3.3.80 \(\int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx\) [280]

Optimal. Leaf size=547 \[ -\frac {b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (a^2 A+20 A b^2-8 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))} \]

[Out]

-b^2*(40*A*a^6*b-84*A*a^4*b^3+69*A*a^2*b^5-20*A*b^7-20*B*a^7+35*B*a^5*b^2-28*B*a^3*b^4+8*B*a*b^6)*arctan((a-b)
^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^6/(a-b)^(7/2)/(a+b)^(7/2)/d+1/2*(A*a^2+20*A*b^2-8*B*a*b)*arctanh(sin(
d*x+c))/a^6/d-1/6*(24*A*a^6*b-146*A*a^4*b^3+167*A*a^2*b^5-60*A*b^7-6*B*a^7+65*B*a^5*b^2-68*B*a^3*b^4+24*B*a*b^
6)*tan(d*x+c)/a^5/(a^2-b^2)^3/d+1/2*(A*a^6-23*A*a^4*b^2+27*A*a^2*b^4-10*A*b^6+12*B*a^5*b-11*B*a^3*b^3+4*B*a*b^
5)*sec(d*x+c)*tan(d*x+c)/a^4/(a^2-b^2)^3/d+1/3*b*(A*b-B*a)*sec(d*x+c)*tan(d*x+c)/a/(a^2-b^2)/d/(a+b*cos(d*x+c)
)^3+1/6*b*(10*A*a^2*b-5*A*b^3-7*B*a^3+2*B*a*b^2)*sec(d*x+c)*tan(d*x+c)/a^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))^2+1/
6*b*(48*A*a^4*b-53*A*a^2*b^3+20*A*b^5-27*B*a^5+20*B*a^3*b^2-8*B*a*b^4)*sec(d*x+c)*tan(d*x+c)/a^3/(a^2-b^2)^3/d
/(a+b*cos(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 4.51, antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3079, 3134, 3080, 3855, 2738, 211} \begin {gather*} \frac {b (A b-a B) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac {\left (a^2 A-8 a b B+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}+\frac {b \left (-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac {b \left (-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\right ) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac {\left (a^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}-\frac {b^2 \left (-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 A b^7\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}-\frac {\left (-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right ) \tan (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

-((b^2*(40*a^6*A*b - 84*a^4*A*b^3 + 69*a^2*A*b^5 - 20*A*b^7 - 20*a^7*B + 35*a^5*b^2*B - 28*a^3*b^4*B + 8*a*b^6
*B)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*(a - b)^(7/2)*(a + b)^(7/2)*d)) + ((a^2*A + 20*A*
b^2 - 8*a*b*B)*ArcTanh[Sin[c + d*x]])/(2*a^6*d) - ((24*a^6*A*b - 146*a^4*A*b^3 + 167*a^2*A*b^5 - 60*A*b^7 - 6*
a^7*B + 65*a^5*b^2*B - 68*a^3*b^4*B + 24*a*b^6*B)*Tan[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) + ((a^6*A - 23*a^4*A*b
^2 + 27*a^2*A*b^4 - 10*A*b^6 + 12*a^5*b*B - 11*a^3*b^3*B + 4*a*b^5*B)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 -
b^2)^3*d) + (b*(A*b - a*B)*Sec[c + d*x]*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (b*(10*a^2
*A*b - 5*A*b^3 - 7*a^3*B + 2*a*b^2*B)*Sec[c + d*x]*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2
) + (b*(48*a^4*A*b - 53*a^2*A*b^3 + 20*A*b^5 - 27*a^5*B + 20*a^3*b^2*B - 8*a*b^4*B)*Sec[c + d*x]*Tan[c + d*x])
/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[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 3079

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(A*b^2 - a*b*B))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c +
d*Sin[e + f*x])^(1 + n)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), I
nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(a*A - b*B)*(b*c - a*d)*(m + 1) + b*d*(A*b - a*B)*
(m + n + 2) + (A*b - a*B)*(a*d*(m + 1) - b*c*(m + 2))*Sin[e + f*x] - b*d*(A*b - a*B)*(m + n + 3)*Sin[e + f*x]^
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 -
d^2, 0] && RationalQ[m] && m < -1 && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n
, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3080

