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3.916 \(\int \frac {\sec ^4(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=465 \[ -\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (a \left (-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\tan (c+d x) \sec (c+d x) \left (-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right )}{2 b^3 d \left (a^2-b^2\right )^2}+\frac {\tan (c+d x) \left (-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right )}{2 b^4 d \left (a^2-b^2\right )^2}-\frac {a \left (12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 4.75, antiderivative size = 465, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4098, 4092, 4082, 3998, 3770, 3831, 2659, 208} \[ \frac {\tan (c+d x) \left (-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+6 a^4 b B-12 a^5 C+a b^4 (5 A-6 C)+2 b^5 B\right )}{2 b^4 d \left (a^2-b^2\right )^2}+\frac {\left (12 a^2 C-6 a b B+2 A b^2+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+15 a^3 b^3 B-6 a^5 b B+12 a^6 C-12 a b^5 B+6 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}-\frac {\tan (c+d x) \sec ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}+\frac {\tan (c+d x) \sec ^2(c+d x) \left (a \left (2 a^2 b B-4 a^3 C+7 a b^2 C-5 b^3 B\right )+3 A b^4\right )}{2 b^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\tan (c+d x) \sec (c+d x) \left (-a^2 b^2 (A-10 C)+3 a^3 b B-6 a^4 C-6 a b^3 B+b^4 (4 A-C)\right )}{2 b^3 d \left (a^2-b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]

[Out]

((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) - (a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^
3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*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*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 2
1*C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - a^2*b^2
*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B -
 a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 + a*(2*a^2*b*B - 5*
b^3*B - 4*a^3*C + 7*a*b^2*C))*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 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 ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \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 \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)-2 \left (A b^2-a b B+2 a^2 C-b^2 C\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \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 {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\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 \left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\right )+b \left (a^2 b B+2 b^3 B+a^3 C-a b^2 (3 A+4 C)\right ) \sec (c+d x)-2 \left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\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 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \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 {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\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 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right )+2 b \left (a^3 b B-4 a b^3 B-2 a^4 C+b^4 (2 A+C)+a^2 b^2 (A+4 C)\right ) \sec (c+d x)+2 \left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\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 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \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 {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\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 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right )+2 \left (a^2-b^2\right )^2 \left (2 A b^2-6 a b B+12 a^2 C+b^2 C\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 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \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 {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\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 \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\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 (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \int \sec (c+d x) \, dx}{2 b^5}\\ &=\frac {\left (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \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 {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\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 \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\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 (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac {\left (6 a^4 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \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 {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\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 \left (6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 C\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 (2 A b^2-6 a b B+12 a^2 C+b^2 C\right ) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac {a \left (2 a^4 A b^2-5 a^2 A b^4+6 A b^6-6 a^5 b B+15 a^3 b^3 B-12 a b^5 B+12 a^6 C-29 a^4 b^2 C+20 a^2 b^4 C\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 b B-11 a^2 b^3 B+2 b^5 B-a^3 b^2 (2 A-21 C)+a b^4 (5 A-6 C)-12 a^5 C\right ) \tan (c+d x)}{2 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a^3 b B-6 a b^3 B-a^2 b^2 (A-10 C)+b^4 (4 A-C)-6 a^4 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \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 {\left (3 A b^4+a \left (2 a^2 b B-5 b^3 B-4 a^3 C+7 a b^2 C\right )\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 [B]  time = 6.43, size = 1124, normalized size = 2.42 \[ \frac {\left (-12 C a^2+6 b B a-2 A b^2-b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^3}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac {\left (12 C a^2-6 b B a+2 A b^2+b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^3}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac {2 a \left (12 C a^6-6 b B a^5+2 A b^2 a^4-29 b^2 C a^4+15 b^3 B a^3-5 A b^4 a^2+20 b^4 C a^2-12 b^5 B a+6 A b^6\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) \sec (c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) (b+a \cos (c+d x))^3}{b^5 \sqrt {a^2-b^2} \left (b^2-a^2\right )^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac {\sec ^3(c+d x) \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right ) \left (-24 C \sin (2 (c+d x)) a^7-12 C \sin (4 (c+d x)) a^7-36 b C \sin (c+d x) a^6+12 b B \sin (2 (c+d x)) a^6-36 b C \sin (3 (c+d x)) a^6+6 b B \sin (4 (c+d x)) a^6+18 b^2 B \sin (c+d x) a^5-4 A b^2 \sin (2 (c+d x)) a^5+26 b^2 C \sin (2 (c+d x)) a^5+18 b^2 B \sin (3 (c+d x)) a^5-2 A b^2 \sin (4 (c+d x)) a^5+21 b^2 C \sin (4 (c+d x)) a^5-6 A b^3 \sin (c+d x) a^4+72 b^3 C \sin (c+d x) a^4-14 b^3 B \sin (2 (c+d x)) a^4-6 A b^3 \sin (3 (c+d x)) a^4+64 b^3 C \sin (3 (c+d x)) a^4-11 b^3 B \sin (4 (c+d x)) a^4-32 b^4 B \sin (c+d x) a^3+10 A b^4 \sin (2 (c+d x)) a^3+20 b^4 C \sin (2 (c+d x)) a^3-32 b^4 B \sin (3 (c+d x)) a^3+5 A b^4 \sin (4 (c+d x)) a^3-6 b^4 C \sin (4 (c+d x)) a^3+12 A b^5 \sin (c+d x) a^2-38 b^5 C \sin (c+d x) a^2-12 b^5 B \sin (2 (c+d x)) a^2+12 A b^5 \sin (3 (c+d x)) a^2-22 b^5 C \sin (3 (c+d x)) a^2+2 b^5 B \sin (4 (c+d x)) a^2+8 b^6 B \sin (c+d x) a-16 b^6 C \sin (2 (c+d x)) a+8 b^6 B \sin (3 (c+d x)) a+8 b^7 C \sin (c+d x)+8 b^7 B \sin (2 (c+d x))\right ) (b+a \cos (c+d x))}{8 b^4 \left (b^2-a^2\right )^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]

[Out]

