\(\int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) [1260]

Optimal result

Integrand size = 29, antiderivative size = 254 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {x}{2 b^2}-\frac {3 \left (a^2-b^2\right ) x}{b^4}-\frac {2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a b^4 d}+\frac {4 \left (2 a^6-3 a^4 b^2+b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^4 \sqrt {a^2-b^2} d}+\frac {2 b \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {2 a \cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 d (a+b \sin (c+d x))} \]

[Out]

-1/2*x/b^2-3*(a^2-b^2)*x/b^4-2*(a^2-b^2)^(3/2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a/b^4/d+2*b*ar
ctanh(cos(d*x+c))/a^3/d-2*a*cos(d*x+c)/b^3/d-cot(d*x+c)/a^2/d+1/2*cos(d*x+c)*sin(d*x+c)/b^2/d-(a^2-b^2)^2*cos(
d*x+c)/a^2/b^3/d/(a+b*sin(d*x+c))+4*(2*a^6-3*a^4*b^2+b^6)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^3
/b^4/d/(a^2-b^2)^(1/2)

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 254, 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, 2718, 2715, 2743, 12, 2739, 632, 210} \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {2 \left (a^2-b^2\right )^{3/2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a b^4 d}-\frac {3 x \left (a^2-b^2\right )}{b^4}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 d (a+b \sin (c+d x))}-\frac {\cot (c+d x)}{a^2 d}+\frac {4 \left (2 a^6-3 a^4 b^2+b^6\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^4 d \sqrt {a^2-b^2}}-\frac {2 a \cos (c+d x)}{b^3 d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b^2 d}-\frac {x}{2 b^2} \]

[In]

Int[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]

[Out]

-1/2*x/b^2 - (3*(a^2 - b^2)*x)/b^4 - (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a
*b^4*d) + (4*(2*a^6 - 3*a^4*b^2 + b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^4*Sqrt[a^2 - b
^2]*d) + (2*b*ArcTanh[Cos[c + d*x]])/(a^3*d) - (2*a*Cos[c + d*x])/(b^3*d) - Cot[c + d*x]/(a^2*d) + (Cos[c + d*
x]*Sin[c + d*x])/(2*b^2*d) - ((a^2 - b^2)^2*Cos[c + d*x])/(a^2*b^3*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 2715

