\(\int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [1261]

Optimal result

Integrand size = 29, antiderivative size = 251 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 a x}{b^3}+\frac {2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {6 \left (a^2-b^2\right )^{3/2} \left (a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {3 \left (a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^2 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {2976, 3855, 3852, 8, 3853, 2718, 2743, 12, 2739, 632, 210} \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \cot (c+d x)}{a^3 d}+\frac {2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}-\frac {6 \left (a^2+b^2\right ) \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 b^3 d}+\frac {3 \left (a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac {2 a x}{b^3}+\frac {\cos (c+d x)}{b^2 d} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 2743

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((a + b*Sin[c + d*x])^(n
+ 1)/(d*(n + 1)*(a^2 - b^2))), x] + Dist[1/((n + 1)*(a^2 - b^2)), Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n +
1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ
erQ[2*n]

Rule 2976

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

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

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 \left (\frac {2 a}{b^3}-\frac {3 \left (a^2-b^2\right ) \csc (c+d x)}{a^4}-\frac {2 b \csc ^2(c+d x)}{a^3}+\frac {\csc ^3(c+d x)}{a^2}-\frac {\sin (c+d x)}{b^2}+\frac {\left (a^2-b^2\right )^3}{a^3 b^3 (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )}{a^4 b^3 (a+b \sin (c+d x))}\right ) \, dx \\ & = \frac {2 a x}{b^3}+\frac {\int \csc ^3(c+d x) \, dx}{a^2}-\frac {\int \sin (c+d x) \, dx}{b^2}-\frac {(2 b) \int \csc ^2(c+d x) \, dx}{a^3}-\frac {\left (3 \left (a^2-b^2\right )\right ) \int \csc (c+d x) \, dx}{a^4}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^3 b^3}-\frac {\left (3 \left (a^2-b^2\right )^2 \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^4 b^3} \\ & = \frac {2 a x}{b^3}+\frac {3 \left (a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac {\int \csc (c+d x) \, dx}{2 a^2}+\frac {\left (a^2-b^2\right )^2 \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^3 b^3}+\frac {(2 b) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}-\frac {\left (6 \left (a^2-b^2\right )^2 \left (a^2+b^2\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^4 b^3 d} \\ & = \frac {2 a x}{b^3}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {3 \left (a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^2 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac {\left (a^2-b^2\right )^2 \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^2 b^3}+\frac {\left (12 \left (a^2-b^2\right )^2 \left (a^2+b^2\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^4 b^3 d} \\ & = \frac {2 a x}{b^3}-\frac {6 \left (a^2-b^2\right )^{3/2} \left (a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {3 \left (a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^2 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}+\frac {\left (2 \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^2 b^3 d} \\ & = \frac {2 a x}{b^3}-\frac {6 \left (a^2-b^2\right )^{3/2} \left (a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {3 \left (a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^2 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}-\frac {\left (4 \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^2 b^3 d} \\ & = \frac {2 a x}{b^3}+\frac {2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {6 \left (a^2-b^2\right )^{3/2} \left (a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^3 d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a^2 d}+\frac {3 \left (a^2-b^2\right ) \text {arctanh}(\cos (c+d x))}{a^4 d}+\frac {\cos (c+d x)}{b^2 d}+\frac {2 b \cot (c+d x)}{a^3 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.47 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.25 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 a (c+d x)}{b^3 d}-\frac {2 \left (a^2-b^2\right )^{3/2} \left (2 a^2+3 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^4 b^3 d}+\frac {\cos (c+d x)}{b^2 d}+\frac {b \cot \left (\frac {1}{2} (c+d x)\right )}{a^3 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^2 d}+\frac {\left (5 a^2-6 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac {\left (-5 a^2+6 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^2 d}+\frac {a^4 \cos (c+d x)-2 a^2 b^2 \cos (c+d x)+b^4 \cos (c+d x)}{a^3 b^2 d (a+b \sin (c+d x))}-\frac {b \tan \left (\frac {1}{2} (c+d x)\right )}{a^3 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {\frac {4 b}{2+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 \left (\frac {-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5} b}{2}+a^{3} b^{3}-\frac {a \,b^{5}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{6}-a^{4} b^{2}-4 a^{2} b^{4}+3 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4} b^{3}}}{d}\)	\(290\)
default	\(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-4 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{3}}+\frac {\frac {4 b}{2+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {1}{8 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-10 a^{2}+12 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {b}{a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {4 \left (\frac {-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {a^{5} b}{2}+a^{3} b^{3}-\frac {a \,b^{5}}{2}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (2 a^{6}-a^{4} b^{2}-4 a^{2} b^{4}+3 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{4} b^{3}}}{d}\)	\(290\)
risch	\(\frac {2 a x}{b^{3}}+\frac {{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}+\frac {{\mathrm e}^{-i \left (d x +c \right )}}{2 d \,b^{2}}+\frac {i \left (-9 i a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}+4 i a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}-2 i a^{5} {\mathrm e}^{i \left (d x +c \right )}+12 i a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+4 i a^{5} {\mathrm e}^{3 i \left (d x +c \right )}-3 i a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}-2 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+10 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+4 i a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-2 i a^{5} {\mathrm e}^{5 i \left (d x +c \right )}-4 a^{2} b^{3}+6 b^{5}+2 a^{4} b +6 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}-12 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} b^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) a^{3} d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{4} d}-\frac {2 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{3}}-\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d b \,a^{2}}+\frac {3 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,a^{4}}+\frac {2 i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{3}}+\frac {i \sqrt {a^{2}-b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d b \,a^{2}}-\frac {3 i \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,a^{4}}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{4} d}\)	\(757\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/4/a^3*(1/2*tan(1/2*d*x+1/2*c)^2*a-4*b*tan(1/2*d*x+1/2*c))+4/b^3*(1/2*b/(1+tan(1/2*d*x+1/2*c)^2)+a*arcta
n(tan(1/2*d*x+1/2*c)))-1/8/a^2/tan(1/2*d*x+1/2*c)^2+1/4/a^4*(-10*a^2+12*b^2)*ln(tan(1/2*d*x+1/2*c))+b/a^3/tan(
1/2*d*x+1/2*c)-4/a^4/b^3*((-1/2*b^2*(a^4-2*a^2*b^2+b^4)*tan(1/2*d*x+1/2*c)-1/2*a^5*b+a^3*b^3-1/2*a*b^5)/(tan(1
/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+1/2*(2*a^6-a^4*b^2-4*a^2*b^4+3*b^6)/(a^2-b^2)^(1/2)*arctan(1/2*(2*
a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (237) = 474\).

