\(\int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [1264]

Optimal result

Integrand size = 21, antiderivative size = 424 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \text {arctanh}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))} \]

[Out]

-2*(a^2-6*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^7/d+1/4*b*(15*a^4-40*a^2*b^2
+24*b^4)*arctanh(cos(d*x+c))/a^7/d-1/15*(38*a^4-135*a^2*b^2+90*b^4)*cot(d*x+c)/a^6/d+1/4*(4*a^4-17*a^2*b^2+12*
b^4)*cot(d*x+c)*csc(d*x+c)/a^5/b/d-1/30*(15*a^4-82*a^2*b^2+60*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^4/b^2/d-1/2*cot(d
*x+c)*csc(d*x+c)/b/d/(a+b*sin(d*x+c))+1/6*a*cot(d*x+c)*csc(d*x+c)^2/b^2/d/(a+b*sin(d*x+c))+1/6*(2*a^4-12*a^2*b
^2+9*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^3/b^2/d/(a+b*sin(d*x+c))+3/10*b*cot(d*x+c)*csc(d*x+c)^3/a^2/d/(a+b*sin(d*x
+c))-1/5*cot(d*x+c)*csc(d*x+c)^4/a/d/(a+b*sin(d*x+c))

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2805, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {2 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^7 d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}+\frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \text {arctanh}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))} \]

[In]

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

[Out]

(-2*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^7*d) + (b*(15*a^4 - 4
0*a^2*b^2 + 24*b^4)*ArcTanh[Cos[c + d*x]])/(4*a^7*d) - ((38*a^4 - 135*a^2*b^2 + 90*b^4)*Cot[c + d*x])/(15*a^6*
d) + ((4*a^4 - 17*a^2*b^2 + 12*b^4)*Cot[c + d*x]*Csc[c + d*x])/(4*a^5*b*d) - ((15*a^4 - 82*a^2*b^2 + 60*b^4)*C
ot[c + d*x]*Csc[c + d*x]^2)/(30*a^4*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(2*b*d*(a + b*Sin[c + d*x])) + (a*Cot
[c + d*x]*Csc[c + d*x]^2)/(6*b^2*d*(a + b*Sin[c + d*x])) + ((2*a^4 - 12*a^2*b^2 + 9*b^4)*Cot[c + d*x]*Csc[c +
d*x]^2)/(6*a^3*b^2*d*(a + b*Sin[c + d*x])) + (3*b*Cot[c + d*x]*Csc[c + d*x]^3)/(10*a^2*d*(a + b*Sin[c + d*x]))
- (Cot[c + d*x]*Csc[c + d*x]^4)/(5*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 2805

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^6, x_Symbol] :> Simp[(-Cos[e + f*x])*(
(a + b*Sin[e + f*x])^(m + 1)/(5*a*f*Sin[e + f*x]^5)), x] + (Dist[1/(20*a^2*b^2*m*(m - 1)), Int[((a + b*Sin[e +
f*x])^m/Sin[e + f*x]^4)*Simp[60*a^4 - 44*a^2*b^2*(m - 1)*m + b^4*m*(m - 1)*(m - 3)*(m - 4) + a*b*m*(20*a^2 -
b^2*m*(m - 1))*Sin[e + f*x] - (40*a^4 + b^4*m*(m - 1)*(m - 2)*(m - 4) - 20*a^2*b^2*(m - 1)*(2*m + 1))*Sin[e +
f*x]^2, x], x], x] + Simp[Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*m*Sin[e + f*x]^2)), x] + Simp[a*Cos[
e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*m*(m - 1)*Sin[e + f*x]^3)), x] - Simp[b*(m - 4)*Cos[e + f*x]*((a
+ b*Sin[e + f*x])^(m + 1)/(20*a^2*f*Sin[e + f*x]^4)), x]) /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0] &
& NeQ[m, 1] && IntegerQ[2*m]

