\(\int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [1265]

Optimal result

Integrand size = 27, antiderivative size = 480 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))} \]

[Out]

2*b*(2*a^2-7*b^2)*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^8/d+1/16*(5*a^6-90*a^4*b^
2+200*a^2*b^4-112*b^6)*arctanh(cos(d*x+c))/a^8/d+1/15*b*(61*a^4-170*a^2*b^2+105*b^4)*cot(d*x+c)/a^7/d-1/16*(27
*a^4-86*a^2*b^2+56*b^4)*cot(d*x+c)*csc(d*x+c)/a^6/d+1/15*(15*a^4-52*a^2*b^2+35*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^
5/b/d-1/24*(16*a^4-61*a^2*b^2+42*b^4)*cot(d*x+c)*csc(d*x+c)^3/a^4/b^2/d-1/3*cot(d*x+c)*csc(d*x+c)^2/b/d/(a+b*s
in(d*x+c))+1/6*a*cot(d*x+c)*csc(d*x+c)^3/b^2/d/(a+b*sin(d*x+c))+1/10*(5*a^4-20*a^2*b^2+14*b^4)*cot(d*x+c)*csc(
d*x+c)^3/a^3/b^2/d/(a+b*sin(d*x+c))+7/30*b*cot(d*x+c)*csc(d*x+c)^4/a^2/d/(a+b*sin(d*x+c))-1/6*cot(d*x+c)*csc(d
*x+c)^5/a/d/(a+b*sin(d*x+c))

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2975, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}+\frac {2 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^8 d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))} \]

[In]

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

[Out]

(2*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^8*d) + ((5*a^6 - 9
0*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*ArcTanh[Cos[c + d*x]])/(16*a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*Cot
[c + d*x])/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*Cot[c + d*x]*Csc[c + d*x])/(16*a^6*d) + ((15*a^4 - 52*
a^2*b^2 + 35*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*Cot[c + d*x]*Csc
[c + d*x]^3)/(24*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*b*d*(a + b*Sin[c + d*x])) + (a*Cot[c + d*x]*Csc
[c + d*x]^3)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*Cot[c + d*x]*Csc[c + d*x]^3)/(10*
a^3*b^2*d*(a + b*Sin[c + d*x])) + (7*b*Cot[c + d*x]*Csc[c + d*x]^4)/(30*a^2*d*(a + b*Sin[c + d*x])) - (Cot[c +
d*x]*Csc[c + d*x]^5)/(6*a*d*(a + b*Sin[c + d*x]))

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 ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^5(c+d x) \left (12 \left (20 a^4-65 a^2 b^2+42 b^4\right )-12 a b \left (5 a^2-2 b^2\right ) \sin (c+d x)-60 \left (3 a^4-10 a^2 b^2+7 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{360 a^2 b^2} \\ & = -\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^5(c+d x) \left (60 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right )-12 a b \left (10 a^4-17 a^2 b^2+7 b^4\right ) \sin (c+d x)-144 \left (5 a^6-25 a^4 b^2+34 a^2 b^4-14 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 a^3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (-288 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right )+36 a b^2 \left (15 a^4-29 a^2 b^2+14 b^4\right ) \sin (c+d x)+180 b \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^4 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (540 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right )-36 a b^3 \left (83 a^4-153 a^2 b^2+70 b^4\right ) \sin (c+d x)-576 b^2 \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4320 a^5 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-576 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right )-36 a b^2 \left (75 a^6-449 a^4 b^2+654 a^2 b^4-280 b^6\right ) \sin (c+d x)+540 b^3 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{8640 a^6 b^2 \left (a^2-b^2\right )} \\ & = \frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-540 b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right )+540 a b^3 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{8640 a^7 b^2 \left (a^2-b^2\right )} \\ & = \frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\left (b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^8}-\frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \int \csc (c+d x) \, dx}{16 a^8} \\ & = \frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}+\frac {\left (2 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^2\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 {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))}-\frac {\left (4 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^2\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 \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \text {arctanh}(\cos (c+d x))}{16 a^8 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \cot (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cot (c+d x) \csc (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{24 a^4 b^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cot (c+d x) \csc ^3(c+d x)}{10 a^3 b^2 d (a+b \sin (c+d x))}+\frac {7 b \cot (c+d x) \csc ^4(c+d x)}{30 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.98 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.93 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {15360 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+480 \left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+480 \left (-5 a^6+90 a^4 b^2-200 a^2 b^4+112 b^6\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {2 a \cot (c+d x) \csc ^6(c+d x) \left (590 a^6-6956 a^4 b^2+15280 a^2 b^4-8400 b^6-8 \left (35 a^6-1289 a^4 b^2+2830 a^2 b^4-1575 b^6\right ) \cos (2 (c+d x))+\left (330 a^6-3844 a^4 b^2+8720 a^2 b^4-5040 b^6\right ) \cos (4 (c+d x))+488 a^4 b^2 \cos (6 (c+d x))-1360 a^2 b^4 \cos (6 (c+d x))+840 b^6 \cos (6 (c+d x))-3942 a^5 b \sin (c+d x)+12620 a^3 b^3 \sin (c+d x)-8400 a b^5 \sin (c+d x)+1967 a^5 b \sin (3 (c+d x))-6590 a^3 b^3 \sin (3 (c+d x))+4200 a b^5 \sin (3 (c+d x))-571 a^5 b \sin (5 (c+d x))+1430 a^3 b^3 \sin (5 (c+d x))-840 a b^5 \sin (5 (c+d x))\right )}{b+a \csc (c+d x)}}{7680 a^8 d} \]

