3.1331 \(\int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx\)

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

[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]  time = 2.21, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \frac {2 b^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^9 d}-\frac {b \left (-161 a^4 b^2+245 a^2 b^4+15 a^6-105 b^6\right ) \cot (c+d x)}{105 a^8 d}+\frac {\left (40 a^6 b^2-240 a^4 b^4+320 a^2 b^6+5 a^8-128 b^8\right ) \tanh ^{-1}(\cos (c+d x))}{128 a^9 d}-\frac {\left (-85 a^2 b^2+48 a^4+40 b^4\right ) \cot (c+d x) \csc ^5(c+d x)}{240 a^3 b^2 d}+\frac {\left (-60 a^2 b^2+35 a^4+28 b^4\right ) \cot (c+d x) \csc ^4(c+d x)}{140 a^4 b d}-\frac {\left (-104 a^2 b^2+59 a^4+48 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{192 a^5 d}+\frac {b \left (-77 a^2 b^2+45 a^4+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{105 a^6 d}+\frac {\left (-88 a^4 b^2+144 a^2 b^4+5 a^6-64 b^6\right ) \cot (c+d x) \csc (c+d x)}{128 a^7 d}+\frac {b \cot (c+d x) \csc ^6(c+d x)}{7 a^2 d}+\frac {a \cot (c+d x) \csc ^5(c+d x)}{5 b^2 d}-\frac {\cot (c+d x) \csc ^7(c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{4 b d} \]

Antiderivative was successfully verified.

[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 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])

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 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 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 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 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 3770

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

Rubi steps

\begin {align*} \int \frac {\cot ^6(c+d x) \csc ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=-\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 ) \tanh ^{-1}(\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 ) \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^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 ) \tanh ^{-1}(\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 ) \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^9 d}\\ &=\frac {2 b^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\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 ) \tanh ^{-1}(\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*}

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Mathematica [A]  time = 3.59, size = 593, normalized size = 1.25 \[ \frac {1720320 b^3 \left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )-6720 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6720 \left (5 a^8+40 a^6 b^2-240 a^4 b^4+320 a^2 b^6-128 b^8\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+a \csc ^8(c+d x) \left (-13895 a^7 \cos (5 (c+d x))-525 a^7 \cos (7 (c+d x))+13440 a^6 b \sin (2 (c+d x))+13440 a^6 b \sin (4 (c+d x))+5760 a^6 b \sin (6 (c+d x))+960 a^6 b \sin (8 (c+d x))-17080 a^5 b^2 \cos (5 (c+d x))+9240 a^5 b^2 \cos (7 (c+d x))+88704 a^4 b^3 \sin (2 (c+d x))-86912 a^4 b^3 \sin (4 (c+d x))+42112 a^4 b^3 \sin (6 (c+d x))-10304 a^4 b^3 \sin (8 (c+d x))+62160 a^3 b^4 \cos (5 (c+d x))-15120 a^3 b^4 \cos (7 (c+d x))-174720 a^2 b^5 \sin (2 (c+d x))+183680 a^2 b^5 \sin (4 (c+d x))-85120 a^2 b^5 \sin (6 (c+d x))+15680 a^2 b^5 \sin (8 (c+d x))-35 \left (895 a^7-904 a^5 b^2+2736 a^3 b^4-1728 a b^6\right ) \cos (3 (c+d x))-35 a \left (1765 a^6+680 a^4 b^2-1392 a^2 b^4+960 b^6\right ) \cos (c+d x)-33600 a b^6 \cos (5 (c+d x))+6720 a b^6 \cos (7 (c+d x))+94080 b^7 \sin (2 (c+d x))-94080 b^7 \sin (4 (c+d x))+40320 b^7 \sin (6 (c+d x))-6720 b^7 \sin (8 (c+d x))\right )}{860160 a^9 d} \]

