(64*a^15
) - (5*a^17*b - 256*a^9*b^9 + 704*a^11*b^7 - 624*a^13*b^5 + 168*a^15*b^3)/(64*a^16) + (b^3*(2*a^2*b - (tan(c/2
+ (d*x)/2)*(384*a^18 - 512*a^16*b^2))/(64*a^15))*(-(a + b)^5*(a - b)^5)^(1/2))/a^9))/a^9 + (b^3*(-(a + b)^5*(
a - b)^5)^(1/2)*((5*a^17*b - 256*a^9*b^9 + 704*a^11*b^7 - 624*a^13*b^5 + 168*a^15*b^3)/(64*a^16) - (tan(c/2 +
(d*x)/2)*(5*a^17 + 512*a^7*b^10 - 1536*a^9*b^8 + 1568*a^11*b^6 - 576*a^13*b^4 + 30*a^15*b^2))/(64*a^15) + (b^3
*(2*a^2*b - (tan(c/2 + (d*x)/2)*(384*a^18 - 512*a^16*b^2))/(64*a^15))*(-(a + b)^5*(a - b)^5)^(1/2))/a^9))/a^9)
)*(-(a + b)^5*(a - b)^5)^(1/2)*2i)/(a^9*d)