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

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 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*} \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {\int \frac {\left (3 a^2 A-5 A b^2+2 a b B-3 a (A b-a B) \cos (c+d x)+4 b (A b-a B) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {\int \frac {\left (2 \left (3 a^4 A-18 a^2 A b^2+10 A b^4+9 a^3 b B-4 a b^3 B\right )-2 a \left (6 a^2 A b-A b^3-3 a^3 B-2 a b^2 B\right ) \cos (c+d x)+3 b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (6 \left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right )-a \left (18 a^4 A b-8 a^2 A b^3+5 A b^5-6 a^5 B-7 a^3 b^2 B-2 a b^4 B\right ) \cos (c+d x)+2 b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (-2 \left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right )+2 a \left (3 a^6 A+27 a^4 A b^2-25 a^2 A b^4+10 A b^6-18 a^5 b B+7 a^3 b^3 B-4 a b^5 B\right ) \cos (c+d x)+6 b \left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^4 \left (a^2-b^2\right )^3}\\ &=-\frac {\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\int \frac {\left (6 \left (a^2-b^2\right )^3 \left (a^2 A+20 A b^2-8 a b B\right )+6 a b \left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^5 \left (a^2-b^2\right )^3}\\ &=-\frac {\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac {\left (a^2 A+20 A b^2-8 a b B\right ) \int \sec (c+d x) \, dx}{2 a^6}-\frac {\left (b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac {\left (a^2 A+20 A b^2-8 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac {\left (b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^6 \left (a^2-b^2\right )^3 d}\\ &=-\frac {b^2 \left (40 a^6 A b-84 a^4 A b^3+69 a^2 A b^5-20 A b^7-20 a^7 B+35 a^5 b^2 B-28 a^3 b^4 B+8 a b^6 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^6 (a-b)^{7/2} (a+b)^{7/2} d}+\frac {\left (a^2 A+20 A b^2-8 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac {\left (24 a^6 A b-146 a^4 A b^3+167 a^2 A b^5-60 A b^7-6 a^7 B+65 a^5 b^2 B-68 a^3 b^4 B+24 a b^6 B\right ) \tan (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}+\frac {\left (a^6 A-23 a^4 A b^2+27 a^2 A b^4-10 A b^6+12 a^5 b B-11 a^3 b^3 B+4 a b^5 B\right ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac {b (A b-a B) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac {b \left (10 a^2 A b-5 A b^3-7 a^3 B+2 a b^2 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac {b \left (48 a^4 A b-53 a^2 A b^3+20 A b^5-27 a^5 B+20 a^3 b^2 B-8 a b^4 B\right ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 5.32, size = 781, normalized size = 1.43 \begin {gather*} \frac {\frac {96 b^2 \left (-40 a^6 A b+84 a^4 A b^3-69 a^2 A b^5+20 A b^7+20 a^7 B-35 a^5 b^2 B+28 a^3 b^4 B-8 a b^6 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}-48 \left (a^2 A+20 A b^2-8 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+48 \left (a^2 A+20 A b^2-8 a b B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 a \left (24 a^{10} A-324 a^8 A b^2+1116 a^6 A b^4-830 a^4 A b^6-61 a^2 A b^8+180 A b^{10}+72 a^9 b B-438 a^7 b^3 B+305 a^5 b^5 B+28 a^3 b^7 B-72 a b^9 B+6 a \left (-20 a^8 A b-9 a^6 A b^3+309 a^4 A b^5-400 a^2 A b^7+150 A b^9+8 a^9 B-6 a^7 b^2 B-135 a^5 b^4 B+163 a^3 b^6 B-60 a b^8 B\right ) \cos (c+d x)+12 b \left (-21 a^8 A b+85 a^6 A b^3-55 a^4 A b^5-19 a^2 A b^7+20 A b^9+6 a^9 B-36 a^7 b^2 B+20 a^5 b^4 B+8 a^3 b^6 B-8 a b^8 B\right ) \cos (2 (c+d x))-138 a^7 A b^3 \cos (3 (c+d x))+738 a^5 A b^5 \cos (3 (c+d x))-840 a^3 A b^7 \cos (3 (c+d x))+300 a A b^9 \cos (3 (c+d x))+36 a^8 b^2 B \cos (3 (c+d x))-318 a^6 b^4 B \cos (3 (c+d x))+342 a^4 b^6 B \cos (3 (c+d x))-120 a^2 b^8 B \cos (3 (c+d x))-24 a^6 A b^4 \cos (4 (c+d x))+146 a^4 A b^6 \cos (4 (c+d x))-167 a^2 A b^8 \cos (4 (c+d x))+60 A b^{10} \cos (4 (c+d x))+6 a^7 b^3 B \cos (4 (c+d x))-65 a^5 b^5 B \cos (4 (c+d x))+68 a^3 b^7 B \cos (4 (c+d x))-24 a b^9 B \cos (4 (c+d x))\right ) \sec (c+d x) \tan (c+d x)}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}}{96 a^6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*Cos[c + d*x])*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