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

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 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 {3 \left (-a^2+b^2\right )}{b^4}-\frac {2 b \csc (c+d x)}{a^3}+\frac {\csc ^2(c+d x)}{a^2}+\frac {2 a \sin (c+d x)}{b^3}-\frac {\sin ^2(c+d x)}{b^2}-\frac {\left (a^2-b^2\right )^3}{a^2 b^4 (a+b \sin (c+d x))^2}+\frac {2 \left (2 a^6-3 a^4 b^2+b^6\right )}{a^3 b^4 (a+b \sin (c+d x))}\right ) \, dx \\ & = -\frac {3 \left (a^2-b^2\right ) x}{b^4}+\frac {\int \csc ^2(c+d x) \, dx}{a^2}+\frac {(2 a) \int \sin (c+d x) \, dx}{b^3}-\frac {\int \sin ^2(c+d x) \, dx}{b^2}-\frac {(2 b) \int \csc (c+d x) \, dx}{a^3}-\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^2 b^4}+\frac {\left (2 \left (2 a^6-3 a^4 b^2+b^6\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^3 b^4} \\ & = -\frac {3 \left (a^2-b^2\right ) x}{b^4}+\frac {2 b \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {2 a \cos (c+d x)}{b^3 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 d (a+b \sin (c+d x))}-\frac {\int 1 \, dx}{2 b^2}-\frac {\left (a^2-b^2\right )^2 \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^2 b^4}-\frac {\text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d}+\frac {\left (4 \left (2 a^6-3 a^4 b^2+b^6\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^3 b^4 d} \\ & = -\frac {x}{2 b^2}-\frac {3 \left (a^2-b^2\right ) x}{b^4}+\frac {2 b \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {2 a \cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 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 b^4}-\frac {\left (8 \left (2 a^6-3 a^4 b^2+b^6\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^3 b^4 d} \\ & = -\frac {x}{2 b^2}-\frac {3 \left (a^2-b^2\right ) x}{b^4}+\frac {4 \left (2 a^6-3 a^4 b^2+b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^4 \sqrt {a^2-b^2} d}+\frac {2 b \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {2 a \cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 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 b^4 d} \\ & = -\frac {x}{2 b^2}-\frac {3 \left (a^2-b^2\right ) x}{b^4}+\frac {4 \left (2 a^6-3 a^4 b^2+b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^4 \sqrt {a^2-b^2} d}+\frac {2 b \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {2 a \cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 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 b^4 d} \\ & = -\frac {x}{2 b^2}-\frac {3 \left (a^2-b^2\right ) x}{b^4}-\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 b^4 d}+\frac {4 \left (2 a^6-3 a^4 b^2+b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^4 \sqrt {a^2-b^2} d}+\frac {2 b \text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {2 a \cos (c+d x)}{b^3 d}-\frac {\cot (c+d x)}{a^2 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^2 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {2 \left (-6 a^2+5 b^2\right ) (c+d x)}{b^4}+\frac {8 \left (a^2-b^2\right )^{3/2} \left (3 a^2+2 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^4}-\frac {8 a \cos (c+d x)}{b^3}-\frac {2 \cot \left (\frac {1}{2} (c+d x)\right )}{a^2}+\frac {8 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}-\frac {8 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}-\frac {4 \left (a^2-b^2\right )^2 \cos (c+d x)}{a^2 b^3 (a+b \sin (c+d x))}+\frac {\sin (2 (c+d x))}{b^2}+\frac {2 \tan \left (\frac {1}{2} (c+d x)\right )}{a^2}}{4 d} \]

[In]

Integrate[(Cos[c + d*x]^4*Cot[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]

[Out]

