\(\int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [1269]

Optimal result

Integrand size = 27, antiderivative size = 399 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {x}{2 b^3}-\frac {3 \left (2 a^2-b^2\right ) x}{b^5}+\frac {\sqrt {a^2-b^2} \left (2 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a b^5 d}-\frac {2 \sqrt {a^2-b^2} \left (5 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a b^5 d}+\frac {2 \left (10 a^6-9 a^4 b^2-b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^5 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (5 a^2+b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))} \]

[Out]

-1/2*x/b^3-3*(2*a^2-b^2)*x/b^5-arctanh(cos(d*x+c))/a^3/d-3*a*cos(d*x+c)/b^4/d+1/2*cos(d*x+c)*sin(d*x+c)/b^3/d+
1/2*(a^2-b^2)^2*cos(d*x+c)/a/b^4/d/(a+b*sin(d*x+c))^2+3/2*(a^2-b^2)*cos(d*x+c)/b^4/d/(a+b*sin(d*x+c))-(a^2-b^2
)*(5*a^2+b^2)*cos(d*x+c)/a^2/b^4/d/(a+b*sin(d*x+c))+2*(10*a^6-9*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^5/d/(a^2-b^2)^(1/2)+(2*a^2+b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))*(a^2-b^
2)^(1/2)/a/b^5/d-2*(5*a^2+b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))*(a^2-b^2)^(1/2)/a/b^5/d

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2976, 3855, 2718, 2715, 8, 2743, 2833, 12, 2739, 632, 210} \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}+\frac {\left (2 a^2+b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a b^5 d}-\frac {2 \left (5 a^2+b^2\right ) \sqrt {a^2-b^2} \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a b^5 d}-\frac {3 x \left (2 a^2-b^2\right )}{b^5}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}-\frac {\left (5 a^2+b^2\right ) \left (a^2-b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}+\frac {2 \left (10 a^6-9 a^4 b^2-b^6\right ) \arctan \left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 b^5 d \sqrt {a^2-b^2}}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\sin (c+d x) \cos (c+d x)}{2 b^3 d}-\frac {x}{2 b^3} \]

[In]

Int[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

-1/2*x/b^3 - (3*(2*a^2 - b^2)*x)/b^5 + (Sqrt[a^2 - b^2]*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2
- b^2]])/(a*b^5*d) - (2*Sqrt[a^2 - b^2]*(5*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*b^
5*d) + (2*(10*a^6 - 9*a^4*b^2 - b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*b^5*Sqrt[a^2 - b^2
]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) - (3*a*Cos[c + d*x])/(b^4*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*b^3*d) + ((
a^2 - b^2)^2*Cos[c + d*x])/(2*a*b^4*d*(a + b*Sin[c + d*x])^2) + (3*(a^2 - b^2)*Cos[c + d*x])/(2*b^4*d*(a + b*S
in[c + d*x])) - ((a^2 - b^2)*(5*a^2 + b^2)*Cos[c + d*x])/(a^2*b^4*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 2833

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(
b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b
^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