((96*b^2*(-40*a^6*A*b + 84*a^4*A*b^3 - 69*a^2*A*b^5 + 20*A*b^7 + 20*a^7*B - 35*a^5*b^2*B + 28*a^3*b^4*B - 8*a*
b^6*B)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - 48*(a^2*A + 20*A*b^2 - 8*a*b
*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(a^2*A + 20*A*b^2 - 8*a*b*B)*Log[Cos[(c + d*x)/2] + Sin[(c +
d*x)/2]] + (2*a*(24*a^10*A - 324*a^8*A*b^2 + 1116*a^6*A*b^4 - 830*a^4*A*b^6 - 61*a^2*A*b^8 + 180*A*b^10 + 72*
a^9*b*B - 438*a^7*b^3*B + 305*a^5*b^5*B + 28*a^3*b^7*B - 72*a*b^9*B + 6*a*(-20*a^8*A*b - 9*a^6*A*b^3 + 309*a^4
*A*b^5 - 400*a^2*A*b^7 + 150*A*b^9 + 8*a^9*B - 6*a^7*b^2*B - 135*a^5*b^4*B + 163*a^3*b^6*B - 60*a*b^8*B)*Cos[c
+ d*x] + 12*b*(-21*a^8*A*b + 85*a^6*A*b^3 - 55*a^4*A*b^5 - 19*a^2*A*b^7 + 20*A*b^9 + 6*a^9*B - 36*a^7*b^2*B +
20*a^5*b^4*B + 8*a^3*b^6*B - 8*a*b^8*B)*Cos[2*(c + d*x)] - 138*a^7*A*b^3*Cos[3*(c + d*x)] + 738*a^5*A*b^5*Cos
[3*(c + d*x)] - 840*a^3*A*b^7*Cos[3*(c + d*x)] + 300*a*A*b^9*Cos[3*(c + d*x)] + 36*a^8*b^2*B*Cos[3*(c + d*x)]
- 318*a^6*b^4*B*Cos[3*(c + d*x)] + 342*a^4*b^6*B*Cos[3*(c + d*x)] - 120*a^2*b^8*B*Cos[3*(c + d*x)] - 24*a^6*A*
b^4*Cos[4*(c + d*x)] + 146*a^4*A*b^6*Cos[4*(c + d*x)] - 167*a^2*A*b^8*Cos[4*(c + d*x)] + 60*A*b^10*Cos[4*(c +
d*x)] + 6*a^7*b^3*B*Cos[4*(c + d*x)] - 65*a^5*b^5*B*Cos[4*(c + d*x)] + 68*a^3*b^7*B*Cos[4*(c + d*x)] - 24*a*b^
9*B*Cos[4*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3))/(96*a^6*d)

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Maple [A]
time = 2.25, size = 672, normalized size = 1.23

method	result	size
derivativedivides	\(\frac {-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -8 A b +2 a B}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2} A +20 A \,b^{2}-8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}-\frac {2 b^{2} \left (\frac {-\frac {\left (30 A \,a^{4} b +6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}-3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}-5 B \,a^{4} b +18 B \,a^{3} b^{2}+2 B \,a^{2} b^{3}-6 B a \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (45 A \,a^{4} b -53 A \,a^{2} b^{3}+18 A \,b^{5}-30 B \,a^{5}+29 B \,a^{3} b^{2}-9 B a \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b -6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}+3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}+5 B \,a^{4} b +18 B \,a^{3} b^{2}-2 B \,a^{2} b^{3}-6 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 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (40 A \,a^{6} b -84 A \,a^{4} b^{3}+69 A \,a^{2} b^{5}-20 A \,b^{7}-20 B \,a^{7}+35 B \,a^{5} b^{2}-28 B \,a^{3} b^{4}+8 B a \,b^{6}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \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^{6}}+\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -8 A b +2 a B}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2} A -20 A \,b^{2}+8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}}{d}\)	\(672\)
default	\(\frac {-\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-a A -8 A b +2 a B}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (a^{2} A +20 A \,b^{2}-8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{6}}-\frac {2 b^{2} \left (\frac {-\frac {\left (30 A \,a^{4} b +6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}-3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}-5 B \,a^{4} b +18 B \,a^{3} b^{2}+2 B \,a^{2} b^{3}-6 B a \,b^{4}\right ) a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 \left (45 A \,a^{4} b -53 A \,a^{2} b^{3}+18 A \,b^{5}-30 B \,a^{5}+29 B \,a^{3} b^{2}-9 B a \,b^{4}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (30 A \,a^{4} b -6 A \,a^{3} b^{2}-34 A \,a^{2} b^{3}+3 A a \,b^{4}+12 A \,b^{5}-20 B \,a^{5}+5 B \,a^{4} b +18 B \,a^{3} b^{2}-2 B \,a^{2} b^{3}-6 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 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (40 A \,a^{6} b -84 A \,a^{4} b^{3}+69 A \,a^{2} b^{5}-20 A \,b^{7}-20 B \,a^{7}+35 B \,a^{5} b^{2}-28 B \,a^{3} b^{4}+8 B a \,b^{6}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \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^{6}}+\frac {A}{2 a^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {-a A -8 A b +2 a B}{2 a^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-a^{2} A -20 A \,b^{2}+8 B a b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{6}}}{d}\)	\(672\)
risch	\(\text {Expression too large to display}\)	\(3144\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2*A/a^4/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(-A*a-8*A*b+2*B*a)/a^5/(tan(1/2*d*x+1/2*c)+1)+1/2*(A*a^2+20*A*b^2
-8*B*a*b)/a^6*ln(tan(1/2*d*x+1/2*c)+1)-2*b^2/a^6*((-1/2*(30*A*a^4*b+6*A*a^3*b^2-34*A*a^2*b^3-3*A*a*b^4+12*A*b^
5-20*B*a^5-5*B*a^4*b+18*B*a^3*b^2+2*B*a^2*b^3-6*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*(45*A*a^4*b-53*A*a^2*b^3+18*A*b^5-30*B*a^5+29*B*a^3*b^2-9*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*(30*A*a^4*b-6*A*a^3*b^2-34*A*a^2*b^3+3*A*a*b^4+12*A*b^5-20*B*a^5+5*B*a^4*b+18*B*a^3*
b^2-2*B*a^2*b^3-6*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))/(a*tan(1/2*d*x+1/2*c)^2-b*t
an(1/2*d*x+1/2*c)^2+a+b)^3+1/2*(40*A*a^6*b-84*A*a^4*b^3+69*A*a^2*b^5-20*A*b^7-20*B*a^7+35*B*a^5*b^2-28*B*a^3*b
^4+8*B*a*b^6)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^
(1/2)))+1/2*A/a^4/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(-A*a-8*A*b+2*B*a)/a^5/(tan(1/2*d*x+1/2*c)-1)+1/2/a^6*(-A*a^2-2
0*A*b^2+8*B*a*b)*ln(tan(1/2*d*x+1/2*c)-1))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(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*b^2-4*a^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1875 vs. \(2 (525) = 1050\).
time = 97.31, size = 3819, normalized size = 6.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