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

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}+\frac {\frac {2 \left (-b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-b a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (3 a^{6}-4 a^{4} b^{2}-a^{2} b^{4}+2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{3} b^{4}}-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+2 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (6 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {1}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\)	\(304\)
default	\(\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}}+\frac {\frac {2 \left (-b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-b a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )\right )}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {2 \left (3 a^{6}-4 a^{4} b^{2}-a^{2} b^{4}+2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{3} b^{4}}-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+2 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (6 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{4}}-\frac {1}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}}{d}\)	\(304\)
risch	\(-\frac {3 x \,a^{2}}{b^{4}}+\frac {5 x}{2 b^{2}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{2} d}-\frac {a \,{\mathrm e}^{i \left (d x +c \right )}}{b^{3} d}-\frac {a \,{\mathrm e}^{-i \left (d x +c \right )}}{b^{3} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{2} d}+\frac {2 i \left (i a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+2 i b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+a^{5} {\mathrm e}^{3 i \left (d x +c \right )}-2 a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-i a^{4} b +2 i a^{2} b^{3}-2 i b^{5}-{\mathrm e}^{i \left (d x +c \right )} a^{5}+2 a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}-3 a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}\right )}{a^{2} d \,b^{4} \left (2 a \,{\mathrm e}^{3 i \left (d x +c \right )}-i b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 a \,{\mathrm e}^{i \left (d x +c \right )}+2 i b \,{\mathrm e}^{2 i \left (d x +c \right )}-i b \right )}-\frac {3 \sqrt {-a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}+\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{2} a}+\frac {2 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a -\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}+\frac {3 \sqrt {-a^{2}+b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{4}}-\frac {\sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,b^{2} a}-\frac {2 \sqrt {-a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a +\sqrt {-a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}-\frac {2 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{3} d}+\frac {2 b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}\)	\(675\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.70 (sec) , antiderivative size = 901, normalized size of antiderivative = 3.55 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\left [-\frac {3 \, a^{4} b^{2} \cos \left (d x + c\right )^{3} + {\left (6 \, a^{5} b - 5 \, a^{3} b^{3}\right )} d x \cos \left (d x + c\right )^{2} - {\left (6 \, a^{5} b - 5 \, a^{3} b^{3}\right )} d x - {\left (3 \, a^{4} b - a^{2} b^{3} - 2 \, b^{5} - {\left (3 \, a^{4} b - a^{2} b^{3} - 2 \, b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{5} - a^{3} b^{2} - 2 \, a b^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - {\left (3 \, a^{4} b^{2} + 2 \, a^{2} b^{4}\right )} \cos \left (d x + c\right ) - 2 \, {\left (b^{6} \cos \left (d x + c\right )^{2} - a b^{5} \sin \left (d x + c\right ) - b^{6}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (b^{6} \cos \left (d x + c\right )^{2} - a b^{5} \sin \left (d x + c\right ) - b^{6}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{3} b^{3} \cos \left (d x + c\right )^{3} + {\left (6 \, a^{6} - 5 \, a^{4} b^{2}\right )} d x + {\left (6 \, a^{5} b - 5 \, a^{3} b^{3} + 4 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{3} b^{5} d \cos \left (d x + c\right )^{2} - a^{4} b^{4} d \sin \left (d x + c\right ) - a^{3} b^{5} d\right )}}, -\frac {3 \, a^{4} b^{2} \cos \left (d x + c\right )^{3} + {\left (6 \, a^{5} b - 5 \, a^{3} b^{3}\right )} d x \cos \left (d x + c\right )^{2} - {\left (6 \, a^{5} b - 5 \, a^{3} b^{3}\right )} d x - 2 \, {\left (3 \, a^{4} b - a^{2} b^{3} - 2 \, b^{5} - {\left (3 \, a^{4} b - a^{2} b^{3} - 2 \, b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, a^{5} - a^{3} b^{2} - 2 \, a b^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (3 \, a^{4} b^{2} + 2 \, a^{2} b^{4}\right )} \cos \left (d x + c\right ) - 2 \, {\left (b^{6} \cos \left (d x + c\right )^{2} - a b^{5} \sin \left (d x + c\right ) - b^{6}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (b^{6} \cos \left (d x + c\right )^{2} - a b^{5} \sin \left (d x + c\right ) - b^{6}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{3} b^{3} \cos \left (d x + c\right )^{3} + {\left (6 \, a^{6} - 5 \, a^{4} b^{2}\right )} d x + {\left (6 \, a^{5} b - 5 \, a^{3} b^{3} + 4 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{3} b^{5} d \cos \left (d x + c\right )^{2} - a^{4} b^{4} d \sin \left (d x + c\right ) - a^{3} b^{5} d\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cos ^{6}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^2/(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 = 384, normalized size of antiderivative = 1.51 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {12 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} + \frac {3 \, {\left (6 \, a^{2} - 5 \, b^{2}\right )} {\left (d x + c\right )}}{b^{4}} + \frac {6 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} b^{3}} - \frac {12 \, {\left (3 \, a^{6} - 4 \, a^{4} b^{2} - a^{2} b^{4} + 2 \, b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{3} b^{4}} - \frac {4 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 21 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2} b^{3}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{3} b^{3}}}{6 \, d} \]

[In]

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

[Out]

