\(\int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx\) [342]

Optimal result

Integrand size = 29, antiderivative size = 411 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=-\frac {(4 A b-a B) x}{a^5}+\frac {b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \]

[Out]

-(4*A*b-B*a)*x/a^5+b*(20*A*a^6*b-35*A*a^4*b^3+28*A*a^2*b^5-8*A*b^7-8*B*a^7+8*B*a^5*b^2-7*B*a^3*b^4+2*B*a*b^6)*
arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^5/(a-b)^(7/2)/(a+b)^(7/2)/d+1/6*(6*A*a^6-65*A*a^4*b^2+68
*A*a^2*b^4-24*A*b^6+26*B*a^5*b-17*B*a^3*b^3+6*B*a*b^5)*sin(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*b*(A*b-B*a)*sin(d*x+c)
/a/(a^2-b^2)/d/(a+b*sec(d*x+c))^3+1/6*b*(9*A*a^2*b-4*A*b^3-6*B*a^3+B*a*b^2)*sin(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*
sec(d*x+c))^2+1/2*b*(12*A*a^4*b-11*A*a^2*b^3+4*A*b^5-6*B*a^5+2*B*a^3*b^2-B*a*b^4)*sin(d*x+c)/a^3/(a^2-b^2)^3/d
/(a+b*sec(d*x+c))

Rubi [A] (verified)

Time = 5.74 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {4115, 4185, 4189, 4004, 3916, 2738, 214} \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=-\frac {x (4 A b-a B)}{a^5}+\frac {b (A b-a B) \sin (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac {b \left (-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right ) \sin (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac {b \left (-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}+\frac {\left (6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right ) \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^3}+\frac {b \left (-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 A b^7\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 d (a-b)^{7/2} (a+b)^{7/2}} \]

[In]

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

[Out]

-(((4*A*b - a*B)*x)/a^5) + (b*(20*a^6*A*b - 35*a^4*A*b^3 + 28*a^2*A*b^5 - 8*A*b^7 - 8*a^7*B + 8*a^5*b^2*B - 7*
a^3*b^4*B + 2*a*b^6*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d
) + ((6*a^6*A - 65*a^4*A*b^2 + 68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B)*Sin[c + d*x])/
(6*a^4*(a^2 - b^2)^3*d) + (b*(A*b - a*B)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b*(9*a^2*
A*b - 4*A*b^3 - 6*a^3*B + a*b^2*B)*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b*(12*a^4*A
*b - 11*a^2*A*b^3 + 4*A*b^5 - 6*a^5*B + 2*a^3*b^2*B - a*b^4*B)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec
[c + d*x]))

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 4115

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

Rule 4185

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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

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 {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac {\int \frac {\cos (c+d x) \left (-3 a^2 A+4 A b^2-a b B+3 a (A b-a B) \sec (c+d x)-3 b (A b-a B) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )} \\ & = \frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {\int \frac {\cos (c+d x) \left (6 a^4 A-23 a^2 A b^2+12 A b^4+8 a^3 b B-3 a b^3 B-2 a \left (6 a^2 A b-A b^3-3 a^3 B-2 a b^2 B\right ) \sec (c+d x)+2 b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (-6 a^6 A+65 a^4 A b^2-68 a^2 A b^4+24 A b^6-26 a^5 b B+17 a^3 b^3 B-6 a b^5 B+a \left (18 a^4 A b-7 a^2 A b^3+4 A b^5-6 a^5 B-8 a^3 b^2 B-a b^4 B\right ) \sec (c+d x)-3 b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3} \\ & = \frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\int \frac {-6 \left (a^2-b^2\right )^3 (4 A b-a B)+3 a b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )^3} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )^3} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 a^5 \left (a^2-b^2\right )^3} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac {\left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 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 \left (a^2-b^2\right )^3 d} \\ & = -\frac {(4 A b-a B) x}{a^5}+\frac {b \left (20 a^6 A b-35 a^4 A b^3+28 a^2 A b^5-8 A b^7-8 a^7 B+8 a^5 b^2 B-7 a^3 b^4 B+2 a b^6 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^5 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (6 a^6 A-65 a^4 A b^2+68 a^2 A b^4-24 A b^6+26 a^5 b B-17 a^3 b^3 B+6 a b^5 B\right ) \sin (c+d x)}{6 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}+\frac {b \left (9 a^2 A b-4 A b^3-6 a^3 B+a b^2 B\right ) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac {b \left (12 a^4 A b-11 a^2 A b^3+4 A b^5-6 a^5 B+2 a^3 b^2 B-a b^4 B\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1205\) vs. \(2(411)=822\).

