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3.328 \(\int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=407 \[ \frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\left (-12 a^2 B+6 a A b-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {a \left (-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\left (-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2}+\frac {\left (-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}+\frac {a^2 \left (-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}} \]

[Out]

-1/2*(6*A*a*b-12*B*a^2-B*b^2)*arctanh(sin(d*x+c))/b^5/d+a^2*(6*A*a^4*b-15*A*a^2*b^3+12*A*b^5-12*B*a^5+29*B*a^3
*b^2-20*B*a*b^4)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^5/(a+b)^(5/2)/d+1/2*(6*A*a^
4*b-11*A*a^2*b^3+2*A*b^5-12*B*a^5+21*B*a^3*b^2-6*B*a*b^4)*tan(d*x+c)/b^4/(a^2-b^2)^2/d-1/2*(3*A*a^3*b-6*A*a*b^
3-6*B*a^4+10*B*a^2*b^2-B*b^4)*sec(d*x+c)*tan(d*x+c)/b^3/(a^2-b^2)^2/d+1/2*a*(A*b-B*a)*sec(d*x+c)^3*tan(d*x+c)/
b/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+1/2*a*(2*A*a^2*b-5*A*b^3-4*B*a^3+7*B*a*b^2)*sec(d*x+c)^2*tan(d*x+c)/b^2/(a^2-
b^2)^2/d/(a+b*sec(d*x+c))

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Rubi [A]  time = 1.96, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {4029, 4098, 4092, 4082, 3998, 3770, 3831, 2659, 208} \[ \frac {\left (-11 a^2 A b^3+6 a^4 A b+21 a^3 b^2 B-12 a^5 B-6 a b^4 B+2 A b^5\right ) \tan (c+d x)}{2 b^4 d \left (a^2-b^2\right )^2}-\frac {\left (-12 a^2 B+6 a A b-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {a^2 \left (-15 a^2 A b^3+6 a^4 A b+29 a^3 b^2 B-12 a^5 B-20 a b^4 B+12 A b^5\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {a (A b-a B) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-4 a^3 B+7 a b^2 B-5 A b^3\right ) \tan (c+d x) \sec ^2(c+d x)}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\left (3 a^3 A b+10 a^2 b^2 B-6 a^4 B-6 a A b^3-b^4 B\right ) \tan (c+d x) \sec (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]

[Out]

-((6*a*A*b - 12*a^2*B - b^2*B)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) + (a^2*(6*a^4*A*b - 15*a^2*A*b^3 + 12*A*b^5 -
12*a^5*B + 29*a^3*b^2*B - 20*a*b^4*B)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*
(a + b)^(5/2)*d) + ((6*a^4*A*b - 11*a^2*A*b^3 + 2*A*b^5 - 12*a^5*B + 21*a^3*b^2*B - 6*a*b^4*B)*Tan[c + d*x])/(
2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*A*b - 6*a*A*b^3 - 6*a^4*B + 10*a^2*b^2*B - b^4*B)*Sec[c + d*x]*Tan[c + d*x])/
(2*b^3*(a^2 - b^2)^2*d) + (a*(A*b - a*B)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^
2) + (a*(2*a^2*A*b - 5*A*b^3 - 4*a^3*B + 7*a*b^2*B)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b
*Sec[c + d*x]))

Rule 208

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

Rule 2659

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3831

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

Rule 3998

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x
_Symbol] :> Dist[B/b, Int[Csc[e + f*x], x], x] + Dist[(A*b - a*B)/b, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x]
, x] /; FreeQ[{a, b, e, f, A, B}, x] && NeQ[A*b - a*B, 0]

Rule 4029

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

Rule 4082

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

Rule 4092

Int[csc[(e_.) + (f_.)*(x_)]^2*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(
e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Simp[(C*Csc[e + f*x]*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m
 + 1))/(b*f*(m + 3)), x] + Dist[1/(b*(m + 3)), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m*Simp[a*C + b*(C*(m + 2)
 + A*(m + 3))*Csc[e + f*x] - (2*a*C - b*B*(m + 3))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m
}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]

Rule 4098

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

Rubi steps

\begin {align*} \int \frac {\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx &=\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\int \frac {\sec ^3(c+d x) \left (3 a (A b-a B)-2 b (A b-a B) \sec (c+d x)-2 \left (a A b-2 a^2 B+b^2 B\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\sec ^2(c+d x) \left (-2 a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right )-b \left (a^2 A b+2 A b^3+a^3 B-4 a b^2 B\right ) \sec (c+d x)+2 \left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (2 a \left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right )-2 b \left (a^3 A b-4 a A b^3-2 a^4 B+4 a^2 b^2 B+b^4 B\right ) \sec (c+d x)-2 \left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (2 a b \left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right )+2 \left (a^2-b^2\right )^2 \left (6 a A b-12 a^2 B-b^2 B\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\left (6 a A b-12 a^2 B-b^2 B\right ) \int \sec (c+d x) \, dx}{2 b^5}+\frac {\left (a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (6 a A b-12 a^2 B-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^6 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (6 a A b-12 a^2 B-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right )\right ) \operatorname {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 )}{b^6 \left (a^2-b^2\right )^2 d}\\ &=-\frac {\left (6 a A b-12 a^2 B-b^2 B\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {a^2 \left (6 a^4 A b-15 a^2 A b^3+12 A b^5-12 a^5 B+29 a^3 b^2 B-20 a b^4 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^5 (a+b)^{5/2} d}+\frac {\left (6 a^4 A b-11 a^2 A b^3+2 A b^5-12 a^5 B+21 a^3 b^2 B-6 a b^4 B\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 A b-6 a A b^3-6 a^4 B+10 a^2 b^2 B-b^4 B\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (A b-a B) \sec ^3(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {a \left (2 a^2 A b-5 A b^3-4 a^3 B+7 a b^2 B\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 3.08, size = 507, normalized size = 1.25 \[ \frac {-8 \left (12 a^2 B-6 a A b+b^2 B\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 \left (12 a^2 B-6 a A b+b^2 B\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {16 a^2 \left (12 a^5 B-6 a^4 A b-29 a^3 b^2 B+15 a^2 A b^3+20 a b^4 B-12 A b^5\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}+\frac {2 b \tan (c+d x) \sec (c+d x) \left (-12 a^7 B \cos (3 (c+d x))+6 a^6 A b \cos (3 (c+d x))-36 a^6 b B+18 a^5 A b^2+21 a^5 b^2 B \cos (3 (c+d x))-11 a^4 A b^3 \cos (3 (c+d x))+68 a^4 b^3 B-32 a^3 A b^4-6 a^3 b^4 B \cos (3 (c+d x))+2 a^2 A b^5 \cos (3 (c+d x))-30 a^2 b^5 B-2 a b \left (18 a^5 B-9 a^4 A b-32 a^3 b^2 B+16 a^2 A b^3+11 a b^4 B-4 A b^5\right ) \cos (2 (c+d x))+\left (-36 a^7 B+18 a^6 A b+47 a^5 b^2 B-25 a^4 A b^3+14 a^3 b^4 B-10 a^2 A b^5-16 a b^6 B+8 A b^7\right ) \cos (c+d x)+8 a A b^6+4 b^7 B\right )}{\left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}}{16 b^5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]

