3.10.17 \(\int \frac {\sec ^3(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [917]

Optimal. Leaf size=323 \[ \frac {(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac {\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 2.13, antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.195, Rules used = {4183, 4175, 4167, 4083, 3855, 3916, 2738, 214} \begin {gather*} -\frac {\tan (c+d x) \sec ^2(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) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{2 b^3 d \left (a^2-b^2\right )}-\frac {a \tan (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}+\frac {\left (6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 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^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 214

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3855

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

Rule 3916

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

Rule 4083

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 4167

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 4175

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

Rule 4183

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 ^3(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 ^2(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 ^2(c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \sec (c+d x)-\left (A b^2-a b B+3 a^2 C-2 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 ^2(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 b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (-b \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right )-\left (a^2-b^2\right ) \left (a^2 b B-2 b^3 B-3 a^3 C+a b^2 (A+4 C)\right ) \sec (c+d x)-b \left (a^2-b^2\right ) \left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(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 b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}-\frac {\int \frac {\sec (c+d x) \left (-b^2 \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right )-2 b \left (a^2-b^2\right )^2 (b B-3 a C) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(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 b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {(b B-3 a C) \int \sec (c+d x) \, dx}{b^4}+\frac {\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(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 b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=\frac {(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(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 b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}+\frac {\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right )^2 d}\\ &=\frac {(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}+\frac {\left (a^2 A b^4+2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+12 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^4 (a+b)^{5/2} d}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^2(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 b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.88, size = 492, normalized size = 1.52 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {8 \left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) \cos (c+d x) (b+a \cos (c+d x))^2}{\left (a^2-b^2\right )^{5/2}}-8 (b B-3 a C) \cos (c+d x) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 (b B-3 a C) \cos (c+d x) (b+a \cos (c+d x))^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 b \left (-3 a^2 A b^4-2 a^5 b B+5 a^3 b^3 B+6 a^6 C-7 a^4 b^2 C-6 a^2 b^4 C+4 b^6 C+2 a b \left (-3 a^3 b B+6 a b^3 B+a^2 b^2 (A-16 C)+9 a^4 C+4 b^4 (-A+C)\right ) \cos (c+d x)+a^2 \left (-3 A b^4-2 a^3 b B+5 a b^3 B+6 a^4 C-11 a^2 b^2 C+2 b^4 C\right ) \cos (2 (c+d x))\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2}\right )}{4 b^4 d (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-8*(2*A*b^6 - 2*a^5*b*B + 5*a^3
*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2
- b^2]]*Cos[c + d*x]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) - 8*(b*B - 3*a*C)*Cos[c + d*x]*(b + a*Cos[c +
d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*(b*B - 3*a*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])^2*Log[Cos
[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*(-3*a^2*A*b^4 - 2*a^5*b*B + 5*a^3*b^3*B + 6*a^6*C - 7*a^4*b^2*C - 6*a
^2*b^4*C + 4*b^6*C + 2*a*b*(-3*a^3*b*B + 6*a*b^3*B + a^2*b^2*(A - 16*C) + 9*a^4*C + 4*b^4*(-A + C))*Cos[c + d*
x] + a^2*(-3*A*b^4 - 2*a^3*b*B + 5*a*b^3*B + 6*a^4*C - 11*a^2*b^2*C + 2*b^4*C)*Cos[2*(c + d*x)])*Sin[c + d*x])
/(a^2 - b^2)^2))/(4*b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)

________________________________________________________________________________________

Maple [A]
time = 1.73, size = 430, normalized size = 1.33

method	result	size
derivativedivides	\(\frac {-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-b B +3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (b B -3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {\left (a A \,b^{3}+4 A \,b^{4}+2 a^{3} b B -a^{2} b^{2} B -6 a \,b^{3} B -4 a^{4} C +a^{3} b C +8 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (a A \,b^{3}-4 A \,b^{4}-2 a^{3} b B -a^{2} b^{2} B +6 a \,b^{3} B +4 a^{4} C +a^{3} b C -8 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (a^{2} A \,b^{4}+2 A \,b^{6}-2 a^{5} b B +5 a^{3} b^{3} B -6 a \,b^{5} B +6 a^{6} C -15 a^{4} b^{2} C +12 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}}{d}\)	\(430\)
default	\(\frac {-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-b B +3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (b B -3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}-\frac {2 \left (\frac {-\frac {\left (a A \,b^{3}+4 A \,b^{4}+2 a^{3} b B -a^{2} b^{2} B -6 a \,b^{3} B -4 a^{4} C +a^{3} b C +8 C \,a^{2} b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (a A \,b^{3}-4 A \,b^{4}-2 a^{3} b B -a^{2} b^{2} B +6 a \,b^{3} B +4 a^{4} C +a^{3} b C -8 C \,a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (a^{2} A \,b^{4}+2 A \,b^{6}-2 a^{5} b B +5 a^{3} b^{3} B -6 a \,b^{5} B +6 a^{6} C -15 a^{4} b^{2} C +12 C \,a^{2} b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}}{d}\)	\(430\)
risch	\(\text {Expression too large to display}\)	\(2188\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^3*(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 de