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 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 (-2 a^2+b^2\right )}{b^5}+\frac {\csc (c+d x)}{a^3}+\frac {3 a \sin (c+d x)}{b^4}-\frac {\sin ^2(c+d x)}{b^3}+\frac {\left (a^2-b^2\right )^3}{a b^5 (a+b \sin (c+d x))^3}-\frac {\left (a^2-b^2\right )^2 \left (5 a^2+b^2\right )}{a^2 b^5 (a+b \sin (c+d x))^2}+\frac {10 a^6-9 a^4 b^2-b^6}{a^3 b^5 (a+b \sin (c+d x))}\right ) \, dx \\ & = -\frac {3 \left (2 a^2-b^2\right ) x}{b^5}+\frac {\int \csc (c+d x) \, dx}{a^3}+\frac {(3 a) \int \sin (c+d x) \, dx}{b^4}-\frac {\int \sin ^2(c+d x) \, dx}{b^3}+\frac {\left (a^2-b^2\right )^3 \int \frac {1}{(a+b \sin (c+d x))^3} \, dx}{a b^5}-\frac {\left (\left (a^2-b^2\right )^2 \left (5 a^2+b^2\right )\right ) \int \frac {1}{(a+b \sin (c+d x))^2} \, dx}{a^2 b^5}+\frac {\left (10 a^6-9 a^4 b^2-b^6\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^3 b^5} \\ & = -\frac {3 \left (2 a^2-b^2\right ) x}{b^5}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}-\frac {\left (a^2-b^2\right ) \left (5 a^2+b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}-\frac {\int 1 \, dx}{2 b^3}-\frac {\left (a^2-b^2\right )^2 \int \frac {-2 a+b \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{2 a b^5}+\frac {\left (\left (a^2-b^2\right )^2 \left (5 a^2+b^2\right )\right ) \int \frac {a}{a+b \sin (c+d x)} \, dx}{a^2 b^5 \left (-a^2+b^2\right )}+\frac {\left (2 \left (10 a^6-9 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^5 d} \\ & = -\frac {x}{2 b^3}-\frac {3 \left (2 a^2-b^2\right ) x}{b^5}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (5 a^2+b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}+\frac {\left (a^2-b^2\right ) \int \frac {2 a^2+b^2}{a+b \sin (c+d x)} \, dx}{2 a b^5}+\frac {\left (\left (a^2-b^2\right )^2 \left (5 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a b^5 \left (-a^2+b^2\right )}-\frac {\left (4 \left (10 a^6-9 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^5 d} \\ & = -\frac {x}{2 b^3}-\frac {3 \left (2 a^2-b^2\right ) x}{b^5}+\frac {2 \left (10 a^6-9 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^5 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (5 a^2+b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}+\frac {\left (\left (a^2-b^2\right ) \left (2 a^2+b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a b^5}+\frac {\left (2 \left (a^2-b^2\right )^2 \left (5 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 b^5 \left (-a^2+b^2\right ) d} \\ & = -\frac {x}{2 b^3}-\frac {3 \left (2 a^2-b^2\right ) x}{b^5}+\frac {2 \left (10 a^6-9 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^5 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (5 a^2+b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}+\frac {\left (\left (a^2-b^2\right ) \left (2 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 b^5 d}+\frac {\left (4 \left (a^2-b^2\right ) \left (5 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 b^5 d} \\ & = -\frac {x}{2 b^3}-\frac {3 \left (2 a^2-b^2\right ) x}{b^5}-\frac {2 \sqrt {a^2-b^2} \left (5 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 b^5 d}+\frac {2 \left (10 a^6-9 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^5 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (5 a^2+b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))}-\frac {\left (2 \left (a^2-b^2\right ) \left (2 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 b^5 d} \\ & = -\frac {x}{2 b^3}-\frac {3 \left (2 a^2-b^2\right ) x}{b^5}+\frac {\sqrt {a^2-b^2} \left (2 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 b^5 d}-\frac {2 \sqrt {a^2-b^2} \left (5 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 b^5 d}+\frac {2 \left (10 a^6-9 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^5 \sqrt {a^2-b^2} d}-\frac {\text {arctanh}(\cos (c+d x))}{a^3 d}-\frac {3 a \cos (c+d x)}{b^4 d}+\frac {\cos (c+d x) \sin (c+d x)}{2 b^3 d}+\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{2 a b^4 d (a+b \sin (c+d x))^2}+\frac {3 \left (a^2-b^2\right ) \cos (c+d x)}{2 b^4 d (a+b \sin (c+d x))}-\frac {\left (a^2-b^2\right ) \left (5 a^2+b^2\right ) \cos (c+d x)}{a^2 b^4 d (a+b \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.71 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.61 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {2 \left (-12 a^2+5 b^2\right ) (c+d x)}{b^5}+\frac {4 \left (12 a^6-11 a^4 b^2+a^2 b^4-2 b^6\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 b^5 \sqrt {a^2-b^2}}-\frac {12 a \cos (c+d x)}{b^4}-\frac {4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}+\frac {4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^3}+\frac {2 \left (a^2-b^2\right )^2 \cos (c+d x)}{a b^4 (a+b \sin (c+d x))^2}+\frac {2 \left (-7 a^4+5 a^2 b^2+2 b^4\right ) \cos (c+d x)}{a^2 b^4 (a+b \sin (c+d x))}+\frac {\sin (2 (c+d x))}{b^3}}{4 d} \]

[In]

Integrate[(Cos[c + d*x]^5*Cot[c + d*x])/(a + b*Sin[c + d*x])^3,x]