Time = 0.86 (sec) , antiderivative size = 1210, normalized size of antiderivative = 4.82 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

[1/4*(8*a^6*d*x*cos(d*x + c)^2 - 8*a^6*d*x + 4*(2*a^5*b - 2*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3 + 2*(2*a^5 + a^3
*b^2 - 3*a*b^4 - (2*a^5 + a^3*b^2 - 3*a*b^4)*cos(d*x + c)^2 + (2*a^4*b + a^2*b^3 - 3*b^5 - (2*a^4*b + a^2*b^3
- 3*b^5)*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)) - 2*(4*a^5*b - 5*a^3*b^3 + 6*a*b^5)*cos(d*x + c) - (5*a^3*b^3 - 6*a*b^5 - (5*a^3*b
^3 - 6*a*b^5)*cos(d*x + c)^2 + (5*a^2*b^4 - 6*b^6 - (5*a^2*b^4 - 6*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*
cos(d*x + c) + 1/2) + (5*a^3*b^3 - 6*a*b^5 - (5*a^3*b^3 - 6*a*b^5)*cos(d*x + c)^2 + (5*a^2*b^4 - 6*b^6 - (5*a^
2*b^4 - 6*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d*x + c) + 1/2) + 2*(4*a^5*b*d*x*cos(d*x + c)^2 + 2*
a^4*b^2*cos(d*x + c)^3 - 4*a^5*b*d*x - (2*a^4*b^2 + 3*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/(a^5*b^3*d*cos(d*x
+ c)^2 - a^5*b^3*d + (a^4*b^4*d*cos(d*x + c)^2 - a^4*b^4*d)*sin(d*x + c)), 1/4*(8*a^6*d*x*cos(d*x + c)^2 - 8*a
^6*d*x + 4*(2*a^5*b - 2*a^3*b^3 + 3*a*b^5)*cos(d*x + c)^3 - 4*(2*a^5 + a^3*b^2 - 3*a*b^4 - (2*a^5 + a^3*b^2 -
3*a*b^4)*cos(d*x + c)^2 + (2*a^4*b + a^2*b^3 - 3*b^5 - (2*a^4*b + a^2*b^3 - 3*b^5)*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*(4*a^5*b - 5*a^3*b^3 + 6*a
*b^5)*cos(d*x + c) - (5*a^3*b^3 - 6*a*b^5 - (5*a^3*b^3 - 6*a*b^5)*cos(d*x + c)^2 + (5*a^2*b^4 - 6*b^6 - (5*a^2
*b^4 - 6*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (5*a^3*b^3 - 6*a*b^5 - (5*a^3*b^3 -
6*a*b^5)*cos(d*x + c)^2 + (5*a^2*b^4 - 6*b^6 - (5*a^2*b^4 - 6*b^6)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(
d*x + c) + 1/2) + 2*(4*a^5*b*d*x*cos(d*x + c)^2 + 2*a^4*b^2*cos(d*x + c)^3 - 4*a^5*b*d*x - (2*a^4*b^2 + 3*a^2*
b^4)*cos(d*x + c))*sin(d*x + c))/(a^5*b^3*d*cos(d*x + c)^2 - a^5*b^3*d + (a^4*b^4*d*cos(d*x + c)^2 - a^4*b^4*d
)*sin(d*x + c))]

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**3/(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.84 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {16 \, {\left (d x + c\right )} a}{b^{3}} - \frac {4 \, {\left (5 \, a^{2} - 6 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} + \frac {30 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 36 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {16 \, {\left (2 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4} + 3 \, 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^{4} b^{3}} + \frac {16 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, 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^{4} b^{2}}}{8 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^3/(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/8*(16*(d*x + c)*a/b^3 - 4*(5*a^2 - 6*b^2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^4 + (a^2*tan(1/2*d*x + 1/2*c)^2 -
8*a*b*tan(1/2*d*x + 1/2*c))/a^4 + (30*a^2*tan(1/2*d*x + 1/2*c)^2 - 36*b^2*tan(1/2*d*x + 1/2*c)^2 + 8*a*b*tan(
1/2*d*x + 1/2*c) - a^2)/(a^4*tan(1/2*d*x + 1/2*c)^2) - 16*(2*a^6 - a^4*b^2 - 4*a^2*b^4 + 3*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^4*b^3) +
16*(a^4*b*tan(1/2*d*x + 1/2*c)^3 - 2*a^2*b^3*tan(1/2*d*x + 1/2*c)^3 + b^5*tan(1/2*d*x + 1/2*c)^3 + 2*a^5*tan(
1/2*d*x + 1/2*c)^2 - 2*a^3*b^2*tan(1/2*d*x + 1/2*c)^2 + a*b^4*tan(1/2*d*x + 1/2*c)^2 + 3*a^4*b*tan(1/2*d*x + 1
/2*c) - 2*a^2*b^3*tan(1/2*d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/2*c) + 2*a^5 - 2*a^3*b^2 + a*b^4)/((a*tan(1/2*d*x
+ 1/2*c)^4 + 2*b*tan(1/2*d*x + 1/2*c)^3 + 2*a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^4*b^2)
)/d