[1/12*(3*((20*B*a^7*b^5 - 40*A*a^6*b^6 - 35*B*a^5*b^7 + 84*A*a^4*b^8 + 28*B*a^3*b^9 - 69*A*a^2*b^10 - 8*B*a*b^
11 + 20*A*b^12)*cos(d*x + c)^5 + 3*(20*B*a^8*b^4 - 40*A*a^7*b^5 - 35*B*a^6*b^6 + 84*A*a^5*b^7 + 28*B*a^4*b^8 -
69*A*a^3*b^9 - 8*B*a^2*b^10 + 20*A*a*b^11)*cos(d*x + c)^4 + 3*(20*B*a^9*b^3 - 40*A*a^8*b^4 - 35*B*a^7*b^5 + 8
4*A*a^6*b^6 + 28*B*a^5*b^7 - 69*A*a^4*b^8 - 8*B*a^3*b^9 + 20*A*a^2*b^10)*cos(d*x + c)^3 + (20*B*a^10*b^2 - 40*
A*a^9*b^3 - 35*B*a^8*b^4 + 84*A*a^7*b^5 + 28*B*a^6*b^6 - 69*A*a^5*b^7 - 8*B*a^4*b^8 + 20*A*a^3*b^9)*cos(d*x +
c)^2)*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x + c)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(d*x
+ c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + a^2)) + 3*((A*a^10*b^3 - 8*B*
a^9*b^4 + 16*A*a^8*b^5 + 32*B*a^7*b^6 - 74*A*a^6*b^7 - 48*B*a^5*b^8 + 116*A*a^4*b^9 + 32*B*a^3*b^10 - 79*A*a^2
*b^11 - 8*B*a*b^12 + 20*A*b^13)*cos(d*x + c)^5 + 3*(A*a^11*b^2 - 8*B*a^10*b^3 + 16*A*a^9*b^4 + 32*B*a^8*b^5 -
74*A*a^7*b^6 - 48*B*a^6*b^7 + 116*A*a^5*b^8 + 32*B*a^4*b^9 - 79*A*a^3*b^10 - 8*B*a^2*b^11 + 20*A*a*b^12)*cos(d
*x + c)^4 + 3*(A*a^12*b - 8*B*a^11*b^2 + 16*A*a^10*b^3 + 32*B*a^9*b^4 - 74*A*a^8*b^5 - 48*B*a^7*b^6 + 116*A*a^
6*b^7 + 32*B*a^5*b^8 - 79*A*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)*cos(d*x + c)^3 + (A*a^13 - 8*B*a^12*b + 16
*A*a^11*b^2 + 32*B*a^10*b^3 - 74*A*a^9*b^4 - 48*B*a^8*b^5 + 116*A*a^7*b^6 + 32*B*a^6*b^7 - 79*A*a^5*b^8 - 8*B*
a^4*b^9 + 20*A*a^3*b^10)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 3*((A*a^10*b^3 - 8*B*a^9*b^4 + 16*A*a^8*b^5 +
32*B*a^7*b^6 - 74*A*a^6*b^7 - 48*B*a^5*b^8 + 116*A*a^4*b^9 + 32*B*a^3*b^10 - 79*A*a^2*b^11 - 8*B*a*b^12 + 20*
A*b^13)*cos(d*x + c)^5 + 3*(A*a^11*b^2 - 8*B*a^10*b^3 + 16*A*a^9*b^4 + 32*B*a^8*b^5 - 74*A*a^7*b^6 - 48*B*a^6*
b^7 + 116*A*a^5*b^8 + 32*B*a^4*b^9 - 79*A*a^3*b^10 - 8*B*a^2*b^11 + 20*A*a*b^12)*cos(d*x + c)^4 + 3*(A*a^12*b
- 8*B*a^11*b^2 + 16*A*a^10*b^3 + 32*B*a^9*b^4 - 74*A*a^8*b^5 - 48*B*a^7*b^6 + 116*A*a^6*b^7 + 32*B*a^5*b^8 - 7
9*A*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)*cos(d*x + c)^3 + (A*a^13 - 8*B*a^12*b + 16*A*a^11*b^2 + 32*B*a^10*
b^3 - 74*A*a^9*b^4 - 48*B*a^8*b^5 + 116*A*a^7*b^6 + 32*B*a^6*b^7 - 79*A*a^5*b^8 - 8*B*a^4*b^9 + 20*A*a^3*b^10)
*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) + 2*(3*A*a^13 - 12*A*a^11*b^2 + 18*A*a^9*b^4 - 12*A*a^7*b^6 + 3*A*a^5*
b^8 + (6*B*a^10*b^3 - 24*A*a^9*b^4 - 71*B*a^8*b^5 + 170*A*a^7*b^6 + 133*B*a^6*b^7 - 313*A*a^5*b^8 - 92*B*a^4*b
^9 + 227*A*a^3*b^10 + 24*B*a^2*b^11 - 60*A*a*b^12)*cos(d*x + c)^4 + 3*(6*B*a^11*b^2 - 23*A*a^10*b^3 - 59*B*a^9