[In]

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

[Out]

(15360*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 480*(5*a^6 - 90*
a^4*b^2 + 200*a^2*b^4 - 112*b^6)*Log[Cos[(c + d*x)/2]] + 480*(-5*a^6 + 90*a^4*b^2 - 200*a^2*b^4 + 112*b^6)*Log
[Sin[(c + d*x)/2]] - (2*a*Cot[c + d*x]*Csc[c + d*x]^6*(590*a^6 - 6956*a^4*b^2 + 15280*a^2*b^4 - 8400*b^6 - 8*(
35*a^6 - 1289*a^4*b^2 + 2830*a^2*b^4 - 1575*b^6)*Cos[2*(c + d*x)] + (330*a^6 - 3844*a^4*b^2 + 8720*a^2*b^4 - 5
040*b^6)*Cos[4*(c + d*x)] + 488*a^4*b^2*Cos[6*(c + d*x)] - 1360*a^2*b^4*Cos[6*(c + d*x)] + 840*b^6*Cos[6*(c +
d*x)] - 3942*a^5*b*Sin[c + d*x] + 12620*a^3*b^3*Sin[c + d*x] - 8400*a*b^5*Sin[c + d*x] + 1967*a^5*b*Sin[3*(c +
d*x)] - 6590*a^3*b^3*Sin[3*(c + d*x)] + 4200*a*b^5*Sin[3*(c + d*x)] - 571*a^5*b*Sin[5*(c + d*x)] + 1430*a^3*b
^3*Sin[5*(c + d*x)] - 840*a*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x]))/(7680*a^8*d)