Mupad [B] (verification not implemented)

Time = 12.66 (sec) , antiderivative size = 4294, normalized size of antiderivative = 17.11 \[ \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

int(cos(c + d*x)^6/(sin(c + d*x)^3*(a + b*sin(c + d*x))^2),x)

[Out]

tan(c/2 + (d*x)/2)^2/(8*a^2*d) + ((tan(c/2 + (d*x)/2)^2*(16*a^4 + 16*b^4 - 17*a^2*b^2))/b^2 - a^2/2 + (tan(c/2
+ (d*x)/2)^4*(32*a^4 + 32*b^4 - 33*a^2*b^2))/(2*b^2) + 3*a*b*tan(c/2 + (d*x)/2) + (4*tan(c/2 + (d*x)/2)^5*(2*
a^4 + 2*b^4 - 3*a^2*b^2))/(a*b) + (tan(c/2 + (d*x)/2)^3*(24*a^4 + 8*b^4 - 9*a^2*b^2))/(a*b))/(d*(4*a^4*tan(c/2
+ (d*x)/2)^2 + 8*a^4*tan(c/2 + (d*x)/2)^4 + 4*a^4*tan(c/2 + (d*x)/2)^6 + 8*a^3*b*tan(c/2 + (d*x)/2)^3 + 8*a^3
*b*tan(c/2 + (d*x)/2)^5)) - (b*tan(c/2 + (d*x)/2))/(a^3*d) - (log(tan(c/2 + (d*x)/2))*(5*a^2 - 6*b^2))/(2*a^4*
d) + (4*a*atan((2560*a)/((4800*a^2)/b - 2560*a*tan(c/2 + (d*x)/2) - 12160*b + (4480*b^3)/a^2 + (11520*b^5)/a^4
- (12096*b^7)/a^6 + (3456*b^9)/a^8 + (6144*b^2*tan(c/2 + (d*x)/2))/a - (5120*a^3*tan(c/2 + (d*x)/2))/b^2 - (2
304*b^4*tan(c/2 + (d*x)/2))/a^3 + (3840*a^5*tan(c/2 + (d*x)/2))/b^4) - (12160*tan(c/2 + (d*x)/2))/((4480*b^2)/
a^2 + (4800*a^2)/b^2 + (11520*b^4)/a^4 - (12096*b^6)/a^6 + (3456*b^8)/a^8 - (2560*a*tan(c/2 + (d*x)/2))/b + (6
144*b*tan(c/2 + (d*x)/2))/a - (2304*b^3*tan(c/2 + (d*x)/2))/a^3 - (5120*a^3*tan(c/2 + (d*x)/2))/b^3 + (3840*a^
5*tan(c/2 + (d*x)/2))/b^5 - 12160) - 6144/(6144*tan(c/2 + (d*x)/2) - (12160*a)/b + (4480*b)/a + (11520*b^3)/a^
3 + (4800*a^3)/b^3 - (12096*b^5)/a^5 + (3456*b^7)/a^7 - (2304*b^2*tan(c/2 + (d*x)/2))/a^2 - (2560*a^2*tan(c/2
+ (d*x)/2))/b^2 - (5120*a^4*tan(c/2 + (d*x)/2))/b^4 + (3840*a^6*tan(c/2 + (d*x)/2))/b^6) + (5120*a^3)/(4800*a^
2*b - 12160*b^3 + (4480*b^5)/a^2 + (11520*b^7)/a^4 - (12096*b^9)/a^6 + (3456*b^11)/a^8 - 5120*a^3*tan(c/2 + (d
*x)/2) - 2560*a*b^2*tan(c/2 + (d*x)/2) + (6144*b^4*tan(c/2 + (d*x)/2))/a + (3840*a^5*tan(c/2 + (d*x)/2))/b^2 -
(2304*b^6*tan(c/2 + (d*x)/2))/a^3) + (2304*b^2)/(4480*a*b + (11520*b^3)/a - (12160*a^3)/b - (12096*b^5)/a^3 +
(4800*a^5)/b^3 + (3456*b^7)/a^5 + 6144*a^2*tan(c/2 + (d*x)/2) - 2304*b^2*tan(c/2 + (d*x)/2) - (2560*a^4*tan(c
/2 + (d*x)/2))/b^2 - (5120*a^6*tan(c/2 + (d*x)/2))/b^4 + (3840*a^8*tan(c/2 + (d*x)/2))/b^6) - (3840*a^5)/(4800
