\(\int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx\) [319]

Optimal result

Integrand size = 31, antiderivative size = 240 \[ \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {2 b^4 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d} \]

[Out]

1/8*(3*A*a^4+4*A*a^2*b^2+8*A*b^4-4*B*a^3*b-8*B*a*b^3)*x/a^5-1/3*(2*a^2+3*b^2)*(A*b-B*a)*sin(d*x+c)/a^4/d+1/8*(
3*A*a^2+4*A*b^2-4*B*a*b)*cos(d*x+c)*sin(d*x+c)/a^3/d-1/3*(A*b-B*a)*cos(d*x+c)^2*sin(d*x+c)/a^2/d+1/4*A*cos(d*x
+c)^3*sin(d*x+c)/a/d-2*b^4*(A*b-B*a)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/d/(a-b)^(1/2)/(a+
b)^(1/2)

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4119, 4189, 4004, 3916, 2738, 214} \[ \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=-\frac {2 b^4 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d \sqrt {a-b} \sqrt {a+b}}-\frac {(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A-4 a b B+4 A b^2\right ) \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac {x \left (3 a^4 A-4 a^3 b B+4 a^2 A b^2-8 a b^3 B+8 A b^4\right )}{8 a^5}+\frac {A \sin (c+d x) \cos ^3(c+d x)}{4 a d} \]

[In]

Int[(Cos[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]

[Out]

((3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B)*x)/(8*a^5) - (2*b^4*(A*b - a*B)*ArcTanh[(Sqrt[a - b
]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - ((2*a^2 + 3*b^2)*(A*b - a*B)*Sin[c + d*x])
/(3*a^4*d) + ((3*a^2*A + 4*A*b^2 - 4*a*b*B)*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Cos[c + d*x]^2
*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*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 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4119

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*n)), x]
+ Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + A*a*(n +
1)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b
- a*B, 0] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rule 4189

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^
(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1
)*((d*Csc[e + f*x])^n/(a*f*n)), x] + Dist[1/(a*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[
a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ
[{a, b, d, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {\cos ^3(c+d x) \left (4 (A b-a B)-3 a A \sec (c+d x)-3 A b \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 a} \\ & = -\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {\cos ^2(c+d x) \left (3 \left (3 a^2 A+4 b (A b-a B)\right )+a (A b+8 a B) \sec (c+d x)-8 b (A b-a B) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^2} \\ & = \frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\int \frac {\cos (c+d x) \left (8 \left (2 a^2+3 b^2\right ) (A b-a B)-a \left (9 a^2 A-4 A b^2+4 a b B\right ) \sec (c+d x)-3 b \left (3 a^2 A+4 A b^2-4 a b B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{24 a^3} \\ & = -\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}+\frac {\int \frac {3 \left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right )+3 a b \left (3 a^2 A+4 A b^2-4 a b B\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{24 a^4} \\ & = \frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (b^4 (A b-a B)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{a^5} \\ & = \frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (b^3 (A b-a B)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{a^5} \\ & = \frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {\left (2 b^3 (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d} \\ & = \frac {\left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) x}{8 a^5}-\frac {2 b^4 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 \sqrt {a-b} \sqrt {a+b} d}-\frac {\left (2 a^2+3 b^2\right ) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac {\left (3 a^2 A+4 A b^2-4 a b B\right ) \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {(A b-a B) \cos ^2(c+d x) \sin (c+d x)}{3 a^2 d}+\frac {A \cos ^3(c+d x) \sin (c+d x)}{4 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.13 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\frac {12 \left (3 a^4 A+4 a^2 A b^2+8 A b^4-4 a^3 b B-8 a b^3 B\right ) (c+d x)+\frac {192 b^4 (A b-a B) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+24 a \left (3 a^2+4 b^2\right ) (-A b+a B) \sin (c+d x)+24 a^2 \left (a^2 A+A b^2-a b B\right ) \sin (2 (c+d x))+8 a^3 (-A b+a B) \sin (3 (c+d x))+3 a^4 A \sin (4 (c+d x))}{96 a^5 d} \]

[In]

Integrate[(Cos[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]