Time = 6.64 (sec) , antiderivative size = 1205, normalized size of antiderivative = 2.93 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\frac {(b+a \cos (c+d x)) \sec ^3(c+d x) (A+B \sec (c+d x)) \left (\frac {24 b \left (-20 a^6 A b+35 a^4 A b^3-28 a^2 A b^5+8 A b^7+8 a^7 B-8 a^5 b^2 B+7 a^3 b^4 B-2 a b^6 B\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^3}{\left (a^2-b^2\right )^{7/2}}+\frac {-144 a^8 A b^2 c+336 a^6 A b^4 c-144 a^4 A b^6 c-144 a^2 A b^8 c+96 A b^{10} c+36 a^9 b B c-84 a^7 b^3 B c+36 a^5 b^5 B c+36 a^3 b^7 B c-24 a b^9 B c-144 a^8 A b^2 d x+336 a^6 A b^4 d x-144 a^4 A b^6 d x-144 a^2 A b^8 d x+96 A b^{10} d x+36 a^9 b B d x-84 a^7 b^3 B d x+36 a^5 b^5 B d x+36 a^3 b^7 B d x-24 a b^9 B d x+18 a \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) (-4 A b+a B) (c+d x) \cos (c+d x)+36 a^2 b \left (a^2-b^2\right )^3 (-4 A b+a B) (c+d x) \cos (2 (c+d x))-24 a^9 A b c \cos (3 (c+d x))+72 a^7 A b^3 c \cos (3 (c+d x))-72 a^5 A b^5 c \cos (3 (c+d x))+24 a^3 A b^7 c \cos (3 (c+d x))+6 a^{10} B c \cos (3 (c+d x))-18 a^8 b^2 B c \cos (3 (c+d x))+18 a^6 b^4 B c \cos (3 (c+d x))-6 a^4 b^6 B c \cos (3 (c+d x))-24 a^9 A b d x \cos (3 (c+d x))+72 a^7 A b^3 d x \cos (3 (c+d x))-72 a^5 A b^5 d x \cos (3 (c+d x))+24 a^3 A b^7 d x \cos (3 (c+d x))+6 a^{10} B d x \cos (3 (c+d x))-18 a^8 b^2 B d x \cos (3 (c+d x))+18 a^6 b^4 B d x \cos (3 (c+d x))-6 a^4 b^6 B d x \cos (3 (c+d x))+18 a^9 A b \sin (c+d x)-90 a^7 A b^3 \sin (c+d x)-135 a^5 A b^5 \sin (c+d x)+228 a^3 A b^7 \sin (c+d x)-96 a A b^9 \sin (c+d x)+36 a^8 b^2 B \sin (c+d x)+72 a^6 b^4 B \sin (c+d x)-57 a^4 b^6 B \sin (c+d x)+24 a^2 b^8 B \sin (c+d x)+6 a^{10} A \sin (2 (c+d x))+18 a^8 A b^2 \sin (2 (c+d x))-300 a^6 A b^4 \sin (2 (c+d x))+336 a^4 A b^6 \sin (2 (c+d x))-120 a^2 A b^8 \sin (2 (c+d x))+120 a^7 b^3 B \sin (2 (c+d x))-90 a^5 b^5 B \sin (2 (c+d x))+30 a^3 b^7 B \sin (2 (c+d x))+18 a^9 A b \sin (3 (c+d x))-114 a^7 A b^3 \sin (3 (c+d x))+125 a^5 A b^5 \sin (3 (c+d x))-44 a^3 A b^7 \sin (3 (c+d x))+36 a^8 b^2 B \sin (3 (c+d x))-32 a^6 b^4 B \sin (3 (c+d x))+11 a^4 b^6 B \sin (3 (c+d x))+3 a^{10} A \sin (4 (c+d x))-9 a^8 A b^2 \sin (4 (c+d x))+9 a^6 A b^4 \sin (4 (c+d x))-3 a^4 A b^6 \sin (4 (c+d x))}{\left (a^2-b^2\right )^3}\right )}{24 a^5 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^4} \]

[In]