*b^4 + 146*A*a^8*b^5 + 110*B*a^7*b^6 - 263*A*a^6*b^7 - 77*B*a^5*b^8 + 190*A*a^4*b^9 + 20*B*a^3*b^10 - 50*A*a^2
*b^11)*cos(d*x + c)^3 + (18*B*a^12*b - 63*A*a^11*b^2 - 132*B*a^10*b^3 + 342*A*a^9*b^4 + 239*B*a^8*b^5 - 590*A*
a^7*b^6 - 169*B*a^6*b^7 + 421*A*a^5*b^8 + 44*B*a^4*b^9 - 110*A*a^3*b^10)*cos(d*x + c)^2 + 3*(2*B*a^13 - 5*A*a^
12*b - 8*B*a^11*b^2 + 20*A*a^10*b^3 + 12*B*a^9*b^4 - 30*A*a^8*b^5 - 8*B*a^7*b^6 + 20*A*a^6*b^7 + 2*B*a^5*b^8 -
5*A*a^4*b^9)*cos(d*x + c))*sin(d*x + c))/((a^14*b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^9 + a^6*b^11)*d*cos(d
*x + c)^5 + 3*(a^15*b^2 - 4*a^13*b^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*d*cos(d*x + c)^4 + 3*(a^16*b - 4*a^1
4*b^3 + 6*a^12*b^5 - 4*a^10*b^7 + a^8*b^9)*d*cos(d*x + c)^3 + (a^17 - 4*a^15*b^2 + 6*a^13*b^4 - 4*a^11*b^6 + a
^9*b^8)*d*cos(d*x + c)^2), 1/12*(6*((20*B*a^7*b^5 - 40*A*a^6*b^6 - 35*B*a^5*b^7 + 84*A*a^4*b^8 + 28*B*a^3*b^9
- 69*A*a^2*b^10 - 8*B*a*b^11 + 20*A*b^12)*cos(d*x + c)^5 + 3*(20*B*a^8*b^4 - 40*A*a^7*b^5 - 35*B*a^6*b^6 + 84*
A*a^5*b^7 + 28*B*a^4*b^8 - 69*A*a^3*b^9 - 8*B*a^2*b^10 + 20*A*a*b^11)*cos(d*x + c)^4 + 3*(20*B*a^9*b^3 - 40*A*
a^8*b^4 - 35*B*a^7*b^5 + 84*A*a^6*b^6 + 28*B*a^5*b^7 - 69*A*a^4*b^8 - 8*B*a^3*b^9 + 20*A*a^2*b^10)*cos(d*x + c
)^3 + (20*B*a^10*b^2 - 40*A*a^9*b^3 - 35*B*a^8*b^4 + 84*A*a^7*b^5 + 28*B*a^6*b^6 - 69*A*a^5*b^7 - 8*B*a^4*b^8
+ 20*A*a^3*b^9)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) +
3*((A*a^10*b^3 - 8*B*a^9*b^4 + 16*A*a^8*b^5 + 32*B*a^7*b^6 - 74*A*a^6*b^7 - 48*B*a^5*b^8 + 116*A*a^4*b^9 + 32
*B*a^3*b^10 - 79*A*a^2*b^11 - 8*B*a*b^12 + 20*A*b^13)*cos(d*x + c)^5 + 3*(A*a^11*b^2 - 8*B*a^10*b^3 + 16*A*a^9
*b^4 + 32*B*a^8*b^5 - 74*A*a^7*b^6 - 48*B*a^6*b^7 + 116*A*a^5*b^8 + 32*B*a^4*b^9 - 79*A*a^3*b^10 - 8*B*a^2*b^1
1 + 20*A*a*b^12)*cos(d*x + c)^4 + 3*(A*a^12*b - 8*B*a^11*b^2 + 16*A*a^10*b^3 + 32*B*a^9*b^4 - 74*A*a^8*b^5 - 4
8*B*a^7*b^6 + 116*A*a^6*b^7 + 32*B*a^5*b^8 - 79*A*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)*cos(d*x + c)^3 + (A*
a^13 - 8*B*a^12*b + 16*A*a^11*b^2 + 32*B*a^10*b^3 - 74*A*a^9*b^4 - 48*B*a^8*b^5 + 116*A*a^7*b^6 + 32*B*a^6*b^7
- 79*A*a^5*b^8 - 8*B*a^4*b^9 + 20*A*a^3*b^10)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 3*((A*a^10*b^3 - 8*B*a^
9*b^4 + 16*A*a^8*b^5 + 32*B*a^7*b^6 - 74*A*a^6*b^7 - 48*B*a^5*b^8 + 116*A*a^4*b^9 + 32*B*a^3*b^10 - 79*A*a^2*b
^11 - 8*B*a*b^12 + 20*A*b^13)*cos(d*x + c)^5 + 3*(A*a^11*b^2 - 8*B*a^10*b^3 + 16*A*a^9*b^4 + 32*B*a^8*b^5 - 74
*A*a^7*b^6 - 48*B*a^6*b^7 + 116*A*a^5*b^8 + 32*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \cos {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)**3/(a+b*cos(d*x+c))**4,x)