[Out]

(12*(3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B)*(c + d*x) + (192*b^4*(A*b - a*B)*ArcTanh[((-a +
b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 24*a*(3*a^2 + 4*b^2)*(-(A*b) + a*B)*Sin[c + d*x] + 24
*a^2*(a^2*A + A*b^2 - a*b*B)*Sin[2*(c + d*x)] + 8*a^3*(-(A*b) + a*B)*Sin[3*(c + d*x)] + 3*a^4*A*Sin[4*(c + d*x
)])/(96*a^5*d)

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {-\frac {2 b^{4} \left (A b -B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (-\frac {5}{8} a^{4} A -A \,a^{3} b -\frac {1}{2} A \,a^{2} b^{2}-A a \,b^{3}+B \,a^{4}+\frac {1}{2} B \,a^{3} b +B \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (\frac {3}{8} a^{4} A -\frac {5}{3} A \,a^{3} b -3 A a \,b^{3}+\frac {5}{3} B \,a^{4}+3 B \,a^{2} b^{2}-\frac {1}{2} A \,a^{2} b^{2}+\frac {1}{2} B \,a^{3} b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {3}{8} a^{4} A +\frac {1}{2} A \,a^{2} b^{2}-\frac {1}{2} B \,a^{3} b -\frac {5}{3} A \,a^{3} b -3 A a \,b^{3}+\frac {5}{3} B \,a^{4}+3 B \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{8} a^{4} A +\frac {1}{2} A \,a^{2} b^{2}-\frac {1}{2} B \,a^{3} b -A \,a^{3} b -A a \,b^{3}+B \,a^{4}+B \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (3 a^{4} A +4 A \,a^{2} b^{2}+8 A \,b^{4}-4 B \,a^{3} b -8 B a \,b^{3}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}}{d}\)	\(381\)
default	\(\frac {-\frac {2 b^{4} \left (A b -B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 \left (\left (-\frac {5}{8} a^{4} A -A \,a^{3} b -\frac {1}{2} A \,a^{2} b^{2}-A a \,b^{3}+B \,a^{4}+\frac {1}{2} B \,a^{3} b +B \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (\frac {3}{8} a^{4} A -\frac {5}{3} A \,a^{3} b -3 A a \,b^{3}+\frac {5}{3} B \,a^{4}+3 B \,a^{2} b^{2}-\frac {1}{2} A \,a^{2} b^{2}+\frac {1}{2} B \,a^{3} b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-\frac {3}{8} a^{4} A +\frac {1}{2} A \,a^{2} b^{2}-\frac {1}{2} B \,a^{3} b -\frac {5}{3} A \,a^{3} b -3 A a \,b^{3}+\frac {5}{3} B \,a^{4}+3 B \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (\frac {5}{8} a^{4} A +\frac {1}{2} A \,a^{2} b^{2}-\frac {1}{2} B \,a^{3} b -A \,a^{3} b -A a \,b^{3}+B \,a^{4}+B \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{4}}+\frac {\left (3 a^{4} A +4 A \,a^{2} b^{2}+8 A \,b^{4}-4 B \,a^{3} b -8 B a \,b^{3}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}}{a^{5}}}{d}\)	\(381\)
risch	\(\frac {3 A x}{8 a}+\frac {x A \,b^{2}}{2 a^{3}}+\frac {x A \,b^{4}}{a^{5}}-\frac {b B x}{2 a^{2}}-\frac {x B \,b^{3}}{a^{4}}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B \,b^{2}}{2 a^{3} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B}{8 a d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B}{8 a d}+\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A b}{8 a^{2} d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{3}}{2 a^{4} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B \,b^{2}}{2 a^{3} d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{3}}{2 a^{4} d}-\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A b}{8 a^{2} d}+\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}-\frac {b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, d \,a^{4}}-\frac {b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) A}{\sqrt {a^{2}-b^{2}}\, d \,a^{5}}+\frac {b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) B}{\sqrt {a^{2}-b^{2}}\, d \,a^{4}}+\frac {A \sin \left (4 d x +4 c \right )}{32 a d}-\frac {\sin \left (3 d x +3 c \right ) A b}{12 a^{2} d}+\frac {\sin \left (3 d x +3 c \right ) B}{12 a d}+\frac {A \sin \left (2 d x +2 c \right )}{4 a d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{2}}{4 a^{3} d}-\frac {\sin \left (2 d x +2 c \right ) B b}{4 a^{2} d}\)	\(627\)