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

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((24*b*(-20*a^6*A*b + 35*a^4*A*b^3 - 28*a^2*A*b^5 +
8*A*b^7 + 8*a^7*B - 8*a^5*b^2*B + 7*a^3*b^4*B - 2*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]
]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (-144*a^8*A*b^2*c + 336*a^6*A*b^4*c - 144*a^4*A*b^6*c - 144*a^2*
A*b^8*c + 96*A*b^10*c + 36*a^9*b*B*c - 84*a^7*b^3*B*c + 36*a^5*b^5*B*c + 36*a^3*b^7*B*c - 24*a*b^9*B*c - 144*a
^8*A*b^2*d*x + 336*a^6*A*b^4*d*x - 144*a^4*A*b^6*d*x - 144*a^2*A*b^8*d*x + 96*A*b^10*d*x + 36*a^9*b*B*d*x - 84
*a^7*b^3*B*d*x + 36*a^5*b^5*B*d*x + 36*a^3*b^7*B*d*x - 24*a*b^9*B*d*x + 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-4*A
*b + a*B)*(c + d*x)*Cos[c + d*x] + 36*a^2*b*(a^2 - b^2)^3*(-4*A*b + a*B)*(c + d*x)*Cos[2*(c + d*x)] - 24*a^9*A
*b*c*Cos[3*(c + d*x)] + 72*a^7*A*b^3*c*Cos[3*(c + d*x)] - 72*a^5*A*b^5*c*Cos[3*(c + d*x)] + 24*a^3*A*b^7*c*Cos
[3*(c + d*x)] + 6*a^10*B*c*Cos[3*(c + d*x)] - 18*a^8*b^2*B*c*Cos[3*(c + d*x)] + 18*a^6*b^4*B*c*Cos[3*(c + d*x)
] - 6*a^4*b^6*B*c*Cos[3*(c + d*x)] - 24*a^9*A*b*d*x*Cos[3*(c + d*x)] + 72*a^7*A*b^3*d*x*Cos[3*(c + d*x)] - 72*
a^5*A*b^5*d*x*Cos[3*(c + d*x)] + 24*a^3*A*b^7*d*x*Cos[3*(c + d*x)] + 6*a^10*B*d*x*Cos[3*(c + d*x)] - 18*a^8*b^
2*B*d*x*Cos[3*(c + d*x)] + 18*a^6*b^4*B*d*x*Cos[3*(c + d*x)] - 6*a^4*b^6*B*d*x*Cos[3*(c + d*x)] + 18*a^9*A*b*S
in[c + d*x] - 90*a^7*A*b^3*Sin[c + d*x] - 135*a^5*A*b^5*Sin[c + d*x] + 228*a^3*A*b^7*Sin[c + d*x] - 96*a*A*b^9
*Sin[c + d*x] + 36*a^8*b^2*B*Sin[c + d*x] + 72*a^6*b^4*B*Sin[c + d*x] - 57*a^4*b^6*B*Sin[c + d*x] + 24*a^2*b^8
*B*Sin[c + d*x] + 6*a^10*A*Sin[2*(c + d*x)] + 18*a^8*A*b^2*Sin[2*(c + d*x)] - 300*a^6*A*b^4*Sin[2*(c + d*x)] +
336*a^4*A*b^6*Sin[2*(c + d*x)] - 120*a^2*A*b^8*Sin[2*(c + d*x)] + 120*a^7*b^3*B*Sin[2*(c + d*x)] - 90*a^5*b^5
*B*Sin[2*(c + d*x)] + 30*a^3*b^7*B*Sin[2*(c + d*x)] + 18*a^9*A*b*Sin[3*(c + d*x)] - 114*a^7*A*b^3*Sin[3*(c + d
*x)] + 125*a^5*A*b^5*Sin[3*(c + d*x)] - 44*a^3*A*b^7*Sin[3*(c + d*x)] + 36*a^8*b^2*B*Sin[3*(c + d*x)] - 32*a^6
*b^4*B*Sin[3*(c + d*x)] + 11*a^4*b^6*B*Sin[3*(c + d*x)] + 3*a^10*A*Sin[4*(c + d*x)] - 9*a^8*A*b^2*Sin[4*(c + d
*x)] + 9*a^6*A*b^4*Sin[4*(c + d*x)] - 3*a^4*A*b^6*Sin[4*(c + d*x)])/(a^2 - b^2)^3))/(24*a^5*d*(B + A*Cos[c + d
*x])*(a + b*Sec[c + d*x])^4)

Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.36

method	result	size
derivativedivides	\(\frac {-\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (4 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}}{d}\)	\(558\)
default	\(\frac {-\frac {2 b \left (\frac {-\frac {\left (20 A \,a^{4} b +5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}-2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}-4 B \,a^{4} b +6 B \,a^{3} b^{2}+B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {2 \left (30 A \,a^{4} b -29 A \,a^{2} b^{3}+9 A \,b^{5}-18 B \,a^{5}+11 B \,a^{3} b^{2}-3 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {\left (20 A \,a^{4} b -5 A \,a^{3} b^{2}-18 A \,a^{2} b^{3}+2 A a \,b^{4}+6 A \,b^{5}-12 B \,a^{5}+4 B \,a^{4} b +6 B \,a^{3} b^{2}-B \,a^{2} b^{3}-2 B a \,b^{4}\right ) a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )^{3}}-\frac {\left (20 A \,a^{6} b -35 A \,a^{4} b^{3}+28 A \,a^{2} b^{5}-8 A \,b^{7}-8 B \,a^{7}+8 B \,a^{5} b^{2}-7 B \,a^{3} b^{4}+2 B a \,b^{6}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{5}}-\frac {2 \left (-\frac {A a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\left (4 A b -B a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a^{5}}}{d}\)	\(558\)
risch	\(\text {Expression too large to display}\)	\(2154\)

[In]

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

[Out]