(2*a*(2*a^4*A*b^2 - 5*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + 12*a^6*C - 29*a^4*b^2*C +
20*a^2*b^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*
Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*
c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((-2*A*b^2 + 6*a*b*B - 12*a^2*C - b^2*C)*(b + a*Cos[c + d*x])^3*Log[Cos[
(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*C
os[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*(b + a*Cos
[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b
^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])*Sec[c +
 d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-6*a^4*A*b^3*Sin[c + d*x] + 12*a^2*A*b^5*Sin[c + d*x] + 18*a^
5*b^2*B*Sin[c + d*x] - 32*a^3*b^4*B*Sin[c + d*x] + 8*a*b^6*B*Sin[c + d*x] - 36*a^6*b*C*Sin[c + d*x] + 72*a^4*b
^3*C*Sin[c + d*x] - 38*a^2*b^5*C*Sin[c + d*x] + 8*b^7*C*Sin[c + d*x] - 4*a^5*A*b^2*Sin[2*(c + d*x)] + 10*a^3*A
*b^4*Sin[2*(c + d*x)] + 12*a^6*b*B*Sin[2*(c + d*x)] - 14*a^4*b^3*B*Sin[2*(c + d*x)] - 12*a^2*b^5*B*Sin[2*(c +
d*x)] + 8*b^7*B*Sin[2*(c + d*x)] - 24*a^7*C*Sin[2*(c + d*x)] + 26*a^5*b^2*C*Sin[2*(c + d*x)] + 20*a^3*b^4*C*Si
n[2*(c + d*x)] - 16*a*b^6*C*Sin[2*(c + d*x)] - 6*a^4*A*b^3*Sin[3*(c + d*x)] + 12*a^2*A*b^5*Sin[3*(c + d*x)] +
18*a^5*b^2*B*Sin[3*(c + d*x)] - 32*a^3*b^4*B*Sin[3*(c + d*x)] + 8*a*b^6*B*Sin[3*(c + d*x)] - 36*a^6*b*C*Sin[3*
(c + d*x)] + 64*a^4*b^3*C*Sin[3*(c + d*x)] - 22*a^2*b^5*C*Sin[3*(c + d*x)] - 2*a^5*A*b^2*Sin[4*(c + d*x)] + 5*
a^3*A*b^4*Sin[4*(c + d*x)] + 6*a^6*b*B*Sin[4*(c + d*x)] - 11*a^4*b^3*B*Sin[4*(c + d*x)] + 2*a^2*b^5*B*Sin[4*(c
 + d*x)] - 12*a^7*C*Sin[4*(c + d*x)] + 21*a^5*b^2*C*Sin[4*(c + d*x)] - 6*a^3*b^4*C*Sin[4*(c + d*x)]))/(8*b^4*(
-a^2 + b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.50, size = 1740, normalized size = 3.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.62, size = 2275, normalized size = 4.89 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^3,x)

[Out]

-1/d/b^3/(tan(1/2*d*x+1/2*c)-1)*B+1/2/d/b^3/(tan(1/2*d*x+1/2*c)-1)*C-1/d/b^3*ln(tan(1/2*d*x+1/2*c)-1)*A+2/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-1/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-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)*C-1/d*a^3/b/(a*ta
n(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-2/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-8/d*a^3/b/(a*tan(1/2*d*x+1/2*c)^2-t
an(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^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-b)^2*tan(1/2*d*x+1/2*c)*C-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*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*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^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+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)*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*C-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*C+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)*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)*C-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))*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))*B+1/2/d/b^3*ln(tan(1/2*d*x+1/2*c)+1)*C-2/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))*A-1/2/d/b^3*ln(tan(1/2*d*x+1/2*c)-1)*C+1/2/d*C
/b^3/(tan(1/2*d*x+1/2*c)-1)^2+5/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))*A-6/d*a*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))*A-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))*C-6/d*a^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+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))*C-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))*C+6/d*a^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-1
/d/b^3/(tan(1/2*d*x+1/2*c)+1)*B+1/2/d/b^3/(tan(1/2*d*x+1/2*c)+1)*C+1/d/b^3*ln(tan(1/2*d*x+1/2*c)+1)*A-1/2/d*C/
b^3/(tan(1/2*d*x+1/2*c)+1)^2+3/d/b^4*ln(tan(1/2*d*x+1/2*c)-1)*B*a+3/d/b^4/(tan(1/2*d*x+1/2*c)+1)*a*C-6/d/b^5*l
n(tan(1/2*d*x+1/2*c)-1)*a^2*C-3/d/b^4*ln(tan(1/2*d*x+1/2*c)+1)*B*a+6/d/b^5*ln(tan(1/2*d*x+1/2*c)+1)*a^2*C+3/d/
b^4/(tan(1/2*d*x+1/2*c)-1)*a*C+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))*B

________________________________________________________________________________________

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)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(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?

________________________________________________________________________________________

mupad [B]  time = 21.04, size = 15937, normalized size = 34.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B/cos(c + d*x) + C/cos(c + d*x)^2)/(cos(c + d*x)^4*(a + b/cos(c + d*x))^3),x)

[Out]