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 (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (12 \left (5 a^4-22 a^2 b^2+15 b^4\right )-4 a b \left (10 a^2-3 b^2\right ) \sin (c+d x)-4 \left (10 a^4-45 a^2 b^2+36 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{120 a^2 b^2} \\ & = -\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (12 \left (15 a^6-97 a^4 b^2+142 a^2 b^4-60 b^6\right )-12 a b \left (5 a^4-8 a^2 b^2+3 b^4\right ) \sin (c+d x)-60 \left (2 a^6-14 a^4 b^2+21 a^2 b^4-9 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-180 b \left (4 a^6-21 a^4 b^2+29 a^2 b^4-12 b^6\right )+12 a b^2 \left (16 a^4-31 a^2 b^2+15 b^4\right ) \sin (c+d x)+24 b \left (15 a^6-97 a^4 b^2+142 a^2 b^4-60 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{360 a^4 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (48 b^2 \left (38 a^6-173 a^4 b^2+225 a^2 b^4-90 b^6\right )-12 a b^3 \left (73 a^4-133 a^2 b^2+60 b^4\right ) \sin (c+d x)-180 b^2 \left (4 a^6-21 a^4 b^2+29 a^2 b^4-12 b^6\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{720 a^5 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-180 b^3 \left (15 a^6-55 a^4 b^2+64 a^2 b^4-24 b^6\right )-180 a b^2 \left (4 a^6-21 a^4 b^2+29 a^2 b^4-12 b^6\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{720 a^6 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-6 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^7}-\frac {\left (b \left (15 a^6-55 a^4 b^2+64 a^2 b^4-24 b^6\right )\right ) \int \csc (c+d x) \, dx}{4 a^7 \left (a^2-b^2\right )} \\ & = \frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \text {arctanh}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}-\frac {\left (2 \left (a^2-6 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^7 d} \\ & = \frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \text {arctanh}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))}+\frac {\left (4 \left (a^2-6 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^7 d} \\ & = -\frac {2 \left (a^2-6 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^7 d}+\frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \text {arctanh}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {1920 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )-240 b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+240 b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \cot (c+d x) \csc ^5(c+d x) \left (196 a^5-735 a^3 b^2+540 a b^4-12 \left (16 a^5-85 a^3 b^2+60 a b^4\right ) \cos (2 (c+d x))+\left (92 a^5-285 a^3 b^2+180 a b^4\right ) \cos (4 (c+d x))+1162 a^4 b \sin (c+d x)-3060 a^2 b^3 \sin (c+d x)+1800 b^5 \sin (c+d x)-562 a^4 b \sin (3 (c+d x))+1470 a^2 b^3 \sin (3 (c+d x))-900 b^5 \sin (3 (c+d x))+76 a^4 b \sin (5 (c+d x))-270 a^2 b^3 \sin (5 (c+d x))+180 b^5 \sin (5 (c+d x))\right )}{b+a \csc (c+d x)}}{960 a^7 d} \]

[In]

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

[Out]

-1/960*(1920*(a^2 - 6*b^2)*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] - 240*b*(15*a^4
- 40*a^2*b^2 + 24*b^4)*Log[Cos[(c + d*x)/2]] + 240*b*(15*a^4 - 40*a^2*b^2 + 24*b^4)*Log[Sin[(c + d*x)/2]] + (2
*a*Cot[c + d*x]*Csc[c + d*x]^5*(196*a^5 - 735*a^3*b^2 + 540*a*b^4 - 12*(16*a^5 - 85*a^3*b^2 + 60*a*b^4)*Cos[2*
(c + d*x)] + (92*a^5 - 285*a^3*b^2 + 180*a*b^4)*Cos[4*(c + d*x)] + 1162*a^4*b*Sin[c + d*x] - 3060*a^2*b^3*Sin[
c + d*x] + 1800*b^5*Sin[c + d*x] - 562*a^4*b*Sin[3*(c + d*x)] + 1470*a^2*b^3*Sin[3*(c + d*x)] - 900*b^5*Sin[3*
(c + d*x)] + 76*a^4*b*Sin[5*(c + d*x)] - 270*a^2*b^3*Sin[5*(c + d*x)] + 180*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[
c + d*x]))/(a^7*d)

Maple [A] (verified)

Time = 0.96 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-\frac {7 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-108 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{6}}-\frac {2 \left (\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (a^{6}-8 a^{4} b^{2}+13 a^{2} b^{4}-6 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{a^{7}}-\frac {1}{160 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+12 b^{2}}{96 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-108 a^{2} b^{2}+80 b^{4}}{32 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (a^{2}-b^{2}\right )}{2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (15 a^{4}-40 a^{2} b^{2}+24 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{7}}}{d}\)	\(465\)
default	\(\frac {\frac {\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{5}-b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-\frac {7 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+4 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+16 a^{3} b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{4}-108 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b^{2}+80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{32 a^{6}}-\frac {2 \left (\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (a^{6}-8 a^{4} b^{2}+13 a^{2} b^{4}-6 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{a^{7}}-\frac {1}{160 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+12 b^{2}}{96 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-108 a^{2} b^{2}+80 b^{4}}{32 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{32 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (a^{2}-b^{2}\right )}{2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (15 a^{4}-40 a^{2} b^{2}+24 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{7}}}{d}\)	\(465\)
risch	\(\text {Expression too large to display}\)	\(1048\)