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{6}-\frac {4 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{2}+3 a^{3} b^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {28 a^{4} b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {32 a^{2} b^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {15 a^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-48 a^{3} b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 a \,b^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-88 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-192 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{7}}-\frac {1}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+12 b^{2}}{256 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-96 a^{2} b^{2}+80 b^{4}}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+360 a^{4} b^{2}-800 a^{2} b^{4}+448 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{8}}+\frac {b}{80 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-8 b^{2}\right )}{48 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-36 a^{2} b^{2}+24 b^{4}\right )}{8 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 b \left (\frac {\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{5} b}{2}-a^{3} b^{3}+\frac {a \,b^{5}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{6}-11 a^{4} b^{2}+16 a^{2} b^{4}-7 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 )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{8}}}{d}\)	\(572\)
default	\(\frac {\frac {\frac {\left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{6}-\frac {4 b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-\frac {3 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5}}{2}+3 a^{3} b^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {28 a^{4} b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {32 a^{2} b^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+\frac {15 a^{5} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-48 a^{3} b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+40 a \,b^{4} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-88 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+288 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{3}-192 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{5}}{64 a^{7}}-\frac {1}{384 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {-6 a^{2}+12 b^{2}}{256 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {15 a^{4}-96 a^{2} b^{2}+80 b^{4}}{128 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-20 a^{6}+360 a^{4} b^{2}-800 a^{2} b^{4}+448 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a^{8}}+\frac {b}{80 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {b \left (7 a^{2}-8 b^{2}\right )}{48 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {b \left (11 a^{4}-36 a^{2} b^{2}+24 b^{4}\right )}{8 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {4 b \left (\frac {\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{5} b}{2}-a^{3} b^{3}+\frac {a \,b^{5}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{6}-11 a^{4} b^{2}+16 a^{2} b^{4}-7 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 )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{8}}}{d}\)	\(572\)
risch	\(\text {Expression too large to display}\)	\(1297\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/64/a^7*(1/6*tan(1/2*d*x+1/2*c)^6*a^5-4/5*b*tan(1/2*d*x+1/2*c)^5*a^4-3/2*tan(1/2*d*x+1/2*c)^4*a^5+3*a^3*
b^2*tan(1/2*d*x+1/2*c)^4+28/3*a^4*b*tan(1/2*d*x+1/2*c)^3-32/3*a^2*b^3*tan(1/2*d*x+1/2*c)^3+15/2*a^5*tan(1/2*d*
x+1/2*c)^2-48*a^3*b^2*tan(1/2*d*x+1/2*c)^2+40*a*b^4*tan(1/2*d*x+1/2*c)^2-88*a^4*b*tan(1/2*d*x+1/2*c)+288*tan(1
/2*d*x+1/2*c)*a^2*b^3-192*tan(1/2*d*x+1/2*c)*b^5)-1/384/a^2/tan(1/2*d*x+1/2*c)^6-1/256*(-6*a^2+12*b^2)/a^4/tan
(1/2*d*x+1/2*c)^4-1/128/a^6*(15*a^4-96*a^2*b^2+80*b^4)/tan(1/2*d*x+1/2*c)^2+1/64/a^8*(-20*a^6+360*a^4*b^2-800*
a^2*b^4+448*b^6)*ln(tan(1/2*d*x+1/2*c))+1/80/a^3*b/tan(1/2*d*x+1/2*c)^5-1/48/a^5*b*(7*a^2-8*b^2)/tan(1/2*d*x+1
/2*c)^3+1/8*b*(11*a^4-36*a^2*b^2+24*b^4)/a^7/tan(1/2*d*x+1/2*c)+4*b/a^8*((1/2*b^2*(a^4-2*a^2*b^2+b^4)*tan(1/2*
d*x+1/2*c)+1/2*a^5*b-a^3*b^3+1/2*a*b^5)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+1/2*(2*a^6-11*a^4*b^
2+16*a^2*b^4-7*b^6)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1252 vs. \(2 (455) = 910\).

