3.3.2 \(\int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [202]

Optimal. Leaf size=492 \[ -\frac {\sqrt {a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))} \]

[Out]

1/8*b*(45*a^4-200*a^2*b^2+168*b^4)*arctanh(cos(d*x+c))/a^8/d-1/30*(91*a^4-645*a^2*b^2+630*b^4)*cot(d*x+c)/a^7/
d+1/8*(8*a^4-79*a^2*b^2+84*b^4)*cot(d*x+c)*csc(d*x+c)/a^6/b/d-1/30*(15*a^4-187*a^2*b^2+210*b^4)*cot(d*x+c)*csc
(d*x+c)^2/a^5/b^2/d-1/3*cot(d*x+c)*csc(d*x+c)/b/d/(a+b*sin(d*x+c))^2+1/12*a*cot(d*x+c)*csc(d*x+c)^2/b^2/d/(a+b
*sin(d*x+c))^2+1/60*(5*a^4-60*a^2*b^2+63*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^3/b^2/d/(a+b*sin(d*x+c))^2+7/20*b*cot(
d*x+c)*csc(d*x+c)^3/a^2/d/(a+b*sin(d*x+c))^2-1/5*cot(d*x+c)*csc(d*x+c)^4/a/d/(a+b*sin(d*x+c))^2+1/12*(4*a^4-54
*a^2*b^2+63*b^4)*cot(d*x+c)*csc(d*x+c)^2/a^4/b^2/d/(a+b*sin(d*x+c))-(2*a^4-29*a^2*b^2+42*b^4)*arctan((b+a*tan(
1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))*(a^2-b^2)^(1/2)/a^8/d