-1/6*(12*b*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - 3*tan(1/2*d*x + 1/2*c)/a^2 + 3*(6*a^2 - 5*b^2)*(d*x + c)/b^4 +
6*(b*tan(1/2*d*x + 1/2*c)^3 + 4*a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c) + 4*a)/((tan(1/2*d*x + 1/2*
c)^2 + 1)^2*b^3) - 12*(3*a^6 - 4*a^4*b^2 - a^2*b^4 + 2*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^3*b^4) - (4*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 1
2*a^4*b*tan(1/2*d*x + 1/2*c)^2 + 21*a^2*b^3*tan(1/2*d*x + 1/2*c)^2 - 4*b^5*tan(1/2*d*x + 1/2*c)^2 - 12*a^5*tan
(1/2*d*x + 1/2*c) + 24*a^3*b^2*tan(1/2*d*x + 1/2*c) - 14*a*b^4*tan(1/2*d*x + 1/2*c) - 3*a^2*b^3)/((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))*a^3*b^3))/d

Mupad [B] (verification not implemented)

Time = 13.57 (sec) , antiderivative size = 5214, normalized size of antiderivative = 20.53 \[ \int \frac {\cos ^4(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

tan(c/2 + (d*x)/2)/(2*a^2*d) - (a + (2*tan(c/2 + (d*x)/2)*(6*a^4 + 3*b^4 - 4*a^2*b^2))/b^3 + (2*tan(c/2 + (d*x
)/2)^5*(6*a^4 + 3*b^4 - 2*a^2*b^2))/b^3 + (4*tan(c/2 + (d*x)/2)^3*(6*a^4 + 3*b^4 - 5*a^2*b^2))/b^3 + (tan(c/2
+ (d*x)/2)^6*(6*a^4 + 4*b^4 - 7*a^2*b^2))/(a*b^2) + (tan(c/2 + (d*x)/2)^2*(18*a^4 + 4*b^4 - 5*a^2*b^2))/(a*b^2
) + (tan(c/2 + (d*x)/2)^4*(24*a^4 + 8*b^4 - 13*a^2*b^2))/(a*b^2))/(d*(6*a^3*tan(c/2 + (d*x)/2)^3 + 6*a^3*tan(c
/2 + (d*x)/2)^5 + 2*a^3*tan(c/2 + (d*x)/2)^7 + 2*a^3*tan(c/2 + (d*x)/2) + 4*a^2*b*tan(c/2 + (d*x)/2)^2 + 8*a^2
*b*tan(c/2 + (d*x)/2)^4 + 4*a^2*b*tan(c/2 + (d*x)/2)^6)) + (atan((((a^2*6i - b^2*5i)*((8*(378*a^15 - 40*a^3*b^
12 + 488*a^5*b^10 - 1158*a^7*b^8 + 541*a^9*b^6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) - ((a^2*6i - b^2*5i
)*(((a^2*6i - b^2*5i)*((((8*(16*a^8*b^13 - 12*a^10*b^11))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^18 - 68*
a^9*b^16 + 8*a^11*b^14))/(a^6*b^12))*(a^2*6i - b^2*5i))/(2*b^4) - (8*(64*a^5*b^14 - 48*a^7*b^12 - 50*a^9*b^10
+ 42*a^11*b^8))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 96*a^8*b^15 - 160*a^10*b^13 +
48*a^12*b^11))/(a^6*b^12)))/(2*b^4) - (8*(48*a^4*b^13 - 64*a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b
^7 - 324*a^12*b^5 + 108*a^14*b^3))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(16*a^3*b^18 + 240*a^5*b^16 + 20*a^7*b^14
- 1696*a^9*b^12 + 2369*a^11*b^10 - 1020*a^13*b^8 + 72*a^15*b^6))/(a^6*b^12)))/(2*b^4) + (8*tan(c/2 + (d*x)/2)
*(32*b^19 - 32*a^2*b^17 + 680*a^4*b^15 - 2560*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2160*
a^14*b^5 - 432*a^16*b^3))/(a^6*b^12))*1i)/(2*b^4) + ((a^2*6i - b^2*5i)*((8*(378*a^15 - 40*a^3*b^12 + 488*a^5*b
^10 - 1158*a^7*b^8 + 541*a^9*b^6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) + ((a^2*6i - b^2*5i)*(((a^2*6i -