[Out]

((16*a^2*(-6*a^4*A*b + 15*a^2*A*b^3 - 12*A*b^5 + 12*a^5*B - 29*a^3*b^2*B + 20*a*b^4*B)*ArcTanh[((-a + b)*Tan[(
c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] - Sin[
(c + d*x)/2]] + 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*(18*a^5*A*b^2
- 32*a^3*A*b^4 + 8*a*A*b^6 - 36*a^6*b*B + 68*a^4*b^3*B - 30*a^2*b^5*B + 4*b^7*B + (18*a^6*A*b - 25*a^4*A*b^3 -
 10*a^2*A*b^5 + 8*A*b^7 - 36*a^7*B + 47*a^5*b^2*B + 14*a^3*b^4*B - 16*a*b^6*B)*Cos[c + d*x] - 2*a*b*(-9*a^4*A*
b + 16*a^2*A*b^3 - 4*A*b^5 + 18*a^5*B - 32*a^3*b^2*B + 11*a*b^4*B)*Cos[2*(c + d*x)] + 6*a^6*A*b*Cos[3*(c + d*x
)] - 11*a^4*A*b^3*Cos[3*(c + d*x)] + 2*a^2*A*b^5*Cos[3*(c + d*x)] - 12*a^7*B*Cos[3*(c + d*x)] + 21*a^5*b^2*B*C
os[3*(c + d*x)] - 6*a^3*b^4*B*Cos[3*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])
^2))/(16*b^5*d)

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fricas [B]  time = 72.77, size = 2444, normalized size = 6.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

[-1/4*(((12*B*a^9 - 6*A*a^8*b - 29*B*a^7*b^2 + 15*A*a^6*b^3 + 20*B*a^5*b^4 - 12*A*a^4*b^5)*cos(d*x + c)^4 + 2*
(12*B*a^8*b - 6*A*a^7*b^2 - 29*B*a^6*b^3 + 15*A*a^5*b^4 + 20*B*a^4*b^5 - 12*A*a^3*b^6)*cos(d*x + c)^3 + (12*B*
a^7*b^2 - 6*A*a^6*b^3 - 29*B*a^5*b^4 + 15*A*a^4*b^5 + 20*B*a^3*b^6 - 12*A*a^2*b^7)*cos(d*x + c)^2)*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)) - ((12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 +
 18*A*a^7*b^3 + 33*B*a^6*b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B*
a^9*b - 6*A*a^8*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 -
B*a*b^9)*cos(d*x + c)^3 + (12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*
b^7 - 9*B*a^2*b^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((12*B*a^10 - 6*A*a^9*b - 35*B
*a^8*b^2 + 18*A*a^7*b^3 + 33*B*a^6*b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4
+ 2*(12*B*a^9*b - 6*A*a^8*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*
a^2*b^8 - B*a*b^9)*cos(d*x + c)^3 + (12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 -
 18*A*a^3*b^7 - 9*B*a^2*b^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(B*a^6*b^4 - 3*B*
a^4*b^6 + 3*B*a^2*b^8 - B*b^10 - (12*B*a^9*b - 6*A*a^8*b^2 - 33*B*a^7*b^3 + 17*A*a^6*b^4 + 27*B*a^5*b^5 - 13*A
*a^4*b^6 - 6*B*a^3*b^7 + 2*A*a^2*b^8)*cos(d*x + c)^3 - (18*B*a^8*b^2 - 9*A*a^7*b^3 - 50*B*a^6*b^4 + 25*A*a^5*b
^5 + 43*B*a^4*b^6 - 20*A*a^3*b^7 - 11*B*a^2*b^8 + 4*A*a*b^9)*cos(d*x + c)^2 - 2*(2*B*a^7*b^3 - A*a^6*b^4 - 6*B
*a^5*b^5 + 3*A*a^4*b^6 + 6*B*a^3*b^7 - 3*A*a^2*b^8 - 2*B*a*b^9 + A*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^5
 - 3*a^6*b^7 + 3*a^4*b^9 - a^2*b^11)*d*cos(d*x + c)^4 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*
x + c)^3 + (a^6*b^7 - 3*a^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2), -1/4*(2*((12*B*a^9 - 6*A*a^8*b - 29*B*
a^7*b^2 + 15*A*a^6*b^3 + 20*B*a^5*b^4 - 12*A*a^4*b^5)*cos(d*x + c)^4 + 2*(12*B*a^8*b - 6*A*a^7*b^2 - 29*B*a^6*
b^3 + 15*A*a^5*b^4 + 20*B*a^4*b^5 - 12*A*a^3*b^6)*cos(d*x + c)^3 + (12*B*a^7*b^2 - 6*A*a^6*b^3 - 29*B*a^5*b^4
+ 15*A*a^4*b^5 + 20*B*a^3*b^6 - 12*A*a^2*b^7)*cos(d*x + c)^2)*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))) - ((12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 + 18*A*a^7*b^3 + 33*B*a^6*
b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B*a^9*b - 6*A*a^8*b^2 - 35*
B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 - B*a*b^9)*cos(d*x + c)^3 +
 (12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*b^7 - 9*B*a^2*b^8 + 6*A*a
*b^9 - B*b^10)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((12*B*a^10 - 6*A*a^9*b - 35*B*a^8*b^2 + 18*A*a^7*b^3 +
 33*B*a^6*b^4 - 18*A*a^5*b^5 - 9*B*a^4*b^6 + 6*A*a^3*b^7 - B*a^2*b^8)*cos(d*x + c)^4 + 2*(12*B*a^9*b - 6*A*a^8
*b^2 - 35*B*a^7*b^3 + 18*A*a^6*b^4 + 33*B*a^5*b^5 - 18*A*a^4*b^6 - 9*B*a^3*b^7 + 6*A*a^2*b^8 - B*a*b^9)*cos(d*
x + c)^3 + (12*B*a^8*b^2 - 6*A*a^7*b^3 - 35*B*a^6*b^4 + 18*A*a^5*b^5 + 33*B*a^4*b^6 - 18*A*a^3*b^7 - 9*B*a^2*b
^8 + 6*A*a*b^9 - B*b^10)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(B*a^6*b^4 - 3*B*a^4*b^6 + 3*B*a^2*b^8 - B
*b^10 - (12*B*a^9*b - 6*A*a^8*b^2 - 33*B*a^7*b^3 + 17*A*a^6*b^4 + 27*B*a^5*b^5 - 13*A*a^4*b^6 - 6*B*a^3*b^7 +
2*A*a^2*b^8)*cos(d*x + c)^3 - (18*B*a^8*b^2 - 9*A*a^7*b^3 - 50*B*a^6*b^4 + 25*A*a^5*b^5 + 43*B*a^4*b^6 - 20*A*
a^3*b^7 - 11*B*a^2*b^8 + 4*A*a*b^9)*cos(d*x + c)^2 - 2*(2*B*a^7*b^3 - A*a^6*b^4 - 6*B*a^5*b^5 + 3*A*a^4*b^6 +
6*B*a^3*b^7 - 3*A*a^2*b^8 - 2*B*a*b^9 + A*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^5 - 3*a^6*b^7 + 3*a^4*b^9
- a^2*b^11)*d*cos(d*x + c)^4 + 2*(a^7*b^6 - 3*a^5*b^8 + 3*a^3*b^10 - a*b^12)*d*cos(d*x + c)^3 + (a^6*b^7 - 3*a
^4*b^9 + 3*a^2*b^11 - b^13)*d*cos(d*x + c)^2)]