[Out]

Integral((A + B*cos(c + d*x))*sec(c + d*x)**3/(a + b*cos(c + d*x))**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1090 vs. \(2 (525) = 1050\).
time = 0.54, size = 1090, normalized size = 1.99 \begin {gather*} -\frac {\frac {6 \, {\left (20 \, B a^{7} b^{2} - 40 \, A a^{6} b^{3} - 35 \, B a^{5} b^{4} + 84 \, A a^{4} b^{5} + 28 \, B a^{3} b^{6} - 69 \, A a^{2} b^{7} - 8 \, B a b^{8} + 20 \, A b^{9}\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 )}}{{\left (a^{12} - 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} - a^{6} b^{6}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, {\left (60 \, B a^{7} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 90 \, A a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, B a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 162 \, A a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, B a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, A a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 117 \, B a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 213 \, A a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, B a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 48 \, A a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 42 \, B a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 81 \, A a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, B a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 36 \, A b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 120 \, B a^{7} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 180 \, A a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 236 \, B a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 392 \, A a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 152 \, B a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 284 \, A a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, A b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, B a^{7} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 90 \, A a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, B a^{6} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 162 \, A a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{5} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, A a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 117 \, B a^{4} b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 213 \, A a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B a^{3} b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, A a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 42 \, B a^{2} b^{8} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 81 \, A a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, B a b^{9} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, A b^{10} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{3}} - \frac {3 \, {\left (A a^{2} - 8 \, B a b + 20 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{6}} + \frac {3 \, {\left (A a^{2} - 8 \, B a b + 20 \, A b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{6}} - \frac {6 \, {\left (A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, A b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A b \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 )}^{2} a^{5}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c))*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(6*(20*B*a^7*b^2 - 40*A*a^6*b^3 - 35*B*a^5*b^4 + 84*A*a^4*b^5 + 28*B*a^3*b^6 - 69*A*a^2*b^7 - 8*B*a*b^8 +
20*A*b^9)*(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^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*sqrt(a^2 - b^2)) + 2*(60*B*a^7*b^3*tan
(1/2*d*x + 1/2*c)^5 - 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 162*A*a^5*b
^5*tan(1/2*d*x + 1/2*c)^5 - 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 117*B*
a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 24*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 +
48*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 42*B*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 81*A*a*b^9*tan(1/2*d*x + 1/2*c)^5
+ 18*B*a*b^9*tan(1/2*d*x + 1/2*c)^5 - 36*A*b^10*tan(1/2*d*x + 1/2*c)^5 + 120*B*a^7*b^3*tan(1/2*d*x + 1/2*c)^3
- 180*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 - 236*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^3 + 392*A*a^4*b^6*tan(1/2*d*x + 1/
2*c)^3 + 152*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 - 284*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 - 36*B*a*b^9*tan(1/2*d*x
+ 1/2*c)^3 + 72*A*b^10*tan(1/2*d*x + 1/2*c)^3 + 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c) - 90*A*a^6*b^4*tan(1/2*d*x +
1/2*c) + 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c) - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c) - 24*B*a^5*b^5*tan(1/2*d*x +
1/2*c) + 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c) - 117*B*a^4*b^6*tan(1/2*d*x + 1/2*c) + 213*A*a^3*b^7*tan(1/2*d*x +
1/2*c) - 24*B*a^3*b^7*tan(1/2*d*x + 1/2*c) + 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c) + 42*B*a^2*b^8*tan(1/2*d*x + 1
/2*c) - 81*A*a*b^9*tan(1/2*d*x + 1/2*c) + 18*B*a*b^9*tan(1/2*d*x + 1/2*c) - 36*A*b^10*tan(1/2*d*x + 1/2*c))/((
a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*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*(
A*a^2 - 8*B*a*b + 20*A*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^6 + 3*(A*a^2 - 8*B*a*b + 20*A*b^2)*log(abs(ta
n(1/2*d*x + 1/2*c) - 1))/a^6 - 6*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan(1/2*d*x + 1/2*c)^3 + 8*A*b*tan(1/2*d*
x + 1/2*c)^3 + A*a*tan(1/2*d*x + 1/2*c) + 2*B*a*tan(1/2*d*x + 1/2*c) - 8*A*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d
*x + 1/2*c)^2 - 1)^2*a^5))/d

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Mupad [B]
time = 13.94, size = 2500, normalized size = 4.57 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*cos(c + d*x))/(cos(c + d*x)^3*(a + b*cos(c + d*x))^4),x)

[Out]