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 3.46, size = 2082, normalized size = 4.37 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [B]  time = 0.26, size = 948, normalized size = 1.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.51, size = 1143, normalized size = 2.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128/d/a^3/tan(1/2*d*x+1/2*c)^4*b^2-5/2/d/a^7*ln(tan(1/2*d*x+1/2*c))*b^6-1/128/d/a^2*b/tan(1/2*d*x+1/2*c)^5-9
/8/d*b^5/a^6/tan(1/2*d*x+1/2*c)-7/96/d/a^4*b^3/tan(1/2*d*x+1/2*c)^3+1/4/d/a^5/tan(1/2*d*x+1/2*c)^2*b^4+1/128/d
/a^2*b*tan(1/2*d*x+1/2*c)^5-3/128/d/a^3*tan(1/2*d*x+1/2*c)^4*b^2-1/4/d/a^5*tan(1/2*d*x+1/2*c)^2*b^4+7/96/d/a^4
*tan(1/2*d*x+1/2*c)^3*b^3+1/2048/d/a*tan(1/2*d*x+1/2*c)^8-1/2048/d/a/tan(1/2*d*x+1/2*c)^8+1/128/a/d*tan(1/2*d*
x+1/2*c)^2-1/128/a/d/tan(1/2*d*x+1/2*c)^2+2/d/a^3*b^3/(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))-5/128/a/d*ln(tan(1/2*d*x+1/2*c))-11/16/d/a^4*b^3*tan(1/2*d*x+1/2*c)+3/128/d/a^2*b/tan(1/2*d*x
+1/2*c)^3-15/128/d*b^2/a^3/tan(1/2*d*x+1/2*c)^2+11/16/d*b^3/a^4/tan(1/2*d*x+1/2*c)-1/384/d/a^3/tan(1/2*d*x+1/2
*c)^6*b^2-1/64/d/a^5/tan(1/2*d*x+1/2*c)^4*b^4-1/8/d/a^7/tan(1/2*d*x+1/2*c)^2*b^6+5/128/d/a^2*tan(1/2*d*x+1/2*c
)*b-5/16/d/a^3*ln(tan(1/2*d*x+1/2*c))*b^2-5/128/d/a^2*b/tan(1/2*d*x+1/2*c)-2/d/a^9*b^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))-3/128/d/a^2*b*tan(1/2*d*x+1/2*c)^3+15/128/d/a^3*b^2*tan(1/2*
d*x+1/2*c)^2+15/8/d/a^5*ln(tan(1/2*d*x+1/2*c))*b^4+1/d/a^9*ln(tan(1/2*d*x+1/2*c))*b^8+1/896/d/a^2*b/tan(1/2*d*
x+1/2*c)^7+1/160/d/a^4*b^3/tan(1/2*d*x+1/2*c)^5+1/24/d/a^6*b^5/tan(1/2*d*x+1/2*c)^3+1/2/d*b^7/a^8/tan(1/2*d*x+
1/2*c)-1/896/d/a^2*b*tan(1/2*d*x+1/2*c)^7+1/384/d/a^3*tan(1/2*d*x+1/2*c)^6*b^2-1/160/d/a^4*tan(1/2*d*x+1/2*c)^
5*b^3+1/64/d/a^5*tan(1/2*d*x+1/2*c)^4*b^4-1/24/d/a^6*tan(1/2*d*x+1/2*c)^3*b^5+1/8/d/a^7*tan(1/2*d*x+1/2*c)^2*b
^6-1/2/d/a^8*b^7*tan(1/2*d*x+1/2*c)+9/8/d/a^6*b^5*tan(1/2*d*x+1/2*c)-1/256/d/a/tan(1/2*d*x+1/2*c)^4-6/d/a^5*b^
5/(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))+6/d/a^7*b^7/(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))+1/256/d/a*tan(1/2*d*x+1/2*c)^4-1/384/d/a*tan(1/2*d*x+1/2*c)^
6+1/384/d/a/tan(1/2*d*x+1/2*c)^6

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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 details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 12.67, size = 1861, normalized size = 3.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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