((tan(c/2 + (d*x)/2)^3*(2*B*b^7 + 36*C*a^7 + 3*C*b^7 - 6*A*a^2*b^5 - 15*A*a^3*b^4 + 3*A*a^4*b^3 + 6*A*a^5*b^2
- 10*B*a^2*b^5 + 16*B*a^3*b^4 + 35*B*a^4*b^3 - 9*B*a^5*b^2 + 5*C*a^2*b^5 + 26*C*a^3*b^4 - 29*C*a^4*b^3 - 67*C*
a^5*b^2 - 4*B*a*b^6 - 18*B*a^6*b - 4*C*a*b^6 + 18*C*a^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) + (tan(c/2 +
 (d*x)/2)^5*(3*C*b^7 - 36*C*a^7 - 2*B*b^7 - 6*A*a^2*b^5 + 15*A*a^3*b^4 + 3*A*a^4*b^3 - 6*A*a^5*b^2 + 10*B*a^2*
b^5 + 16*B*a^3*b^4 - 35*B*a^4*b^3 - 9*B*a^5*b^2 + 5*C*a^2*b^5 - 26*C*a^3*b^4 - 29*C*a^4*b^3 + 67*C*a^5*b^2 - 4
*B*a*b^6 + 18*B*a^6*b + 4*C*a*b^6 + 18*C*a^6*b))/((a + b)^2*(b^6 - 2*a*b^5 + a^2*b^4)) - (tan(c/2 + (d*x)/2)^7
*(C*b^6 - 12*C*a^6 - 2*B*b^6 + 6*A*a^2*b^4 + A*a^3*b^3 - 2*A*a^4*b^2 + 4*B*a^2*b^4 - 12*B*a^3*b^3 - 3*B*a^4*b^
2 - 8*C*a^2*b^4 - 10*C*a^3*b^3 + 23*C*a^4*b^2 + 2*B*a*b^5 + 6*B*a^5*b + 5*C*a*b^5 + 6*C*a^5*b))/((a*b^4 - b^5)
*(a + b)^2) + (tan(c/2 + (d*x)/2)*(2*B*b^6 - 12*C*a^6 + C*b^6 + 6*A*a^2*b^4 - A*a^3*b^3 - 2*A*a^4*b^2 - 4*B*a^
2*b^4 - 12*B*a^3*b^3 + 3*B*a^4*b^2 - 8*C*a^2*b^4 + 10*C*a^3*b^3 + 23*C*a^4*b^2 + 2*B*a*b^5 + 6*B*a^5*b - 5*C*a
*b^5 - 6*C*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(((((((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*
A*a^5*b^16 - 8*A*a^6*b^15 + 36*A*a^7*b^14 + 4*A*a^8*b^13 - 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 - 156*
B*a^4*b^17 - 84*B*a^5*b^16 + 192*B*a^6*b^15 + 48*B*a^7*b^14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b^11
+ 28*C*a^2*b^19 - 80*C*a^3*b^18 - 120*C*a^4*b^17 + 276*C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^
8*b^13 + 212*C*a^9*b^12 + 24*C*a^10*b^11 - 48*C*a^11*b^10 - 24*A*a*b^20 - 24*B*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) - (8*tan(c/2 + (d*x)/2)*(6*C*a^2 + b^2*(A
+ C/2) - 3*B*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)))*(6*C*a^2 + b^2*(A + C/2) - 3*B*a*b))/b^5 - (8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 28
8*C^2*a^14 + C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a*b^13 - 288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*
A^2*a^4*b^10 - 48*A^2*a^5*b^9 + 57*A^2*a^6*b^8 + 32*A^2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*
b^4 + 36*B^2*a^2*b^12 - 72*B^2*a^3*b^11 + 36*B^2*a^4*b^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^8 - 432*B^2*a^7*b^
7 + 441*B^2*a^8*b^6 + 288*B^2*a^9*b^5 - 288*B^2*a^10*b^4 - 72*B^2*a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12
 - 40*C^2*a^3*b^11 + 74*C^2*a^4*b^10 - 108*C^2*a^5*b^9 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 -
1538*C^2*a^9*b^5 + 1538*C^2*a^10*b^4 + 1104*C^2*a^11*b^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 - 24*A*B*a*b^13 - 8*
A*C*a*b^13 - 12*B*C*a*b^13 - 288*B*C*a^13*b + 48*A*B*a^2*b^12 - 72*A*B*a^3*b^11 - 192*A*B*a^4*b^10 + 252*A*B*a
^5*b^9 + 288*A*B*a^6*b^8 - 318*A*B*a^7*b^7 - 192*A*B*a^8*b^6 + 192*A*B*a^9*b^5 + 48*A*B*a^10*b^4 - 48*A*B*a^11
*b^3 + 36*A*C*a^2*b^12 - 64*A*C*a^3*b^11 + 104*A*C*a^4*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^6*b^8 - 544*A*C*a^7*
b^7 + 598*A*C*a^8*b^6 + 376*A*C*a^9*b^5 - 376*A*C*a^10*b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b^2 + 24*B*C*a^2*b^
12 - 108*B*C*a^3*b^11 + 192*B*C*a^4*b^10 - 72*B*C*a^5*b^9 - 1008*B*C*a^6*b^8 + 984*B*C*a^7*b^7 + 1632*B*C*a^8*
b^6 - 1650*B*C*a^9*b^5 - 1128*B*C*a^10*b^4 + 1128*B*C*a^11*b^3 + 288*B*C*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))*(6*C*a^2 + b^2*(A + C/2) - 3*B*a*b)*1i)/b^5 - (
((((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 - 8*A*a^6*b^15 + 36
*A*a^7*b^14 + 4*A*a^8*b^13 - 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 - 156*B*a^4*b^17 - 84*B*a^5*b^16 + 1
92*B*a^6*b^15 + 48*B*a^7*b^14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b^11 + 28*C*a^2*b^19 - 80*C*a^3*b^1
8 - 120*C*a^4*b^17 + 276*C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^8*b^13 + 212*C*a^9*b^12 + 24*C
*a^10*b^11 - 48*C*a^11*b^10 - 24*A*a*b^20 - 24*B*a*b^20))/(a*b^18 + b^19 - 3*a^2*b^17 - 3*a^3*b^16 + 3*a^4*b^1
5 + 3*a^5*b^14 - a^6*b^13 - a^7*b^12) + (8*tan(c/2 + (d*x)/2)*(6*C*a^2 + b^2*(A + C/2) - 3*B*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)))*(