1/d*(-2*b/a^5*((-1/2*(20*A*a^4*b+5*A*a^3*b^2-18*A*a^2*b^3-2*A*a*b^4+6*A*b^5-12*B*a^5-4*B*a^4*b+6*B*a^3*b^2+B*a
^2*b^3-2*B*a*b^4)*a*b/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5+2/3*(30*A*a^4*b-29*A*a^2*b^3+9*A*b^
5-18*B*a^5+11*B*a^3*b^2-3*B*a*b^4)*a*b/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3-1/2*(20*A*a^4*b-5*
A*a^3*b^2-18*A*a^2*b^3+2*A*a*b^4+6*A*b^5-12*B*a^5+4*B*a^4*b+6*B*a^3*b^2-B*a^2*b^3-2*B*a*b^4)*a*b/(a+b)/(a^3-3*
a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3-1/2*(20*A*a^6*b-3
5*A*a^4*b^3+28*A*a^2*b^5-8*A*b^7-8*B*a^7+8*B*a^5*b^2-7*B*a^3*b^4+2*B*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((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*(-A*a*tan(1/2*d*x+1/2*c)/(1+tan(1
/2*d*x+1/2*c)^2)+(4*A*b-B*a)*arctan(tan(1/2*d*x+1/2*c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1251 vs. \(2 (393) = 786\).