________________________________________________________________________________________

Rubi [A]
time = 1.39, antiderivative size = 492, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2805, 3134, 3080, 3855, 2739, 632, 210} \begin {gather*} \frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac {\sqrt {a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[a^2 - b^2]*(2*a^4 - 29*a^2*b^2 + 42*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^8*d)) +
(b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*ArcTanh[Cos[c + d*x]])/(8*a^8*d) - ((91*a^4 - 645*a^2*b^2 + 630*b^4)*Cot[c
+ d*x])/(30*a^7*d) + ((8*a^4 - 79*a^2*b^2 + 84*b^4)*Cot[c + d*x]*Csc[c + d*x])/(8*a^6*b*d) - ((15*a^4 - 187*a
^2*b^2 + 210*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(30*a^5*b^2*d) - (Cot[c + d*x]*Csc[c + d*x])/(3*b*d*(a + b*Sin[
c + d*x])^2) + (a*Cot[c + d*x]*Csc[c + d*x]^2)/(12*b^2*d*(a + b*Sin[c + d*x])^2) + ((5*a^4 - 60*a^2*b^2 + 63*b
^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(60*a^3*b^2*d*(a + b*Sin[c + d*x])^2) + (7*b*Cot[c + d*x]*Csc[c + d*x]^3)/(20
*a^2*d*(a + b*Sin[c + d*x])^2) - (Cot[c + d*x]*Csc[c + d*x]^4)/(5*a*d*(a + b*Sin[c + d*x])^2) + ((4*a^4 - 54*a
^2*b^2 + 63*b^4)*Cot[c + d*x]*Csc[c + d*x]^2)/(12*a^4*b^2*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*} \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^4(c+d x) \left (12 \left (5 a^4-44 a^2 b^2+42 b^4\right )-12 a b \left (5 a^2-3 b^2\right ) \sin (c+d x)-20 \left (2 a^4-20 a^2 b^2+21 b^4\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{240 a^2 b^2}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^4(c+d x) \left (24 \left (10 a^6-114 a^4 b^2+209 a^2 b^4-105 b^6\right )-8 a b \left (20 a^4-41 a^2 b^2+21 b^4\right ) \sin (c+d x)-32 \left (5 a^6-65 a^4 b^2+123 a^2 b^4-63 b^6\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{480 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^4(c+d x) \left (48 \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right )-24 a b \left (10 a^2-21 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-120 \left (a^2-b^2\right )^2 \left (4 a^4-54 a^2 b^2+63 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{480 a^4 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (-360 b \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right )+24 a b^2 \left (62 a^2-105 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)+96 b \left (a^2-b^2\right )^2 \left (15 a^4-187 a^2 b^2+210 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{1440 a^5 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (96 b^2 \left (a^2-b^2\right )^2 \left (91 a^4-645 a^2 b^2+630 b^4\right )-24 a b^3 \left (311 a^2-420 b^2\right ) \left (a^2-b^2\right )^2 \sin (c+d x)-360 b^2 \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2880 a^6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-360 b^3 \left (a^2-b^2\right )^2 \left (45 a^4-200 a^2 b^2+168 b^4\right )-360 a b^2 \left (a^2-b^2\right )^2 \left (8 a^4-79 a^2 b^2+84 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2880 a^7 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^8}-\frac {\left (b \left (45 a^4-200 a^2 b^2+168 