*a^2*b^3 - 12160*b^5 + (4480*b^7)/a^2 + (11520*b^9)/a^4 - (12096*b^11)/a^6 + (3456*b^13)/a^8 + 3840*a^5*tan(c/
2 + (d*x)/2) - 2560*a*b^4*tan(c/2 + (d*x)/2) - 5120*a^3*b^2*tan(c/2 + (d*x)/2) + (6144*b^6*tan(c/2 + (d*x)/2))
/a - (2304*b^8*tan(c/2 + (d*x)/2))/a^3) + (4800*a^2*tan(c/2 + (d*x)/2))/(4800*a^2 - 12160*b^2 + (4480*b^4)/a^2
+ (11520*b^6)/a^4 - (12096*b^8)/a^6 + (3456*b^10)/a^8 + (6144*b^3*tan(c/2 + (d*x)/2))/a - (5120*a^3*tan(c/2 +
(d*x)/2))/b - (2304*b^5*tan(c/2 + (d*x)/2))/a^3 + (3840*a^5*tan(c/2 + (d*x)/2))/b^3 - 2560*a*b*tan(c/2 + (d*x
)/2)) + (11520*b^3*tan(c/2 + (d*x)/2))/(4480*a^2*b + 11520*b^3 - (12160*a^4)/b - (12096*b^5)/a^2 + (4800*a^6)/
b^3 + (3456*b^7)/a^4 + 6144*a^3*tan(c/2 + (d*x)/2) - 2304*a*b^2*tan(c/2 + (d*x)/2) - (2560*a^5*tan(c/2 + (d*x)
/2))/b^2 - (5120*a^7*tan(c/2 + (d*x)/2))/b^4 + (3840*a^9*tan(c/2 + (d*x)/2))/b^6) - (12096*b^5*tan(c/2 + (d*x)
/2))/(4480*a^4*b - 12096*b^5 + 11520*a^2*b^3 - (12160*a^6)/b + (3456*b^7)/a^2 + (4800*a^8)/b^3 + 6144*a^5*tan(
c/2 + (d*x)/2) - 2304*a^3*b^2*tan(c/2 + (d*x)/2) - (2560*a^7*tan(c/2 + (d*x)/2))/b^2 - (5120*a^9*tan(c/2 + (d*
x)/2))/b^4 + (3840*a^11*tan(c/2 + (d*x)/2))/b^6) + (3456*b^7*tan(c/2 + (d*x)/2))/(4480*a^6*b + 3456*b^7 - 1209
6*a^2*b^5 + 11520*a^4*b^3 - (12160*a^8)/b + (4800*a^10)/b^3 + 6144*a^7*tan(c/2 + (d*x)/2) - 2304*a^5*b^2*tan(c
/2 + (d*x)/2) - (2560*a^9*tan(c/2 + (d*x)/2))/b^2 - (5120*a^11*tan(c/2 + (d*x)/2))/b^4 + (3840*a^13*tan(c/2 +
(d*x)/2))/b^6) + (4480*b*tan(c/2 + (d*x)/2))/(4480*b + 6144*a*tan(c/2 + (d*x)/2) - (12160*a^2)/b + (11520*b^3)
/a^2 + (4800*a^4)/b^3 - (12096*b^5)/a^4 + (3456*b^7)/a^6 - (2304*b^2*tan(c/2 + (d*x)/2))/a - (2560*a^3*tan(c/2
+ (d*x)/2))/b^2 - (5120*a^5*tan(c/2 + (d*x)/2))/b^4 + (3840*a^7*tan(c/2 + (d*x)/2))/b^6)))/(b^3*d) - (atan(((
(2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(56*a^14 + 36*a^4*b^10 - 96*a^6*b^8 + 232*a^8*b^6 - 224*a^10
*b^4))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(54*b^18 - 189*a^2*b^16 + 180*a^4*b^14 + 70*a^6*b^12 + 194*a^8*b^10
- 653*a^10*b^8 + 292*a^12*b^6 + 120*a^14*b^4 - 64*a^16*b^2))/(a^9*b^8) + ((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^
