3.2.91 \(\int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [191]

Optimal. Leaf size=424 \[ -\frac {2 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^7 d}+\frac {b \left (15 a^4-40 a^2 b^2+24 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{4 a^7 d}-\frac {\left (38 a^4-135 a^2 b^2+90 b^4\right ) \cot (c+d x)}{15 a^6 d}+\frac {\left (4 a^4-17 a^2 b^2+12 b^4\right ) \cot (c+d x) \csc (c+d x)}{4 a^5 b d}-\frac {\left (15 a^4-82 a^2 b^2+60 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{30 a^4 b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 b d (a+b \sin (c+d x))}+\frac {a \cot (c+d x) \csc ^2(c+d x)}{6 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 a^4-12 a^2 b^2+9 b^4\right ) \cot (c+d x) \csc ^2(c+d x)}{6 a^3 b^2 d (a+b \sin (c+d x))}+\frac {3 b \cot (c+d x) \csc ^3(c+d x)}{10 a^2 d (a+b \sin (c+d x))}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d (a+b \sin (c+d x))} \]

[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]
time = 1.00, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 10, 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 {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} \text {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 ) \tanh ^{-1}(\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))} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {\cot ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=-\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 ) \tanh ^{-1}(\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 ) \tanh ^{-1}(\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} \tan ^{-1}\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 ) \tanh ^{-1}(\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*}

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Mathematica [A]
time = 1.04, size = 361, normalized size = 0.85 \begin {gather*} -\frac {1920 \left (a^2-6 b^2\right ) \left (a^2-b^2\right )^{3/2} \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [A]
time = 0.63, size = 465, normalized size = 1.10 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/32/a^6*(1/5*a^4*tan(1/2*d*x+1/2*c)^5-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+4*a^2*b^2*
tan(1/2*d*x+1/2*c)^3+16*b*a^3*tan(1/2*d*x+1/2*c)^2-16*a*b^3*tan(1/2*d*x+1/2*c)^2+22*a^4*tan(1/2*d*x+1/2*c)-108
*a^2*b^2*tan(1/2*d*x+1/2*c)+80*b^4*tan(1/2*d*x+1/2*c))-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))/(a*tan(1/2*d*x+1/2*c)^2+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*b/a^3/tan(1/2*
d*x+1/2*c)^4-1/2/a^5*b*(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)))

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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))^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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 964 vs. \(2 (401) = 802\).
time = 0.68, size = 2011, normalized size = 4.74 \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))^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))]

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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 )^{2}}\, 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))**2,x)

[Out]

Integral(cot(c + d*x)**6/(a + b*sin(c + d*x))**2, x)

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Giac [A]
time = 9.68, size = 596, normalized size = 1.41 \begin {gather*} -\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} \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))^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

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Mupad [B]
time = 6.91, size = 1424, normalized size = 3.36 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,a^2\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {1}{4\,a^2}+\frac {b^2}{2\,a^4}-\frac {4\,b\,\left (\frac {b\,\left (64\,a^2+128\,b^2\right )}{256\,a^5}-\frac {b}{8\,a^3}+\frac {4\,b\,\left (\frac {64\,a^2+128\,b^2}{1024\,a^4}+\frac {5}{32\,a^2}-\frac {b^2}{2\,a^4}\right )}{a}\right )}{a}+\frac {\left (64\,a^2+128\,b^2\right )\,\left (\frac {64\,a^2+128\,b^2}{1024\,a^4}+\frac {5}{32\,a^2}-\frac {b^2}{2\,a^4}\right )}{32\,a^2}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {64\,a^2+128\,b^2}{3072\,a^4}+\frac {5}{96\,a^2}-\frac {b^2}{6\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {31\,a^4\,b}{3}-8\,a^2\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {59\,a^5}{3}-72\,a^3\,b^2+48\,a\,b^4\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (124\,a^4\,b-360\,a^2\,b^3+224\,b^5\right )+\frac {a^5}{5}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {32\,a^5}{15}-2\,a^3\,b^2\right )-\frac {3\,a^4\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (11\,a^6-22\,a^4\,b^2-24\,a^2\,b^4+32\,b^6\right )}{a}}{d\,\left (32\,a^7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,a^7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+64\,b\,a^6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {b\,\left (64\,a^2+128\,b^2\right )}{512\,a^5}-\frac {b}{16\,a^3}+\frac {2\,b\,\left (\frac {64\,a^2+128\,b^2}{1024\,a^4}+\frac {5}{32\,a^2}-\frac {b^2}{2\,a^4}\right )}{a}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (15\,a^4\,b-40\,a^2\,b^3+24\,b^5\right )}{4\,a^7\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,a^3\,d}-\frac {\mathrm {atan}\left (\frac {\frac {\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {4\,a^{13}-47\,a^{11}\,b^2+92\,a^9\,b^4-48\,a^7\,b^6}{2\,a^{12}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,a^{11}\,b-134\,a^9\,b^3+208\,a^7\,b^5-96\,a^5\,b^7\right )}{2\,a^{11}}+\frac {\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^{14}-16\,a^{12}\,b^2\right )}{2\,a^{11}}\right )\,\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^7}\right )\,1{}\mathrm {i}}{a^7}+\frac {\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {4\,a^{13}-47\,a^{11}\,b^2+92\,a^9\,b^4-48\,a^7\,b^6}{2\,a^{12}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,a^{11}\,b-134\,a^9\,b^3+208\,a^7\,b^5-96\,a^5\,b^7\right )}{2\,a^{11}}-\frac {\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^{14}-16\,a^{12}\,b^2\right )}{2\,a^{11}}\right )\,\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^7}\right )\,1{}\mathrm {i}}{a^7}}{\frac {15\,a^{10}\,b-160\,a^8\,b^3+539\,a^6\,b^5-802\,a^4\,b^7+552\,a^2\,b^9-144\,b^{11}}{a^{12}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,a^{10}-98\,a^8\,b^2+400\,a^6\,b^4-682\,a^4\,b^6+516\,a^2\,b^8-144\,b^{10}\right )}{a^{11}}-\frac {\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {4\,a^{13}-47\,a^{11}\,b^2+92\,a^9\,b^4-48\,a^7\,b^6}{2\,a^{12}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,a^{11}\,b-134\,a^9\,b^3+208\,a^7\,b^5-96\,a^5\,b^7\right )}{2\,a^{11}}+\frac {\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^{14}-16\,a^{12}\,b^2\right )}{2\,a^{11}}\right )\,\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^7}\right )}{a^7}+\frac {\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (\frac {4\,a^{13}-47\,a^{11}\,b^2+92\,a^9\,b^4-48\,a^7\,b^6}{2\,a^{12}}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,a^{11}\,b-134\,a^9\,b^3+208\,a^7\,b^5-96\,a^5\,b^7\right )}{2\,a^{11}}-\frac {\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (12\,a^{14}-16\,a^{12}\,b^2\right )}{2\,a^{11}}\right )\,\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^7}\right )}{a^7}}\right )\,\left (a^2-6\,b^2\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,2{}\mathrm {i}}{a^7\,d} \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))^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)

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