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^8}\\ &=\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}-\frac {\left (\left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^8 d}\\ &=\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 \left (a^2-b^2\right ) \left (2 a^4-29 a^2 b^2+42 b^4\right )\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 {\sqrt {a^2-b^2} \left (2 a^4-29 a^2 b^2+42 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^8 d}+\frac {b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^8 d}-\frac {\left (91 a^4-645 a^2 b^2+630 b^4\right ) \cot (c+d x)}{30 a^7 d}+\frac {\left (8 a^4-79 a^2 b^2+84 b^4\right ) \cot (c+d x) \csc (c+d x)}{8 a^6 b d}-\frac {\left (15 a^4-187 a^2 b^2+210 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^5 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{3 b d (a+b \sin (c+d x))^2}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{12 b^2 d (a+b \sin (c+d x))^2}+\frac {\left (5 a^4-60 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{60 a^3 b^2 d (a+b \sin (c+d x))^2}+\frac {7 b \cot (c+d x) \csc ^3(c+d x)}{20 a^2 d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))^2}+\frac {\left (4 a^4-54 a^2 b^2+63 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{12 a^4 b^2 d (a+b \sin (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 1.13, size = 448, normalized size = 0.91 \begin {gather*} \frac {-\frac {3840 \left (2 a^6-31 a^4 b^2+71 a^2 b^4-42 b^6\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+480 b \left (45 a^4-200 a^2 b^2+168 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-480 b \left (45 a^4-200 a^2 b^2+168 b^4\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 (-784 a^6+3256 a^4 b^2+7860 a^2 b^4-12600 b^6+2 \left (384 a^6-2131 a^4 b^2-6315 a^2 b^4+9450 b^6\right ) \cos (2 (c+d x))+\left (-368 a^6+824 a^4 b^2+6060 a^2 b^4-7560 b^6\right ) \cos (4 (c+d x))+182 a^4 b^2 \cos (6 (c+d x))-1290 a^2 b^4 \cos (6 (c+d x))+1260 b^6 \cos (6 (c+d x))-8156 a^5 b \sin (c+d x)+42270 a^3 b^3 \sin (c+d x)-37800 a b^5 \sin (c+d x)+3956 a^5 b \sin (3 (c+d x))-20715 a^3 b^3 \sin (3 (c+d x))+18900 a b^5 \sin (3 (c+d x))-608 a^5 b \sin (5 (c+d x))+3975 a^3 b^3 \sin (5 (c+d x))-3780 a b^5 \sin (5 (c+d x))\right )}{(b+a \csc (c+d x))^2}}{3840 a^8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-3840*(2*a^6 - 31*a^4*b^2 + 71*a^2*b^4 - 42*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2
- b^2] + 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)*Log[Cos[(c + d*x)/2]] - 480*b*(45*a^4 - 200*a^2*b^2 + 168*b^4)
*Log[Sin[(c + d*x)/2]] + (2*a*Cot[c + d*x]*Csc[c + d*x]^6*(-784*a^6 + 3256*a^4*b^2 + 7860*a^2*b^4 - 12600*b^6
+ 2*(384*a^6 - 2131*a^4*b^2 - 6315*a^2*b^4 + 9450*b^6)*Cos[2*(c + d*x)] + (-368*a^6 + 824*a^4*b^2 + 6060*a^2*b
^4 - 7560*b^6)*Cos[4*(c + d*x)] + 182*a^4*b^2*Cos[6*(c + d*x)] - 1290*a^2*b^4*Cos[6*(c + d*x)] + 1260*b^6*Cos[
6*(c + d*x)] - 8156*a^5*b*Sin[c + d*x] + 42270*a^3*b^3*Sin[c + d*x] - 37800*a*b^5*Sin[c + d*x] + 3956*a^5*b*Si
n[3*(c + d*x)] - 20715*a^3*b^3*Sin[3*(c + d*x)] + 18900*a*b^5*Sin[3*(c + d*x)] - 608*a^5*b*Sin[5*(c + d*x)] +
3975*a^3*b^3*Sin[5*(c + d*x)] - 3780*a*b^5*Sin[5*(c + d*x)]))/(b + a*Csc[c + d*x])^2)/(3840*a^8*d)