3)^(1/2)*((16*(72*a^2*b^14 - 174*a^4*b^12 + 95*a^6*b^10 + 42*a^8*b^8 - 35*a^10*b^6 + 32*a^12*b^4 - 24*a^14*b^2
))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(18*a^4*b^16 + 60*a^6*b^14 - 220*a^8*b^12 + 132*a^10*b^10 + 202*a^12*b^8
- 200*a^14*b^6 + 16*a^16*b^4))/(a^9*b^8) + ((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(48*a^6*b^12 -
76*a^8*b^10 + 15*a^10*b^8 + 14*a^12*b^6))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(96*a^6*b^16 - 182*a^8*b^14 + 73*
a^10*b^12 + 30*a^12*b^10 - 16*a^14*b^8))/(a^9*b^8) + (((16*(8*a^10*b^10 - 6*a^12*b^8))/(a^8*b^5) + (16*tan(c/2
+ (d*x)/2)*(32*a^10*b^14 - 34*a^12*b^12 + 4*a^14*b^10))/(a^9*b^8))*(2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/
2))/(a^4*b^3)))/(a^4*b^3)))/(a^4*b^3))*1i)/(a^4*b^3) + ((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(56*
a^14 + 36*a^4*b^10 - 96*a^6*b^8 + 232*a^8*b^6 - 224*a^10*b^4))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(54*b^18 - 1
89*a^2*b^16 + 180*a^4*b^14 + 70*a^6*b^12 + 194*a^8*b^10 - 653*a^10*b^8 + 292*a^12*b^6 + 120*a^14*b^4 - 64*a^16
*b^2))/(a^9*b^8) - ((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(72*a^2*b^14 - 174*a^4*b^12 + 95*a^6*b^1
0 + 42*a^8*b^8 - 35*a^10*b^6 + 32*a^12*b^4 - 24*a^14*b^2))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(18*a^4*b^16 + 6
0*a^6*b^14 - 220*a^8*b^12 + 132*a^10*b^10 + 202*a^12*b^8 - 200*a^14*b^6 + 16*a^16*b^4))/(a^9*b^8) - ((2*a^2 +
3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(48*a^6*b^12 - 76*a^8*b^10 + 15*a^10*b^8 + 14*a^12*b^6))/(a^8*b^5) +
(16*tan(c/2 + (d*x)/2)*(96*a^6*b^16 - 182*a^8*b^14 + 73*a^10*b^12 + 30*a^12*b^10 - 16*a^14*b^8))/(a^9*b^8) - (
((16*(8*a^10*b^10 - 6*a^12*b^8))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(32*a^10*b^14 - 34*a^12*b^12 + 4*a^14*b^10
))/(a^9*b^8))*(2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^4*b^3)))/(a^4*b^3)))/(a^4*b^3))*1i)/(a^4*b^3))/
((32*(140*a^12 - 108*b^12 + 378*a^2*b^10 - 648*a^4*b^8 + 556*a^6*b^6 - 318*a^10*b^2))/(a^8*b^5) - (32*tan(c/2
+ (d*x)/2)*(432*a^8*b^8 - 256*a^16 - 960*a^10*b^6 + 368*a^12*b^4 + 416*a^14*b^2))/(a^9*b^8) - ((2*a^2 + 3*b^2)
*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(56*a^14 + 36*a^4*b^10 - 96*a^6*b^8 + 232*a^8*b^6 - 224*a^10*b^4))/(a^8*b^5