((tan(c/2 + (d*x)/2)*(A*a^8 + 20*A*b^8 + 2*B*a^8 - 59*A*a^2*b^6 - 27*A*a^3*b^5 + 57*A*a^4*b^4 + 21*A*a^5*b^3 -
11*A*a^6*b^2 - 4*B*a^2*b^6 + 24*B*a^3*b^5 + 11*B*a^4*b^4 - 26*B*a^5*b^3 - 6*B*a^6*b^2 + 10*A*a*b^7 - 7*A*a^7*
b - 8*B*a*b^7 + 2*B*a^7*b))/(a^5*(a + b)*(a - b)^3) + (2*tan(c/2 + (d*x)/2)^5*(9*A*a^10 + 180*A*b^10 - 611*A*a
^2*b^8 + 740*A*a^4*b^6 - 324*A*a^6*b^4 + 36*A*a^8*b^2 + 248*B*a^3*b^7 - 320*B*a^5*b^5 + 132*B*a^7*b^3 - 72*B*a
*b^9 - 18*B*a^9*b))/(3*a^5*(a + b)^3*(a - b)^3) + (tan(c/2 + (d*x)/2)^9*(A*a^8 + 20*A*b^8 - 2*B*a^8 - 59*A*a^2
*b^6 + 27*A*a^3*b^5 + 57*A*a^4*b^4 - 21*A*a^5*b^3 - 11*A*a^6*b^2 + 4*B*a^2*b^6 + 24*B*a^3*b^5 - 11*B*a^4*b^4 -
26*B*a^5*b^3 + 6*B*a^6*b^2 - 10*A*a*b^7 + 7*A*a^7*b - 8*B*a*b^7 + 2*B*a^7*b))/(a^5*(a + b)^3*(a - b)) + (2*ta
n(c/2 + (d*x)/2)^3*(6*A*a^9 - 120*A*b^9 + 6*B*a^9 + 364*A*a^2*b^7 + 71*A*a^3*b^6 - 369*A*a^4*b^5 - 45*A*a^5*b^
4 + 111*A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 - 148*B*a^3*b^6 - 29*B*a^4*b^5 + 159*B*a^5*b^4 + 18*B*a^6*b^3 -
30*B*a^7*b^2 - 30*A*a*b^8 - 21*A*a^8*b + 48*B*a*b^8 - 6*B*a^8*b))/(3*a^5*(a + b)^2*(a - b)^3) + (2*tan(c/2 +
(d*x)/2)^7*(6*A*a^9 + 120*A*b^9 - 6*B*a^9 - 364*A*a^2*b^7 + 71*A*a^3*b^6 + 369*A*a^4*b^5 - 45*A*a^5*b^4 - 111*
A*a^6*b^3 + 3*A*a^7*b^2 + 12*B*a^2*b^7 + 148*B*a^3*b^6 - 29*B*a^4*b^5 - 159*B*a^5*b^4 + 18*B*a^6*b^3 + 30*B*a^
7*b^2 - 30*A*a*b^8 + 21*A*a^8*b - 48*B*a*b^8 - 6*B*a^8*b))/(3*a^5*(a + b)^3*(a - b)^2))/(d*(tan(c/2 + (d*x)/2)
^4*(6*a*b^2 - 6*a^2*b - 2*a^3 + 10*b^3) - tan(c/2 + (d*x)/2)^2*(9*a*b^2 + 3*a^2*b - a^3 + 5*b^3) + tan(c/2 + (
d*x)/2)^6*(6*a*b^2 + 6*a^2*b - 2*a^3 - 10*b^3) + 3*a*b^2 + 3*a^2*b + a^3 + b^3 + tan(c/2 + (d*x)/2)^10*(3*a*b^
2 - 3*a^2*b + a^3 - b^3) + tan(c/2 + (d*x)/2)^8*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3))) + (atan(((((8*tan(c/2 + (d
*x)/2)*(800*A^2*a*b^17 - 800*A^2*b^18 - A^2*a^18 + 2*A^2*a^17*b + 4720*A^2*a^2*b^16 - 4720*A^2*a^3*b^15 - 1152
2*A^2*a^4*b^14 + 11522*A^2*a^5*b^13 + 14837*A^2*a^6*b^12 - 14812*A^2*a^7*b^11 - 10385*A^2*a^8*b^10 + 10430*A^2
*a^9*b^9 + 3325*A^2*a^10*b^8 - 3640*A^2*a^11*b^7 + 45*A^2*a^12*b^6 + 350*A^2*a^13*b^5 - 209*A^2*a^14*b^4 + 68*
A^2*a^15*b^3 - 35*A^2*a^16*b^2 - 128*B^2*a^2*b^16 + 128*B^2*a^3*b^15 + 768*B^2*a^4*b^14 - 768*B^2*a^5*b^13 - 1
920*B^2*a^6*b^12 + 1920*B^2*a^7*b^11 + 2600*B^2*a^8*b^10 - 2560*B^2*a^9*b^9 - 2025*B^2*a^10*b^8 + 1920*B^2*a^1
1*b^7 + 824*B^2*a^12*b^6 - 768*B^2*a^13*b^5 - 80*B^2*a^14*b^4 + 128*B^2*a^15*b^3 - 64*B^2*a^16*b^2 + 640*A*B*a
*b^17 + 16*A*B*a^17*b - 640*A*B*a^2*b^16 - 3808*A*B*a^3*b^15 + 3808*A*B*a^4*b^14 + 9408*A*B*a^5*b^13 - 9408*A*