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Maple [A]
time = 0.84, size = 559, normalized size = 1.14

method	result	size
derivativedivides	\(\frac {\frac {\frac {a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{3}+8 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 b \,a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-216 a^{2} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+240 b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{7}}-\frac {1}{160 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+24 b^{2}}{96 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-216 a^{2} b^{2}+240 b^{4}}{32 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{64 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (3 a^{2}-5 b^{2}\right )}{4 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (45 a^{4}-200 a^{2} b^{2}+168 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{8}}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{5} b^{2}-\frac {19}{2} a^{3} b^{4}+7 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{6}-9 a^{4} b^{2}-21 a^{2} b^{4}+26 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{4}-49 a^{2} b^{2}+38 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{4}-17 a^{2} b^{2}+13 b^{4}\right )}{2}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{6}-31 a^{4} b^{2}+71 a^{2} b^{4}-42 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}\)	\(559\)
default	\(\frac {\frac {\frac {a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {3 b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{2}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{3}+8 a^{2} b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 b \,a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-40 a \,b^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+22 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-216 a^{2} b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+240 b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 a^{7}}-\frac {1}{160 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {-7 a^{2}+24 b^{2}}{96 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {22 a^{4}-216 a^{2} b^{2}+240 b^{4}}{32 a^{7} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{64 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b \left (3 a^{2}-5 b^{2}\right )}{4 a^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b \left (45 a^{4}-200 a^{2} b^{2}+168 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{8}}-\frac {2 \left (\frac {\left (\frac {5}{2} a^{5} b^{2}-\frac {19}{2} a^{3} b^{4}+7 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {b \left (4 a^{6}-9 a^{4} b^{2}-21 a^{2} b^{4}+26 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+\frac {a \,b^{2} \left (11 a^{4}-49 a^{2} b^{2}+38 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+\frac {a^{2} b \left (4 a^{4}-17 a^{2} b^{2}+13 b^{4}\right )}{2}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (2 a^{6}-31 a^{4} b^{2}+71 a^{2} b^{4}-42 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}\)	\(559\)
risch	\(\text {Expression too large to display}\)	\(1254\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/32/a^7*(1/5*a^4*tan(1/2*d*x+1/2*c)^5-3/2*b*tan(1/2*d*x+1/2*c)^4*a^3-7/3*tan(1/2*d*x+1/2*c)^3*a^4+8*a^2*
b^2*tan(1/2*d*x+1/2*c)^3+24*b*a^3*tan(1/2*d*x+1/2*c)^2-40*a*b^3*tan(1/2*d*x+1/2*c)^2+22*a^4*tan(1/2*d*x+1/2*c)
-216*a^2*b^2*tan(1/2*d*x+1/2*c)+240*b^4*tan(1/2*d*x+1/2*c))-1/160/a^3/tan(1/2*d*x+1/2*c)^5-1/96*(-7*a^2+24*b^2
)/a^5/tan(1/2*d*x+1/2*c)^3-1/32*(22*a^4-216*a^2*b^2+240*b^4)/a^7/tan(1/2*d*x+1/2*c)+3/64*b/a^4/tan(1/2*d*x+1/2
*c)^4-1/4/a^6*b*(3*a^2-5*b^2)/tan(1/2*d*x+1/2*c)^2-1/8/a^8*b*(45*a^4-200*a^2*b^2+168*b^4)*ln(tan(1/2*d*x+1/2*c
))-2/a^8*(((5/2*a^5*b^2-19/2*a^3*b^4+7*a*b^6)*tan(1/2*d*x+1/2*c)^3+1/2*b*(4*a^6-9*a^4*b^2-21*a^2*b^4+26*b^6)*t
an(1/2*d*x+1/2*c)^2+1/2*a*b^2*(11*a^4-49*a^2*b^2+38*b^4)*tan(1/2*d*x+1/2*c)+1/2*a^2*b*(4*a^4-17*a^2*b^2+13*b^4
))/(a*tan(1/2*d*x+1/2*c)^2+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(2*a^6-31*a^4*b^2+71*a^2*b^4-42*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))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (467) = 934\).