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1079 vs. \(2 (311) = 622\).
time = 169.57, size = 2217, normalized size = 6.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(((6*C*a^8 - 2*B*a^7*b - 15*C*a^6*b^2 + 5*B*a^5*b^3 + (A + 12*C)*a^4*b^4 - 6*B*a^3*b^5 + 2*A*a^2*b^6)*cos
(d*x + c)^3 + 2*(6*C*a^7*b - 2*B*a^6*b^2 - 15*C*a^5*b^3 + 5*B*a^4*b^4 + (A + 12*C)*a^3*b^5 - 6*B*a^2*b^6 + 2*A
*a*b^7)*cos(d*x + c)^2 + (6*C*a^6*b^2 - 2*B*a^5*b^3 - 15*C*a^4*b^4 + 5*B*a^3*b^5 + (A + 12*C)*a^2*b^6 - 6*B*a*
b^7 + 2*A*b^8)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 + 2*sqrt(a
^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) -
2*((3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*cos(d
*x + c)^3 + 2*(3*C*a^8*b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a^2*b^7 + B
*a*b^8)*cos(d*x + c)^2 + (3*C*a^7*b^2 - B*a^6*b^3 - 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*
C*a*b^8 + B*b^9)*cos(d*x + c))*log(sin(d*x + c) + 1) + 2*((3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C
*a^5*b^4 - 3*B*a^4*b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*cos(d*x + c)^3 + 2*(3*C*a^8*b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*
B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a^2*b^7 + B*a*b^8)*cos(d*x + c)^2 + (3*C*a^7*b^2 - B*a^6*b^3 - 9*C
*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) +
2*(2*C*a^6*b^3 - 6*C*a^4*b^5 + 6*C*a^2*b^7 - 2*C*b^9 + (6*C*a^8*b - 2*B*a^7*b^2 - 17*C*a^6*b^3 + 7*B*a^5*b^4
- (3*A - 13*C)*a^4*b^5 - 5*B*a^3*b^6 + (3*A - 2*C)*a^2*b^7)*cos(d*x + c)^2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 + (A -
25*C)*a^5*b^4 + 9*B*a^4*b^5 - 5*(A - 4*C)*a^3*b^6 - 6*B*a^2*b^7 + 4*(A - C)*a*b^8)*cos(d*x + c))*sin(d*x + c)
)/((a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d*cos(d*x + c)^3 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11
)*d*cos(d*x + c)^2 + (a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*d*cos(d*x + c)), 1/2*(((6*C*a^8 - 2*B*a^7*b - 1
5*C*a^6*b^2 + 5*B*a^5*b^3 + (A + 12*C)*a^4*b^4 - 6*B*a^3*b^5 + 2*A*a^2*b^6)*cos(d*x + c)^3 + 2*(6*C*a^7*b - 2*
B*a^6*b^2 - 15*C*a^5*b^3 + 5*B*a^4*b^4 + (A + 12*C)*a^3*b^5 - 6*B*a^2*b^6 + 2*A*a*b^7)*cos(d*x + c)^2 + (6*C*a
^6*b^2 - 2*B*a^5*b^3 - 15*C*a^4*b^4 + 5*B*a^3*b^5 + (A + 12*C)*a^2*b^6 - 6*B*a*b^7 + 2*A*b^8)*cos(d*x + c))*sq
rt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - ((3*C*a^9 - B*a^8*b
- 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*cos(d*x + c)^3 + 2*(3*C*a^
8*b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a^2*b^7 + B*a*b^8)*cos(d*x + c)^
2 + (3*C*a^7*b^2 - B*a^6*b^3 - 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)*cos(
d*x + c))*log(sin(d*x + c) + 1) + ((3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5
- 3*C*a^3*b^6 + B*a^2*b^7)*cos(d*x + c)^3 + 2*(3*C*a^8*b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5
- 3*B*a^3*b^6 - 3*C*a^2*b^7 + B*a*b^8)*cos(d*x + c)^2 + (3*C*a^7*b^2 - B*a^6*b^3 - 9*C*a^5*b^4 + 3*B*a^4*b^5
+ 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)*cos(d*x + c))*log(-sin(d*x + c) + 1) + (2*C*a^6*b^3 - 6*C*a^4
*b^5 + 6*C*a^2*b^7 - 2*C*b^9 + (6*C*a^8*b - 2*B*a^7*b^2 - 17*C*a^6*b^3 + 7*B*a^5*b^4 - (3*A - 13*C)*a^4*b^5 -
5*B*a^3*b^6 + (3*A - 2*C)*a^2*b^7)*cos(d*x + c)^2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 + (A - 25*C)*a^5*b^4 + 9*B*a^4*
b^5 - 5*(A - 4*C)*a^3*b^6 - 6*B*a^2*b^7 + 4*(A - C)*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^8*b^4 - 3*a^6*b^6 +
3*a^4*b^8 - a^2*b^10)*d*cos(d*x + c)^3 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*d*cos(d*x + c)^2 + (a^6
*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*d*cos(d*x + c))]