Time = 0.53 (sec) , antiderivative size = 2560, normalized size of antiderivative = 6.23 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/12*(12*(B*a^12 - 4*A*a^11*b - 4*B*a^10*b^2 + 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^7*b^5 - 4*B*a^6*b^6 + 16*A
*a^5*b^7 + B*a^4*b^8 - 4*A*a^3*b^9)*d*x*cos(d*x + c)^3 + 36*(B*a^11*b - 4*A*a^10*b^2 - 4*B*a^9*b^3 + 16*A*a^8*
b^4 + 6*B*a^7*b^5 - 24*A*a^6*b^6 - 4*B*a^5*b^7 + 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*d*x*cos(d*x + c)^2 +
36*(B*a^10*b^2 - 4*A*a^9*b^3 - 4*B*a^8*b^4 + 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 - 4*B*a^4*b^8 + 16*A*a
^3*b^9 + B*a^2*b^10 - 4*A*a*b^11)*d*x*cos(d*x + c) + 12*(B*a^9*b^3 - 4*A*a^8*b^4 - 4*B*a^7*b^5 + 16*A*a^6*b^6
+ 6*B*a^5*b^7 - 24*A*a^4*b^8 - 4*B*a^3*b^9 + 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*d*x - 3*(8*B*a^7*b^4 - 20*A*
a^6*b^5 - 8*B*a^5*b^6 + 35*A*a^4*b^7 + 7*B*a^3*b^8 - 28*A*a^2*b^9 - 2*B*a*b^10 + 8*A*b^11 + (8*B*a^10*b - 20*A
*a^9*b^2 - 8*B*a^8*b^3 + 35*A*a^7*b^4 + 7*B*a^6*b^5 - 28*A*a^5*b^6 - 2*B*a^4*b^7 + 8*A*a^3*b^8)*cos(d*x + c)^3
+ 3*(8*B*a^9*b^2 - 20*A*a^8*b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 + 7*B*a^5*b^6 - 28*A*a^4*b^7 - 2*B*a^3*b^8 + 8*A
*a^2*b^9)*cos(d*x + c)^2 + 3*(8*B*a^8*b^3 - 20*A*a^7*b^4 - 8*B*a^6*b^5 + 35*A*a^5*b^6 + 7*B*a^4*b^7 - 28*A*a^3
*b^8 - 2*B*a^2*b^9 + 8*A*a*b^10)*cos(d*x + c))*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)) + 2*(6*A*a^9*b^3 + 26*B*a^8*b^4 - 71*A*a^7*b^5 - 43*B*a^6*b^6 + 133*A*a^5*b^7 + 23*B*a^4*b^8 -
92*A*a^3*b^9 - 6*B*a^2*b^10 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)
*cos(d*x + c)^3 + (18*A*a^11*b + 36*B*a^10*b^2 - 132*A*a^9*b^3 - 68*B*a^8*b^4 + 239*A*a^7*b^5 + 43*B*a^6*b^6 -
169*A*a^5*b^7 - 11*B*a^4*b^8 + 44*A*a^3*b^9)*cos(d*x + c)^2 + 3*(6*A*a^10*b^2 + 20*B*a^9*b^3 - 59*A*a^8*b^4 -
35*B*a^7*b^5 + 110*A*a^6*b^6 + 20*B*a^5*b^7 - 77*A*a^4*b^8 - 5*B*a^3*b^9 + 20*A*a^2*b^10)*cos(d*x + c))*sin(d
*x + c))/((a^16 - 4*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8*b^8)*d*cos(d*x + c)^3 + 3*(a^15*b - 4*a^13*b^3 +
6*a^11*b^5 - 4*a^9*b^7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(a^14*b^2 - 4*a^12*b^4 + 6*a^10*b^6 - 4*a^8*b^8 + a^6*b
^10)*d*cos(d*x + c) + (a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4*a^7*b^9 + a^5*b^11)*d), 1/6*(6*(B*a^12 - 4*A*a^11
*b - 4*B*a^10*b^2 + 16*A*a^9*b^3 + 6*B*a^8*b^4 - 24*A*a^7*b^5 - 4*B*a^6*b^6 + 16*A*a^5*b^7 + B*a^4*b^8 - 4*A*a
^3*b^9)*d*x*cos(d*x + c)^3 + 18*(B*a^11*b - 4*A*a^10*b^2 - 4*B*a^9*b^3 + 16*A*a^8*b^4 + 6*B*a^7*b^5 - 24*A*a^6
*b^6 - 4*B*a^5*b^7 + 16*A*a^4*b^8 + B*a^3*b^9 - 4*A*a^2*b^10)*d*x*cos(d*x + c)^2 + 18*(B*a^10*b^2 - 4*A*a^9*b^
3 - 4*B*a^8*b^4 + 16*A*a^7*b^5 + 6*B*a^6*b^6 - 24*A*a^5*b^7 - 4*B*a^4*b^8 + 16*A*a^3*b^9 + B*a^2*b^10 - 4*A*a*
b^11)*d*x*cos(d*x + c) + 6*(B*a^9*b^3 - 4*A*a^8*b^4 - 4*B*a^7*b^5 + 16*A*a^6*b^6 + 6*B*a^5*b^7 - 24*A*a^4*b^8
- 4*B*a^3*b^9 + 16*A*a^2*b^10 + B*a*b^11 - 4*A*b^12)*d*x - 3*(8*B*a^7*b^4 - 20*A*a^6*b^5 - 8*B*a^5*b^6 + 35*A*
a^4*b^7 + 7*B*a^3*b^8 - 28*A*a^2*b^9 - 2*B*a*b^10 + 8*A*b^11 + (8*B*a^10*b - 20*A*a^9*b^2 - 8*B*a^8*b^3 + 35*A
*a^7*b^4 + 7*B*a^6*b^5 - 28*A*a^5*b^6 - 2*B*a^4*b^7 + 8*A*a^3*b^8)*cos(d*x + c)^3 + 3*(8*B*a^9*b^2 - 20*A*a^8*
b^3 - 8*B*a^7*b^4 + 35*A*a^6*b^5 + 7*B*a^5*b^6 - 28*A*a^4*b^7 - 2*B*a^3*b^8 + 8*A*a^2*b^9)*cos(d*x + c)^2 + 3*
(8*B*a^8*b^3 - 20*A*a^7*b^4 - 8*B*a^6*b^5 + 35*A*a^5*b^6 + 7*B*a^4*b^7 - 28*A*a^3*b^8 - 2*B*a^2*b^9 + 8*A*a*b^
10)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) +
(6*A*a^9*b^3 + 26*B*a^8*b^4 - 71*A*a^7*b^5 - 43*B*a^6*b^6 + 133*A*a^5*b^7 + 23*B*a^4*b^8 - 92*A*a^3*b^9 - 6*B
*a^2*b^10 + 24*A*a*b^11 + 6*(A*a^12 - 4*A*a^10*b^2 + 6*A*a^8*b^4 - 4*A*a^6*b^6 + A*a^4*b^8)*cos(d*x + c)^3 + (
18*A*a^11*b + 36*B*a^10*b^2 - 132*A*a^9*b^3 - 68*B*a^8*b^4 + 239*A*a^7*b^5 + 43*B*a^6*b^6 - 169*A*a^5*b^7 - 11
*B*a^4*b^8 + 44*A*a^3*b^9)*cos(d*x + c)^2 + 3*(6*A*a^10*b^2 + 20*B*a^9*b^3 - 59*A*a^8*b^4 - 35*B*a^7*b^5 + 110
*A*a^6*b^6 + 20*B*a^5*b^7 - 77*A*a^4*b^8 - 5*B*a^3*b^9 + 20*A*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^16 - 4
*a^14*b^2 + 6*a^12*b^4 - 4*a^10*b^6 + a^8*b^8)*d*cos(d*x + c)^3 + 3*(a^15*b - 4*a^13*b^3 + 6*a^11*b^5 - 4*a^9*
b^7 + a^7*b^9)*d*cos(d*x + c)^2 + 3*(a^14*b^2 - 4*a^12*b^4 + 6*a^10*b^6 - 4*a^8*b^8 + a^6*b^10)*d*cos(d*x + c)
+ (a^13*b^3 - 4*a^11*b^5 + 6*a^9*b^7 - 4*a^7*b^9 + a^5*b^11)*d)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^4,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. 966 vs. \(2 (393) = 786\).