time = 0.72, size = 2571, normalized size = 5.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/240*(8*(91*a^5*b^2 - 645*a^3*b^4 + 630*a*b^6)*cos(d*x + c)^7 - 4*(92*a^7 + 67*a^5*b^2 - 3450*a^3*b^4 + 378
0*a*b^6)*cos(d*x + c)^5 + 40*(14*a^7 - 37*a^5*b^2 - 303*a^3*b^4 + 378*a*b^6)*cos(d*x + c)^3 - 60*(2*(2*a^5*b -
29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^6 - 4*a^5*b + 58*a^3*b^3 - 84*a*b^5 - 6*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)
*cos(d*x + c)^4 + 6*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 29*a^2*b^4 + 42*b^6)*cos(
d*x + c)^6 - 2*a^6 + 27*a^4*b^2 - 13*a^2*b^4 - 42*b^6 - (2*a^6 - 23*a^4*b^2 - 45*a^2*b^4 + 126*b^6)*cos(d*x +
c)^4 + (4*a^6 - 52*a^4*b^2 - 3*a^2*b^4 + 126*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)) - 60*(4*a^7 - 17*a^5*b^2 - 58*a^3*b^4 + 84
*a*b^6)*cos(d*x + c) + 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos
(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^
6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7 - (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(
d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^
2*b^5 + 504*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*(90*a^5*b^2 - 400*a^3*b^4 + 33
6*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*c
os(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5
+ 168*b^7 - (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*
b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*
cos(d*x + c) + 1/2) - 2*((608*a^6*b - 3975*a^4*b^3 + 3780*a^2*b^5)*cos(d*x + c)^5 - 5*(289*a^6*b - 1632*a^4*b^
3 + 1512*a^2*b^5)*cos(d*x + c)^3 + 15*(53*a^6*b - 279*a^4*b^3 + 252*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*a^
9*b*d*cos(d*x + c)^6 - 6*a^9*b*d*cos(d*x + c)^4 + 6*a^9*b*d*cos(d*x + c)^2 - 2*a^9*b*d + (a^8*b^2*d*cos(d*x +
c)^6 - (a^10 + 3*a^8*b^2)*d*cos(d*x + c)^4 + (2*a^10 + 3*a^8*b^2)*d*cos(d*x + c)^2 - (a^10 + a^8*b^2)*d)*sin(d
*x + c)), -1/240*(8*(91*a^5*b^2 - 645*a^3*b^4 + 630*a*b^6)*cos(d*x + c)^7 - 4*(92*a^7 + 67*a^5*b^2 - 3450*a^3*
b^4 + 3780*a*b^6)*cos(d*x + c)^5 + 40*(14*a^7 - 37*a^5*b^2 - 303*a^3*b^4 + 378*a*b^6)*cos(d*x + c)^3 - 120*(2*
(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^6 - 4*a^5*b + 58*a^3*b^3 - 84*a*b^5 - 6*(2*a^5*b - 29*a^3*b^3 +
42*a*b^5)*cos(d*x + c)^4 + 6*(2*a^5*b - 29*a^3*b^3 + 42*a*b^5)*cos(d*x + c)^2 + ((2*a^4*b^2 - 29*a^2*b^4 + 42
*b^6)*cos(d*x + c)^6 - 2*a^6 + 27*a^4*b^2 - 13*a^2*b^4 - 42*b^6 - (2*a^6 - 23*a^4*b^2 - 45*a^2*b^4 + 126*b^6)*
cos(d*x + c)^4 + (4*a^6 - 52*a^4*b^2 - 3*a^2*b^4 + 126*b^6)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*arct
an(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - 60*(4*a^7 - 17*a^5*b^2 - 58*a^3*b^4 + 84*a*b^6)*cos
(d*x + c) + 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6
+ 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x
+ c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7 - (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(d*x + c)^6
+ (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*b^7)*cos(d*x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504
*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 15*(90*a^5*b^2 - 400*a^3*b^4 + 336*a*b^6 - 2
*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^6 + 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)
^4 - 6*(45*a^5*b^2 - 200*a^3*b^4 + 168*a*b^6)*cos(d*x + c)^2 + (45*a^6*b - 155*a^4*b^3 - 32*a^2*b^5 + 168*b^7
- (45*a^4*b^3 - 200*a^2*b^5 + 168*b^7)*cos(d*x + c)^6 + (45*a^6*b - 65*a^4*b^3 - 432*a^2*b^5 + 504*b^7)*cos(d*
x + c)^4 - (90*a^6*b - 265*a^4*b^3 - 264*a^2*b^5 + 504*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c
) + 1/2) - 2*((608*a^6*b - 3975*a^4*b^3 + 3780*a^2*b^5)*cos(d*x + c)^5 - 5*(289*a^6*b - 1632*a^4*b^3 + 1512*a^
2*b^5)*cos(d*x + c)^3 + 15*(53*a^6*b - 279*a^4*b^3 + 252*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(2*a^9*b*d*cos(d
*x + c)^6 - 6*a^9*b*d*cos(d*x + c)^4 + 6*a^9*b*d*cos(d*x + c)^2 - 2*a^9*b*d + (a^8*b^2*d*cos(d*x + c)^6 - (a^1
0 + 3*a^8*b^2)*d*cos(d*x + c)^4 + (2*a^10 + 3*a^8*b^2)*d*cos(d*x + c)^2 - (a^10 + a^8*b^2)*d)*sin(d*x + c))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{6}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**6/(a+b*sin(d*x+c))**3,x)

