\(\int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1090]

Optimal result

Integrand size = 27, antiderivative size = 269 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{3/2} d}+\frac {\left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]

[Out]

b*(6*a^4-19*a^2*b^2+12*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^5/(a^2-b^2)^(3/2)/d+1/2*(a^2-12
*b^2)*arctanh(cos(d*x+c))/a^5/d+1/2*b*(11*a^2-12*b^2)*cot(d*x+c)/a^4/(a^2-b^2)/d-1/2*(5*a^2-6*b^2)*cot(d*x+c)*
csc(d*x+c)/a^3/(a^2-b^2)/d+1/2*cot(d*x+c)*csc(d*x+c)/a/d/(a+b*sin(d*x+c))^2+1/2*(3*a^2-4*b^2)*cot(d*x+c)*csc(d
*x+c)/a^2/(a^2-b^2)/d/(a+b*sin(d*x+c))

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2968, 3135, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {\left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 d \left (a^2-b^2\right )}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d \left (a^2-b^2\right )}+\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{3/2}}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2} \]

[In]

Int[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

(b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) +
((a^2 - 12*b^2)*ArcTanh[Cos[c + d*x]])/(2*a^5*d) + (b*(11*a^2 - 12*b^2)*Cot[c + d*x])/(2*a^4*(a^2 - b^2)*d) -
((5*a^2 - 6*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^3*(a^2 - b^2)*d) + (Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b
*Sin[c + d*x])^2) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d*x])/(2*a^2*(a^2 - 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 2968

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

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 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + 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] + Dist[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[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C
)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n +
3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, 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] && L
tQ[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}& = \int \frac {\csc ^3(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx \\ & = \frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )} \\ & = \frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (2 \left (5 a^4-11 a^2 b^2+6 b^4\right )-a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2} \\ & = -\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 b \left (11 a^4-23 a^2 b^2+12 b^4\right )-2 a \left (a^4-3 a^2 b^2+2 b^4\right ) \sin (c+d x)+2 b \left (5 a^4-11 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2} \\ & = \frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-2 \left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2+2 a b \left (5 a^4-11 a^2 b^2+6 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2} \\ & = \frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (a^2-12 b^2\right ) \int \csc (c+d x) \, dx}{2 a^5}+\frac {\left (b \left (6 a^4-19 a^2 b^2+12 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )} \\ & = \frac {\left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (b \left (6 a^4-19 a^2 b^2+12 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^5 \left (a^2-b^2\right ) d} \\ & = \frac {\left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (2 b \left (6 a^4-19 a^2 b^2+12 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^5 \left (a^2-b^2\right ) d} \\ & = \frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{3/2} d}+\frac {\left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.23 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{3/2} d}+\frac {3 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\left (a^2-12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\left (-a^2+12 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {b^2 \cos (c+d x)}{2 a^3 d (a+b \sin (c+d x))^2}+\frac {5 a^2 b^2 \cos (c+d x)-6 b^4 \cos (c+d x)}{2 a^4 (a-b) (a+b) d (a+b \sin (c+d x))}-\frac {3 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \]

[In]

Integrate[(Cot[c + d*x]^2*Csc[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

(b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2
- b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) + (3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) + ((a^2 -
12*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + ((-a^2 + 12*b^2)*Log[Sin[(c + d*x)/2]])/(2*a^5*d) + Sec[(c + d*x)/
2]^2/(8*a^3*d) + (b^2*Cos[c + d*x])/(2*a^3*d*(a + b*Sin[c + d*x])^2) + (5*a^2*b^2*Cos[c + d*x] - 6*b^4*Cos[c +
d*x])/(2*a^4*(a - b)*(a + b)*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2])/(2*a^4*d)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.28