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giac [B]  time = 0.50, size = 1391, normalized size = 3.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

-1/2*(2*(12*B*a^7 - 6*A*a^6*b - 29*B*a^5*b^2 + 15*A*a^4*b^3 + 20*B*a^3*b^4 - 12*A*a^2*b^5)*(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^4*b^5 - 2*a^2*b^7 + b^9)*sqrt(-a^2 + b^2)) - 2*(12*B*a^7*tan(1/2*d*x + 1/2*c)^7 - 6*A*a^6*b*tan(1/2*d*x +
1/2*c)^7 - 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^7 + 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^7 - 17*B*a^5*b^2*tan(1/2*d*x +
 1/2*c)^7 + 9*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 + 33*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^7 - 16*A*a^3*b^4*tan(1/2*d*
x + 1/2*c)^7 - 2*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^7 + 2*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^7 - 13*B*a^2*b^5*tan(1/2*
d*x + 1/2*c)^7 + 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^7 + 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^7 - 2*A*b^7*tan(1/2*d*x + 1
/2*c)^7 + B*b^7*tan(1/2*d*x + 1/2*c)^7 - 36*B*a^7*tan(1/2*d*x + 1/2*c)^5 + 18*A*a^6*b*tan(1/2*d*x + 1/2*c)^5 +
 18*B*a^6*b*tan(1/2*d*x + 1/2*c)^5 - 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^5 + 67*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^5
- 35*A*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 - 29*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^5 + 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c
)^5 - 26*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^5 + 10*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^5 + 5*B*a^2*b^5*tan(1/2*d*x + 1/
2*c)^5 - 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^5 + 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^5 - 2*A*b^7*tan(1/2*d*x + 1/2*c)^5
+ 3*B*b^7*tan(1/2*d*x + 1/2*c)^5 + 36*B*a^7*tan(1/2*d*x + 1/2*c)^3 - 18*A*a^6*b*tan(1/2*d*x + 1/2*c)^3 + 18*B*
a^6*b*tan(1/2*d*x + 1/2*c)^3 - 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 67*B*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 + 35*A
*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 - 29*B*a^4*b^3*tan(1/2*d*x + 1/2*c)^3 + 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 +
26*B*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 - 10*A*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^2*b^5*tan(1/2*d*x + 1/2*c)^3
 - 4*A*a*b^6*tan(1/2*d*x + 1/2*c)^3 - 4*B*a*b^6*tan(1/2*d*x + 1/2*c)^3 + 2*A*b^7*tan(1/2*d*x + 1/2*c)^3 + 3*B*
b^7*tan(1/2*d*x + 1/2*c)^3 - 12*B*a^7*tan(1/2*d*x + 1/2*c) + 6*A*a^6*b*tan(1/2*d*x + 1/2*c) - 18*B*a^6*b*tan(1
/2*d*x + 1/2*c) + 9*A*a^5*b^2*tan(1/2*d*x + 1/2*c) + 17*B*a^5*b^2*tan(1/2*d*x + 1/2*c) - 9*A*a^4*b^3*tan(1/2*d
*x + 1/2*c) + 33*B*a^4*b^3*tan(1/2*d*x + 1/2*c) - 16*A*a^3*b^4*tan(1/2*d*x + 1/2*c) + 2*B*a^3*b^4*tan(1/2*d*x
+ 1/2*c) - 2*A*a^2*b^5*tan(1/2*d*x + 1/2*c) - 13*B*a^2*b^5*tan(1/2*d*x + 1/2*c) + 4*A*a*b^6*tan(1/2*d*x + 1/2*
c) - 4*B*a*b^6*tan(1/2*d*x + 1/2*c) + 2*A*b^7*tan(1/2*d*x + 1/2*c) + B*b^7*tan(1/2*d*x + 1/2*c))/((a^4*b^4 - 2
*a^2*b^6 + b^8)*(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b)^2)
- (12*B*a^2 - 6*A*a*b + B*b^2)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^5 + (12*B*a^2 - 6*A*a*b + B*b^2)*log(abs(t
an(1/2*d*x + 1/2*c) - 1))/b^5)/d