Time = 1.05 (sec) , antiderivative size = 2588, normalized size of antiderivative = 5.39 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[1/480*(32*(61*a^5*b^2 - 170*a^3*b^4 + 105*a*b^6)*cos(d*x + c)^7 + 2*(165*a^7 - 3386*a^5*b^2 + 8440*a^3*b^4 -
5040*a*b^6)*cos(d*x + c)^5 - 80*(5*a^7 - 94*a^5*b^2 + 218*a^3*b^4 - 126*a*b^6)*cos(d*x + c)^3 + 240*((2*a^5*b
- 9*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^6 - 2*a^5*b + 9*a^3*b^3 - 7*a*b^5 - 3*(2*a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(
d*x + c)^4 + 3*(2*a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 9*a^2*b^4 + 7*b^6)*cos(d*x + c)^
6 - 2*a^4*b^2 + 9*a^2*b^4 - 7*b^6 - 3*(2*a^4*b^2 - 9*a^2*b^4 + 7*b^6)*cos(d*x + c)^4 + 3*(2*a^4*b^2 - 9*a^2*b^
4 + 7*b^6)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x +
c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a
*b*sin(d*x + c) - a^2 - b^2)) + 30*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c) - 15*(5*a^7 - 9
0*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6 - (5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^6 + 3*(5*a^7
- 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^4 - 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos
(d*x + c)^2 + (5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7 - (5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*c
os(d*x + c)^6 + 3*(5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 90*a^4*b^3 + 20
0*a^2*b^5 - 112*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 15*(5*a^7 - 90*a^5*b^2 + 200*
a^3*b^4 - 112*a*b^6 - (5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^6 + 3*(5*a^7 - 90*a^5*b^2 +
200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^4 - 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^2 + (5
*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7 - (5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^6 +
3*(5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*
b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*((571*a^6*b - 1430*a^4*b^3 + 840*a^2*b^5)*
cos(d*x + c)^5 - 40*(23*a^6*b - 68*a^4*b^3 + 42*a^2*b^5)*cos(d*x + c)^3 + 15*(27*a^6*b - 86*a^4*b^3 + 56*a^2*b
^5)*cos(d*x + c))*sin(d*x + c))/(a^9*d*cos(d*x + c)^6 - 3*a^9*d*cos(d*x + c)^4 + 3*a^9*d*cos(d*x + c)^2 - a^9*
d + (a^8*b*d*cos(d*x + c)^6 - 3*a^8*b*d*cos(d*x + c)^4 + 3*a^8*b*d*cos(d*x + c)^2 - a^8*b*d)*sin(d*x + c)), 1/
480*(32*(61*a^5*b^2 - 170*a^3*b^4 + 105*a*b^6)*cos(d*x + c)^7 + 2*(165*a^7 - 3386*a^5*b^2 + 8440*a^3*b^4 - 504
0*a*b^6)*cos(d*x + c)^5 - 80*(5*a^7 - 94*a^5*b^2 + 218*a^3*b^4 - 126*a*b^6)*cos(d*x + c)^3 - 480*((2*a^5*b - 9
*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^6 - 2*a^5*b + 9*a^3*b^3 - 7*a*b^5 - 3*(2*a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(d*x
+ c)^4 + 3*(2*a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 9*a^2*b^4 + 7*b^6)*cos(d*x + c)^6 -
2*a^4*b^2 + 9*a^2*b^4 - 7*b^6 - 3*(2*a^4*b^2 - 9*a^2*b^4 + 7*b^6)*cos(d*x + c)^4 + 3*(2*a^4*b^2 - 9*a^2*b^4 +
7*b^6)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x +
c))) + 30*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c) - 15*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 -
112*a*b^6 - (5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^6 + 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b
^4 - 112*a*b^6)*cos(d*x + c)^4 - 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^2 + (5*a^6*b -
90*a^4*b^3 + 200*a^2*b^5 - 112*b^7 - (5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^6 + 3*(5*a^6*
b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(
d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 15*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6 - (5*
a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^6 + 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*
cos(d*x + c)^4 - 3*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*cos(d*x + c)^2 + (5*a^6*b - 90*a^4*b^3 + 200
*a^2*b^5 - 112*b^7 - (5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^6 + 3*(5*a^6*b - 90*a^4*b^3 +
200*a^2*b^5 - 112*b^7)*cos(d*x + c)^4 - 3*(5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*cos(d*x + c)^2)*sin(
d*x + c))*log(-1/2*cos(d*x + c) + 1/2) - 2*((571*a^6*b - 1430*a^4*b^3 + 840*a^2*b^5)*cos(d*x + c)^5 - 40*(23*a
^6*b - 68*a^4*b^3 + 42*a^2*b^5)*cos(d*x + c)^3 + 15*(27*a^6*b - 86*a^4*b^3 + 56*a^2*b^5)*cos(d*x + c))*sin(d*x
+ c))/(a^9*d*cos(d*x + c)^6 - 3*a^9*d*cos(d*x + c)^4 + 3*a^9*d*cos(d*x + c)^2 - a^9*d + (a^8*b*d*cos(d*x + c)
^6 - 3*a^8*b*d*cos(d*x + c)^4 + 3*a^8*b*d*cos(d*x + c)^2 - a^8*b*d)*sin(d*x + c))]

Sympy [F(-1)]

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

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**7/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c))^2,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 = 0.39 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.53 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^7/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