[Out]

((2*(-12*a^2 + 5*b^2)*(c + d*x))/b^5 + (4*(12*a^6 - 11*a^4*b^2 + a^2*b^4 - 2*b^6)*ArcTan[(b + a*Tan[(c + d*x)/
2])/Sqrt[a^2 - b^2]])/(a^3*b^5*Sqrt[a^2 - b^2]) - (12*a*Cos[c + d*x])/b^4 - (4*Log[Cos[(c + d*x)/2]])/a^3 + (4
*Log[Sin[(c + d*x)/2]])/a^3 + (2*(a^2 - b^2)^2*Cos[c + d*x])/(a*b^4*(a + b*Sin[c + d*x])^2) + (2*(-7*a^4 + 5*a
^2*b^2 + 2*b^4)*Cos[c + d*x])/(a^2*b^4*(a + b*Sin[c + d*x])) + Sin[2*(c + d*x)]/b^3)/(4*d)

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+3 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (12 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {\frac {2 \left (\left (-\frac {5}{2} a^{5} b^{2}+\frac {1}{2} a^{3} b^{4}+2 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 b \left (2 a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}-2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \,b^{2} \left (19 a^{4}-11 a^{2} b^{2}-8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {3 a^{2} b \left (2 a^{4}-a^{2} b^{2}-b^{4}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (12 a^{6}-11 a^{4} b^{2}+a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{3} b^{5}}}{d}\)	\(359\)
default	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}-\frac {2 \left (\frac {\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2}+3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2}+3 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (12 a^{2}-5 b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {\frac {2 \left (\left (-\frac {5}{2} a^{5} b^{2}+\frac {1}{2} a^{3} b^{4}+2 a \,b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {3 b \left (2 a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}-2 b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}-\frac {a \,b^{2} \left (19 a^{4}-11 a^{2} b^{2}-8 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}-\frac {3 a^{2} b \left (2 a^{4}-a^{2} b^{2}-b^{4}\right )}{2}\right )}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (12 a^{6}-11 a^{4} b^{2}+a^{2} b^{4}-2 b^{6}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}}{a^{3} b^{5}}}{d}\)	\(359\)
risch	\(-\frac {6 x \,a^{2}}{b^{5}}+\frac {5 x}{2 b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {3 a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {i \left (-8 i a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+7 i a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+i a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+20 i a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-13 i a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}-7 i a \,b^{5} {\mathrm e}^{i \left (d x +c \right )}+14 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-9 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-7 a^{4} b^{2}+5 a^{2} b^{4}+2 b^{6}\right )}{\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} a^{2} d \,b^{5}}+\frac {6 i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}+a \right )}{b}\right )}{d \,b^{5}}+\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 )}{2 d \,b^{3} a}+\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^{3}}-\frac {6 i \sqrt {a^{2}-b^{2}}\, a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \left (\sqrt {a^{2}-b^{2}}-a \right )}{b}\right )}{d \,b^{5}}-\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 )}{2 d \,b^{3} a}-\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^{3}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}\)	\(676\)

[In]

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

[Out]

1/d*(1/a^3*ln(tan(1/2*d*x+1/2*c))-2/b^5*((1/2*tan(1/2*d*x+1/2*c)^3*b^2+3*tan(1/2*d*x+1/2*c)^2*a*b-1/2*tan(1/2*
d*x+1/2*c)*b^2+3*a*b)/(1+tan(1/2*d*x+1/2*c)^2)^2+1/2*(12*a^2-5*b^2)*arctan(tan(1/2*d*x+1/2*c)))+2/a^3/b^5*(((-
5/2*a^5*b^2+1/2*a^3*b^4+2*a*b^6)*tan(1/2*d*x+1/2*c)^3-3/2*b*(2*a^6+3*a^4*b^2-3*a^2*b^4-2*b^6)*tan(1/2*d*x+1/2*
c)^2-1/2*a*b^2*(19*a^4-11*a^2*b^2-8*b^4)*tan(1/2*d*x+1/2*c)-3/2*a^2*b*(2*a^4-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)^2+1/2*(12*a^6-11*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))))