[In]

int(cos(d*x+c)^4*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-2*b^4*(A*b-B*a)/a^5/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a-b)*(a+b))^(1/2))+2/a^5*(((-
5/8*a^4*A-A*a^3*b-1/2*A*a^2*b^2-A*a*b^3+B*a^4+1/2*B*a^3*b+B*a^2*b^2)*tan(1/2*d*x+1/2*c)^7+(3/8*a^4*A-5/3*A*a^3
*b-3*A*a*b^3+5/3*B*a^4+3*B*a^2*b^2-1/2*A*a^2*b^2+1/2*B*a^3*b)*tan(1/2*d*x+1/2*c)^5+(-3/8*a^4*A+1/2*A*a^2*b^2-1
/2*B*a^3*b-5/3*A*a^3*b-3*A*a*b^3+5/3*B*a^4+3*B*a^2*b^2)*tan(1/2*d*x+1/2*c)^3+(5/8*a^4*A+1/2*A*a^2*b^2-1/2*B*a^
3*b-A*a^3*b-A*a*b^3+B*a^4+B*a^2*b^2)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^4+1/8*(3*A*a^4+4*A*a^2*b^2+8
*A*b^4-4*B*a^3*b-8*B*a*b^3)*arctan(tan(1/2*d*x+1/2*c))))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.85 \[ \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\left [\frac {3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} - 4 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} + 8 \, B a b^{5} - 8 \, A b^{6}\right )} d x - 12 \, {\left (B a b^{4} - A b^{5}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) + {\left (16 \, B a^{6} - 16 \, A a^{5} b + 8 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3} - 24 \, B a^{2} b^{4} + 24 \, A a b^{5} + 6 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (B a^{6} - A a^{5} b - B a^{4} b^{2} + A a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} + 4 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}, \frac {3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} - 4 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4} + 8 \, B a b^{5} - 8 \, A b^{6}\right )} d x + 24 \, {\left (B a b^{4} - A b^{5}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) + {\left (16 \, B a^{6} - 16 \, A a^{5} b + 8 \, B a^{4} b^{2} - 8 \, A a^{3} b^{3} - 24 \, B a^{2} b^{4} + 24 \, A a b^{5} + 6 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (B a^{6} - A a^{5} b - B a^{4} b^{2} + A a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (3 \, A a^{6} - 4 \, B a^{5} b + A a^{4} b^{2} + 4 \, B a^{3} b^{3} - 4 \, A a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{7} - a^{5} b^{2}\right )} d}\right ] \]

[In]

integrate(cos(d*x+c)^4*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x, algorithm="fricas")

[Out]

[1/24*(3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 - 4*B*a^3*b^3 + 4*A*a^2*b^4 + 8*B*a*b^5 - 8*A*b^6)*d*x - 12*(B*a*b^4
- A*b^5)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*
x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) + (16*B*a^6 - 16*A*a^
5*b + 8*B*a^4*b^2 - 8*A*a^3*b^3 - 24*B*a^2*b^4 + 24*A*a*b^5 + 6*(A*a^6 - A*a^4*b^2)*cos(d*x + c)^3 + 8*(B*a^6
- A*a^5*b - B*a^4*b^2 + A*a^3*b^3)*cos(d*x + c)^2 + 3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 + 4*B*a^3*b^3 - 4*A*a^2
*b^4)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^5*b^2)*d), 1/24*(3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 - 4*B*a^3*b^3
+ 4*A*a^2*b^4 + 8*B*a*b^5 - 8*A*b^6)*d*x + 24*(B*a*b^4 - A*b^5)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*c
os(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (16*B*a^6 - 16*A*a^5*b + 8*B*a^4*b^2 - 8*A*a^3*b^3 - 24*B*a^2*b
^4 + 24*A*a*b^5 + 6*(A*a^6 - A*a^4*b^2)*cos(d*x + c)^3 + 8*(B*a^6 - A*a^5*b - B*a^4*b^2 + A*a^3*b^3)*cos(d*x +
c)^2 + 3*(3*A*a^6 - 4*B*a^5*b + A*a^4*b^2 + 4*B*a^3*b^3 - 4*A*a^2*b^4)*cos(d*x + c))*sin(d*x + c))/((a^7 - a^
5*b^2)*d)]