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 705 vs. \(2 (311) = 622\).
time = 0.60, size = 705, normalized size = 2.18 \begin {gather*} \frac {\frac {{\left (6 \, C a^{6} - 2 \, B a^{5} b - 15 \, C a^{4} b^{2} + 5 \, B a^{3} b^{3} + A a^{2} b^{4} + 12 \, C a^{2} b^{4} - 6 \, B a b^{5} + 2 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {4 \, C a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}^{2}} - \frac {{\left (3 \, C a - B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} + \frac {{\left (3 \, C a - B b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} b^{3}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((6*C*a^6 - 2*B*a^5*b - 15*C*a^4*b^2 + 5*B*a^3*b^3 + A*a^2*b^4 + 12*C*a^2*b^4 - 6*B*a*b^5 + 2*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*tan(1/2*d*x + 1/2*c))/sqrt(-a^2
+ b^2)))/((a^4*b^4 - 2*a^2*b^6 + b^8)*sqrt(-a^2 + b^2)) - (4*C*a^6*tan(1/2*d*x + 1/2*c)^3 - 2*B*a^5*b*tan(1/2
*d*x + 1/2*c)^3 - 5*C*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 3*B*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - 7*C*a^4*b^2*tan(1/2*
d*x + 1/2*c)^3 - A*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*C*a^3*b^3*tan(1/2*d
*x + 1/2*c)^3 - 3*A*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 - 6*B*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*A*a*b^5*tan(1/2*d*
x + 1/2*c)^3 - 4*C*a^6*tan(1/2*d*x + 1/2*c) + 2*B*a^5*b*tan(1/2*d*x + 1/2*c) - 5*C*a^5*b*tan(1/2*d*x + 1/2*c)
+ 3*B*a^4*b^2*tan(1/2*d*x + 1/2*c) + 7*C*a^4*b^2*tan(1/2*d*x + 1/2*c) - A*a^3*b^3*tan(1/2*d*x + 1/2*c) - 5*B*a
^3*b^3*tan(1/2*d*x + 1/2*c) + 8*C*a^3*b^3*tan(1/2*d*x + 1/2*c) + 3*A*a^2*b^4*tan(1/2*d*x + 1/2*c) - 6*B*a^2*b^
4*tan(1/2*d*x + 1/2*c) + 4*A*a*b^5*tan(1/2*d*x + 1/2*c))/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a*tan(1/2*d*x + 1/2*c)^
2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^2) - (3*C*a - B*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^4 + (3*C*a - B*b
)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^4 - 2*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2 - 1)*b^3))/d