b^2*5i)*((8*(64*a^5*b^14 - 48*a^7*b^12 - 50*a^9*b^10 + 42*a^11*b^8))/(a^6*b^8) + (((8*(16*a^8*b^13 - 12*a^10*b
^11))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^18 - 68*a^9*b^16 + 8*a^11*b^14))/(a^6*b^12))*(a^2*6i - b^2*5
i))/(2*b^4) - (8*tan(c/2 + (d*x)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 96*a^8*b^15 - 160*a^10*b^13 + 48*a^12*b^11)
)/(a^6*b^12)))/(2*b^4) - (8*(48*a^4*b^13 - 64*a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b^7 - 324*a^12*
b^5 + 108*a^14*b^3))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(16*a^3*b^18 + 240*a^5*b^16 + 20*a^7*b^14 - 1696*a^9*b^
12 + 2369*a^11*b^10 - 1020*a^13*b^8 + 72*a^15*b^6))/(a^6*b^12)))/(2*b^4) + (8*tan(c/2 + (d*x)/2)*(32*b^19 - 32
*a^2*b^17 + 680*a^4*b^15 - 2560*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2160*a^14*b^5 - 432
*a^16*b^3))/(a^6*b^12))*1i)/(2*b^4))/(((a^2*6i - b^2*5i)*((8*(378*a^15 - 40*a^3*b^12 + 488*a^5*b^10 - 1158*a^7
*b^8 + 541*a^9*b^6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) - ((a^2*6i - b^2*5i)*(((a^2*6i - b^2*5i)*((((8*
(16*a^8*b^13 - 12*a^10*b^11))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^18 - 68*a^9*b^16 + 8*a^11*b^14))/(a^
6*b^12))*(a^2*6i - b^2*5i))/(2*b^4) - (8*(64*a^5*b^14 - 48*a^7*b^12 - 50*a^9*b^10 + 42*a^11*b^8))/(a^6*b^8) +
(8*tan(c/2 + (d*x)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 96*a^8*b^15 - 160*a^10*b^13 + 48*a^12*b^11))/(a^6*b^12)))
/(2*b^4) - (8*(48*a^4*b^13 - 64*a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b^7 - 324*a^12*b^5 + 108*a^14
*b^3))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(16*a^3*b^18 + 240*a^5*b^16 + 20*a^7*b^14 - 1696*a^9*b^12 + 2369*a^11
*b^10 - 1020*a^13*b^8 + 72*a^15*b^6))/(a^6*b^12)))/(2*b^4) + (8*tan(c/2 + (d*x)/2)*(32*b^19 - 32*a^2*b^17 + 68
0*a^4*b^15 - 2560*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2160*a^14*b^5 - 432*a^16*b^3))/(a
^6*b^12)))/(2*b^4) - ((a^2*6i - b^2*5i)*((8*(378*a^15 - 40*a^3*b^12 + 488*a^5*b^10 - 1158*a^7*b^8 + 541*a^9*b^
6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) + ((a^2*6i - b^2*5i)*(((a^2*6i - b^2*5i)*((8*(64*a^5*b^14 - 48*a
^7*b^12 - 50*a^9*b^10 + 42*a^11*b^8))/(a^6*b^8) + (((8*(16*a^8*b^13 - 12*a^10*b^11))/(a^6*b^8) + (8*tan(c/2 +
(d*x)/2)*(64*a^7*b^18 - 68*a^9*b^16 + 8*a^11*b^14))/(a^6*b^12))*(a^2*6i - b^2*5i))/(2*b^4) - (8*tan(c/2 + (d*x
)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 96*a^8*b^15 - 160*a^10*b^13 + 48*a^12*b^11))/(a^6*b^12)))/(2*b^4) - (8*(48
*a^4*b^13 - 64*a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b^7 - 324*a^12*b^5 + 108*a^14*b^3))/(a^6*b^8)
+ (8*tan(c/2 + (d*x)/2)*(16*a^3*b^18 + 240*a^5*b^16 + 20*a^7*b^14 - 1696*a^9*b^12 + 2369*a^11*b^10 - 1020*a^13