Sympy [F]

\[ \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{4}{\left (c + d x \right )}}{a + b \sec {\left (c + d x \right )}}\, dx \]

[In]

integrate(cos(d*x+c)**4*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x)

[Out]

Integral((A + B*sec(c + d*x))*cos(c + d*x)**4/(a + b*sec(c + d*x)), x)

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)^4*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),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*a^2-4*b^2>0)', see `assume?`
for more de

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (221) = 442\).

Time = 0.33 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.68 \[ \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\frac {\frac {3 \, {\left (3 \, A a^{4} - 4 \, B a^{3} b + 4 \, A a^{2} b^{2} - 8 \, B a b^{3} + 8 \, A b^{4}\right )} {\left (d x + c\right )}}{a^{5}} + \frac {48 \, {\left (B a b^{4} - A b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{5}} - \frac {2 \, {\left (15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a^{4}}}{24 \, d} \]

[In]

integrate(cos(d*x+c)^4*(A+B*sec(d*x+c))/(a+b*sec(d*x+c)),x, algorithm="giac")

[Out]

1/24*(3*(3*A*a^4 - 4*B*a^3*b + 4*A*a^2*b^2 - 8*B*a*b^3 + 8*A*b^4)*(d*x + c)/a^5 + 48*(B*a*b^4 - A*b^5)*(pi*flo
or(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a
^2 + b^2)))/(sqrt(-a^2 + b^2)*a^5) - 2*(15*A*a^3*tan(1/2*d*x + 1/2*c)^7 - 24*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 24
*A*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 12*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 24*B
*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 24*A*b^3*tan(1/2*d*x + 1/2*c)^7 - 9*A*a^3*tan(1/2*d*x + 1/2*c)^5 - 40*B*a^3*ta
n(1/2*d*x + 1/2*c)^5 + 40*A*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 12*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 12*A*a*b^2*tan(
1/2*d*x + 1/2*c)^5 - 72*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 72*A*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*A*a^3*tan(1/2*d*x
+ 1/2*c)^3 - 40*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 40*A*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 12*B*a^2*b*tan(1/2*d*x + 1
/2*c)^3 - 12*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 72*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 72*A*b^3*tan(1/2*d*x + 1/2*c
)^3 - 15*A*a^3*tan(1/2*d*x + 1/2*c) - 24*B*a^3*tan(1/2*d*x + 1/2*c) + 24*A*a^2*b*tan(1/2*d*x + 1/2*c) + 12*B*a
^2*b*tan(1/2*d*x + 1/2*c) - 12*A*a*b^2*tan(1/2*d*x + 1/2*c) - 24*B*a*b^2*tan(1/2*d*x + 1/2*c) + 24*A*b^3*tan(1
/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^4*a^4))/d

Mupad [B] (verification not implemented)

Time = 21.66 (sec) , antiderivative size = 5903, normalized size of antiderivative = 24.60 \[ \int \frac {\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx=\text {Too large to display} \]

[In]

int((cos(c + d*x)^4*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x)),x)

[Out]