Time = 0.40 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.35 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/3*(3*(8*B*a^7*b - 20*A*a^6*b^2 - 8*B*a^5*b^3 + 35*A*a^4*b^4 + 7*B*a^3*b^5 - 28*A*a^2*b^6 - 2*B*a*b^7 + 8*A*
b^8)*(pi*floor(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)))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*sqrt(-a^2 + b^2)) + (36*B*a^7*b^2*tan(1/2*d*
x + 1/2*c)^5 - 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 105*A*a^5*b^4*tan(1
/2*d*x + 1/2*c)^5 - 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 + 24*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 45*B*a^4*b^5*ta
n(1/2*d*x + 1/2*c)^5 - 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 24*A*a^2*b^
7*tan(1/2*d*x + 1/2*c)^5 - 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 + 42*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 6*B*a*b^8
*tan(1/2*d*x + 1/2*c)^5 - 18*A*b^9*tan(1/2*d*x + 1/2*c)^5 - 72*B*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^6*b^
3*tan(1/2*d*x + 1/2*c)^3 + 116*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 236*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 - 56*B*
a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 152*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 + 12*B*a*b^8*tan(1/2*d*x + 1/2*c)^3 - 36
*A*b^9*tan(1/2*d*x + 1/2*c)^3 + 36*B*a^7*b^2*tan(1/2*d*x + 1/2*c) - 60*A*a^6*b^3*tan(1/2*d*x + 1/2*c) + 60*B*a
^6*b^3*tan(1/2*d*x + 1/2*c) - 105*A*a^5*b^4*tan(1/2*d*x + 1/2*c) - 6*B*a^5*b^4*tan(1/2*d*x + 1/2*c) + 24*A*a^4
*b^5*tan(1/2*d*x + 1/2*c) - 45*B*a^4*b^5*tan(1/2*d*x + 1/2*c) + 117*A*a^3*b^6*tan(1/2*d*x + 1/2*c) - 6*B*a^3*b
^6*tan(1/2*d*x + 1/2*c) + 24*A*a^2*b^7*tan(1/2*d*x + 1/2*c) + 15*B*a^2*b^7*tan(1/2*d*x + 1/2*c) - 42*A*a*b^8*t
an(1/2*d*x + 1/2*c) + 6*B*a*b^8*tan(1/2*d*x + 1/2*c) - 18*A*b^9*tan(1/2*d*x + 1/2*c))/((a^10 - 3*a^8*b^2 + 3*a
^6*b^4 - a^4*b^6)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) - 3*(B*a - 4*A*b)*(d*x + c)
/a^5 - 6*A*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 + 1)*a^4))/d

Mupad [B] (verification not implemented)