*b^8 + 72*a^15*b^6))/(a^6*b^12)))/(2*b^4) + (8*tan(c/2 + (d*x)/2)*(32*b^19 - 32*a^2*b^17 + 680*a^4*b^15 - 2560
*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2160*a^14*b^5 - 432*a^16*b^3))/(a^6*b^12)))/(2*b^4
) + (16*(80*b^13 - 756*a^12*b - 576*a^2*b^11 + 1056*a^4*b^9 + 214*a^6*b^7 - 2448*a^8*b^5 + 2430*a^10*b^3))/(a^
6*b^8) - (16*tan(c/2 + (d*x)/2)*(1600*a^5*b^12 - 2592*a^17 - 5840*a^7*b^10 + 4504*a^9*b^8 + 8160*a^11*b^6 - 17
064*a^13*b^4 + 11232*a^15*b^2))/(a^6*b^12)))*(a^2*6i - b^2*5i)*1i)/(b^4*d) - (2*b*log(tan(c/2 + (d*x)/2)))/(a^
3*d) + (atan((((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(378*a^15 - 40*a^3*b^12 + 488*a^5*b^10 - 1158*
a^7*b^8 + 541*a^9*b^6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(32*b^19 - 32*a^2*b^
17 + 680*a^4*b^15 - 2560*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2160*a^14*b^5 - 432*a^16*b
^3))/(a^6*b^12) + ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2)*(16*a^3*b^18 + 240*a^5*
b^16 + 20*a^7*b^14 - 1696*a^9*b^12 + 2369*a^11*b^10 - 1020*a^13*b^8 + 72*a^15*b^6))/(a^6*b^12) - (8*(48*a^4*b^
13 - 64*a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b^7 - 324*a^12*b^5 + 108*a^14*b^3))/(a^6*b^8) + ((3*a
^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(64*a^5*b^14 - 48*a^7*b^12 - 50*a^9*b^10 + 42*a^11*b^8))/(a^6*b^8
) - (8*tan(c/2 + (d*x)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 96*a^8*b^15 - 160*a^10*b^13 + 48*a^12*b^11))/(a^6*b^1
2) + (((8*(16*a^8*b^13 - 12*a^10*b^11))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^18 - 68*a^9*b^16 + 8*a^11*
b^14))/(a^6*b^12))*(3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^3*b^4)))/(a^3*b^4)))/(a^3*b^4))*1i)/(a^3*b
^4) + ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(378*a^15 - 40*a^3*b^12 + 488*a^5*b^10 - 1158*a^7*b^8
+ 541*a^9*b^6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(32*b^19 - 32*a^2*b^17 + 680
*a^4*b^15 - 2560*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2160*a^14*b^5 - 432*a^16*b^3))/(a^
6*b^12) - ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2)*(16*a^3*b^18 + 240*a^5*b^16 + 2
0*a^7*b^14 - 1696*a^9*b^12 + 2369*a^11*b^10 - 1020*a^13*b^8 + 72*a^15*b^6))/(a^6*b^12) - (8*(48*a^4*b^13 - 64*
a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b^7 - 324*a^12*b^5 + 108*a^14*b^3))/(a^6*b^8) + ((3*a^2 + 2*b
^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 96*a^8*b^15 - 160*a^10*
b^13 + 48*a^12*b^11))/(a^6*b^12) - (8*(64*a^5*b^14 - 48*a^7*b^12 - 50*a^9*b^10 + 42*a^11*b^8))/(a^6*b^8) + (((
8*(16*a^8*b^13 - 12*a^10*b^11))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^18 - 68*a^9*b^16 + 8*a^11*b^14))/(