[In]

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

[Out]

1/d*(1/32/a^6*(1/5*tan(1/2*d*x+1/2*c)^5*a^4-b*tan(1/2*d*x+1/2*c)^4*a^3-7/3*a^4*tan(1/2*d*x+1/2*c)^3+4*a^2*b^2*
tan(1/2*d*x+1/2*c)^3+16*a^3*b*tan(1/2*d*x+1/2*c)^2-16*a*b^3*tan(1/2*d*x+1/2*c)^2+22*tan(1/2*d*x+1/2*c)*a^4-108
*tan(1/2*d*x+1/2*c)*a^2*b^2+80*tan(1/2*d*x+1/2*c)*b^4)-2/a^7*((b^2*(a^4-2*a^2*b^2+b^4)*tan(1/2*d*x+1/2*c)+b*a*
(a^4-2*a^2*b^2+b^4))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+(a^6-8*a^4*b^2+13*a^2*b^4-6*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)))-1/160/a^2/tan(1/2*d*x+1/2*c)^5-1/96*(-7*a^
2+12*b^2)/a^4/tan(1/2*d*x+1/2*c)^3-1/32*(22*a^4-108*a^2*b^2+80*b^4)/a^6/tan(1/2*d*x+1/2*c)+1/32/a^3*b/tan(1/2*
d*x+1/2*c)^4-1/2*b/a^5*(a^2-b^2)/tan(1/2*d*x+1/2*c)^2-1/4/a^7*b*(15*a^4-40*a^2*b^2+24*b^4)*ln(tan(1/2*d*x+1/2*
c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 964 vs. \(2 (401) = 802\).

Time = 0.81 (sec) , antiderivative size = 2011, normalized size of antiderivative = 4.74 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/120*(2*(92*a^6 - 285*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^5 - 40*(7*a^6 - 27*a^4*b^2 + 18*a^2*b^4)*cos(d*x +
c)^3 + 60*((a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^6 - a^4*b + 7*a^2*b^3 - 6*b^5 - 3*(a^4*b - 7*a^2*b^3 + 6*
b^5)*cos(d*x + c)^4 + 3*(a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^2 - (a^5 - 7*a^3*b^2 + 6*a*b^4 + (a^5 - 7*a^3
*b^2 + 6*a*b^4)*cos(d*x + c)^4 - 2*(a^5 - 7*a^3*b^2 + 6*a*b^4)*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*(4*a^6 - 17*a^4*b^2 + 12
*a^2*b^4)*cos(d*x + c) + 15*((15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40*a^2*b^4 - 24*
b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^
2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(15*a^5*b - 40*a
^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*((15*a^4*b^2 - 40*a^2*b^4 +
24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40*a^2*b^4 - 24*b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^
4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (15*a^5*b - 40*a
^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/
2*cos(d*x + c) + 1/2) + 2*(4*(38*a^5*b - 135*a^3*b^3 + 90*a*b^5)*cos(d*x + c)^5 - 5*(79*a^5*b - 228*a^3*b^3 +
144*a*b^5)*cos(d*x + c)^3 + 15*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a^7*b*d*cos(d*x
+ c)^6 - 3*a^7*b*d*cos(d*x + c)^4 + 3*a^7*b*d*cos(d*x + c)^2 - a^7*b*d - (a^8*d*cos(d*x + c)^4 - 2*a^8*d*cos(
d*x + c)^2 + a^8*d)*sin(d*x + c)), 1/120*(2*(92*a^6 - 285*a^4*b^2 + 180*a^2*b^4)*cos(d*x + c)^5 - 40*(7*a^6 -
27*a^4*b^2 + 18*a^2*b^4)*cos(d*x + c)^3 + 120*((a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^6 - a^4*b + 7*a^2*b^3
- 6*b^5 - 3*(a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^4 + 3*(a^4*b - 7*a^2*b^3 + 6*b^5)*cos(d*x + c)^2 - (a^5 -
7*a^3*b^2 + 6*a*b^4 + (a^5 - 7*a^3*b^2 + 6*a*b^4)*cos(d*x + c)^4 - 2*(a^5 - 7*a^3*b^2 + 6*a*b^4)*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*(4*a^6 - 1
7*a^4*b^2 + 12*a^2*b^4)*cos(d*x + c) + 15*((15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40
*a^2*b^4 - 24*b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)
*cos(d*x + c)^2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(1
5*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*((15*a^4*b^2 -
40*a^2*b^4 + 24*b^6)*cos(d*x + c)^6 - 15*a^4*b^2 + 40*a^2*b^4 - 24*b^6 - 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)
*cos(d*x + c)^4 + 3*(15*a^4*b^2 - 40*a^2*b^4 + 24*b^6)*cos(d*x + c)^2 - (15*a^5*b - 40*a^3*b^3 + 24*a*b^5 + (1
5*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^4 - 2*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c)^2)*sin(d*x
+ c))*log(-1/2*cos(d*x + c) + 1/2) + 2*(4*(38*a^5*b - 135*a^3*b^3 + 90*a*b^5)*cos(d*x + c)^5 - 5*(79*a^5*b -
228*a^3*b^3 + 144*a*b^5)*cos(d*x + c)^3 + 15*(15*a^5*b - 40*a^3*b^3 + 24*a*b^5)*cos(d*x + c))*sin(d*x + c))/(a
^7*b*d*cos(d*x + c)^6 - 3*a^7*b*d*cos(d*x + c)^4 + 3*a^7*b*d*cos(d*x + c)^2 - a^7*b*d - (a^8*d*cos(d*x + c)^4
- 2*a^8*d*cos(d*x + c)^2 + a^8*d)*sin(d*x + c))]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^6/(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 = 0.38 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.41 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {120 \, {\left (15 \, a^{4} b - 40 \, a^{2} b^{3} + 24 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{7}} + \frac {960 \, {\left (a^{6} - 8 \, a^{4} b^{2} + 13 \, a^{2} b^{4} - 6 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{7}} + \frac {960 \, {\left (a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{7}} - \frac {3 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{7} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, a^{5} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 330 \, a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1620 \, a^{6} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1200 \, a^{4} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{10}} - \frac {4110 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 10960 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6576 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 330 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1620 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1200 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{5}}{a^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]