[Out]

Integral(cot(c + d*x)**6/(a + b*sin(c + d*x))**3, x)

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Giac [A]
time = 11.78, size = 731, normalized size = 1.49 \begin {gather*} -\frac {\frac {120 \, {\left (45 \, a^{4} b - 200 \, a^{2} b^{3} + 168 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{8}} + \frac {960 \, {\left (2 \, a^{6} - 31 \, a^{4} b^{2} + 71 \, a^{2} b^{4} - 42 \, 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^{8}} + \frac {960 \, {\left (5 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 19 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{6} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 26 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 49 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 38 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{6} b - 17 \, a^{4} b^{3} + 13 \, a^{2} 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 )}^{2} a^{8}} - \frac {12330 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 54800 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 46032 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 660 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 6480 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 7200 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 720 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1200 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 70 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 45 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{5}}{a^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - \frac {6 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{11} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 70 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 240 \, a^{10} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 720 \, a^{11} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1200 \, a^{9} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 660 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6480 \, a^{10} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7200 \, a^{8} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{15}}}{960 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^6/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/960*(120*(45*a^4*b - 200*a^2*b^3 + 168*b^5)*log(abs(tan(1/2*d*x + 1/2*c)))/a^8 + 960*(2*a^6 - 31*a^4*b^2 +
71*a^2*b^4 - 42*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^8) + 960*(5*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 19*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 14
*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 4*a^6*b*tan(1/2*d*x + 1/2*c)^2 - 9*a^4*b^3*tan(1/2*d*x + 1/2*c)^2 - 21*a^2*b^5
*tan(1/2*d*x + 1/2*c)^2 + 26*b^7*tan(1/2*d*x + 1/2*c)^2 + 11*a^5*b^2*tan(1/2*d*x + 1/2*c) - 49*a^3*b^4*tan(1/2
*d*x + 1/2*c) + 38*a*b^6*tan(1/2*d*x + 1/2*c) + 4*a^6*b - 17*a^4*b^3 + 13*a^2*b^5)/((a*tan(1/2*d*x + 1/2*c)^2
+ 2*b*tan(1/2*d*x + 1/2*c) + a)^2*a^8) - (12330*a^4*b*tan(1/2*d*x + 1/2*c)^5 - 54800*a^2*b^3*tan(1/2*d*x + 1/2
*c)^5 + 46032*b^5*tan(1/2*d*x + 1/2*c)^5 - 660*a^5*tan(1/2*d*x + 1/2*c)^4 + 6480*a^3*b^2*tan(1/2*d*x + 1/2*c)^
4 - 7200*a*b^4*tan(1/2*d*x + 1/2*c)^4 - 720*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 1200*a^2*b^3*tan(1/2*d*x + 1/2*c)^3
+ 70*a^5*tan(1/2*d*x + 1/2*c)^2 - 240*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + 45*a^4*b*tan(1/2*d*x + 1/2*c) - 6*a^5)
/(a^8*tan(1/2*d*x + 1/2*c)^5) - (6*a^12*tan(1/2*d*x + 1/2*c)^5 - 45*a^11*b*tan(1/2*d*x + 1/2*c)^4 - 70*a^12*ta
n(1/2*d*x + 1/2*c)^3 + 240*a^10*b^2*tan(1/2*d*x + 1/2*c)^3 + 720*a^11*b*tan(1/2*d*x + 1/2*c)^2 - 1200*a^9*b^3*
tan(1/2*d*x + 1/2*c)^2 + 660*a^12*tan(1/2*d*x + 1/2*c) - 6480*a^10*b^2*tan(1/2*d*x + 1/2*c) + 7200*a^8*b^4*tan
(1/2*d*x + 1/2*c))/a^15)/d

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Mupad [B]
time = 7.30, size = 1614, normalized size = 3.28 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^6/(a + b*sin(c + d*x))^3,x)

[Out]