((tan(c/2 + (d*x)/2)*(5*A*a^3 - 8*A*b^3 + 8*B*a^3 + 4*A*a*b^2 - 8*A*a^2*b + 8*B*a*b^2 - 4*B*a^2*b))/(4*a^4) -
(tan(c/2 + (d*x)/2)^7*(5*A*a^3 + 8*A*b^3 - 8*B*a^3 + 4*A*a*b^2 + 8*A*a^2*b - 8*B*a*b^2 - 4*B*a^2*b))/(4*a^4) -
(tan(c/2 + (d*x)/2)^3*(9*A*a^3 + 72*A*b^3 - 40*B*a^3 - 12*A*a*b^2 + 40*A*a^2*b - 72*B*a*b^2 + 12*B*a^2*b))/(1
2*a^4) + (tan(c/2 + (d*x)/2)^5*(9*A*a^3 - 72*A*b^3 + 40*B*a^3 - 12*A*a*b^2 - 40*A*a^2*b + 72*B*a*b^2 + 12*B*a^
2*b))/(12*a^4))/(d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)
/2)^8 + 1)) - (atan(((((((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*
b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (tan(c
/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*
a^3*b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5) + (tan(c/2 + (d*
x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^
2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^
2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b
^3 + 16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 4
64*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A
*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i)*1i)/(8*a^5) - (((((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5
+ 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2
- 12*A*a^15*b - 16*B*a^15*b)/a^12 + (tan(c/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(A*a^4*3i
+ A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3
*8i - B*a^3*b*4i))/(8*a^5) - (tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b -
256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 8
1*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B
^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*
b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^
3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i)*1i)/(8*a^5))/((64
*A^3*b^14 - 96*A^3*a*b^13 + 96*A^3*a^2*b^12 - 104*A^3*a^3*b^11 + 104*A^3*a^4*b^10 - 88*A^3*a^5*b^9 + 48*A^3*a^
6*b^8 - 33*A^3*a^7*b^7 + 18*A^3*a^8*b^6 - 9*A^3*a^9*b^5 - 64*B^3*a^3*b^11 + 96*B^3*a^4*b^10 - 96*B^3*a^5*b^9 +
80*B^3*a^6*b^8 - 32*B^3*a^7*b^7 + 16*B^3*a^8*b^6 - 192*A^2*B*a*b^13 + 192*A*B^2*a^2*b^12 - 288*A*B^2*a^3*b^11
+ 288*A*B^2*a^4*b^10 - 264*A*B^2*a^5*b^9 + 168*A*B^2*a^6*b^8 - 120*A*B^2*a^7*b^7 + 48*A*B^2*a^8*b^6 - 24*A*B^
2*a^9*b^5 + 288*A^2*B*a^2*b^12 - 288*A^2*B*a^3*b^11 + 288*A^2*B*a^4*b^10 - 240*A^2*B*a^5*b^9 + 192*A^2*B*a^6*b
^8 - 96*A^2*B*a^7*b^7 + 57*A^2*B*a^8*b^6 - 18*A^2*B*a^9*b^5 + 9*A^2*B*a^10*b^4)/a^12 + (((((12*A*a^16 + 32*A*a
^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a
^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (tan(c/2 + (d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 2
56*a^11*b^2)*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i +
A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5) + (tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a
*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^
5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 +
256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*
B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a
^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*
b*4i))/(8*a^5) + (((((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2
- 32*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 + (tan(c/2 +
(d*x)/2)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2)*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*
b*4i))/(16*a^13))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5) - (tan(c/2 + (d*x)/2
)*(9*A^2*a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^
4*b^7 + 256*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^
9 + 256*B^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 +
16*B^2*a^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A
*B*a^5*b^6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8))*(A*a^4*3i + A*b^4