6*C*a^2 + b^2*(A + C/2) - 3*B*a*b))/b^5 + (8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*
a*b^13 - 2*C^2*a*b^13 - 288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5*b^9
+ 57*A^2*a^6*b^8 + 32*A^2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*b^4 + 36*B^2*a^2*b^12 - 72*B^2
*a^3*b^11 + 36*B^2*a^4*b^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^8 - 432*B^2*a^7*b^7 + 441*B^2*a^8*b^6 + 288*B^2*
a^9*b^5 - 288*B^2*a^10*b^4 - 72*B^2*a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C^2*a^
4*b^10 - 108*C^2*a^5*b^9 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C^2*a^9*b^5 + 1538*C^2*a^
10*b^4 + 1104*C^2*a^11*b^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 - 24*A*B*a*b^13 - 8*A*C*a*b^13 - 12*B*C*a*b^13 - 2
88*B*C*a^13*b + 48*A*B*a^2*b^12 - 72*A*B*a^3*b^11 - 192*A*B*a^4*b^10 + 252*A*B*a^5*b^9 + 288*A*B*a^6*b^8 - 318
*A*B*a^7*b^7 - 192*A*B*a^8*b^6 + 192*A*B*a^9*b^5 + 48*A*B*a^10*b^4 - 48*A*B*a^11*b^3 + 36*A*C*a^2*b^12 - 64*A*
C*a^3*b^11 + 104*A*C*a^4*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^6*b^8 - 544*A*C*a^7*b^7 + 598*A*C*a^8*b^6 + 376*A*
C*a^9*b^5 - 376*A*C*a^10*b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b^2 + 24*B*C*a^2*b^12 - 108*B*C*a^3*b^11 + 192*B*
C*a^4*b^10 - 72*B*C*a^5*b^9 - 1008*B*C*a^6*b^8 + 984*B*C*a^7*b^7 + 1632*B*C*a^8*b^6 - 1650*B*C*a^9*b^5 - 1128*
B*C*a^10*b^4 + 1128*B*C*a^11*b^3 + 288*B*C*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))*(6*C*a^2 + b^2*(A + C/2) - 3*B*a*b)*1i)/b^5)/((8*(1728*C^3*a^15 + 24*A^3*a*b^
14 - 864*C^3*a^14*b + 48*A^3*a^2*b^13 - 68*A^3*a^3*b^12 - 52*A^3*a^4*b^11 + 72*A^3*a^5*b^10 + 26*A^3*a^6*b^9 -
 36*A^3*a^7*b^8 - 4*A^3*a^8*b^7 + 8*A^3*a^9*b^6 - 432*B^3*a^4*b^11 - 432*B^3*a^5*b^10 + 1404*B^3*a^6*b^9 + 756
*B^3*a^7*b^8 - 1728*B^3*a^8*b^7 - 486*B^3*a^9*b^6 + 972*B^3*a^10*b^5 + 108*B^3*a^11*b^4 - 216*B^3*a^12*b^3 + 2
0*C^3*a^3*b^12 - 20*C^3*a^4*b^11 + 411*C^3*a^5*b^10 - 11*C^3*a^6*b^9 + 1314*C^3*a^7*b^8 + 2326*C^3*a^8*b^7 - 7
829*C^3*a^9*b^6 - 4770*C^3*a^10*b^5 + 11700*C^3*a^11*b^4 + 3456*C^3*a^12*b^3 - 7344*C^3*a^13*b^2 + 6*A*C^2*a*b
^14 + 24*A^2*C*a*b^14 - 2592*B*C^2*a^14*b + 504*A*B^2*a^3*b^12 + 648*A*B^2*a^4*b^11 - 1548*A*B^2*a^5*b^10 - 97
2*A*B^2*a^6*b^9 + 1800*A*B^2*a^7*b^8 + 558*A*B^2*a^8*b^7 - 972*A*B^2*a^9*b^6 - 108*A*B^2*a^10*b^5 + 216*A*B^2*
a^11*b^4 - 192*A^2*B*a^2*b^13 - 312*A^2*B*a^3*b^12 + 564*A^2*B*a^4*b^11 + 396*A^2*B*a^5*b^10 - 624*A^2*B*a^6*b
^9 - 210*A^2*B*a^7*b^8 + 324*A^2*B*a^8*b^7 + 36*A^2*B*a^9*b^6 - 72*A^2*B*a^10*b^5 - 6*A*C^2*a^2*b^13 + 207*A*C
^2*a^3*b^12 + 33*A*C^2*a^4*b^11 + 1158*A*C^2*a^5*b^10 + 1974*A*C^2*a^6*b^9 - 4977*A*C^2*a^7*b^8 - 3405*A*C^2*a
^8*b^7 + 6486*A*C^2*a^9*b^6 + 2088*A*C^2*a^10*b^5 - 3744*A*C^2*a^11*b^4 - 432*A*C^2*a^12*b^3 + 864*A*C^2*a^13*
b^2 + 12*A^2*C*a^2*b^13 + 300*A^2*C*a^3*b^12 + 552*A^2*C*a^4*b^11 - 1020*A^2*C*a^5*b^10 - 747*A^2*C*a^6*b^9 +
1188*A^2*C*a^7*b^8 + 408*A^2*C*a^8*b^7 - 636*A^2*C*a^9*b^6 - 72*A^2*C*a^10*b^5 + 144*A^2*C*a^11*b^4 - 12*B*C^2
*a^2*b^13 + 12*B*C^2*a^3*b^12 - 489*B*C^2*a^4*b^11 + 9*B*C^2*a^5*b^10 - 2892*B*C^2*a^6*b^9 - 3972*B*C^2*a^7*b^
8 + 13347*B*C^2*a^8*b^7 + 7767*B*C^2*a^9*b^6 - 18594*B*C^2*a^10*b^5 - 5400*B*C^2*a^11*b^4 + 11232*B*C^2*a^12*b
^3 + 1296*B*C^2*a^13*b^2 + 144*B^2*C*a^3*b^12 + 1980*B^2*C*a^5*b^10 + 2268*B^2*C*a^6*b^9 - 7524*B^2*C*a^7*b^8
- 4203*B^2*C*a^8*b^7 + 9828*B^2*C*a^9*b^6 + 2808*B^2*C*a^10*b^5 - 5724*B^2*C*a^11*b^4 - 648*B^2*C*a^12*b^3 + 1
296*B^2*C*a^13*b^2 - 120*A*B*C*a^2*b^13 - 24*A*B*C*a^3*b^12 - 1560*A*B*C*a^4*b^11 - 2268*A*B*C*a^5*b^10 + 5568
*A*B*C*a^6*b^9 + 3642*A*B*C*a^7*b^8 - 6840*A*B*C*a^8*b^7 - 2160*A*B*C*a^9*b^6 + 3816*A*B*C*a^10*b^5 + 432*A*B*
C*a^11*b^4 - 864*A*B*C*a^12*b^3))/(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^1
3 - a^7*b^12) + (((((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 -
8*A*a^6*b^15 + 36*A*a^7*b^14 + 4*A*a^8*b^13 - 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 - 156*B*a^4*b^17 -
84*B*a^5*b^16 + 192*B*a^6*b^15 + 48*B*a^7*b^14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b^11 + 28*C*a^2*b^
19 - 80*C*a^3*b^18 - 120*C*a^4*b^17 + 276*C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^8*b^13 + 212*