Time = 27.60 (sec) , antiderivative size = 7863, normalized size of antiderivative = 19.13 \[ \int \frac {\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(log(tan(c/2 + (d*x)/2) - 1i)*(4*A*b - B*a)*1i)/(a^5*d) - ((tan(c/2 + (d*x)/2)*(2*A*a^7 - 8*A*b^7 + 24*A*a^2*b
^5 + 11*A*a^3*b^4 - 26*A*a^4*b^3 - 6*A*a^5*b^2 + B*a^2*b^5 - 6*B*a^3*b^4 - 4*B*a^4*b^3 + 12*B*a^5*b^2 - 4*A*a*
b^6 + 2*A*a^6*b + 2*B*a*b^6))/((a + b)*(3*a^6*b - a^7 + a^4*b^3 - 3*a^5*b^2)) - (tan(c/2 + (d*x)/2)^3*(18*A*a^
8 + 72*A*b^8 - 236*A*a^2*b^6 + 47*A*a^3*b^5 + 273*A*a^4*b^4 - 60*A*a^5*b^3 - 72*A*a^6*b^2 + 3*B*a^2*b^6 + 59*B
*a^3*b^5 - 14*B*a^4*b^4 - 96*B*a^5*b^3 + 36*B*a^6*b^2 - 12*A*a*b^7 - 18*B*a*b^7))/(3*(a + b)^2*(3*a^6*b - a^7
+ a^4*b^3 - 3*a^5*b^2)) + (tan(c/2 + (d*x)/2)^7*(24*A*a^2*b^5 - 8*A*b^7 - 2*A*a^7 - 11*A*a^3*b^4 - 26*A*a^4*b^
3 + 6*A*a^5*b^2 - B*a^2*b^5 - 6*B*a^3*b^4 + 4*B*a^4*b^3 + 12*B*a^5*b^2 + 4*A*a*b^6 + 2*A*a^6*b + 2*B*a*b^6))/(
(a^4*b - a^5)*(a + b)^3) + (tan(c/2 + (d*x)/2)^5*(18*A*a^8 + 72*A*b^8 - 236*A*a^2*b^6 - 47*A*a^3*b^5 + 273*A*a
^4*b^4 + 60*A*a^5*b^3 - 72*A*a^6*b^2 - 3*B*a^2*b^6 + 59*B*a^3*b^5 + 14*B*a^4*b^4 - 96*B*a^5*b^3 - 36*B*a^6*b^2
+ 12*A*a*b^7 - 18*B*a*b^7))/(3*(a^4*b - a^5)*(a + b)^3*(a - b)))/(d*(3*a*b^2 + 3*a^2*b - tan(c/2 + (d*x)/2)^4
*(6*a^2*b - 6*b^3) + tan(c/2 + (d*x)/2)^2*(6*a*b^2 - 2*a^3 + 4*b^3) + tan(c/2 + (d*x)/2)^6*(2*a^3 - 6*a*b^2 +
4*b^3) + a^3 + b^3 - tan(c/2 + (d*x)/2)^8*(3*a*b^2 - 3*a^2*b + a^3 - b^3))) - (log(tan(c/2 + (d*x)/2) + 1i)*(A
*b*4i - B*a*1i))/(a^5*d) - (b*atan(((b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 - 128*A^2*a*b^15 - 8*
B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 +
2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^
4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 +
117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5
- 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b + 64*A*B*a^2*b^14 + 384*
A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 -
1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2
))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 +
10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) + (b*((a + b)^7*(a - b)^7)^(1/2)*((8*(4*B*a^24 + 16*A*a^10*b^14 - 8*A*a
^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b
^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^
12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 1
10*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b))/(a^22*b + a^23 - a
^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^2
0*b^3 - 5*a^21*b^2) - (4*b*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 +
35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*b^6)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*
a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^1
9*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 -
35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^
12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^
5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*b^6))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*
a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(8*A*b^7 + 8*B*
a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*b^6)*1i)/(2*(a^19 - a^5*b^1
4 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)) + (b*((8*tan(c/2 + (d*x)
/2)*(128*A^2*b^16 + 4*B^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^
2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 -
824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 -
8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 16
0*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 - 64
*A*B*a*b^15 - 32*A*B*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A
*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 3
84*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a
^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (b*((a + b)^7*(a
- b)^7)^(1/2)*((8*(4*B*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b
^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72
*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9
+ 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*
b^2 - 16*A*a^23*b - 16*B*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b
^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (4*b*tan(c/2 + (d*x)/2)*((a + b)^7*(
a - b)^7)^(1/2)*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*
B*a*b^6)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 +
160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 -
a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a
^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*
b^3 - 5*a^17*b^2)))*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b
- 2*B*a*b^6))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^1
7*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*
b^2 - 20*A*a^6*b - 2*B*a*b^6)*1i)/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 +
21*a^15*b^4 - 7*a^17*b^2)))/((16*(256*A^3*b^16 - 128*A^3*a*b^15 - 16*B^3*a^15*b - 1664*A^3*a^2*b^14 + 800*A^3
*a^3*b^13 + 4576*A^3*a^4*b^12 - 2176*A^3*a^5*b^11 - 6944*A^3*a^6*b^10 + 3204*A^3*a^7*b^9 + 6176*A^3*a^8*b^8 -
2560*A^3*a^9*b^7 - 3040*A^3*a^10*b^6 + 960*A^3*a^11*b^5 + 640*A^3*a^12*b^4 - 4*B^3*a^3*b^13 + 2*B^3*a^4*b^12 +
26*B^3*a^5*b^11 - 11*B^3*a^6*b^10 - 70*B^3*a^7*b^9 + 34*B^3*a^8*b^8 + 110*B^3*a^9*b^7 - 66*B^3*a^10*b^6 - 110
*B^3*a^11*b^5 + 64*B^3*a^12*b^4 + 64*B^3*a^13*b^3 - 48*B^3*a^14*b^2 - 192*A^2*B*a*b^15 + 48*A*B^2*a^2*b^14 - 2
4*A*B^2*a^3*b^13 - 312*A*B^2*a^4*b^12 + 138*A*B^2*a^5*b^11 + 846*A*B^2*a^6*b^10 - 408*A*B^2*a^7*b^9 - 1314*A*B