Fricas [A] (verification not implemented)

none

Time = 0.72 (sec) , antiderivative size = 1007, normalized size of antiderivative = 2.52 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[-1/4*(8*a^4*b^3*cos(d*x + c)^3 + 2*(12*a^5*b^2 - 5*a^3*b^4)*d*x*cos(d*x + c)^2 - 2*(12*a^7 + 7*a^5*b^2 - 5*a^
3*b^4)*d*x + (12*a^6 + 13*a^4*b^2 + 3*a^2*b^4 + 2*b^6 - (12*a^4*b^2 + a^2*b^4 + 2*b^6)*cos(d*x + c)^2 + 2*(12*
a^5*b + a^3*b^3 + 2*a*b^5)*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*(12*a^6*b + a^4*b^3 - 3*a^2*b^5)*cos(d*x + c) + 2*(b^7*cos(d*x + c)^2 - 2*a*
b^6*sin(d*x + c) - a^2*b^5 - b^7)*log(1/2*cos(d*x + c) + 1/2) - 2*(b^7*cos(d*x + c)^2 - 2*a*b^6*sin(d*x + c) -
a^2*b^5 - b^7)*log(-1/2*cos(d*x + c) + 1/2) - 2*(a^3*b^4*cos(d*x + c)^3 + 2*(12*a^6*b - 5*a^4*b^3)*d*x + 2*(9
*a^5*b^2 - 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sin(d*x + c))/(a^3*b^7*d*cos(d*x + c)^2 - 2*a^4*b^6*d*sin(d*x + c)
- (a^5*b^5 + a^3*b^7)*d), -1/2*(4*a^4*b^3*cos(d*x + c)^3 + (12*a^5*b^2 - 5*a^3*b^4)*d*x*cos(d*x + c)^2 - (12*
a^7 + 7*a^5*b^2 - 5*a^3*b^4)*d*x - (12*a^6 + 13*a^4*b^2 + 3*a^2*b^4 + 2*b^6 - (12*a^4*b^2 + a^2*b^4 + 2*b^6)*c
os(d*x + c)^2 + 2*(12*a^5*b + a^3*b^3 + 2*a*b^5)*sin(d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*sin(d*x + c) + b)/(s
qrt(a^2 - b^2)*cos(d*x + c))) - (12*a^6*b + a^4*b^3 - 3*a^2*b^5)*cos(d*x + c) + (b^7*cos(d*x + c)^2 - 2*a*b^6*
sin(d*x + c) - a^2*b^5 - b^7)*log(1/2*cos(d*x + c) + 1/2) - (b^7*cos(d*x + c)^2 - 2*a*b^6*sin(d*x + c) - a^2*b
^5 - b^7)*log(-1/2*cos(d*x + c) + 1/2) - (a^3*b^4*cos(d*x + c)^3 + 2*(12*a^6*b - 5*a^4*b^3)*d*x + 2*(9*a^5*b^2
- 3*a^3*b^4 - a*b^6)*cos(d*x + c))*sin(d*x + c))/(a^3*b^7*d*cos(d*x + c)^2 - 2*a^4*b^6*d*sin(d*x + c) - (a^5*
b^5 + a^3*b^7)*d)]

Sympy [F]

\[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cos ^{6}{\left (c + d x \right )} \csc {\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)/(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 [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.59 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {{\left (12 \, a^{2} - 5 \, b^{2}\right )} {\left (d x + c\right )}}{b^{5}} + \frac {2 \, {\left (12 \, a^{6} - 11 \, 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^{5}} - \frac {2 \, {\left (6 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 13 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 9 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 6 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 54 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 39 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 21 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 20 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 23 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 42 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 11 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a^{6} - 3 \, a^{4} b^{2} - 3 \, a^{2} 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 )}^{2} a^{3} b^{4}}}{2 \, d} \]

[In]

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

[Out]