-1/1920*(120*(5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*log(abs(tan(1/2*d*x + 1/2*c)))/a^8 - 3840*(2*a^6*b -
11*a^4*b^3 + 16*a^2*b^5 - 7*b^7)*(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) - 3840*(a^4*b^3*tan(1/2*d*x + 1/2*c) - 2*a^2*b^5*tan(1/2*d*x + 1/2*
c) + b^7*tan(1/2*d*x + 1/2*c) + a^5*b^2 - 2*a^3*b^4 + a*b^6)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/
2*c) + a)*a^8) - (5*a^10*tan(1/2*d*x + 1/2*c)^6 - 24*a^9*b*tan(1/2*d*x + 1/2*c)^5 - 45*a^10*tan(1/2*d*x + 1/2*
c)^4 + 90*a^8*b^2*tan(1/2*d*x + 1/2*c)^4 + 280*a^9*b*tan(1/2*d*x + 1/2*c)^3 - 320*a^7*b^3*tan(1/2*d*x + 1/2*c)
^3 + 225*a^10*tan(1/2*d*x + 1/2*c)^2 - 1440*a^8*b^2*tan(1/2*d*x + 1/2*c)^2 + 1200*a^6*b^4*tan(1/2*d*x + 1/2*c)
^2 - 2640*a^9*b*tan(1/2*d*x + 1/2*c) + 8640*a^7*b^3*tan(1/2*d*x + 1/2*c) - 5760*a^5*b^5*tan(1/2*d*x + 1/2*c))/
a^12 - (1470*a^6*tan(1/2*d*x + 1/2*c)^6 - 26460*a^4*b^2*tan(1/2*d*x + 1/2*c)^6 + 58800*a^2*b^4*tan(1/2*d*x + 1
/2*c)^6 - 32928*b^6*tan(1/2*d*x + 1/2*c)^6 + 2640*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 8640*a^3*b^3*tan(1/2*d*x + 1/
2*c)^5 + 5760*a*b^5*tan(1/2*d*x + 1/2*c)^5 - 225*a^6*tan(1/2*d*x + 1/2*c)^4 + 1440*a^4*b^2*tan(1/2*d*x + 1/2*c
)^4 - 1200*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 280*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 320*a^3*b^3*tan(1/2*d*x + 1/2*c
)^3 + 45*a^6*tan(1/2*d*x + 1/2*c)^2 - 90*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 + 24*a^5*b*tan(1/2*d*x + 1/2*c) - 5*a^
6)/(a^8*tan(1/2*d*x + 1/2*c)^6))/d

Mupad [B] (verification not implemented)

Time = 12.11 (sec) , antiderivative size = 1810, normalized size of antiderivative = 3.77 \[ \int \frac {\cot ^6(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^7*(a + b*sin(c + d*x))^2),x)

[Out]