) + (16*tan(c/2 + (d*x)/2)*(54*b^18 - 189*a^2*b^16 + 180*a^4*b^14 + 70*a^6*b^12 + 194*a^8*b^10 - 653*a^10*b^8
+ 292*a^12*b^6 + 120*a^14*b^4 - 64*a^16*b^2))/(a^9*b^8) + ((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(
72*a^2*b^14 - 174*a^4*b^12 + 95*a^6*b^10 + 42*a^8*b^8 - 35*a^10*b^6 + 32*a^12*b^4 - 24*a^14*b^2))/(a^8*b^5) +
(16*tan(c/2 + (d*x)/2)*(18*a^4*b^16 + 60*a^6*b^14 - 220*a^8*b^12 + 132*a^10*b^10 + 202*a^12*b^8 - 200*a^14*b^6
+ 16*a^16*b^4))/(a^9*b^8) + ((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(48*a^6*b^12 - 76*a^8*b^10 + 1
5*a^10*b^8 + 14*a^12*b^6))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(96*a^6*b^16 - 182*a^8*b^14 + 73*a^10*b^12 + 30*
a^12*b^10 - 16*a^14*b^8))/(a^9*b^8) + (((16*(8*a^10*b^10 - 6*a^12*b^8))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(32
*a^10*b^14 - 34*a^12*b^12 + 4*a^14*b^10))/(a^9*b^8))*(2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^4*b^3)))
/(a^4*b^3)))/(a^4*b^3)))/(a^4*b^3) + ((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(56*a^14 + 36*a^4*b^10
- 96*a^6*b^8 + 232*a^8*b^6 - 224*a^10*b^4))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(54*b^18 - 189*a^2*b^16 + 180*
a^4*b^14 + 70*a^6*b^12 + 194*a^8*b^10 - 653*a^10*b^8 + 292*a^12*b^6 + 120*a^14*b^4 - 64*a^16*b^2))/(a^9*b^8) -
((2*a^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((16*(72*a^2*b^14 - 174*a^4*b^12 + 95*a^6*b^10 + 42*a^8*b^8 - 3
5*a^10*b^6 + 32*a^12*b^4 - 24*a^14*b^2))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(18*a^4*b^16 + 60*a^6*b^14 - 220*a
^8*b^12 + 132*a^10*b^10 + 202*a^12*b^8 - 200*a^14*b^6 + 16*a^16*b^4))/(a^9*b^8) - ((2*a^2 + 3*b^2)*(-(a + b)^3
*(a - b)^3)^(1/2)*((16*(48*a^6*b^12 - 76*a^8*b^10 + 15*a^10*b^8 + 14*a^12*b^6))/(a^8*b^5) + (16*tan(c/2 + (d*x
)/2)*(96*a^6*b^16 - 182*a^8*b^14 + 73*a^10*b^12 + 30*a^12*b^10 - 16*a^14*b^8))/(a^9*b^8) - (((16*(8*a^10*b^10
- 6*a^12*b^8))/(a^8*b^5) + (16*tan(c/2 + (d*x)/2)*(32*a^10*b^14 - 34*a^12*b^12 + 4*a^14*b^10))/(a^9*b^8))*(2*a
^2 + 3*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^4*b^3)))/(a^4*b^3)))/(a^4*b^3)))/(a^4*b^3)))*(2*a^2 + 3*b^2)*(-(a
+ b)^3*(a - b)^3)^(1/2)*2i)/(a^4*b^3*d)