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {2 b \left (\frac {\frac {a \,b^{2} \left (7 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {b \left (6 a^{4}+5 a^{2} b^{2}-14 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {a \,b^{2} \left (17 a^{2}-20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (6 a^{2}-7 b^{2}\right )}{2 a^{2}-2 b^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{4}-19 a^{2} b^{2}+12 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{5}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(345\)
default	\(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {2 b \left (\frac {\frac {a \,b^{2} \left (7 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {b \left (6 a^{4}+5 a^{2} b^{2}-14 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {a \,b^{2} \left (17 a^{2}-20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (6 a^{2}-7 b^{2}\right )}{2 a^{2}-2 b^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{4}-19 a^{2} b^{2}+12 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{5}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(345\)
risch	\(\frac {-30 i b \,{\mathrm e}^{2 i \left (d x +c \right )} a^{4}-36 i b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+44 i a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-5 a^{3} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+6 a \,b^{4} {\mathrm e}^{7 i \left (d x +c \right )}+5 i a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-15 i a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-14 i a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}+4 a^{5} {\mathrm e}^{5 i \left (d x +c \right )}+45 a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-54 a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+11 i a^{2} b^{3}+12 i b^{5} {\mathrm e}^{6 i \left (d x +c \right )}-i a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{5} {\mathrm e}^{3 i \left (d x +c \right )}-87 a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+90 a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-12 i b^{5}+36 i b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+39 a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}-42 a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right ) a^{4} d}-\frac {3 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {19 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {6 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}+\frac {3 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {19 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {6 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{5} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{5} d}\)	\(976\)

[In]

int(cos(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/4/a^4*(1/2*tan(1/2*d*x+1/2*c)^2*a-6*b*tan(1/2*d*x+1/2*c))+2*b/a^5*((1/2*a*b^2*(7*a^2-8*b^2)/(a^2-b^2)*t
an(1/2*d*x+1/2*c)^3+1/2*b*(6*a^4+5*a^2*b^2-14*b^4)/(a^2-b^2)*tan(1/2*d*x+1/2*c)^2+1/2*a*b^2*(17*a^2-20*b^2)/(a
^2-b^2)*tan(1/2*d*x+1/2*c)+1/2*a^2*b*(6*a^2-7*b^2)/(a^2-b^2))/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a
)^2+1/2*(6*a^4-19*a^2*b^2+12*b^4)/(a^2-b^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))-1/
8/a^3/tan(1/2*d*x+1/2*c)^2+1/4/a^5*(-2*a^2+24*b^2)*ln(tan(1/2*d*x+1/2*c))+3/2*b/a^4/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. 919 vs. \(2 (254) = 508\).

Time = 0.79 (sec) , antiderivative size = 1922, normalized size of antiderivative = 7.14 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/4*(2*(17*a^6*b^2 - 35*a^4*b^4 + 18*a^2*b^6)*cos(d*x + c)^3 - (6*a^6*b - 13*a^4*b^3 - 7*a^2*b^5 + 12*b^7 +
(6*a^4*b^3 - 19*a^2*b^5 + 12*b^7)*cos(d*x + c)^4 - (6*a^6*b - 7*a^4*b^3 - 26*a^2*b^5 + 24*b^7)*cos(d*x + c)^2
+ 2*(6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6 - (6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))*sqr
t(-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)) + 2*(a^8 - 19*
a^6*b^2 + 36*a^4*b^4 - 18*a^2*b^6)*cos(d*x + c) - (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*
b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6 - 24*b^8)*
cos(d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*
cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b
^8 + (a^6*b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6
- 24*b^8)*cos(d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 -
12*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*((11*a^5*b^3 - 23*a^3*b^5 + 12*a*b^7)
*cos(d*x + c)^3 - (4*a^7*b + 3*a^5*b^3 - 19*a^3*b^5 + 12*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2 - 2*a^7*
b^4 + a^5*b^6)*d*cos(d*x + c)^4 - (a^11 - 3*a^7*b^4 + 2*a^5*b^6)*d*cos(d*x + c)^2 + (a^11 - a^9*b^2 - a^7*b^4
+ a^5*b^6)*d - 2*((a^10*b - 2*a^8*b^3 + a^6*b^5)*d*cos(d*x + c)^2 - (a^10*b - 2*a^8*b^3 + a^6*b^5)*d)*sin(d*x
+ c)), -1/4*(2*(17*a^6*b^2 - 35*a^4*b^4 + 18*a^2*b^6)*cos(d*x + c)^3 + 2*(6*a^6*b - 13*a^4*b^3 - 7*a^2*b^5 + 1
2*b^7 + (6*a^4*b^3 - 19*a^2*b^5 + 12*b^7)*cos(d*x + c)^4 - (6*a^6*b - 7*a^4*b^3 - 26*a^2*b^5 + 24*b^7)*cos(d*x
+ c)^2 + 2*(6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6 - (6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6)*cos(d*x + c)^2)*sin(d*x +
c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) + 2*(a^8 - 19*a^6*b^2 + 36*a
^4*b^4 - 18*a^2*b^6)*cos(d*x + c) - (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*b^2 - 14*a^4*b
^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6 - 24*b^8)*cos(d*x + c)^2
+ 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*cos(d*x + c)^2
)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*b^2
- 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6 - 24*b^8)*cos(
d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*cos(
d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*((11*a^5*b^3 - 23*a^3*b^5 + 12*a*b^7)*cos(d*x + c)^
3 - (4*a^7*b + 3*a^5*b^3 - 19*a^3*b^5 + 12*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2 - 2*a^7*b^4 + a^5*b^6)
*d*cos(d*x + c)^4 - (a^11 - 3*a^7*b^4 + 2*a^5*b^6)*d*cos(d*x + c)^2 + (a^11 - a^9*b^2 - a^7*b^4 + a^5*b^6)*d -
2*((a^10*b - 2*a^8*b^3 + a^6*b^5)*d*cos(d*x + c)^2 - (a^10*b - 2*a^8*b^3 + a^6*b^5)*d)*sin(d*x + c))]