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maple [B]  time = 0.88, size = 1599, normalized size = 3.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x)

[Out]

-3/d/b^4*ln(tan(1/2*d*x+1/2*c)+1)*A*a+6/d/b^5*ln(tan(1/2*d*x+1/2*c)+1)*a^2*B-10/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c
)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+1/d*a^4/b^2/(a*tan(1/2*d*x+1/2*
c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+4/d*a^5/b^3/(a*tan(1/2*d*x+1/2
*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-4/d*a^5/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan
(1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+8/d*a^3/b/(a*tan(1/2*d*x+1/2*c)^2-tan(
1/2*d*x+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+1/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan
(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A+6/d*a^6/b^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+
1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-1/d*a^5/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x
+1/2*c)^2*b-a-b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-6/d*a^6/b^4/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*
x+1/2*c)^2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-1/d*a^5/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^
2*b-a-b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B+10/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)
^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-8/d*a^3/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b-a-b)^2/(a+b)/(a
-b)^2*tan(1/2*d*x+1/2*c)*A-20/d*a^3/b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)
/((a-b)*(a+b))^(1/2))*B-15/d*a^4/b^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/
((a-b)*(a+b))^(1/2))*A+1/2/d*B/b^3/(tan(1/2*d*x+1/2*c)-1)^2-1/d/b^3/(tan(1/2*d*x+1/2*c)-1)*A+1/2/d/b^3/(tan(1/
2*d*x+1/2*c)-1)*B-1/2/d/b^3*ln(tan(1/2*d*x+1/2*c)-1)*B-1/2/d*B/b^3/(tan(1/2*d*x+1/2*c)+1)^2-1/d/b^3/(tan(1/2*d
*x+1/2*c)+1)*A+1/2/d/b^3/(tan(1/2*d*x+1/2*c)+1)*B+1/2/d/b^3*ln(tan(1/2*d*x+1/2*c)+1)*B-12/d*a^7/b^5/(a^4-2*a^2
*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+29/d*a^5/b^3/(a^4-2*a^2*
b^2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+6/d*a^6/b^4/(a^4-2*a^2*b^
2+b^4)/((a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+3/d/b^4*ln(tan(1/2*d*x+1/2*
c)-1)*A*a+3/d/b^4/(tan(1/2*d*x+1/2*c)-1)*a*B+3/d/b^4/(tan(1/2*d*x+1/2*c)+1)*a*B+12/d*a^2/(a^4-2*a^2*b^2+b^4)/(
(a-b)*(a+b))^(1/2)*arctanh(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-6/d/b^5*ln(tan(1/2*d*x+1/2*c)-1)*a^
2*B

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
 for more details)Is 4*a^2-4*b^2 positive or negative?

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mupad [B]  time = 14.23, size = 10533, normalized size = 25.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B/cos(c + d*x))/(cos(c + d*x)^5*(a + b/cos(c + d*x))^3),x)

[Out]