[In]

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

[Out]

-1/480*(120*(15*a^4*b - 40*a^2*b^3 + 24*b^5)*log(abs(tan(1/2*d*x + 1/2*c)))/a^7 + 960*(a^6 - 8*a^4*b^2 + 13*a^
2*b^4 - 6*b^6)*(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^7) + 960*(a^4*b^2*tan(1/2*d*x + 1/2*c) - 2*a^2*b^4*tan(1/2*d*x + 1/2*c) + b^6*tan(1/2*d*x
+ 1/2*c) + a^5*b - 2*a^3*b^3 + a*b^5)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^7) - (3*a^
8*tan(1/2*d*x + 1/2*c)^5 - 15*a^7*b*tan(1/2*d*x + 1/2*c)^4 - 35*a^8*tan(1/2*d*x + 1/2*c)^3 + 60*a^6*b^2*tan(1/
2*d*x + 1/2*c)^3 + 240*a^7*b*tan(1/2*d*x + 1/2*c)^2 - 240*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 + 330*a^8*tan(1/2*d*x
+ 1/2*c) - 1620*a^6*b^2*tan(1/2*d*x + 1/2*c) + 1200*a^4*b^4*tan(1/2*d*x + 1/2*c))/a^10 - (4110*a^4*b*tan(1/2*
d*x + 1/2*c)^5 - 10960*a^2*b^3*tan(1/2*d*x + 1/2*c)^5 + 6576*b^5*tan(1/2*d*x + 1/2*c)^5 - 330*a^5*tan(1/2*d*x
+ 1/2*c)^4 + 1620*a^3*b^2*tan(1/2*d*x + 1/2*c)^4 - 1200*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 240*a^4*b*tan(1/2*d*x +
1/2*c)^3 + 240*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + 35*a^5*tan(1/2*d*x + 1/2*c)^2 - 60*a^3*b^2*tan(1/2*d*x + 1/2*
c)^2 + 15*a^4*b*tan(1/2*d*x + 1/2*c) - 3*a^5)/(a^7*tan(1/2*d*x + 1/2*c)^5))/d

Mupad [B] (verification not implemented)