Sympy [F]

\[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

integrate(cos(d*x+c)**2*csc(d*x+c)**3/(a+b*sin(d*x+c))**3,x)

[Out]

Integral(cos(c + d*x)**2*csc(c + d*x)**3/(a + b*sin(c + d*x))**3, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^3/(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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (254) = 508\).

Time = 0.52 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.96 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {8 \, {\left (6 \, a^{4} b - 19 \, a^{2} b^{3} + 12 \, b^{5}\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 )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 26 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 20 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 32 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 53 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 64 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 28 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 112 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 68 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 76 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{6} + a^{4} b^{2}}{{\left (a^{7} - a^{5} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2}} - \frac {4 \, {\left (a^{2} - 12 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{8 \, d} \]

[In]

integrate(cos(d*x+c)^2*csc(d*x+c)^3/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/8*(8*(6*a^4*b - 19*a^2*b^3 + 12*b^5)*(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)))/((a^7 - a^5*b^2)*sqrt(a^2 - b^2)) + (2*a^6*tan(1/2*d*x + 1/2*c)^6 - 26*a^4*b^2*tan(1
/2*d*x + 1/2*c)^6 + 24*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 + 20*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 60*a^3*b^3*tan(1/2*d
*x + 1/2*c)^5 + 32*a*b^5*tan(1/2*d*x + 1/2*c)^5 + 3*a^6*tan(1/2*d*x + 1/2*c)^4 + 53*a^4*b^2*tan(1/2*d*x + 1/2*
c)^4 - 64*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 16*b^6*tan(1/2*d*x + 1/2*c)^4 + 28*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 6
0*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 112*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 68*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 76*a
^2*b^4*tan(1/2*d*x + 1/2*c)^2 + 8*a^5*b*tan(1/2*d*x + 1/2*c) - 8*a^3*b^3*tan(1/2*d*x + 1/2*c) - a^6 + a^4*b^2)
/((a^7 - a^5*b^2)*(a*tan(1/2*d*x + 1/2*c)^3 + 2*b*tan(1/2*d*x + 1/2*c)^2 + a*tan(1/2*d*x + 1/2*c))^2) - 4*(a^2
- 12*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 + (a^3*tan(1/2*d*x + 1/2*c)^2 - 12*a^2*b*tan(1/2*d*x + 1/2*c))/a
^6)/d

Mupad [B] (verification not implemented)

Time = 12.38 (sec) , antiderivative size = 1906, normalized size of antiderivative = 7.09 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^2/(sin(c + d*x)^3*(a + b*sin(c + d*x))^3),x)

[Out]