((tan(c/2 + (d*x)/2)^3*(2*A*b^7 + 36*B*a^7 + 3*B*b^7 - 10*A*a^2*b^5 + 16*A*a^3*b^4 + 35*A*a^4*b^3 - 9*A*a^5*b^
2 + 5*B*a^2*b^5 + 26*B*a^3*b^4 - 29*B*a^4*b^3 - 67*B*a^5*b^2 - 4*A*a*b^6 - 18*A*a^6*b - 4*B*a*b^6 + 18*B*a^6*b
))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) + (tan(c/2 + (d*x)/2)^5*(3*B*b^7 - 36*B*a^7 - 2*A*b^7 + 10*A*a^2*b^5
+ 16*A*a^3*b^4 - 35*A*a^4*b^3 - 9*A*a^5*b^2 + 5*B*a^2*b^5 - 26*B*a^3*b^4 - 29*B*a^4*b^3 + 67*B*a^5*b^2 - 4*A*a
*b^6 + 18*A*a^6*b + 4*B*a*b^6 + 18*B*a^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^7*(B*
b^6 - 12*B*a^6 - 2*A*b^6 + 4*A*a^2*b^4 - 12*A*a^3*b^3 - 3*A*a^4*b^2 - 8*B*a^2*b^4 - 10*B*a^3*b^3 + 23*B*a^4*b^
2 + 2*A*a*b^5 + 6*A*a^5*b + 5*B*a*b^5 + 6*B*a^5*b))/((a*b^4 - b^5)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(2*A*b^6 -
 12*B*a^6 + B*b^6 - 4*A*a^2*b^4 - 12*A*a^3*b^3 + 3*A*a^4*b^2 - 8*B*a^2*b^4 + 10*B*a^3*b^3 + 23*B*a^4*b^2 + 2*A
*a*b^5 + 6*A*a^5*b - 5*B*a*b^5 - 6*B*a^5*b))/((a + b)*(b^6 - 2*a*b^5 + a^2*b^4)))/(d*(2*a*b + tan(c/2 + (d*x)/
2)^4*(6*a^2 - 2*b^2) - tan(c/2 + (d*x)/2)^2*(4*a*b + 4*a^2) + tan(c/2 + (d*x)/2)^6*(4*a*b - 4*a^2) + tan(c/2 +
 (d*x)/2)^8*(a^2 - 2*a*b + b^2) + a^2 + b^2)) - (atan(((((8*tan(c/2 + (d*x)/2)*(288*B^2*a^14 + B^2*b^14 - 2*B^
2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^2*b^12 - 72*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^
6*b^8 - 432*A^2*a^7*b^7 + 441*A^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^2*a^11*b^3 + 72*A^2*a^12
*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b^11 + 74*B^2*a^4*b^10 - 108*B^2*a^5*b^9 + 18*B^2*a^6*b^8 + 872*B^2*a^7*b^
7 - 827*B^2*a^8*b^6 - 1538*B^2*a^9*b^5 + 1538*B^2*a^10*b^4 + 1104*B^2*a^11*b^3 - 1104*B^2*a^12*b^2 - 12*A*B*a*
b^13 - 288*A*B*a^13*b + 24*A*B*a^2*b^12 - 108*A*B*a^3*b^11 + 192*A*B*a^4*b^10 - 72*A*B*a^5*b^9 - 1008*A*B*a^6*
b^8 + 984*A*B*a^7*b^7 + 1632*A*B*a^8*b^6 - 1650*A*B*a^9*b^5 - 1128*A*B*a^10*b^4 + 1128*A*B*a^11*b^3 + 288*A*B*
a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) - (((4*(4*B
*b^21 + 48*A*a^2*b^19 + 72*A*a^3*b^18 - 156*A*a^4*b^17 - 84*A*a^5*b^16 + 192*A*a^6*b^15 + 48*A*a^7*b^14 - 108*
A*a^8*b^13 - 12*A*a^9*b^12 + 24*A*a^10*b^11 + 28*B*a^2*b^19 - 80*B*a^3*b^18 - 120*B*a^4*b^17 + 276*B*a^5*b^16
+ 164*B*a^6*b^15 - 360*B*a^7*b^14 - 100*B*a^8*b^13 + 212*B*a^9*b^12 + 24*B*a^10*b^11 - 48*B*a^11*b^10 - 24*A*a
*b^20))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) - (4*tan(c/2
 + (d*x)/2)*(12*B*a^2 + B*b^2 - 6*A*a*b)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48
*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*a^10*b^10))/(b^5*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^
12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*(12*B*a^2 + B*b^2 - 6*A*a*b))/(2*b^5))*(12*B*a^2 + B*b^2 -
 6*A*a*b)*1i)/(2*b^5) + (((8*tan(c/2 + (d*x)/2)*(288*B^2*a^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*
A^2*a^2*b^12 - 72*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^6*b^8 - 432*A^2*a^7*b^7 + 441*A
^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^2*a^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2
*a^3*b^11 + 74*B^2*a^4*b^10 - 108*B^2*a^5*b^9 + 18*B^2*a^6*b^8 + 872*B^2*a^7*b^7 - 827*B^2*a^8*b^6 - 1538*B^2*
a^9*b^5 + 1538*B^2*a^10*b^4 + 1104*B^2*a^11*b^3 - 1104*B^2*a^12*b^2 - 12*A*B*a*b^13 - 288*A*B*a^13*b + 24*A*B*
a^2*b^12 - 108*A*B*a^3*b^11 + 192*A*B*a^4*b^10 - 72*A*B*a^5*b^9 - 1008*A*B*a^6*b^8 + 984*A*B*a^7*b^7 + 1632*A*
B*a^8*b^6 - 1650*A*B*a^9*b^5 - 1128*A*B*a^10*b^4 + 1128*A*B*a^11*b^3 + 288*A*B*a^12*b^2))/(a*b^14 + b^15 - 3*a
^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) + (((4*(4*B*b^21 + 48*A*a^2*b^19 + 72*A*a^
3*b^18 - 156*A*a^4*b^17 - 84*A*a^5*b^16 + 192*A*a^6*b^15 + 48*A*a^7*b^14 - 108*A*a^8*b^13 - 12*A*a^9*b^12 + 24
*A*a^10*b^11 + 28*B*a^2*b^19 - 80*B*a^3*b^18 - 120*B*a^4*b^17 + 276*B*a^5*b^16 + 164*B*a^6*b^15 - 360*B*a^7*b^
14 - 100*B*a^8*b^13 + 212*B*a^9*b^12 + 24*B*a^10*b^11 - 48*B*a^11*b^10 - 24*A*a*b^20))/(a*b^18 + b^19 - 3*a^2*
b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) + (4*tan(c/2 + (d*x)/2)*(12*B*a^2 + B*b^2 -
 6*A*a*b)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^
8*b^12 + 8*a^9*b^11 - 8*a^10*b^10))/(b^5*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 -
a^6*b^9 - a^7*b^8)))*(12*B*a^2 + B*b^2 - 6*A*a*b))/(2*b^5))*(12*B*a^2 + B*b^2 - 6*A*a*b)*1i)/(2*b^5))/((8*(172
8*B^3*a^15 - 864*B^3*a^14*b - 432*A^3*a^4*b^11 - 432*A^3*a^5*b^10 + 1404*A^3*a^6*b^9 + 756*A^3*a^7*b^8 - 1728*