C*a^9*b^12 + 24*C*a^10*b^11 - 48*C*a^11*b^10 - 24*A*a*b^20 - 24*B*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) - (8*tan(c/2 + (d*x)/2)*(6*C*a^2 + b^2*(A + C/2) - 3*B*
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^1
2 + 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)))*(6*C*a^2 + b^2*(A + C/2) - 3*B*a*b))/b^5 - (8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 288*C^2*a^14 +
C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a*b^13 - 288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*A^2*a^4*b^10
- 48*A^2*a^5*b^9 + 57*A^2*a^6*b^8 + 32*A^2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*b^4 + 36*B^2*
a^2*b^12 - 72*B^2*a^3*b^11 + 36*B^2*a^4*b^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^8 - 432*B^2*a^7*b^7 + 441*B^2*a
^8*b^6 + 288*B^2*a^9*b^5 - 288*B^2*a^10*b^4 - 72*B^2*a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12 - 40*C^2*a^3
*b^11 + 74*C^2*a^4*b^10 - 108*C^2*a^5*b^9 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C^2*a^9*
b^5 + 1538*C^2*a^10*b^4 + 1104*C^2*a^11*b^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 - 24*A*B*a*b^13 - 8*A*C*a*b^13 -
12*B*C*a*b^13 - 288*B*C*a^13*b + 48*A*B*a^2*b^12 - 72*A*B*a^3*b^11 - 192*A*B*a^4*b^10 + 252*A*B*a^5*b^9 + 288*
A*B*a^6*b^8 - 318*A*B*a^7*b^7 - 192*A*B*a^8*b^6 + 192*A*B*a^9*b^5 + 48*A*B*a^10*b^4 - 48*A*B*a^11*b^3 + 36*A*C
*a^2*b^12 - 64*A*C*a^3*b^11 + 104*A*C*a^4*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^6*b^8 - 544*A*C*a^7*b^7 + 598*A*C
*a^8*b^6 + 376*A*C*a^9*b^5 - 376*A*C*a^10*b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b^2 + 24*B*C*a^2*b^12 - 108*B*C*
a^3*b^11 + 192*B*C*a^4*b^10 - 72*B*C*a^5*b^9 - 1008*B*C*a^6*b^8 + 984*B*C*a^7*b^7 + 1632*B*C*a^8*b^6 - 1650*B*
C*a^9*b^5 - 1128*B*C*a^10*b^4 + 1128*B*C*a^11*b^3 + 288*B*C*a^12*b^2))/(a*b^14 + b^15 - 3*a^2*b^13 - 3*a^3*b^1
2 + 3*a^4*b^11 + 3*a^5*b^10 - a^6*b^9 - a^7*b^8))*(6*C*a^2 + b^2*(A + C/2) - 3*B*a*b))/b^5 + (((((4*(8*A*b^21
+ 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 - 8*A*a^6*b^15 + 36*A*a^7*b^14 + 4*
A*a^8*b^13 - 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 - 156*B*a^4*b^17 - 84*B*a^5*b^16 + 192*B*a^6*b^15 +
48*B*a^7*b^14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b^11 + 28*C*a^2*b^19 - 80*C*a^3*b^18 - 120*C*a^4*b^
17 + 276*C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^8*b^13 + 212*C*a^9*b^12 + 24*C*a^10*b^11 - 48*
C*a^11*b^10 - 24*A*a*b^20 - 24*B*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) + (8*tan(c/2 + (d*x)/2)*(6*C*a^2 + b^2*(A + C/2) - 3*B*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)))*(6*C*a^2 + b^2*(A
 + C/2) - 3*B*a*b))/b^5 + (8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a
*b^13 - 288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5*b^9 + 57*A^2*a^6*b^8
 + 32*A^2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*b^4 + 36*B^2*a^2*b^12 - 72*B^2*a^3*b^11 + 36*B
^2*a^4*b^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^8 - 432*B^2*a^7*b^7 + 441*B^2*a^8*b^6 + 288*B^2*a^9*b^5 - 288*B^
2*a^10*b^4 - 72*B^2*a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C^2*a^4*b^10 - 108*C^2
*a^5*b^9 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C^2*a^9*b^5 + 1538*C^2*a^10*b^4 + 1104*C^
2*a^11*b^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 - 24*A*B*a*b^13 - 8*A*C*a*b^13 - 12*B*C*a*b^13 - 288*B*C*a^13*b +
48*A*B*a^2*b^12 - 72*A*B*a^3*b^11 - 192*A*B*a^4*b^10 + 252*A*B*a^5*b^9 + 288*A*B*a^6*b^8 - 318*A*B*a^7*b^7 - 1
92*A*B*a^8*b^6 + 192*A*B*a^9*b^5 + 48*A*B*a^10*b^4 - 48*A*B*a^11*b^3 + 36*A*C*a^2*b^12 - 64*A*C*a^3*b^11 + 104
*A*C*a^4*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^6*b^8 - 544*A*C*a^7*b^7 + 598*A*C*a^8*b^6 + 376*A*C*a^9*b^5 - 376*
A*C*a^10*b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b^2 + 24*B*C*a^2*b^12 - 108*B*C*a^3*b^11 + 192*B*C*a^4*b^10 - 72*
B*C*a^5*b^9 - 1008*B*C*a^6*b^8 + 984*B*C*a^7*b^7 + 1632*B*C*a^8*b^6 - 1650*B*C*a^9*b^5 - 1128*B*C*a^10*b^4 + 1
128*B*C*a^11*b^3 + 288*B*C*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))*(6*C*a^2 + b^2*(A + C/2) - 3*B*a*b))/b^5))*(6*C*a^2 + b^2*(A + C/2) - 3*B*a*b)*2i)/(b^5*d) -
(a*atan(((a*((a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*a
*b^13 - 2*C^2*a*b^13 - 288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5*b^9 +