1/2*(2*log(abs(tan(1/2*d*x + 1/2*c)))/a^3 - (12*a^2 - 5*b^2)*(d*x + c)/b^5 + 2*(12*a^6 - 11*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)))/(sqr
t(a^2 - b^2)*a^3*b^5) - 2*(6*a^5*b*tan(1/2*d*x + 1/2*c)^7 - a^3*b^3*tan(1/2*d*x + 1/2*c)^7 - 4*a*b^5*tan(1/2*d
*x + 1/2*c)^7 + 12*a^6*tan(1/2*d*x + 1/2*c)^6 + 13*a^4*b^2*tan(1/2*d*x + 1/2*c)^6 - 9*a^2*b^4*tan(1/2*d*x + 1/
2*c)^6 - 6*b^6*tan(1/2*d*x + 1/2*c)^6 + 54*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 9*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 - 1
6*a*b^5*tan(1/2*d*x + 1/2*c)^5 + 36*a^6*tan(1/2*d*x + 1/2*c)^4 + 39*a^4*b^2*tan(1/2*d*x + 1/2*c)^4 - 21*a^2*b^
4*tan(1/2*d*x + 1/2*c)^4 - 12*b^6*tan(1/2*d*x + 1/2*c)^4 + 90*a^5*b*tan(1/2*d*x + 1/2*c)^3 - 27*a^3*b^3*tan(1/
2*d*x + 1/2*c)^3 - 20*a*b^5*tan(1/2*d*x + 1/2*c)^3 + 36*a^6*tan(1/2*d*x + 1/2*c)^2 + 23*a^4*b^2*tan(1/2*d*x +
1/2*c)^2 - 15*a^2*b^4*tan(1/2*d*x + 1/2*c)^2 - 6*b^6*tan(1/2*d*x + 1/2*c)^2 + 42*a^5*b*tan(1/2*d*x + 1/2*c) -
11*a^3*b^3*tan(1/2*d*x + 1/2*c) - 8*a*b^5*tan(1/2*d*x + 1/2*c) + 12*a^6 - 3*a^4*b^2 - 3*a^2*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)^2*a^3*
b^4))/d

Mupad [B] (verification not implemented)

Time = 14.30 (sec) , antiderivative size = 5354, normalized size of antiderivative = 13.42 \[ \int \frac {\cos ^5(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