a^6*b^12))*(3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^3*b^4)))/(a^3*b^4)))/(a^3*b^4))*1i)/(a^3*b^4))/((1
6*(80*b^13 - 756*a^12*b - 576*a^2*b^11 + 1056*a^4*b^9 + 214*a^6*b^7 - 2448*a^8*b^5 + 2430*a^10*b^3))/(a^6*b^8)
- (16*tan(c/2 + (d*x)/2)*(1600*a^5*b^12 - 2592*a^17 - 5840*a^7*b^10 + 4504*a^9*b^8 + 8160*a^11*b^6 - 17064*a^
13*b^4 + 11232*a^15*b^2))/(a^6*b^12) - ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(378*a^15 - 40*a^3*b^
12 + 488*a^5*b^10 - 1158*a^7*b^8 + 541*a^9*b^6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) + (8*tan(c/2 + (d*x
)/2)*(32*b^19 - 32*a^2*b^17 + 680*a^4*b^15 - 2560*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2
160*a^14*b^5 - 432*a^16*b^3))/(a^6*b^12) + ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2
)*(16*a^3*b^18 + 240*a^5*b^16 + 20*a^7*b^14 - 1696*a^9*b^12 + 2369*a^11*b^10 - 1020*a^13*b^8 + 72*a^15*b^6))/(
a^6*b^12) - (8*(48*a^4*b^13 - 64*a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b^7 - 324*a^12*b^5 + 108*a^1
4*b^3))/(a^6*b^8) + ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(64*a^5*b^14 - 48*a^7*b^12 - 50*a^9*b^10
+ 42*a^11*b^8))/(a^6*b^8) - (8*tan(c/2 + (d*x)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 96*a^8*b^15 - 160*a^10*b^13
+ 48*a^12*b^11))/(a^6*b^12) + (((8*(16*a^8*b^13 - 12*a^10*b^11))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^1
8 - 68*a^9*b^16 + 8*a^11*b^14))/(a^6*b^12))*(3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^3*b^4)))/(a^3*b^4
)))/(a^3*b^4)))/(a^3*b^4) + ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*(378*a^15 - 40*a^3*b^12 + 488*a^
5*b^10 - 1158*a^7*b^8 + 541*a^9*b^6 + 1008*a^11*b^4 - 1215*a^13*b^2))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(32*b^
19 - 32*a^2*b^17 + 680*a^4*b^15 - 2560*a^6*b^13 + 2502*a^8*b^11 + 1216*a^10*b^9 - 3564*a^12*b^7 + 2160*a^14*b^
5 - 432*a^16*b^3))/(a^6*b^12) - ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2)*(16*a^3*b
^18 + 240*a^5*b^16 + 20*a^7*b^14 - 1696*a^9*b^12 + 2369*a^11*b^10 - 1020*a^13*b^8 + 72*a^15*b^6))/(a^6*b^12) -
(8*(48*a^4*b^13 - 64*a^2*b^15 + 100*a^6*b^11 - 184*a^8*b^9 + 315*a^10*b^7 - 324*a^12*b^5 + 108*a^14*b^3))/(a^
6*b^8) + ((3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*((8*tan(c/2 + (d*x)/2)*(136*a^6*b^17 - 128*a^4*b^19 + 9
6*a^8*b^15 - 160*a^10*b^13 + 48*a^12*b^11))/(a^6*b^12) - (8*(64*a^5*b^14 - 48*a^7*b^12 - 50*a^9*b^10 + 42*a^11
*b^8))/(a^6*b^8) + (((8*(16*a^8*b^13 - 12*a^10*b^11))/(a^6*b^8) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^18 - 68*a^9*
b^16 + 8*a^11*b^14))/(a^6*b^12))*(3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2))/(a^3*b^4)))/(a^3*b^4)))/(a^3*b^
4)))/(a^3*b^4)))*(3*a^2 + 2*b^2)*(-(a + b)^3*(a - b)^3)^(1/2)*2i)/(a^3*b^4*d)