tan(c/2 + (d*x)/2)^5/(160*a^3*d) - (tan(c/2 + (d*x)/2)^3*((a^2 + 4*b^2)/(32*a^5) + 1/(24*a^3) - (3*b^2)/(8*a^5
)))/d + (tan(c/2 + (d*x)/2)*(1/(8*a^3) - (3*(a^2 + 4*b^2))/(32*a^5) - (6*b*((6*b*((3*(a^2 + 4*b^2))/(32*a^5) +
1/(8*a^3) - (9*b^2)/(8*a^5)))/a - (384*a^2*b + 256*b^3)/(1024*a^6) + (9*b*(a^2 + 4*b^2))/(16*a^6)))/a + (3*(a
^2 + 4*b^2)*((3*(a^2 + 4*b^2))/(32*a^5) + 1/(8*a^3) - (9*b^2)/(8*a^5)))/a^2 + (3*b*(384*a^2*b + 256*b^3))/(512
*a^7)))/d - (tan(c/2 + (d*x)/2)^3*((187*a^5*b)/15 - 14*a^3*b^3) + a^6/5 + tan(c/2 + (d*x)/2)^4*((263*a^6)/15 +
112*a^2*b^4 - (358*a^4*b^2)/3) + tan(c/2 + (d*x)/2)^5*(1216*a*b^5 + (1519*a^5*b)/6 - 1360*a^3*b^3) - tan(c/2
+ (d*x)/2)^2*((29*a^6)/15 - (14*a^4*b^2)/5) + tan(c/2 + (d*x)/2)^8*(22*a^6 + 448*b^6 - 368*a^2*b^4 - 56*a^4*b^
2) + tan(c/2 + (d*x)/2)^6*((125*a^6)/3 + 2176*b^6 - 2112*a^2*b^4 + 112*a^4*b^2) + (8*tan(c/2 + (d*x)/2)^7*(30*
a^6*b + 104*b^7 + 36*a^2*b^5 - 149*a^4*b^3))/a - (7*a^5*b*tan(c/2 + (d*x)/2))/10)/(d*(32*a^9*tan(c/2 + (d*x)/2
)^5 + 32*a^9*tan(c/2 + (d*x)/2)^9 + tan(c/2 + (d*x)/2)^7*(64*a^9 + 128*a^7*b^2) + 128*a^8*b*tan(c/2 + (d*x)/2)
^6 + 128*a^8*b*tan(c/2 + (d*x)/2)^8)) + (tan(c/2 + (d*x)/2)^2*((3*b*((3*(a^2 + 4*b^2))/(32*a^5) + 1/(8*a^3) -
(9*b^2)/(8*a^5)))/a - (384*a^2*b + 256*b^3)/(2048*a^6) + (9*b*(a^2 + 4*b^2))/(32*a^6)))/d - (log(tan(c/2 + (d*
x)/2))*(45*a^4*b + 168*b^5 - 200*a^2*b^3))/(8*a^8*d) - (3*b*tan(c/2 + (d*x)/2)^4)/(64*a^4*d) - (atan((((-(a +
b)*(a - b))^(1/2)*(a^4 + 21*b^4 - (29*a^2*b^2)/2)*((2*a^14 - 84*a^8*b^6 + 121*a^10*b^4 - (169*a^12*b^2)/4)/a^1
4 + (tan(c/2 + (d*x)/2)*(61*a^12*b - 672*a^6*b^7 + 1136*a^8*b^5 - 538*a^10*b^3))/(4*a^13) + ((-(a + b)*(a - b)
)^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^16 - 32*a^14*b^2))/(4*a^13))*(a^4 + 21*b^4 - (29*a^2*b^2)/2))/a^8
)*1i)/a^8 + ((-(a + b)*(a - b))^(1/2)*(a^4 + 21*b^4 - (29*a^2*b^2)/2)*((2*a^14 - 84*a^8*b^6 + 121*a^10*b^4 - (
169*a^12*b^2)/4)/a^14 + (tan(c/2 + (d*x)/2)*(61*a^12*b - 672*a^6*b^7 + 1136*a^8*b^5 - 538*a^10*b^3))/(4*a^13)
- ((-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^16 - 32*a^14*b^2))/(4*a^13))*(a^4 + 21*b^4 -
(29*a^2*b^2)/2))/a^8)*1i)/a^8)/(((45*a^10*b)/2 - 1764*b^11 + 5082*a^2*b^9 - (10649*a^4*b^7)/2 + (9731*a^6*b^5)
/4 - (1795*a^8*b^3)/4)/a^14 + (tan(c/2 + (d*x)/2)*(16*a^10 - 3528*b^10 + 9282*a^2*b^8 - 8549*a^4*b^6 + 3185*a^
6*b^4 - 406*a^8*b^2))/(2*a^13) - ((-(a + b)*(a - b))^(1/2)*(a^4 + 21*b^4 - (29*a^2*b^2)/2)*((2*a^14 - 84*a^8*b
^6 + 121*a^10*b^4 - (169*a^12*b^2)/4)/a^14 + (tan(c/2 + (d*x)/2)*(61*a^12*b - 672*a^6*b^7 + 1136*a^8*b^5 - 538
*a^10*b^3))/(4*a^13) + ((-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^16 - 32*a^14*b^2))/(4*a^
13))*(a^4 + 21*b^4 - (29*a^2*b^2)/2))/a^8))/a^8 + ((-(a + b)*(a - b))^(1/2)*(a^4 + 21*b^4 - (29*a^2*b^2)/2)*((
2*a^14 - 84*a^8*b^6 + 121*a^10*b^4 - (169*a^12*b^2)/4)/a^14 + (tan(c/2 + (d*x)/2)*(61*a^12*b - 672*a^6*b^7 + 1
136*a^8*b^5 - 538*a^10*b^3))/(4*a^13) - ((-(a + b)*(a - b))^(1/2)*(2*a^2*b - (tan(c/2 + (d*x)/2)*(24*a^16 - 32
*a^14*b^2))/(4*a^13))*(a^4 + 21*b^4 - (29*a^2*b^2)/2))/a^8))/a^8))*(-(a + b)*(a - b))^(1/2)*(a^4 + 21*b^4 - (2
9*a^2*b^2)/2)*2i)/(a^8*d)

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