^2*a^8*b^8 + 726*A*B^2*a^9*b^7 + 1266*A*B^2*a^10*b^6 - 690*A*B^2*a^11*b^5 - 702*A*B^2*a^12*b^4 + 408*A*B^2*a^1
3*b^3 + 168*A*B^2*a^14*b^2 + 96*A^2*B*a^2*b^14 + 1248*A^2*B*a^3*b^13 - 576*A^2*B*a^4*b^12 - 3408*A^2*B*a^5*b^1
1 + 1632*A^2*B*a^6*b^10 + 5232*A^2*B*a^7*b^9 - 2649*A^2*B*a^8*b^8 - 4848*A^2*B*a^9*b^7 + 2376*A^2*B*a^10*b^6 +
2544*A^2*B*a^11*b^5 - 1104*A^2*B*a^12*b^4 - 576*A^2*B*a^13*b^3))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a
^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) + (b*(
(8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 - 128*A^2*a*b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*
a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1
920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 +
8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 16
4*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44
*B^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b + 64*A*B*a^2*b^14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*
B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 59
2*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B*a^14*b^2))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10
+ 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) +
(b*((a + b)^7*(a - b)^7)^(1/2)*((8*(4*B*a^24 + 16*A*a^10*b^14 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b
^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 1
90*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13 + 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^1
0 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^18*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^2
1*b^3 - 12*B*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b))/(a^22*b + a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^
15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^4 - 5*a^20*b^3 - 5*a^21*b^2) - (4*b*tan(c/2 + (d*
x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2
- 20*A*a^6*b - 2*B*a*b^6)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10
+ 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6 + 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^
22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*
(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*
a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5
*b^2 - 20*A*a^6*b - 2*B*a*b^6))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 2
1*a^15*b^4 - 7*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*
a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*b^6))/(2*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8
- 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)) - (b*((8*tan(c/2 + (d*x)/2)*(128*A^2*b^16 + 4*B^2*a^16 - 128*A^2*a*
b^15 - 8*B^2*a^15*b - 768*A^2*a^2*b^14 + 768*A^2*a^3*b^13 + 1920*A^2*a^4*b^12 - 1920*A^2*a^5*b^11 - 2600*A^2*a
^6*b^10 + 2560*A^2*a^7*b^9 + 2025*A^2*a^8*b^8 - 1920*A^2*a^9*b^7 - 824*A^2*a^10*b^6 + 768*A^2*a^11*b^5 + 80*A^
2*a^12*b^4 - 128*A^2*a^13*b^3 + 64*A^2*a^14*b^2 + 8*B^2*a^2*b^14 - 8*B^2*a^3*b^13 - 48*B^2*a^4*b^12 + 48*B^2*a
^5*b^11 + 117*B^2*a^6*b^10 - 120*B^2*a^7*b^9 - 164*B^2*a^8*b^8 + 160*B^2*a^9*b^7 + 156*B^2*a^10*b^6 - 120*B^2*
a^11*b^5 - 92*B^2*a^12*b^4 + 48*B^2*a^13*b^3 + 44*B^2*a^14*b^2 - 64*A*B*a*b^15 - 32*A*B*a^15*b + 64*A*B*a^2*b^
14 + 384*A*B*a^3*b^13 - 384*A*B*a^4*b^12 - 948*A*B*a^5*b^11 + 960*A*B*a^6*b^10 + 1306*A*B*a^7*b^9 - 1280*A*B*a
^8*b^8 - 1128*A*B*a^9*b^7 + 960*A*B*a^10*b^6 + 592*A*B*a^11*b^5 - 384*A*B*a^12*b^4 - 160*A*B*a^13*b^3 + 64*A*B
*a^14*b^2))/(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^
14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2) - (b*((a + b)^7*(a - b)^7)^(1/2)*((8*(4*B*a^24 + 16*A*a^10*b^1
4 - 8*A*a^11*b^13 - 104*A*a^12*b^12 + 50*A*a^13*b^11 + 286*A*a^14*b^10 - 126*A*a^15*b^9 - 434*A*a^16*b^8 + 174
*A*a^17*b^7 + 386*A*a^18*b^6 - 146*A*a^19*b^5 - 190*A*a^20*b^4 + 72*A*a^21*b^3 + 40*A*a^22*b^2 - 4*B*a^11*b^13
+ 2*B*a^12*b^12 + 26*B*a^13*b^11 - 14*B*a^14*b^10 - 70*B*a^15*b^9 + 30*B*a^16*b^8 + 110*B*a^17*b^7 - 30*B*a^1
8*b^6 - 110*B*a^19*b^5 + 20*B*a^20*b^4 + 64*B*a^21*b^3 - 12*B*a^22*b^2 - 16*A*a^23*b - 16*B*a^23*b))/(a^22*b +
a^23 - a^12*b^11 - a^13*b^10 + 5*a^14*b^9 + 5*a^15*b^8 - 10*a^16*b^7 - 10*a^17*b^6 + 10*a^18*b^5 + 10*a^19*b^
4 - 5*a^20*b^3 - 5*a^21*b^2) + (4*b*tan(c/2 + (d*x)/2)*((a + b)^7*(a - b)^7)^(1/2)*(8*A*b^7 + 8*B*a^7 - 28*A*a
^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*b^6)*(8*a^23*b - 8*a^10*b^14 + 8*a^11*b
^13 + 48*a^12*b^12 - 48*a^13*b^11 - 120*a^14*b^10 + 120*a^15*b^9 + 160*a^16*b^8 - 160*a^17*b^7 - 120*a^18*b^6
+ 120*a^19*b^5 + 48*a^20*b^4 - 48*a^21*b^3 - 8*a^22*b^2))/((a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^
11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)*(a^18*b + a^19 - a^8*b^11 - a^9*b^10 + 5*a^10*b^9 + 5*a^11*b^
8 - 10*a^12*b^7 - 10*a^13*b^6 + 10*a^14*b^5 + 10*a^15*b^4 - 5*a^16*b^3 - 5*a^17*b^2)))*(8*A*b^7 + 8*B*a^7 - 28
*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*b^6))/(2*(a^19 - a^5*b^14 + 7*a^7*b
^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2)))*((a + b)^7*(a - b)^7)^(1/2)*(8*A*b
^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*b^6))/(2*(a^19 - a
^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2))))*((a + b)^7*(a -
b)^7)^(1/2)*(8*A*b^7 + 8*B*a^7 - 28*A*a^2*b^5 + 35*A*a^4*b^3 + 7*B*a^3*b^4 - 8*B*a^5*b^2 - 20*A*a^6*b - 2*B*a*
b^6)*1i)/(d*(a^19 - a^5*b^14 + 7*a^7*b^12 - 21*a^9*b^10 + 35*a^11*b^8 - 35*a^13*b^6 + 21*a^15*b^4 - 7*a^17*b^2
))