log(tan(c/2 + (d*x)/2))/(a^3*d) + ((3*(b^4 - 4*a^4 + a^2*b^2))/(a*b^4) + (tan(c/2 + (d*x)/2)*(8*b^4 - 42*a^4 +
11*a^2*b^2))/(a^2*b^3) - (3*tan(c/2 + (d*x)/2)^4*(12*a^6 - 4*b^6 - 7*a^2*b^4 + 13*a^4*b^2))/(a^3*b^4) - (tan(
c/2 + (d*x)/2)^6*(12*a^6 - 6*b^6 - 9*a^2*b^4 + 13*a^4*b^2))/(a^3*b^4) - (tan(c/2 + (d*x)/2)^2*(36*a^6 - 6*b^6
- 15*a^2*b^4 + 23*a^4*b^2))/(a^3*b^4) + (tan(c/2 + (d*x)/2)^7*(4*b^4 - 6*a^4 + a^2*b^2))/(a^2*b^3) + (tan(c/2
+ (d*x)/2)^5*(16*b^4 - 54*a^4 + 9*a^2*b^2))/(a^2*b^3) + (tan(c/2 + (d*x)/2)^3*(20*b^4 - 90*a^4 + 27*a^2*b^2))/
(a^2*b^3))/(d*(tan(c/2 + (d*x)/2)^2*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2)^6*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2
)^4*(6*a^2 + 8*b^2) + a^2*tan(c/2 + (d*x)/2)^8 + a^2 + 12*a*b*tan(c/2 + (d*x)/2)^3 + 12*a*b*tan(c/2 + (d*x)/2)
^5 + 4*a*b*tan(c/2 + (d*x)/2)^7 + 4*a*b*tan(c/2 + (d*x)/2))) - (atan((((a^2*12i - b^2*5i)*((4*(20*a^3*b^12 - 6
048*a^15 + 332*a^5*b^10 - 1947*a^7*b^8 + 3979*a^9*b^6 - 7038*a^11*b^4 + 10800*a^13*b^2))/(a^6*b^11) - ((a^2*12
i - b^2*5i)*((4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a^6*b^12 + 32*a^8*b^10 - 1110*a^10*b^8 + 1872*a^12*b^6 - 864*
a^14*b^4))/(a^6*b^11) - ((a^2*12i - b^2*5i)*((4*(64*a^5*b^16 - 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11*b^10))/(a
^6*b^11) - (((4*(32*a^8*b^16 - 24*a^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 - 68*a^9*b^20 +
8*a^11*b^18))/(a^6*b^16))*(a^2*12i - b^2*5i))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(64*a^4*b^22 - 68*a^6*b^20 + 192
*a^8*b^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16)))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(4*a^3*b^20 - 180*a^5*
b^18 + 725*a^7*b^16 - 3454*a^9*b^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*b^16)))/(2*b^5) +
(8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 + 445*a^4*b^16 - 2390*a^6*b^14 + 5544*a^8*b^12 - 9958*a^10*b^10 + 1
5912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b^4))/(a^6*b^16))*1i)/(2*b^5) + ((a^2*12i - b^2*5i)*((4*(20*a^3*b^1
2 - 6048*a^15 + 332*a^5*b^10 - 1947*a^7*b^8 + 3979*a^9*b^6 - 7038*a^11*b^4 + 10800*a^13*b^2))/(a^6*b^11) + ((a
^2*12i - b^2*5i)*(((a^2*12i - b^2*5i)*((4*(64*a^5*b^16 - 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11*b^10))/(a^6*b^1
1) + (((4*(32*a^8*b^16 - 24*a^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 - 68*a^9*b^20 + 8*a^11
*b^18))/(a^6*b^16))*(a^2*12i - b^2*5i))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(64*a^4*b^22 - 68*a^6*b^20 + 192*a^8*b
^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16)))/(2*b^5) + (4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a^6*b^12 + 32*
a^8*b^10 - 1110*a^10*b^8 + 1872*a^12*b^6 - 864*a^14*b^4))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*a^3*b^20 - 180
*a^5*b^18 + 725*a^7*b^16 - 3454*a^9*b^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*b^16)))/(2*b^
5) + (8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 + 445*a^4*b^16 - 2390*a^6*b^14 + 5544*a^8*b^12 - 9958*a^10*b^1
0 + 15912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b^4))/(a^6*b^16))*1i)/(2*b^5))/(((a^2*12i - b^2*5i)*((4*(20*a^
3*b^12 - 6048*a^15 + 332*a^5*b^10 - 1947*a^7*b^8 + 3979*a^9*b^6 - 7038*a^11*b^4 + 10800*a^13*b^2))/(a^6*b^11)
- ((a^2*12i - b^2*5i)*((4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a^6*b^12 + 32*a^8*b^10 - 1110*a^10*b^8 + 1872*a^12*
b^6 - 864*a^14*b^4))/(a^6*b^11) - ((a^2*12i - b^2*5i)*((4*(64*a^5*b^16 - 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11
*b^10))/(a^6*b^11) - (((4*(32*a^8*b^16 - 24*a^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 - 68*a