A^3*a^8*b^7 - 486*A^3*a^9*b^6 + 972*A^3*a^10*b^5 + 108*A^3*a^11*b^4 - 216*A^3*a^12*b^3 + 20*B^3*a^3*b^12 - 20*
B^3*a^4*b^11 + 411*B^3*a^5*b^10 - 11*B^3*a^6*b^9 + 1314*B^3*a^7*b^8 + 2326*B^3*a^8*b^7 - 7829*B^3*a^9*b^6 - 47
70*B^3*a^10*b^5 + 11700*B^3*a^11*b^4 + 3456*B^3*a^12*b^3 - 7344*B^3*a^13*b^2 - 2592*A*B^2*a^14*b - 12*A*B^2*a^
2*b^13 + 12*A*B^2*a^3*b^12 - 489*A*B^2*a^4*b^11 + 9*A*B^2*a^5*b^10 - 2892*A*B^2*a^6*b^9 - 3972*A*B^2*a^7*b^8 +
 13347*A*B^2*a^8*b^7 + 7767*A*B^2*a^9*b^6 - 18594*A*B^2*a^10*b^5 - 5400*A*B^2*a^11*b^4 + 11232*A*B^2*a^12*b^3
+ 1296*A*B^2*a^13*b^2 + 144*A^2*B*a^3*b^12 + 1980*A^2*B*a^5*b^10 + 2268*A^2*B*a^6*b^9 - 7524*A^2*B*a^7*b^8 - 4
203*A^2*B*a^8*b^7 + 9828*A^2*B*a^9*b^6 + 2808*A^2*B*a^10*b^5 - 5724*A^2*B*a^11*b^4 - 648*A^2*B*a^12*b^3 + 1296
*A^2*B*a^13*b^2))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) -
(((8*tan(c/2 + (d*x)/2)*(288*B^2*a^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^2*b^12 - 72*A^2*a^
3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^6*b^8 - 432*A^2*a^7*b^7 + 441*A^2*a^8*b^6 + 288*A^2*a^9
*b^5 - 288*A^2*a^10*b^4 - 72*A^2*a^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b^11 + 74*B^2*a^4*b
^10 - 108*B^2*a^5*b^9 + 18*B^2*a^6*b^8 + 872*B^2*a^7*b^7 - 827*B^2*a^8*b^6 - 1538*B^2*a^9*b^5 + 1538*B^2*a^10*
b^4 + 1104*B^2*a^11*b^3 - 1104*B^2*a^12*b^2 - 12*A*B*a*b^13 - 288*A*B*a^13*b + 24*A*B*a^2*b^12 - 108*A*B*a^3*b
^11 + 192*A*B*a^4*b^10 - 72*A*B*a^5*b^9 - 1008*A*B*a^6*b^8 + 984*A*B*a^7*b^7 + 1632*A*B*a^8*b^6 - 1650*A*B*a^9
*b^5 - 1128*A*B*a^10*b^4 + 1128*A*B*a^11*b^3 + 288*A*B*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3
*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) - (((4*(4*B*b^21 + 48*A*a^2*b^19 + 72*A*a^3*b^18 - 156*A*a^4*b^17
- 84*A*a^5*b^16 + 192*A*a^6*b^15 + 48*A*a^7*b^14 - 108*A*a^8*b^13 - 12*A*a^9*b^12 + 24*A*a^10*b^11 + 28*B*a^2*
b^19 - 80*B*a^3*b^18 - 120*B*a^4*b^17 + 276*B*a^5*b^16 + 164*B*a^6*b^15 - 360*B*a^7*b^14 - 100*B*a^8*b^13 + 21
2*B*a^9*b^12 + 24*B*a^10*b^11 - 48*B*a^11*b^10 - 24*A*a*b^20))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^
4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) - (4*tan(c/2 + (d*x)/2)*(12*B*a^2 + B*b^2 - 6*A*a*b)*(8*a*b^19 - 8*
a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*
a^10*b^10))/(b^5*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*(12
*B*a^2 + B*b^2 - 6*A*a*b))/(2*b^5))*(12*B*a^2 + B*b^2 - 6*A*a*b))/(2*b^5) + (((8*tan(c/2 + (d*x)/2)*(288*B^2*a
^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^2*b^12 - 72*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2
*a^5*b^9 - 288*A^2*a^6*b^8 - 432*A^2*a^7*b^7 + 441*A^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^2*a
^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b^11 + 74*B^2*a^4*b^10 - 108*B^2*a^5*b^9 + 18*B^2*a^6
*b^8 + 872*B^2*a^7*b^7 - 827*B^2*a^8*b^6 - 1538*B^2*a^9*b^5 + 1538*B^2*a^10*b^4 + 1104*B^2*a^11*b^3 - 1104*B^2
*a^12*b^2 - 12*A*B*a*b^13 - 288*A*B*a^13*b + 24*A*B*a^2*b^12 - 108*A*B*a^3*b^11 + 192*A*B*a^4*b^10 - 72*A*B*a^
5*b^9 - 1008*A*B*a^6*b^8 + 984*A*B*a^7*b^7 + 1632*A*B*a^8*b^6 - 1650*A*B*a^9*b^5 - 1128*A*B*a^10*b^4 + 1128*A*
B*a^11*b^3 + 288*A*B*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 -
 a^7*b^8) + (((4*(4*B*b^21 + 48*A*a^2*b^19 + 72*A*a^3*b^18 - 156*A*a^4*b^17 - 84*A*a^5*b^16 + 192*A*a^6*b^15 +
 48*A*a^7*b^14 - 108*A*a^8*b^13 - 12*A*a^9*b^12 + 24*A*a^10*b^11 + 28*B*a^2*b^19 - 80*B*a^3*b^18 - 120*B*a^4*b
^17 + 276*B*a^5*b^16 + 164*B*a^6*b^15 - 360*B*a^7*b^14 - 100*B*a^8*b^13 + 212*B*a^9*b^12 + 24*B*a^10*b^11 - 48
*B*a^11*b^10 - 24*A*a*b^20))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a
^7*b^12) + (4*tan(c/2 + (d*x)/2)*(12*B*a^2 + B*b^2 - 6*A*a*b)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^
16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*a^10*b^10))/(b^5*(a*b^14 + b^15 -
3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*(12*B*a^2 + B*b^2 - 6*A*a*b))/(2*b^5)
)*(12*B*a^2 + B*b^2 - 6*A*a*b))/(2*b^5)))*(12*B*a^2 + B*b^2 - 6*A*a*b)*1i)/(b^5*d) - (a^2*atan(((a^2*((a + b)^
5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(288*B^2*a^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^
2*b^12 - 72*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^6*b^8 - 432*A^2*a^7*b^7 + 441*A^2*a^8
*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^2*a^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b