 57*A^2*a^6*b^8 + 32*A^2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*b^4 + 36*B^2*a^2*b^12 - 72*B^2*
a^3*b^11 + 36*B^2*a^4*b^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^8 - 432*B^2*a^7*b^7 + 441*B^2*a^8*b^6 + 288*B^2*a
^9*b^5 - 288*B^2*a^10*b^4 - 72*B^2*a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C^2*a^4
*b^10 - 108*C^2*a^5*b^9 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C^2*a^9*b^5 + 1538*C^2*a^1
0*b^4 + 1104*C^2*a^11*b^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 - 24*A*B*a*b^13 - 8*A*C*a*b^13 - 12*B*C*a*b^13 - 28
8*B*C*a^13*b + 48*A*B*a^2*b^12 - 72*A*B*a^3*b^11 - 192*A*B*a^4*b^10 + 252*A*B*a^5*b^9 + 288*A*B*a^6*b^8 - 318*
A*B*a^7*b^7 - 192*A*B*a^8*b^6 + 192*A*B*a^9*b^5 + 48*A*B*a^10*b^4 - 48*A*B*a^11*b^3 + 36*A*C*a^2*b^12 - 64*A*C
*a^3*b^11 + 104*A*C*a^4*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^6*b^8 - 544*A*C*a^7*b^7 + 598*A*C*a^8*b^6 + 376*A*C
*a^9*b^5 - 376*A*C*a^10*b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b^2 + 24*B*C*a^2*b^12 - 108*B*C*a^3*b^11 + 192*B*C
*a^4*b^10 - 72*B*C*a^5*b^9 - 1008*B*C*a^6*b^8 + 984*B*C*a^7*b^7 + 1632*B*C*a^8*b^6 - 1650*B*C*a^9*b^5 - 1128*B
*C*a^10*b^4 + 1128*B*C*a^11*b^3 + 288*B*C*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*((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 -
 72*A*a^5*b^16 - 8*A*a^6*b^15 + 36*A*a^7*b^14 + 4*A*a^8*b^13 - 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 -
156*B*a^4*b^17 - 84*B*a^5*b^16 + 192*B*a^6*b^15 + 48*B*a^7*b^14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b
^11 + 28*C*a^2*b^19 - 80*C*a^3*b^18 - 120*C*a^4*b^17 + 276*C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*
C*a^8*b^13 + 212*C*a^9*b^12 + 24*C*a^10*b^11 - 48*C*a^11*b^10 - 24*A*a*b^20 - 24*B*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*tan(c/2 + (d*x)/2)*((a + b)^5*(
a - b)^5)^(1/2)*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 -
 12*B*a*b^5 - 6*B*a^5*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^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)*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b
^2 - 12*B*a*b^5 - 6*B*a^5*b))/(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)))*(6*A*
b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5
*b)*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*((a + b)^5*(a - b)^5)^(
1/2)*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a*b^13 - 288*C^2*a^13
*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5*b^9 + 57*A^2*a^6*b^8 + 32*A^2*a^7*b^7 -
32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*b^4 + 36*B^2*a^2*b^12 - 72*B^2*a^3*b^11 + 36*B^2*a^4*b^10 + 288*B^
2*a^5*b^9 - 288*B^2*a^6*b^8 - 432*B^2*a^7*b^7 + 441*B^2*a^8*b^6 + 288*B^2*a^9*b^5 - 288*B^2*a^10*b^4 - 72*B^2*
a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C^2*a^4*b^10 - 108*C^2*a^5*b^9 + 18*C^2*a^
6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C^2*a^9*b^5 + 1538*C^2*a^10*b^4 + 1104*C^2*a^11*b^3 - 1104*C^
2*a^12*b^2 + 4*A*C*b^14 - 24*A*B*a*b^13 - 8*A*C*a*b^13 - 12*B*C*a*b^13 - 288*B*C*a^13*b + 48*A*B*a^2*b^12 - 72
*A*B*a^3*b^11 - 192*A*B*a^4*b^10 + 252*A*B*a^5*b^9 + 288*A*B*a^6*b^8 - 318*A*B*a^7*b^7 - 192*A*B*a^8*b^6 + 192
*A*B*a^9*b^5 + 48*A*B*a^10*b^4 - 48*A*B*a^11*b^3 + 36*A*C*a^2*b^12 - 64*A*C*a^3*b^11 + 104*A*C*a^4*b^10 + 336*
A*C*a^5*b^9 - 444*A*C*a^6*b^8 - 544*A*C*a^7*b^7 + 598*A*C*a^8*b^6 + 376*A*C*a^9*b^5 - 376*A*C*a^10*b^4 - 96*A*
C*a^11*b^3 + 96*A*C*a^12*b^2 + 24*B*C*a^2*b^12 - 108*B*C*a^3*b^11 + 192*B*C*a^4*b^10 - 72*B*C*a^5*b^9 - 1008*B
*C*a^6*b^8 + 984*B*C*a^7*b^7 + 1632*B*C*a^8*b^6 - 1650*B*C*a^9*b^5 - 1128*B*C*a^10*b^4 + 1128*B*C*a^11*b^3 + 2
88*B*C*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
*((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 - 8*A*a^6*b^15 + 36*
A*a^7*b^14 + 4*A*a^8*b^13 - 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 - 156*B*a^4*b^17 - 84*B*a^5*b^16 + 19
2*B*a^6*b^15 + 48*B*a^7*b^14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b^11 + 28*C*a^2*b^19 - 80*C*a^3*b^18
 - 120*C*a^4*b^17 + 276*C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^8*b^13 + 212*C*a^9*b^12 + 24*C*