tan(c/2 + (d*x)/2)^2/(8*a^3*d) - (a^3/2 - (2*tan(c/2 + (d*x)/2)^5*(3*a^4*b - 16*b^5 + 11*a^2*b^3))/(a^2 - b^2)
- (2*tan(c/2 + (d*x)/2)^3*(5*a^4*b - 52*b^5 + 41*a^2*b^3))/(a^2 - b^2) - 4*a^2*b*tan(c/2 + (d*x)/2) + (tan(c/
2 + (d*x)/2)^2*(50*a*b^4 + a^5 - 47*a^3*b^2))/(a^2 - b^2) + (tan(c/2 + (d*x)/2)^4*(a^6 + 112*b^6 + 8*a^2*b^4 -
97*a^4*b^2))/(2*a*(a^2 - b^2)))/(d*(4*a^6*tan(c/2 + (d*x)/2)^2 + 4*a^6*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)
/2)^4*(8*a^6 + 16*a^4*b^2) + 16*a^5*b*tan(c/2 + (d*x)/2)^3 + 16*a^5*b*tan(c/2 + (d*x)/2)^5)) - (3*b*tan(c/2 +
(d*x)/2))/(2*a^4*d) - (log(tan(c/2 + (d*x)/2))*(a^2 - 12*b^2))/(2*a^5*d) - (b*atan(((b*(-(a + b)^3*(a - b)^3)^
(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*((7*a^9*b + 24*a^5*b^5 - 32*a^7*b^3)/(a^10 - a^8*b^2) - (tan(c/2 + (d*x)/2
)*(a^11 + 48*a^3*b^8 - 124*a^5*b^6 + 103*a^7*b^4 - 28*a^9*b^2))/(a^11 + a^7*b^4 - 2*a^9*b^2) + (b*((2*a^12*b -
2*a^10*b^3)/(a^10 - a^8*b^2) - (tan(c/2 + (d*x)/2)*(6*a^14 - 8*a^8*b^6 + 22*a^10*b^4 - 20*a^12*b^2))/(a^11 +
a^7*b^4 - 2*a^9*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2))/(2*(a^11 - a^5*b^6 + 3*a^7*b
^4 - 3*a^9*b^2)))*1i)/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)) - (b*(-(a + b)^3*(a - b)^3)^(1/2)*(6*a^4 +
12*b^4 - 19*a^2*b^2)*((tan(c/2 + (d*x)/2)*(a^11 + 48*a^3*b^8 - 124*a^5*b^6 + 103*a^7*b^4 - 28*a^9*b^2))/(a^11
+ a^7*b^4 - 2*a^9*b^2) - (7*a^9*b + 24*a^5*b^5 - 32*a^7*b^3)/(a^10 - a^8*b^2) + (b*((2*a^12*b - 2*a^10*b^3)/(a
^10 - a^8*b^2) - (tan(c/2 + (d*x)/2)*(6*a^14 - 8*a^8*b^6 + 22*a^10*b^4 - 20*a^12*b^2))/(a^11 + a^7*b^4 - 2*a^9
*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2))/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)
))*1i)/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)))/((6*a^6*b - 144*b^7 + 240*a^2*b^5 - 91*a^4*b^3)/(a^10 - a
^8*b^2) + (2*tan(c/2 + (d*x)/2)*(72*b^8 - 174*a^2*b^6 + 131*a^4*b^4 - 30*a^6*b^2))/(a^11 + a^7*b^4 - 2*a^9*b^2
) + (b*(-(a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*((7*a^9*b + 24*a^5*b^5 - 32*a^7*b^3)/(a^10 -
a^8*b^2) - (tan(c/2 + (d*x)/2)*(a^11 + 48*a^3*b^8 - 124*a^5*b^6 + 103*a^7*b^4 - 28*a^9*b^2))/(a^11 + a^7*b^4
- 2*a^9*b^2) + (b*((2*a^12*b - 2*a^10*b^3)/(a^10 - a^8*b^2) - (tan(c/2 + (d*x)/2)*(6*a^14 - 8*a^8*b^6 + 22*a^1
0*b^4 - 20*a^12*b^2))/(a^11 + a^7*b^4 - 2*a^9*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)
)/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))))/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)) + (b*(-(a + b)^3
*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*((tan(c/2 + (d*x)/2)*(a^11 + 48*a^3*b^8 - 124*a^5*b^6 + 103*a^
7*b^4 - 28*a^9*b^2))/(a^11 + a^7*b^4 - 2*a^9*b^2) - (7*a^9*b + 24*a^5*b^5 - 32*a^7*b^3)/(a^10 - a^8*b^2) + (b*
((2*a^12*b - 2*a^10*b^3)/(a^10 - a^8*b^2) - (tan(c/2 + (d*x)/2)*(6*a^14 - 8*a^8*b^6 + 22*a^10*b^4 - 20*a^12*b^
2))/(a^11 + a^7*b^4 - 2*a^9*b^2))*(-(a + b)^3*(a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2))/(2*(a^11 - a^5*b
^6 + 3*a^7*b^4 - 3*a^9*b^2))))/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))))*(-(a + b)^3*(a - b)^3)^(1/2)*(6*
a^4 + 12*b^4 - 19*a^2*b^2)*1i)/(d*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2))