^9*b^20 + 8*a^11*b^18))/(a^6*b^16))*(a^2*12i - b^2*5i))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(64*a^4*b^22 - 68*a^6*
b^20 + 192*a^8*b^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16)))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(4*a^3*b^20
- 180*a^5*b^18 + 725*a^7*b^16 - 3454*a^9*b^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*b^16)))/
(2*b^5) + (8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 + 445*a^4*b^16 - 2390*a^6*b^14 + 5544*a^8*b^12 - 9958*a^1
0*b^10 + 15912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b^4))/(a^6*b^16)))/(2*b^5) - ((a^2*12i - b^2*5i)*((4*(20*
a^3*b^12 - 6048*a^15 + 332*a^5*b^10 - 1947*a^7*b^8 + 3979*a^9*b^6 - 7038*a^11*b^4 + 10800*a^13*b^2))/(a^6*b^11
) + ((a^2*12i - b^2*5i)*(((a^2*12i - b^2*5i)*((4*(64*a^5*b^16 - 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11*b^10))/(
a^6*b^11) + (((4*(32*a^8*b^16 - 24*a^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 - 68*a^9*b^20 +
8*a^11*b^18))/(a^6*b^16))*(a^2*12i - b^2*5i))/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(64*a^4*b^22 - 68*a^6*b^20 + 19
2*a^8*b^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16)))/(2*b^5) + (4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a^6*b^1
2 + 32*a^8*b^10 - 1110*a^10*b^8 + 1872*a^12*b^6 - 864*a^14*b^4))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*a^3*b^2
0 - 180*a^5*b^18 + 725*a^7*b^16 - 3454*a^9*b^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*b^16))
)/(2*b^5) + (8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 + 445*a^4*b^16 - 2390*a^6*b^14 + 5544*a^8*b^12 - 9958*a
^10*b^10 + 15912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b^4))/(a^6*b^16)))/(2*b^5) - (8*(20*b^12 - 6048*a^12 +
132*a^2*b^10 - 837*a^4*b^8 + 2107*a^6*b^6 - 6174*a^8*b^4 + 10800*a^10*b^2))/(a^6*b^11) + (16*tan(c/2 + (d*x)/2
)*(41472*a^17 + 1100*a^5*b^12 - 7030*a^7*b^10 + 21386*a^9*b^8 - 55200*a^11*b^6 + 108864*a^13*b^4 - 110592*a^15
*b^2))/(a^6*b^16)))*(a^2*12i - b^2*5i)*1i)/(b^5*d) + (atan((((-(a + b)*(a - b))^(1/2)*(6*a^4 + b^4 + (a^2*b^2)
/2)*((4*(20*a^3*b^12 - 6048*a^15 + 332*a^5*b^10 - 1947*a^7*b^8 + 3979*a^9*b^6 - 7038*a^11*b^4 + 10800*a^13*b^2
))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 + 445*a^4*b^16 - 2390*a^6*b^14 + 5544*a^8*b^12 - 99
58*a^10*b^10 + 15912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b^4))/(a^6*b^16) + ((-(a + b)*(a - b))^(1/2)*(6*a^4
+ b^4 + (a^2*b^2)/2)*((4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a^6*b^12 + 32*a^8*b^10 - 1110*a^10*b^8 + 1872*a^12*
b^6 - 864*a^14*b^4))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*a^3*b^20 - 180*a^5*b^18 + 725*a^7*b^16 - 3454*a^9*b
^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*b^16) + ((-(a + b)*(a - b))^(1/2)*((4*(64*a^5*b^16
- 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11*b^10))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^4*b^22 - 68*a^6*b^20
+ 192*a^8*b^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16) + ((-(a + b)*(a - b))^(1/2)*((4*(32*a^8*b^16 - 24*a
^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 - 68*a^9*b^20 + 8*a^11*b^18))/(a^6*b^16))*(6*a^4 +
b^4 + (a^2*b^2)/2))/(a^3*b^5))*(6*a^4 + b^4 + (a^2*b^2)/2))/(a^3*b^5)))/(a^3*b^5))*1i)/(a^3*b^5) + ((-(a + b)*
(a - b))^(1/2)*(6*a^4 + b^4 + (a^2*b^2)/2)*((4*(20*a^3*b^12 - 6048*a^15 + 332*a^5*b^10 - 1947*a^7*b^8 + 3979*a
^9*b^6 - 7038*a^11*b^4 + 10800*a^13*b^2))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 + 445*a^4*b^
16 - 2390*a^6*b^14 + 5544*a^8*b^12 - 9958*a^10*b^10 + 15912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b^4))/(a^6*b