*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*a^3*b*4i))/(8*a^5)))*(A*a^4*3i + A*b^4*8i + A*a^2*b^2*4i - B*a*b^3*8i - B*
a^3*b*4i)*1i)/(4*a^5*d) - (b^4*atan(((b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11
- 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A
^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a
^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b
^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 -
368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) + (b^4*((a + b)*(a - b))^(1/2)*
(A*b - B*a)*((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a
^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (b^4*tan(c/2 + (d*
x)/2)*((a + b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2)))
)/(a^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2) + (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*
a^11 - 128*A^2*b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 2
56*A^2*a^5*b^6 - 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B
^2*a^3*b^8 - 256*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a
^9*b^2 + 256*A*B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^
6 - 368*A*B*a^6*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) - (b^4*((a + b)*(a - b))^(1
/2)*(A*b - B*a)*((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32
*B*a^11*b^5 + 48*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 + (b^4*tan(c/2 +
(d*x)/2)*((a + b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^
2))))/(a^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2))/((64*A^3*b^14 - 96*A^3*a*b^13 + 96*A^3*a^2*b^12 - 104*A^3*a^3*b^11
+ 104*A^3*a^4*b^10 - 88*A^3*a^5*b^9 + 48*A^3*a^6*b^8 - 33*A^3*a^7*b^7 + 18*A^3*a^8*b^6 - 9*A^3*a^9*b^5 - 64*B
^3*a^3*b^11 + 96*B^3*a^4*b^10 - 96*B^3*a^5*b^9 + 80*B^3*a^6*b^8 - 32*B^3*a^7*b^7 + 16*B^3*a^8*b^6 - 192*A^2*B*
a*b^13 + 192*A*B^2*a^2*b^12 - 288*A*B^2*a^3*b^11 + 288*A*B^2*a^4*b^10 - 264*A*B^2*a^5*b^9 + 168*A*B^2*a^6*b^8
- 120*A*B^2*a^7*b^7 + 48*A*B^2*a^8*b^6 - 24*A*B^2*a^9*b^5 + 288*A^2*B*a^2*b^12 - 288*A^2*B*a^3*b^11 + 288*A^2*
B*a^4*b^10 - 240*A^2*B*a^5*b^9 + 192*A^2*B*a^6*b^8 - 96*A^2*B*a^7*b^7 + 57*A^2*B*a^8*b^6 - 18*A^2*B*a^9*b^5 +
9*A^2*B*a^10*b^4)/a^12 + (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*b
^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6 -
216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 256
*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*B
*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6*
b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) + (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*
((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 48
*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 - (b^4*tan(c/2 + (d*x)/2)*((a +
b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2))))/(a^7 - a^5
*b^2)))/(a^7 - a^5*b^2) - (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)*((tan(c/2 + (d*x)/2)*(9*A^2*a^11 - 128*A^2*
b^11 + 256*A^2*a*b^10 - 27*A^2*a^10*b - 256*A^2*a^2*b^9 + 256*A^2*a^3*b^8 - 256*A^2*a^4*b^7 + 256*A^2*a^5*b^6
- 216*A^2*a^6*b^5 + 136*A^2*a^7*b^4 - 81*A^2*a^8*b^3 + 51*A^2*a^9*b^2 - 128*B^2*a^2*b^9 + 256*B^2*a^3*b^8 - 25
6*B^2*a^4*b^7 + 256*B^2*a^5*b^6 - 208*B^2*a^6*b^5 + 112*B^2*a^7*b^4 - 48*B^2*a^8*b^3 + 16*B^2*a^9*b^2 + 256*A*
B*a*b^10 - 24*A*B*a^10*b - 512*A*B*a^2*b^9 + 512*A*B*a^3*b^8 - 512*A*B*a^4*b^7 + 464*A*B*a^5*b^6 - 368*A*B*a^6
*b^5 + 264*A*B*a^7*b^4 - 152*A*B*a^8*b^3 + 72*A*B*a^9*b^2))/(2*a^8) - (b^4*((a + b)*(a - b))^(1/2)*(A*b - B*a)
*((12*A*a^16 + 32*A*a^10*b^6 - 48*A*a^11*b^5 + 16*A*a^12*b^4 - 4*A*a^13*b^3 + 4*A*a^14*b^2 - 32*B*a^11*b^5 + 4
8*B*a^12*b^4 - 16*B*a^13*b^3 + 16*B*a^14*b^2 - 12*A*a^15*b - 16*B*a^15*b)/a^12 + (b^4*tan(c/2 + (d*x)/2)*((a +
b)*(a - b))^(1/2)*(A*b - B*a)*(128*a^12*b + 128*a^10*b^3 - 256*a^11*b^2))/(2*a^8*(a^7 - a^5*b^2))))/(a^7 - a^
5*b^2)))/(a^7 - a^5*b^2)))*((a + b)*(a - b))^(1/2)*(A*b - B*a)*2i)/(d*(a^7 - a^5*b^2))