^11 + 74*B^2*a^4*b^10 - 108*B^2*a^5*b^9 + 18*B^2*a^6*b^8 + 872*B^2*a^7*b^7 - 827*B^2*a^8*b^6 - 1538*B^2*a^9*b^
5 + 1538*B^2*a^10*b^4 + 1104*B^2*a^11*b^3 - 1104*B^2*a^12*b^2 - 12*A*B*a*b^13 - 288*A*B*a^13*b + 24*A*B*a^2*b^
12 - 108*A*B*a^3*b^11 + 192*A*B*a^4*b^10 - 72*A*B*a^5*b^9 - 1008*A*B*a^6*b^8 + 984*A*B*a^7*b^7 + 1632*A*B*a^8*
b^6 - 1650*A*B*a^9*b^5 - 1128*A*B*a^10*b^4 + 1128*A*B*a^11*b^3 + 288*A*B*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^1
3 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) - (a^2*((4*(4*B*b^21 + 48*A*a^2*b^19 + 72*A*a^3*
b^18 - 156*A*a^4*b^17 - 84*A*a^5*b^16 + 192*A*a^6*b^15 + 48*A*a^7*b^14 - 108*A*a^8*b^13 - 12*A*a^9*b^12 + 24*A
*a^10*b^11 + 28*B*a^2*b^19 - 80*B*a^3*b^18 - 120*B*a^4*b^17 + 276*B*a^5*b^16 + 164*B*a^6*b^15 - 360*B*a^7*b^14
 - 100*B*a^8*b^13 + 212*B*a^9*b^12 + 24*B*a^10*b^11 - 48*B*a^11*b^10 - 24*A*a*b^20))/(a*b^18 + b^19 - 3*a^2*b^
17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) - (4*a^2*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b
)^5)^(1/2)*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4)*(8*a*b^19 - 8*a^2*b^18
 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*a^10*b^1
0))/((b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3
*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*((a + b)^5*(a - b)^5)^(1/2)*(12*A*b^5 - 12*B*a^5 - 15*A
*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^
7 - a^10*b^5)))*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4)*1i)/(2*(b^15 - 5*
a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)) + (a^2*((a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 +
(d*x)/2)*(288*B^2*a^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^2*b^12 - 72*A^2*a^3*b^11 + 36*A^2
*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^6*b^8 - 432*A^2*a^7*b^7 + 441*A^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*
a^10*b^4 - 72*A^2*a^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b^11 + 74*B^2*a^4*b^10 - 108*B^2*a
^5*b^9 + 18*B^2*a^6*b^8 + 872*B^2*a^7*b^7 - 827*B^2*a^8*b^6 - 1538*B^2*a^9*b^5 + 1538*B^2*a^10*b^4 + 1104*B^2*
a^11*b^3 - 1104*B^2*a^12*b^2 - 12*A*B*a*b^13 - 288*A*B*a^13*b + 24*A*B*a^2*b^12 - 108*A*B*a^3*b^11 + 192*A*B*a
^4*b^10 - 72*A*B*a^5*b^9 - 1008*A*B*a^6*b^8 + 984*A*B*a^7*b^7 + 1632*A*B*a^8*b^6 - 1650*A*B*a^9*b^5 - 1128*A*B
*a^10*b^4 + 1128*A*B*a^11*b^3 + 288*A*B*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a
^5*b^10 - a^6*b^9 - a^7*b^8) + (a^2*((4*(4*B*b^21 + 48*A*a^2*b^19 + 72*A*a^3*b^18 - 156*A*a^4*b^17 - 84*A*a^5*
b^16 + 192*A*a^6*b^15 + 48*A*a^7*b^14 - 108*A*a^8*b^13 - 12*A*a^9*b^12 + 24*A*a^10*b^11 + 28*B*a^2*b^19 - 80*B
*a^3*b^18 - 120*B*a^4*b^17 + 276*B*a^5*b^16 + 164*B*a^6*b^15 - 360*B*a^7*b^14 - 100*B*a^8*b^13 + 212*B*a^9*b^1
2 + 24*B*a^10*b^11 - 48*B*a^11*b^10 - 24*A*a*b^20))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*
a^5*b^14 - a^6*b^13 - a^7*b^12) + (4*a^2*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(12*A*b^5 - 12*B*a^5 -
 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48
*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*a^10*b^10))/((b^15 - 5*a^2*b^13 + 10*a^4*
b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 -
 a^6*b^9 - a^7*b^8)))*((a + b)^5*(a - b)^5)^(1/2)*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4
*b - 20*B*a*b^4))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*A*b^5 - 12*B*
a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4)*1i)/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b
^9 + 5*a^8*b^7 - a^10*b^5)))/((8*(1728*B^3*a^15 - 864*B^3*a^14*b - 432*A^3*a^4*b^11 - 432*A^3*a^5*b^10 + 1404*
A^3*a^6*b^9 + 756*A^3*a^7*b^8 - 1728*A^3*a^8*b^7 - 486*A^3*a^9*b^6 + 972*A^3*a^10*b^5 + 108*A^3*a^11*b^4 - 216
*A^3*a^12*b^3 + 20*B^3*a^3*b^12 - 20*B^3*a^4*b^11 + 411*B^3*a^5*b^10 - 11*B^3*a^6*b^9 + 1314*B^3*a^7*b^8 + 232
6*B^3*a^8*b^7 - 7829*B^3*a^9*b^6 - 4770*B^3*a^10*b^5 + 11700*B^3*a^11*b^4 + 3456*B^3*a^12*b^3 - 7344*B^3*a^13*
b^2 - 2592*A*B^2*a^14*b - 12*A*B^2*a^2*b^13 + 12*A*B^2*a^3*b^12 - 489*A*B^2*a^4*b^11 + 9*A*B^2*a^5*b^10 - 2892
*A*B^2*a^6*b^9 - 3972*A*B^2*a^7*b^8 + 13347*A*B^2*a^8*b^7 + 7767*A*B^2*a^9*b^6 - 18594*A*B^2*a^10*b^5 - 5400*A
*B^2*a^11*b^4 + 11232*A*B^2*a^12*b^3 + 1296*A*B^2*a^13*b^2 + 144*A^2*B*a^3*b^12 + 1980*A^2*B*a^5*b^10 + 2268*A
^2*B*a^6*b^9 - 7524*A^2*B*a^7*b^8 - 4203*A^2*B*a^8*b^7 + 9828*A^2*B*a^9*b^6 + 2808*A^2*B*a^10*b^5 - 5724*A^2*B
*a^11*b^4 - 648*A^2*B*a^12*b^3 + 1296*A^2*B*a^13*b^2))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 +