B*a^6*b^12 - 12430*A*B*a^7*b^11 + 12320*A*B*a^8*b^10 + 9200*A*B*a^9*b^9 - 8960*A*B*a^10*b^8 - 3360*A*B*a^11*b^
7 + 3360*A*B*a^12*b^6 + 144*A*B*a^13*b^5 - 448*A*B*a^14*b^4 + 240*A*B*a^15*b^3 - 32*A*B*a^16*b^2))/(a^20*b + a
^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4
- 5*a^18*b^3 - 5*a^19*b^2) + (((4*(4*A*a^27 - 80*A*a^12*b^15 + 40*A*a^13*b^14 + 516*A*a^14*b^13 - 248*A*a^15*b
^12 - 1404*A*a^16*b^11 + 640*A*a^17*b^10 + 2076*A*a^18*b^9 - 896*A*a^19*b^8 - 1764*A*a^20*b^7 + 724*A*a^21*b^6
+ 816*A*a^22*b^5 - 316*A*a^23*b^4 - 160*A*a^24*b^3 + 52*A*a^25*b^2 + 32*B*a^13*b^14 - 16*B*a^14*b^13 - 208*B*
a^15*b^12 + 100*B*a^16*b^11 + 572*B*a^17*b^10 - 252*B*a^18*b^9 - 868*B*a^19*b^8 + 348*B*a^20*b^7 + 772*B*a^21*
b^6 - 292*B*a^22*b^5 - 380*B*a^23*b^4 + 144*B*a^24*b^3 + 80*B*a^25*b^2 - 32*B*a^26*b))/(a^25*b + a^26 - a^15*b
^11 - a^16*b^10 + 5*a^17*b^9 + 5*a^18*b^8 - 10*a^19*b^7 - 10*a^20*b^6 + 10*a^21*b^5 + 10*a^22*b^4 - 5*a^23*b^3
- 5*a^24*b^2) - (4*tan(c/2 + (d*x)/2)*(A*a^2 + 20*A*b^2 - 8*B*a*b)*(8*a^25*b - 8*a^12*b^14 + 8*a^13*b^13 + 48
*a^14*b^12 - 48*a^15*b^11 - 120*a^16*b^10 + 120*a^17*b^9 + 160*a^18*b^8 - 160*a^19*b^7 - 120*a^20*b^6 + 120*a^
21*b^5 + 48*a^22*b^4 - 48*a^23*b^3 - 8*a^24*b^2))/(a^6*(a^20*b + a^21 - a^10*b^11 - a^11*b^10 + 5*a^12*b^9 + 5
*a^13*b^8 - 10*a^14*b^7 - 10*a^15*b^6 + 10*a^16*b^5 + 10*a^17*b^4 - 5*a^18*b^3 - 5*a^19*b^2)))*(A*a^2 + 20*A*b
^2 - 8*B*a*b))/(2*a^6))*(A*a^2 + 20*A*b^2 - 8*B*a*b)*1i)/(2*a^6) + (((8*tan(c/2 + (d*x)/2)*(800*A^2*a*b^17 - 8
00*A^2*b^18 - A^2*a^18 + 2*A^2*a^17*b + 4720*A^2*a^2*b^16 - 4720*A^2*a^3*b^15 - 11522*A^2*a^4*b^14 + 11522*A^2
*a^5*b^13 + 14837*A^2*a^6*b^12 - 14812*A^2*a^7*b^11 - 10385*A^2*a^8*b^10 + 10430*A^2*a^9*b^9 + 3325*A^2*a^10*b
^8 - 3640*A^2*a^11*b^7 + 45*A^2*a^12*b^6 + 350*A^2*a^13*b^5 - 209*A^2*a^14*b^4 + 68*A^2*a^15*b^3 - 35*A^2*a^16
*b^2 - 128*B^2*a^2*b^16 + 128*B^2*a^3*b^15 + 768*B^2*a^4*b^14 - 768*B^2*a^5*b^13 - 1920*B^2*a^6*b^12 + 1920*B^
2*a^7*b^11 + 2600*B^2*a^8*b^10 - 2560*B^2*a^9*b^9 - 2025*B^2*a^10*b^8 + 1920*B^2*a^11*b^7 + 824*B^2*a^12*b^6 -
768*B^2*a^13*b^5 - 80*B^2*a^14*b^4 + 128*B^2*a^15*b^3 - 64*B^2*a^16*b^2 + 640*A*B*a*b^17 + 16*A*B*a^17*b - 64
0*A*B*a^2*b^16 - 3808*A*B*a^3*b^15 + 3808*A*B*a^4*b^14 + 9408*A*B*a^5*b^13 - 9408*A*B*a^6*b^12 - 12430*A*B*a^7
*b^11 + 12320*A*B*a^8*b^10 + 9200*A*B*a^9*b^9 - 8960*A*B*a^10*b^8 - 3360*A*B*a^11*b^7 + 3360*A*B*a^12*b^6 + 14
4*A*B*a^13*b^5 - 448*A*B*a^14*b^4 + 240*A*B*a^1...

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