a^10*b^11 - 48*C*a^11*b^10 - 24*A*a*b^20 - 24*B*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*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^6 + 12*C*a^6
 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*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^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)*(6*A*b^6 + 12*C
*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*b))/(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)))*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*
a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*b)*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*C^3*a^15 + 24*A^3*a*b^14 - 864*C^3*a^14*b + 48*A^3*a
^2*b^13 - 68*A^3*a^3*b^12 - 52*A^3*a^4*b^11 + 72*A^3*a^5*b^10 + 26*A^3*a^6*b^9 - 36*A^3*a^7*b^8 - 4*A^3*a^8*b^
7 + 8*A^3*a^9*b^6 - 432*B^3*a^4*b^11 - 432*B^3*a^5*b^10 + 1404*B^3*a^6*b^9 + 756*B^3*a^7*b^8 - 1728*B^3*a^8*b^
7 - 486*B^3*a^9*b^6 + 972*B^3*a^10*b^5 + 108*B^3*a^11*b^4 - 216*B^3*a^12*b^3 + 20*C^3*a^3*b^12 - 20*C^3*a^4*b^
11 + 411*C^3*a^5*b^10 - 11*C^3*a^6*b^9 + 1314*C^3*a^7*b^8 + 2326*C^3*a^8*b^7 - 7829*C^3*a^9*b^6 - 4770*C^3*a^1
0*b^5 + 11700*C^3*a^11*b^4 + 3456*C^3*a^12*b^3 - 7344*C^3*a^13*b^2 + 6*A*C^2*a*b^14 + 24*A^2*C*a*b^14 - 2592*B
*C^2*a^14*b + 504*A*B^2*a^3*b^12 + 648*A*B^2*a^4*b^11 - 1548*A*B^2*a^5*b^10 - 972*A*B^2*a^6*b^9 + 1800*A*B^2*a
^7*b^8 + 558*A*B^2*a^8*b^7 - 972*A*B^2*a^9*b^6 - 108*A*B^2*a^10*b^5 + 216*A*B^2*a^11*b^4 - 192*A^2*B*a^2*b^13
- 312*A^2*B*a^3*b^12 + 564*A^2*B*a^4*b^11 + 396*A^2*B*a^5*b^10 - 624*A^2*B*a^6*b^9 - 210*A^2*B*a^7*b^8 + 324*A
^2*B*a^8*b^7 + 36*A^2*B*a^9*b^6 - 72*A^2*B*a^10*b^5 - 6*A*C^2*a^2*b^13 + 207*A*C^2*a^3*b^12 + 33*A*C^2*a^4*b^1
1 + 1158*A*C^2*a^5*b^10 + 1974*A*C^2*a^6*b^9 - 4977*A*C^2*a^7*b^8 - 3405*A*C^2*a^8*b^7 + 6486*A*C^2*a^9*b^6 +
2088*A*C^2*a^10*b^5 - 3744*A*C^2*a^11*b^4 - 432*A*C^2*a^12*b^3 + 864*A*C^2*a^13*b^2 + 12*A^2*C*a^2*b^13 + 300*
A^2*C*a^3*b^12 + 552*A^2*C*a^4*b^11 - 1020*A^2*C*a^5*b^10 - 747*A^2*C*a^6*b^9 + 1188*A^2*C*a^7*b^8 + 408*A^2*C
*a^8*b^7 - 636*A^2*C*a^9*b^6 - 72*A^2*C*a^10*b^5 + 144*A^2*C*a^11*b^4 - 12*B*C^2*a^2*b^13 + 12*B*C^2*a^3*b^12
- 489*B*C^2*a^4*b^11 + 9*B*C^2*a^5*b^10 - 2892*B*C^2*a^6*b^9 - 3972*B*C^2*a^7*b^8 + 13347*B*C^2*a^8*b^7 + 7767
*B*C^2*a^9*b^6 - 18594*B*C^2*a^10*b^5 - 5400*B*C^2*a^11*b^4 + 11232*B*C^2*a^12*b^3 + 1296*B*C^2*a^13*b^2 + 144
*B^2*C*a^3*b^12 + 1980*B^2*C*a^5*b^10 + 2268*B^2*C*a^6*b^9 - 7524*B^2*C*a^7*b^8 - 4203*B^2*C*a^8*b^7 + 9828*B^
2*C*a^9*b^6 + 2808*B^2*C*a^10*b^5 - 5724*B^2*C*a^11*b^4 - 648*B^2*C*a^12*b^3 + 1296*B^2*C*a^13*b^2 - 120*A*B*C
*a^2*b^13 - 24*A*B*C*a^3*b^12 - 1560*A*B*C*a^4*b^11 - 2268*A*B*C*a^5*b^10 + 5568*A*B*C*a^6*b^9 + 3642*A*B*C*a^
7*b^8 - 6840*A*B*C*a^8*b^7 - 2160*A*B*C*a^9*b^6 + 3816*A*B*C*a^10*b^5 + 432*A*B*C*a^11*b^4 - 864*A*B*C*a^12*b^
3))/(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*((a + b)^5*
(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a*b^13 -
288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*A^2*a^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5*b^9 + 57*A^2*a^6*b^8 + 32*A^
2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9*b^5 + 8*A^2*a^10*b^4 + 36*B^2*a^2*b^12 - 72*B^2*a^3*b^11 + 36*B^2*a^4*b
^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^8 - 432*B^2*a^7*b^7 + 441*B^2*a^8*b^6 + 288*B^2*a^9*b^5 - 288*B^2*a^10*b
^4 - 72*B^2*a^11*b^3 + 72*B^2*a^12*b^2 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C^2*a^4*b^10 - 108*C^2*a^5*b^9
 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 - 827*C^2*a^8*b^6 - 1538*C^2*a^9*b^5 + 1538*C^2*a^10*b^4 + 1104*C^2*a^11*b
^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 - 24*A*B*a*b^13 - 8*A*C*a*b^13 - 12*B*C*a*b^13 - 288*B*C*a^13*b + 48*A*B*a
^2*b^12 - 72*A*B*a^3*b^11 - 192*A*B*a^4*b^10 + 252*A*B*a^5*b^9 + 288*A*B*a^6*b^8 - 318*A*B*a^7*b^7 - 192*A*B*a
^8*b^6 + 192*A*B*a^9*b^5 + 48*A*B*a^10*b^4 - 48*A*B*a^11*b^3 + 36*A*C*a^2*b^12 - 64*A*C*a^3*b^11 + 104*A*C*a^4
*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^6*b^8 - 544*A*C*a^7*b^7 + 598*A*C*a^8*b^6 + 376*A*C*a^9*b^5 - 376*A*C*a^10
*b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b^2 + 24*B*C*a^2*b^12 - 108*B*C*a^3*b^11 + 192*B*C*a^4*b^10 - 72*B*C*a^5*