^16) - ((-(a + b)*(a - b))^(1/2)*(6*a^4 + b^4 + (a^2*b^2)/2)*((4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a^6*b^12 + 3
2*a^8*b^10 - 1110*a^10*b^8 + 1872*a^12*b^6 - 864*a^14*b^4))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*a^3*b^20 - 1
80*a^5*b^18 + 725*a^7*b^16 - 3454*a^9*b^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*b^16) - ((-
(a + b)*(a - b))^(1/2)*((4*(64*a^5*b^16 - 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11*b^10))/(a^6*b^11) + (8*tan(c/2
+ (d*x)/2)*(64*a^4*b^22 - 68*a^6*b^20 + 192*a^8*b^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16) - ((-(a + b)
*(a - b))^(1/2)*((4*(32*a^8*b^16 - 24*a^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 - 68*a^9*b^2
0 + 8*a^11*b^18))/(a^6*b^16))*(6*a^4 + b^4 + (a^2*b^2)/2))/(a^3*b^5))*(6*a^4 + b^4 + (a^2*b^2)/2))/(a^3*b^5)))
/(a^3*b^5))*1i)/(a^3*b^5))/((8*(20*b^12 - 6048*a^12 + 132*a^2*b^10 - 837*a^4*b^8 + 2107*a^6*b^6 - 6174*a^8*b^4
+ 10800*a^10*b^2))/(a^6*b^11) - (16*tan(c/2 + (d*x)/2)*(41472*a^17 + 1100*a^5*b^12 - 7030*a^7*b^10 + 21386*a^
9*b^8 - 55200*a^11*b^6 + 108864*a^13*b^4 - 110592*a^15*b^2))/(a^6*b^16) + ((-(a + b)*(a - b))^(1/2)*(6*a^4 + b
^4 + (a^2*b^2)/2)*((4*(20*a^3*b^12 - 6048*a^15 + 332*a^5*b^10 - 1947*a^7*b^8 + 3979*a^9*b^6 - 7038*a^11*b^4 +
10800*a^13*b^2))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 + 445*a^4*b^16 - 2390*a^6*b^14 + 5544
*a^8*b^12 - 9958*a^10*b^10 + 15912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b^4))/(a^6*b^16) + ((-(a + b)*(a - b)
)^(1/2)*(6*a^4 + b^4 + (a^2*b^2)/2)*((4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a^6*b^12 + 32*a^8*b^10 - 1110*a^10*b^
8 + 1872*a^12*b^6 - 864*a^14*b^4))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*a^3*b^20 - 180*a^5*b^18 + 725*a^7*b^1
6 - 3454*a^9*b^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*b^16) + ((-(a + b)*(a - b))^(1/2)*((
4*(64*a^5*b^16 - 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11*b^10))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^4*b^22
- 68*a^6*b^20 + 192*a^8*b^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16) + ((-(a + b)*(a - b))^(1/2)*((4*(32*a
^8*b^16 - 24*a^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 - 68*a^9*b^20 + 8*a^11*b^18))/(a^6*b^
16))*(6*a^4 + b^4 + (a^2*b^2)/2))/(a^3*b^5))*(6*a^4 + b^4 + (a^2*b^2)/2))/(a^3*b^5)))/(a^3*b^5)))/(a^3*b^5) -
((-(a + b)*(a - b))^(1/2)*(6*a^4 + b^4 + (a^2*b^2)/2)*((4*(20*a^3*b^12 - 6048*a^15 + 332*a^5*b^10 - 1947*a^7*b
^8 + 3979*a^9*b^6 - 7038*a^11*b^4 + 10800*a^13*b^2))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*b^20 - 4*a^2*b^18 +
445*a^4*b^16 - 2390*a^6*b^14 + 5544*a^8*b^12 - 9958*a^10*b^10 + 15912*a^12*b^8 - 12960*a^14*b^6 + 3456*a^16*b
^4))/(a^6*b^16) - ((-(a + b)*(a - b))^(1/2)*(6*a^4 + b^4 + (a^2*b^2)/2)*((4*(32*a^2*b^16 - 24*a^4*b^14 + 160*a
^6*b^12 + 32*a^8*b^10 - 1110*a^10*b^8 + 1872*a^12*b^6 - 864*a^14*b^4))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(4*a
^3*b^20 - 180*a^5*b^18 + 725*a^7*b^16 - 3454*a^9*b^14 + 6506*a^11*b^12 - 3840*a^13*b^10 + 288*a^15*b^8))/(a^6*
b^16) - ((-(a + b)*(a - b))^(1/2)*((4*(64*a^5*b^16 - 48*a^7*b^14 + 160*a^9*b^12 - 168*a^11*b^10))/(a^6*b^11) +
(8*tan(c/2 + (d*x)/2)*(64*a^4*b^22 - 68*a^6*b^20 + 192*a^8*b^18 - 280*a^10*b^16 + 96*a^12*b^14))/(a^6*b^16) -
((-(a + b)*(a - b))^(1/2)*((4*(32*a^8*b^16 - 24*a^10*b^14))/(a^6*b^11) + (8*tan(c/2 + (d*x)/2)*(64*a^7*b^22 -
68*a^9*b^20 + 8*a^11*b^18))/(a^6*b^16))*(6*a^4 + b^4 + (a^2*b^2)/2))/(a^3*b^5))*(6*a^4 + b^4 + (a^2*b^2)/2))/
(a^3*b^5)))/(a^3*b^5)))/(a^3*b^5)))*(-(a + b)*(a - b))^(1/2)*(6*a^4 + b^4 + (a^2*b^2)/2)*2i)/(a^3*b^5*d)