 3*a^5*b^14 - a^6*b^13 - a^7*b^12) - (a^2*((a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(288*B^2*a^14 + B
^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^13*b + 36*A^2*a^2*b^12 - 72*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^
9 - 288*A^2*a^6*b^8 - 432*A^2*a^7*b^7 + 441*A^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^2*a^11*b^3
 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12 - 40*B^2*a^3*b^11 + 74*B^2*a^4*b^10 - 108*B^2*a^5*b^9 + 18*B^2*a^6*b^8 +
872*B^2*a^7*b^7 - 827*B^2*a^8*b^6 - 1538*B^2*a^9*b^5 + 1538*B^2*a^10*b^4 + 1104*B^2*a^11*b^3 - 1104*B^2*a^12*b
^2 - 12*A*B*a*b^13 - 288*A*B*a^13*b + 24*A*B*a^2*b^12 - 108*A*B*a^3*b^11 + 192*A*B*a^4*b^10 - 72*A*B*a^5*b^9 -
 1008*A*B*a^6*b^8 + 984*A*B*a^7*b^7 + 1632*A*B*a^8*b^6 - 1650*A*B*a^9*b^5 - 1128*A*B*a^10*b^4 + 1128*A*B*a^11*
b^3 + 288*A*B*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^
8) - (a^2*((4*(4*B*b^21 + 48*A*a^2*b^19 + 72*A*a^3*b^18 - 156*A*a^4*b^17 - 84*A*a^5*b^16 + 192*A*a^6*b^15 + 48
*A*a^7*b^14 - 108*A*a^8*b^13 - 12*A*a^9*b^12 + 24*A*a^10*b^11 + 28*B*a^2*b^19 - 80*B*a^3*b^18 - 120*B*a^4*b^17
 + 276*B*a^5*b^16 + 164*B*a^6*b^15 - 360*B*a^7*b^14 - 100*B*a^8*b^13 + 212*B*a^9*b^12 + 24*B*a^10*b^11 - 48*B*
a^11*b^10 - 24*A*a*b^20))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*
b^12) - (4*a^2*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b
^2 + 6*A*a^4*b - 20*B*a*b^4)*(8*a*b^19 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 -
32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^11 - 8*a^10*b^10))/((b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*
b^7 - a^10*b^5)*(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*((a
+ b)^5*(a - b)^5)^(1/2)*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4))/(2*(b^15
 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*
a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)) +
 (a^2*((a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(288*B^2*a^14 + B^2*b^14 - 2*B^2*a*b^13 - 288*B^2*a^1
3*b + 36*A^2*a^2*b^12 - 72*A^2*a^3*b^11 + 36*A^2*a^4*b^10 + 288*A^2*a^5*b^9 - 288*A^2*a^6*b^8 - 432*A^2*a^7*b^
7 + 441*A^2*a^8*b^6 + 288*A^2*a^9*b^5 - 288*A^2*a^10*b^4 - 72*A^2*a^11*b^3 + 72*A^2*a^12*b^2 + 21*B^2*a^2*b^12
 - 40*B^2*a^3*b^11 + 74*B^2*a^4*b^10 - 108*B^2*a^5*b^9 + 18*B^2*a^6*b^8 + 872*B^2*a^7*b^7 - 827*B^2*a^8*b^6 -
1538*B^2*a^9*b^5 + 1538*B^2*a^10*b^4 + 1104*B^2*a^11*b^3 - 1104*B^2*a^12*b^2 - 12*A*B*a*b^13 - 288*A*B*a^13*b
+ 24*A*B*a^2*b^12 - 108*A*B*a^3*b^11 + 192*A*B*a^4*b^10 - 72*A*B*a^5*b^9 - 1008*A*B*a^6*b^8 + 984*A*B*a^7*b^7
+ 1632*A*B*a^8*b^6 - 1650*A*B*a^9*b^5 - 1128*A*B*a^10*b^4 + 1128*A*B*a^11*b^3 + 288*A*B*a^12*b^2))/(a*b^14 + b
^15 - 3*a^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8) + (a^2*((4*(4*B*b^21 + 48*A*a^2*b
^19 + 72*A*a^3*b^18 - 156*A*a^4*b^17 - 84*A*a^5*b^16 + 192*A*a^6*b^15 + 48*A*a^7*b^14 - 108*A*a^8*b^13 - 12*A*
a^9*b^12 + 24*A*a^10*b^11 + 28*B*a^2*b^19 - 80*B*a^3*b^18 - 120*B*a^4*b^17 + 276*B*a^5*b^16 + 164*B*a^6*b^15 -
 360*B*a^7*b^14 - 100*B*a^8*b^13 + 212*B*a^9*b^12 + 24*B*a^10*b^11 - 48*B*a^11*b^10 - 24*A*a*b^20))/(a*b^18 +
b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^15 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) + (4*a^2*tan(c/2 + (d*x)/2)*((
a + b)^5*(a - b)^5)^(1/2)*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4)*(8*a*b^
19 - 8*a^2*b^18 - 32*a^3*b^17 + 32*a^4*b^16 + 48*a^5*b^15 - 48*a^6*b^14 - 32*a^7*b^13 + 32*a^8*b^12 + 8*a^9*b^
11 - 8*a^10*b^10))/((b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5)*(a*b^14 + b^15 - 3*a
^2*b^13 - 3*a^3*b^12 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8)))*((a + b)^5*(a - b)^5)^(1/2)*(12*A*b^5 -
12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4))/(2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6
*b^9 + 5*a^8*b^7 - a^10*b^5)))*(12*A*b^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4))/(
2*(b^15 - 5*a^2*b^13 + 10*a^4*b^11 - 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5))))*((a + b)^5*(a - b)^5)^(1/2)*(12*A*b
^5 - 12*B*a^5 - 15*A*a^2*b^3 + 29*B*a^3*b^2 + 6*A*a^4*b - 20*B*a*b^4)*1i)/(d*(b^15 - 5*a^2*b^13 + 10*a^4*b^11
- 10*a^6*b^9 + 5*a^8*b^7 - a^10*b^5))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**5*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**3,x)

[Out]

Integral((A + B*sec(c + d*x))*sec(c + d*x)**5/(a + b*sec(c + d*x))**3, x)

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