b^9 - 1008*B*C*a^6*b^8 + 984*B*C*a^7*b^7 + 1632*B*C*a^8*b^6 - 1650*B*C*a^9*b^5 - 1128*B*C*a^10*b^4 + 1128*B*C*
a^11*b^3 + 288*B*C*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*((4*(8*A*b^21 + 4*C*b^21 - 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 - 8*A*a^
6*b^15 + 36*A*a^7*b^14 + 4*A*a^8*b^13 - 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 - 156*B*a^4*b^17 - 84*B*a
^5*b^16 + 192*B*a^6*b^15 + 48*B*a^7*b^14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b^11 + 28*C*a^2*b^19 - 8
0*C*a^3*b^18 - 120*C*a^4*b^17 + 276*C*a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^8*b^13 + 212*C*a^9*
b^12 + 24*C*a^10*b^11 - 48*C*a^11*b^10 - 24*A*a*b^20 - 24*B*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*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^
6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*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^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)*(6*
A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a
^5*b))/(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)))*(6*A*b^6 + 12*C*a^6 - 5*A*a^
2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*b))/(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*((a + b)^5*(a - b)^5)^(1/2)*((8*tan(c/2 + (d*x)/2
)*(4*A^2*b^14 + 288*C^2*a^14 + C^2*b^14 - 8*A^2*a*b^13 - 2*C^2*a*b^13 - 288*C^2*a^13*b + 24*A^2*a^2*b^12 + 32*
A^2*a^3*b^11 - 52*A^2*a^4*b^10 - 48*A^2*a^5*b^9 + 57*A^2*a^6*b^8 + 32*A^2*a^7*b^7 - 32*A^2*a^8*b^6 - 8*A^2*a^9
*b^5 + 8*A^2*a^10*b^4 + 36*B^2*a^2*b^12 - 72*B^2*a^3*b^11 + 36*B^2*a^4*b^10 + 288*B^2*a^5*b^9 - 288*B^2*a^6*b^
8 - 432*B^2*a^7*b^7 + 441*B^2*a^8*b^6 + 288*B^2*a^9*b^5 - 288*B^2*a^10*b^4 - 72*B^2*a^11*b^3 + 72*B^2*a^12*b^2
 + 21*C^2*a^2*b^12 - 40*C^2*a^3*b^11 + 74*C^2*a^4*b^10 - 108*C^2*a^5*b^9 + 18*C^2*a^6*b^8 + 872*C^2*a^7*b^7 -
827*C^2*a^8*b^6 - 1538*C^2*a^9*b^5 + 1538*C^2*a^10*b^4 + 1104*C^2*a^11*b^3 - 1104*C^2*a^12*b^2 + 4*A*C*b^14 -
24*A*B*a*b^13 - 8*A*C*a*b^13 - 12*B*C*a*b^13 - 288*B*C*a^13*b + 48*A*B*a^2*b^12 - 72*A*B*a^3*b^11 - 192*A*B*a^
4*b^10 + 252*A*B*a^5*b^9 + 288*A*B*a^6*b^8 - 318*A*B*a^7*b^7 - 192*A*B*a^8*b^6 + 192*A*B*a^9*b^5 + 48*A*B*a^10
*b^4 - 48*A*B*a^11*b^3 + 36*A*C*a^2*b^12 - 64*A*C*a^3*b^11 + 104*A*C*a^4*b^10 + 336*A*C*a^5*b^9 - 444*A*C*a^6*
b^8 - 544*A*C*a^7*b^7 + 598*A*C*a^8*b^6 + 376*A*C*a^9*b^5 - 376*A*C*a^10*b^4 - 96*A*C*a^11*b^3 + 96*A*C*a^12*b
^2 + 24*B*C*a^2*b^12 - 108*B*C*a^3*b^11 + 192*B*C*a^4*b^10 - 72*B*C*a^5*b^9 - 1008*B*C*a^6*b^8 + 984*B*C*a^7*b
^7 + 1632*B*C*a^8*b^6 - 1650*B*C*a^9*b^5 - 1128*B*C*a^10*b^4 + 1128*B*C*a^11*b^3 + 288*B*C*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*((4*(8*A*b^21 + 4*C*b^21
- 16*A*a^2*b^19 + 68*A*a^3*b^18 + 12*A*a^4*b^17 - 72*A*a^5*b^16 - 8*A*a^6*b^15 + 36*A*a^7*b^14 + 4*A*a^8*b^13
- 8*A*a^9*b^12 + 48*B*a^2*b^19 + 72*B*a^3*b^18 - 156*B*a^4*b^17 - 84*B*a^5*b^16 + 192*B*a^6*b^15 + 48*B*a^7*b^
14 - 108*B*a^8*b^13 - 12*B*a^9*b^12 + 24*B*a^10*b^11 + 28*C*a^2*b^19 - 80*C*a^3*b^18 - 120*C*a^4*b^17 + 276*C*
a^5*b^16 + 164*C*a^6*b^15 - 360*C*a^7*b^14 - 100*C*a^8*b^13 + 212*C*a^9*b^12 + 24*C*a^10*b^11 - 48*C*a^11*b^10
 - 24*A*a*b^20 - 24*B*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*tan(c/2 + (d*x)/2)*((a + b)^5*(a - b)^5)^(1/2)*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b
^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*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^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)*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a
^4*b^2 + 15*B*a^3*b^3 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*b))/(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)))*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3 + 2
0*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*b))/(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)*(6*A*b^6 + 12*C*a^6 - 5*A*a^2*b^4 + 2*A*a^4*b^2 + 15*B*a^3*b^3
 + 20*C*a^2*b^4 - 29*C*a^4*b^2 - 12*B*a*b^5 - 6*B*a^5*b)*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 \[ \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**3,x)

[Out]

Integral((A + B*sec(c + d*x) + C*sec(c + d*x)**2)*sec(c + d*x)**4/(a + b*sec(c + d*x))**3, x)

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