tan(c/2 + (d*x)/2)^6/(384*a^2*d) - (tan(c/2 + (d*x)/2)^4*((128*a^2 + 256*b^2)/(16384*a^4) + 1/(64*a^2) - b^2/(
16*a^4)))/d + (tan(c/2 + (d*x)/2)^2*(3/(128*a^2) + b^2/(8*a^4) - (2*b*((b*(128*a^2 + 256*b^2))/(1024*a^5) - b/
(16*a^3) + (4*b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/a))/a + ((128*a^2 + 256*b^2)*((12
8*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/(128*a^2)))/d + (tan(c/2 + (d*x)/2)*(b/(16*a^3) - ((1
28*a^2 + 256*b^2)*((b*(128*a^2 + 256*b^2))/(1024*a^5) - b/(16*a^3) + (4*b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/
(16*a^2) - b^2/(4*a^4)))/a))/(64*a^2) + (4*b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/a -
(4*b*(3/(64*a^2) + b^2/(4*a^4) - (4*b*((b*(128*a^2 + 256*b^2))/(1024*a^5) - b/(16*a^3) + (4*b*((128*a^2 + 256*
b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/a))/a + ((128*a^2 + 256*b^2)*((128*a^2 + 256*b^2)/(4096*a^4) + 1/
(16*a^2) - b^2/(4*a^4)))/(64*a^2)))/a))/d + (tan(c/2 + (d*x)/2)^3*((b*(128*a^2 + 256*b^2))/(3072*a^5) - b/(48*
a^3) + (4*b*((128*a^2 + 256*b^2)/(4096*a^4) + 1/(16*a^2) - b^2/(4*a^4)))/(3*a)))/d - (tan(c/2 + (d*x)/2)^3*((8
3*a^5*b)/15 - (14*a^3*b^3)/3) + a^6/6 + tan(c/2 + (d*x)/2)^4*(6*a^6 + (56*a^2*b^4)/3 - (79*a^4*b^2)/3) - tan(c
/2 + (d*x)/2)^5*(112*a*b^5 + (191*a^5*b)/3 - (544*a^3*b^3)/3) - tan(c/2 + (d*x)/2)^2*((4*a^6)/3 - (7*a^4*b^2)/
5) + tan(c/2 + (d*x)/2)^6*((15*a^6)/2 - 512*b^6 + 872*a^2*b^4 - 352*a^4*b^2) - (8*tan(c/2 + (d*x)/2)^7*(11*a^6
*b + 16*b^7 - 8*a^2*b^5 - 20*a^4*b^3))/a - (7*a^5*b*tan(c/2 + (d*x)/2))/15)/(d*(64*a^8*tan(c/2 + (d*x)/2)^6 +
64*a^8*tan(c/2 + (d*x)/2)^8 + 128*a^7*b*tan(c/2 + (d*x)/2)^7)) - (b*tan(c/2 + (d*x)/2)^5)/(80*a^3*d) - (log(ta
n(c/2 + (d*x)/2))*(5*a^6 - 112*b^6 + 200*a^2*b^4 - 90*a^4*b^2))/(16*a^8*d) + (b*atan(((b*(2*a^2 - 7*b^2)*(-(a
+ b)^3*(a - b)^3)^(1/2)*((tan(c/2 + (d*x)/2)*(5*a^14 + 448*a^6*b^8 - 1024*a^8*b^6 + 732*a^10*b^4 - 164*a^12*b^
2))/(8*a^13) - (37*a^14*b - 224*a^8*b^7 + 456*a^10*b^5 - 266*a^12*b^3)/(8*a^14) + (b*(2*a^2*b - (tan(c/2 + (d*
x)/2)*(48*a^16 - 64*a^14*b^2))/(8*a^13))*(2*a^2 - 7*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/a^8)*1i)/a^8 - (b*(2*a^
2 - 7*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((37*a^14*b - 224*a^8*b^7 + 456*a^10*b^5 - 266*a^12*b^3)/(8*a^14) - (t
an(c/2 + (d*x)/2)*(5*a^14 + 448*a^6*b^8 - 1024*a^8*b^6 + 732*a^10*b^4 - 164*a^12*b^2))/(8*a^13) + (b*(2*a^2*b
- (tan(c/2 + (d*x)/2)*(48*a^16 - 64*a^14*b^2))/(8*a^13))*(2*a^2 - 7*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/a^8)*1i
)/a^8)/((10*a^12*b + 784*b^13 - 3192*a^2*b^11 + 5062*a^4*b^9 - 3899*a^6*b^7 + 1470*a^8*b^5 - 235*a^10*b^3)/(4*
a^14) + (tan(c/2 + (d*x)/2)*(784*b^12 - 2996*a^2*b^10 + 4362*a^4*b^8 - 2980*a^6*b^6 + 938*a^8*b^4 - 108*a^10*b
^2))/(4*a^13) + (b*(2*a^2 - 7*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((tan(c/2 + (d*x)/2)*(5*a^14 + 448*a^6*b^8 - 1
024*a^8*b^6 + 732*a^10*b^4 - 164*a^12*b^2))/(8*a^13) - (37*a^14*b - 224*a^8*b^7 + 456*a^10*b^5 - 266*a^12*b^3)
/(8*a^14) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^16 - 64*a^14*b^2))/(8*a^13))*(2*a^2 - 7*b^2)*(-(a + b)^3*(
a - b)^3)^(1/2))/a^8))/a^8 + (b*(2*a^2 - 7*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((37*a^14*b - 224*a^8*b^7 + 456*a
^10*b^5 - 266*a^12*b^3)/(8*a^14) - (tan(c/2 + (d*x)/2)*(5*a^14 + 448*a^6*b^8 - 1024*a^8*b^6 + 732*a^10*b^4 - 1
64*a^12*b^2))/(8*a^13) + (b*(2*a^2*b - (tan(c/2 + (d*x)/2)*(48*a^16 - 64*a^14*b^2))/(8*a^13))*(2*a^2 - 7*b^2)*
(-(a + b)^3*(a - b)^3)^(1/2))/a^8))/a^8))*(2*a^2 - 7*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*2i)/(a^8*d)