________________________________________________________________________________________

Mupad [B]
time = 17.61, size = 2500, normalized size = 7.74 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(6*C*a^5 + 2*C*b^5 + A*a^2*b^3 + 6*B*a^2*b^3 - B*a^3*b^2 - 4*C*a^2*b^3 - 12*C*a^3*b^2 - 4
*A*a*b^4 - 2*B*a^4*b + 2*C*a*b^4 + 3*C*a^4*b))/((a + b)*(b^5 - 2*a*b^4 + a^2*b^3)) - (tan(c/2 + (d*x)/2)^5*(2*
C*b^5 - 6*C*a^5 + A*a^2*b^3 - 6*B*a^2*b^3 - B*a^3*b^2 - 4*C*a^2*b^3 + 12*C*a^3*b^2 + 4*A*a*b^4 + 2*B*a^4*b - 2
*C*a*b^4 + 3*C*a^4*b))/((a*b^3 - b^4)*(a + b)^2) + (2*tan(c/2 + (d*x)/2)^3*(2*C*b^6 - 6*C*a^6 + 3*A*a^2*b^4 -
5*B*a^3*b^3 - 6*C*a^2*b^4 + 13*C*a^4*b^2 + 2*B*a^5*b))/(b*(a*b^2 - b^3)*(a + b)^2*(a - b)))/(d*(2*a*b - tan(c/
2 + (d*x)/2)^2*(2*a*b + 3*a^2 - b^2) - tan(c/2 + (d*x)/2)^6*(a^2 - 2*a*b + b^2) + a^2 + b^2 - tan(c/2 + (d*x)/
2)^4*(2*a*b - 3*a^2 + b^2))) + (atan((((B*b - 3*C*a)*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 4*B^2*b^12 + 72*C^2*
a^12 - 8*B^2*a*b^11 - 72*C^2*a^11*b + 4*A^2*a^2*b^10 + A^2*a^4*b^8 + 24*B^2*a^2*b^10 + 32*B^2*a^3*b^9 - 52*B^2
*a^4*b^8 - 48*B^2*a^5*b^7 + 57*B^2*a^6*b^6 + 32*B^2*a^7*b^5 - 32*B^2*a^8*b^4 - 8*B^2*a^9*b^3 + 8*B^2*a^10*b^2
+ 36*C^2*a^2*b^10 - 72*C^2*a^3*b^9 + 36*C^2*a^4*b^8 + 288*C^2*a^5*b^7 - 288*C^2*a^6*b^6 - 432*C^2*a^7*b^5 + 44
1*C^2*a^8*b^4 + 288*C^2*a^9*b^3 - 288*C^2*a^10*b^2 - 24*A*B*a*b^11 - 24*B*C*a*b^11 - 48*B*C*a^11*b + 8*A*B*a^3
*b^9 + 2*A*B*a^5*b^7 - 4*A*B*a^7*b^5 + 48*A*C*a^2*b^10 - 36*A*C*a^4*b^8 - 6*A*C*a^6*b^6 + 12*A*C*a^8*b^4 + 48*
B*C*a^2*b^10 - 72*B*C*a^3*b^9 - 192*B*C*a^4*b^8 + 252*B*C*a^5*b^7 + 288*B*C*a^6*b^6 - 318*B*C*a^7*b^5 - 192*B*
C*a^8*b^4 + 192*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b
^8 - a^6*b^7 - a^7*b^6) + ((B*b - 3*C*a)*((8*(4*A*b^18 + 4*B*b^18 - 6*A*a^2*b^16 + 6*A*a^3*b^15 + 2*A*a^6*b^12
- 2*A*a^7*b^11 - 8*B*a^2*b^16 + 34*B*a^3*b^15 + 6*B*a^4*b^14 - 36*B*a^5*b^13 - 4*B*a^6*b^12 + 18*B*a^7*b^11 +
2*B*a^8*b^10 - 4*B*a^9*b^9 + 24*C*a^2*b^16 + 36*C*a^3*b^15 - 78*C*a^4*b^14 - 42*C*a^5*b^13 + 96*C*a^6*b^12 +
24*C*a^7*b^11 - 54*C*a^8*b^10 - 6*C*a^9*b^9 + 12*C*a^10*b^8 - 4*A*a*b^17 - 12*B*a*b^17 - 12*C*a*b^17))/(a*b^15
+ b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (8*tan(c/2 + (d*x)/2)*(B*b
- 3*C*a)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^
8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/(b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*
b^7 - a^7*b^6))))/b^4)*1i)/b^4 + ((B*b - 3*C*a)*((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 4*B^2*b^12 + 72*C^2*a^12