Time = 11.86 (sec) , antiderivative size = 1424, normalized size of antiderivative = 3.36 \[ \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

tan(c/2 + (d*x)/2)^5/(160*a^2*d) + (tan(c/2 + (d*x)/2)*(1/(4*a^2) + b^2/(2*a^4) - (4*b*((b*(64*a^2 + 128*b^2))
/(256*a^5) - b/(8*a^3) + (4*b*((64*a^2 + 128*b^2)/(1024*a^4) + 5/(32*a^2) - b^2/(2*a^4)))/a))/a + ((64*a^2 + 1
28*b^2)*((64*a^2 + 128*b^2)/(1024*a^4) + 5/(32*a^2) - b^2/(2*a^4)))/(32*a^2)))/d - (tan(c/2 + (d*x)/2)^3*((64*
a^2 + 128*b^2)/(3072*a^4) + 5/(96*a^2) - b^2/(6*a^4)))/d - (tan(c/2 + (d*x)/2)^3*((31*a^4*b)/3 - 8*a^2*b^3) +
tan(c/2 + (d*x)/2)^4*(48*a*b^4 + (59*a^5)/3 - 72*a^3*b^2) + tan(c/2 + (d*x)/2)^5*(124*a^4*b + 224*b^5 - 360*a^
2*b^3) + a^5/5 - tan(c/2 + (d*x)/2)^2*((32*a^5)/15 - 2*a^3*b^2) - (3*a^4*b*tan(c/2 + (d*x)/2))/5 + (2*tan(c/2
+ (d*x)/2)^6*(11*a^6 + 32*b^6 - 24*a^2*b^4 - 22*a^4*b^2))/a)/(d*(32*a^7*tan(c/2 + (d*x)/2)^5 + 32*a^7*tan(c/2
+ (d*x)/2)^7 + 64*a^6*b*tan(c/2 + (d*x)/2)^6)) + (tan(c/2 + (d*x)/2)^2*((b*(64*a^2 + 128*b^2))/(512*a^5) - b/(
16*a^3) + (2*b*((64*a^2 + 128*b^2)/(1024*a^4) + 5/(32*a^2) - b^2/(2*a^4)))/a))/d - (log(tan(c/2 + (d*x)/2))*(1
5*a^4*b + 24*b^5 - 40*a^2*b^3))/(4*a^7*d) - (b*tan(c/2 + (d*x)/2)^4)/(32*a^3*d) - (atan((((a^2 - 6*b^2)*(-(a +
b)^3*(a - b)^3)^(1/2)*((4*a^13 - 48*a^7*b^6 + 92*a^9*b^4 - 47*a^11*b^2)/(2*a^12) + (tan(c/2 + (d*x)/2)*(23*a^
11*b - 96*a^5*b^7 + 208*a^7*b^5 - 134*a^9*b^3))/(2*a^11) + ((2*a^2*b - (tan(c/2 + (d*x)/2)*(12*a^14 - 16*a^12*
b^2))/(2*a^11))*(a^2 - 6*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/a^7)*1i)/a^7 + ((a^2 - 6*b^2)*(-(a + b)^3*(a - b)^
3)^(1/2)*((4*a^13 - 48*a^7*b^6 + 92*a^9*b^4 - 47*a^11*b^2)/(2*a^12) + (tan(c/2 + (d*x)/2)*(23*a^11*b - 96*a^5*
b^7 + 208*a^7*b^5 - 134*a^9*b^3))/(2*a^11) - ((2*a^2*b - (tan(c/2 + (d*x)/2)*(12*a^14 - 16*a^12*b^2))/(2*a^11)
)*(a^2 - 6*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/a^7)*1i)/a^7)/((15*a^10*b - 144*b^11 + 552*a^2*b^9 - 802*a^4*b^7
+ 539*a^6*b^5 - 160*a^8*b^3)/a^12 + (tan(c/2 + (d*x)/2)*(8*a^10 - 144*b^10 + 516*a^2*b^8 - 682*a^4*b^6 + 400*
a^6*b^4 - 98*a^8*b^2))/a^11 - ((a^2 - 6*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((4*a^13 - 48*a^7*b^6 + 92*a^9*b^4 -
47*a^11*b^2)/(2*a^12) + (tan(c/2 + (d*x)/2)*(23*a^11*b - 96*a^5*b^7 + 208*a^7*b^5 - 134*a^9*b^3))/(2*a^11) +
((2*a^2*b - (tan(c/2 + (d*x)/2)*(12*a^14 - 16*a^12*b^2))/(2*a^11))*(a^2 - 6*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))
/a^7))/a^7 + ((a^2 - 6*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((4*a^13 - 48*a^7*b^6 + 92*a^9*b^4 - 47*a^11*b^2)/(2*
a^12) + (tan(c/2 + (d*x)/2)*(23*a^11*b - 96*a^5*b^7 + 208*a^7*b^5 - 134*a^9*b^3))/(2*a^11) - ((2*a^2*b - (tan(
c/2 + (d*x)/2)*(12*a^14 - 16*a^12*b^2))/(2*a^11))*(a^2 - 6*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/a^7))/a^7))*(a^2
- 6*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*2i)/(a^7*d)