- 8*B^2*a*b^11 - 72*C^2*a^11*b + 4*A^2*a^2*b^10 + A^2*a^4*b^8 + 24*B^2*a^2*b^10 + 32*B^2*a^3*b^9 - 52*B^2*a^4*
b^8 - 48*B^2*a^5*b^7 + 57*B^2*a^6*b^6 + 32*B^2*a^7*b^5 - 32*B^2*a^8*b^4 - 8*B^2*a^9*b^3 + 8*B^2*a^10*b^2 + 36*
C^2*a^2*b^10 - 72*C^2*a^3*b^9 + 36*C^2*a^4*b^8 + 288*C^2*a^5*b^7 - 288*C^2*a^6*b^6 - 432*C^2*a^7*b^5 + 441*C^2
*a^8*b^4 + 288*C^2*a^9*b^3 - 288*C^2*a^10*b^2 - 24*A*B*a*b^11 - 24*B*C*a*b^11 - 48*B*C*a^11*b + 8*A*B*a^3*b^9
+ 2*A*B*a^5*b^7 - 4*A*B*a^7*b^5 + 48*A*C*a^2*b^10 - 36*A*C*a^4*b^8 - 6*A*C*a^6*b^6 + 12*A*C*a^8*b^4 + 48*B*C*a
^2*b^10 - 72*B*C*a^3*b^9 - 192*B*C*a^4*b^8 + 252*B*C*a^5*b^7 + 288*B*C*a^6*b^6 - 318*B*C*a^7*b^5 - 192*B*C*a^8
*b^4 + 192*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 -
a^6*b^7 - a^7*b^6) - ((B*b - 3*C*a)*((8*(4*A*b^18 + 4*B*b^18 - 6*A*a^2*b^16 + 6*A*a^3*b^15 + 2*A*a^6*b^12 - 2*
A*a^7*b^11 - 8*B*a^2*b^16 + 34*B*a^3*b^15 + 6*B*a^4*b^14 - 36*B*a^5*b^13 - 4*B*a^6*b^12 + 18*B*a^7*b^11 + 2*B*
a^8*b^10 - 4*B*a^9*b^9 + 24*C*a^2*b^16 + 36*C*a^3*b^15 - 78*C*a^4*b^14 - 42*C*a^5*b^13 + 96*C*a^6*b^12 + 24*C*
a^7*b^11 - 54*C*a^8*b^10 - 6*C*a^9*b^9 + 12*C*a^10*b^8 - 4*A*a*b^17 - 12*B*a*b^17 - 12*C*a*b^17))/(a*b^15 + b^
16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (8*tan(c/2 + (d*x)/2)*(B*b - 3*
C*a)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^1
0 + 8*a^9*b^9 - 8*a^10*b^8))/(b^4*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 -
a^7*b^6))))/b^4)*1i)/b^4)/((16*(108*C^3*a^12 + 4*A*B^2*b^12 - 4*A^2*B*b^12 - 12*B^3*a*b^11 - 54*C^3*a^11*b -
24*B^3*a^2*b^10 + 34*B^3*a^3*b^9 + 26*B^3*a^4*b^8 - 36*B^3*a^5*b^7 - 13*B^3*a^6*b^6 + 18*B^3*a^7*b^5 + 2*B^3*a
^8*b^4 - 4*B^3*a^9*b^3 + 216*C^3*a^4*b^8 + 216*C^3*a^5*b^7 - 702*C^3*a^6*b^6 - 378*C^3*a^7*b^5 + 864*C^3*a^8*b
^4 + 243*C^3*a^9*b^3 - 486*C^3*a^10*b^2 + 20*A*B^2*a*b^11 + 12*A^2*C*a*b^11 - 108*B*C^2*a^11*b - 6*A*B^2*a^2*b
^10 - 2*A*B^2*a^3*b^9 - 2*A*B^2*a^5*b^7 + 2*A*B^2*a^6*b^6 + 2*A*B^2*a^7*b^5 - 4*A^2*B*a^2*b^10 - A^2*B*a^4*b^8
+ 36*A*C^2*a^2*b^10 + 108*A*C^2*a^3*b^9 - 54*A*C^2*a^4*b^8 - 54*A*C^2*a^5*b^7 - 18*A*C^2*a^7*b^5 + 18*A*C^2*a
^8*b^4 + 18*A*C^2*a^9*b^3 + 12*A^2*C*a^3*b^9 + 3*A^2*C*a^5*b^7 - 252*B*C^2*a^3*b^9 - 324*B*C^2*a^4*b^8 + 774*B
*C^2*a^5*b^7 + 486*B*C^2*a^6*b^6 - 900*B*C^2*a^7*b^5 - 279*B*C^2*a^8*b^4 + 486*B*C^2*a^9*b^3 + 54*B*C^2*a^10*b
^2 + 96*B^2*C*a^2*b^10 + 156*B^2*C*a^3*b^9 - 282*B^2*C*a^4*b^8 - 198*B^2*C*a^5*b^7 + 312*B^2*C*a^6*b^6 + 105*B
^2*C*a^7*b^5 - 162*B^2*C*a^8*b^4 - 18*B^2*C*a^9...

________________________________________________________________________________________