3.7.96 \(\int \frac {\cos (c+d x) (A+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [696]

Optimal. Leaf size=266 \[ -\frac {3 A b x}{a^4}-\frac {\left (15 a^2 A b^4-6 A b^6-2 a^6 C-a^4 b^2 (12 A+C)\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 (a-b)^{5/2} (a+b)^{5/2} d}-\frac {\left (11 a^2 A b^2-6 A b^4-a^4 (2 A-3 C)\right ) \sin (c+d x)}{2 a^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {\left (3 A b^4-2 a^4 C-a^2 b^2 (6 A+C)\right ) \sin (c+d x)}{2 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

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

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Rubi [A]
time = 0.69, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4186, 4185, 4189, 4004, 3916, 2738, 214} \begin {gather*} -\frac {3 A b x}{a^4}+\frac {\left (a^2 C+A b^2\right ) \sin (c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\left (-2 a^4 C-a^2 b^2 (6 A+C)+3 A b^4\right ) \sin (c+d x)}{2 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {\left (-2 a^6 C-a^4 b^2 (12 A+C)+15 a^2 A b^4-6 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac {\left (-\left (a^4 (2 A-3 C)\right )+11 a^2 A b^2-6 A b^4\right ) \sin (c+d x)}{2 a^3 d \left (a^2-b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*A*b*x)/a^4 - ((15*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2
])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((11*a^2*A*b^2 - 6*A*b^4 - a^4*(2*A - 3*C))*Sin[c + d*x
])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*A
*b^4 - 2*a^4*C - a^2*b^2*(6*A + C))*Sin[c + d*x])/(2*a^2*(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 3916

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

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 4185

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

Rule 4186

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

Rule 4189

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 6.83, size = 902, normalized size = 3.39 \begin {gather*} \frac {(b+a \cos (c+d x)) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (-\frac {8 i \left (-15 a^2 A b^4+6 A b^6+2 a^6 C+a^4 b^2 (12 A+C)\right ) \text {ArcTan}\left (\frac {(i \cos (c)+\sin (c)) \left (a \sin (c)+(-b+a \cos (c)) \tan \left (\frac {d x}{2}\right )\right )}{\sqrt {a^2-b^2} \sqrt {(\cos (c)-i \sin (c))^2}}\right ) (b+a \cos (c+d x))^2 (\cos (c)-i \sin (c))}{\left (a^2-b^2\right )^{5/2} \sqrt {(\cos (c)-i \sin (c))^2}}+\frac {\sec (c) \left (-12 A b \left (a^2-b^2\right )^2 \left (a^2+2 b^2\right ) d x \cos (c)-24 a A b^2 \left (a^2-b^2\right )^2 d x \cos (d x)-24 a^5 A b^2 d x \cos (2 c+d x)+48 a^3 A b^4 d x \cos (2 c+d x)-24 a A b^6 d x \cos (2 c+d x)-6 a^6 A b d x \cos (c+2 d x)+12 a^4 A b^3 d x \cos (c+2 d x)-6 a^2 A b^5 d x \cos (c+2 d x)-6 a^6 A b d x \cos (3 c+2 d x)+12 a^4 A b^3 d x \cos (3 c+2 d x)-6 a^2 A b^5 d x \cos (3 c+2 d x)+16 a^4 A b^3 \sin (c)+22 a^2 A b^5 \sin (c)-20 A b^7 \sin (c)+8 a^6 b C \sin (c)+14 a^4 b^3 C \sin (c)-4 a^2 b^5 C \sin (c)+a^7 A \sin (d x)+2 a^5 A b^2 \sin (d x)-53 a^3 A b^4 \sin (d x)+32 a A b^6 \sin (d x)-22 a^5 b^2 C \sin (d x)+4 a^3 b^4 C \sin (d x)+a^7 A \sin (2 c+d x)+2 a^5 A b^2 \sin (2 c+d x)+11 a^3 A b^4 \sin (2 c+d x)-8 a A b^6 \sin (2 c+d x)+10 a^5 b^2 C \sin (2 c+d x)-4 a^3 b^4 C \sin (2 c+d x)+4 a^6 A b \sin (c+2 d x)-24 a^4 A b^3 \sin (c+2 d x)+14 a^2 A b^5 \sin (c+2 d x)-8 a^6 b C \sin (c+2 d x)+2 a^4 b^3 C \sin (c+2 d x)+4 a^6 A b \sin (3 c+2 d x)-8 a^4 A b^3 \sin (3 c+2 d x)+4 a^2 A b^5 \sin (3 c+2 d x)+a^7 A \sin (2 c+3 d x)-2 a^5 A b^2 \sin (2 c+3 d x)+a^3 A b^4 \sin (2 c+3 d x)+a^7 A \sin (4 c+3 d x)-2 a^5 A b^2 \sin (4 c+3 d x)+a^3 A b^4 \sin (4 c+3 d x)\right )}{\left (a^2-b^2\right )^2}\right )}{4 a^4 d (A+2 C+A \cos (2 (c+d x))) (a+b \sec (c+d x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Cos[c + d*x]*(A + 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]*(A + C*Sec[c + d*x]^2)*(((-8*I)*(-15*a^2*A*b^4 + 6*A*b^6 + 2*a^6*C + a^4*b^
2*(12*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Co
s[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^
2]) + (Sec[c]*(-12*A*b*(a^2 - b^2)^2*(a^2 + 2*b^2)*d*x*Cos[c] - 24*a*A*b^2*(a^2 - b^2)^2*d*x*Cos[d*x] - 24*a^5
*A*b^2*d*x*Cos[2*c + d*x] + 48*a^3*A*b^4*d*x*Cos[2*c + d*x] - 24*a*A*b^6*d*x*Cos[2*c + d*x] - 6*a^6*A*b*d*x*Co
s[c + 2*d*x] + 12*a^4*A*b^3*d*x*Cos[c + 2*d*x] - 6*a^2*A*b^5*d*x*Cos[c + 2*d*x] - 6*a^6*A*b*d*x*Cos[3*c + 2*d*
x] + 12*a^4*A*b^3*d*x*Cos[3*c + 2*d*x] - 6*a^2*A*b^5*d*x*Cos[3*c + 2*d*x] + 16*a^4*A*b^3*Sin[c] + 22*a^2*A*b^5
*Sin[c] - 20*A*b^7*Sin[c] + 8*a^6*b*C*Sin[c] + 14*a^4*b^3*C*Sin[c] - 4*a^2*b^5*C*Sin[c] + a^7*A*Sin[d*x] + 2*a
^5*A*b^2*Sin[d*x] - 53*a^3*A*b^4*Sin[d*x] + 32*a*A*b^6*Sin[d*x] - 22*a^5*b^2*C*Sin[d*x] + 4*a^3*b^4*C*Sin[d*x]
+ a^7*A*Sin[2*c + d*x] + 2*a^5*A*b^2*Sin[2*c + d*x] + 11*a^3*A*b^4*Sin[2*c + d*x] - 8*a*A*b^6*Sin[2*c + d*x]
+ 10*a^5*b^2*C*Sin[2*c + d*x] - 4*a^3*b^4*C*Sin[2*c + d*x] + 4*a^6*A*b*Sin[c + 2*d*x] - 24*a^4*A*b^3*Sin[c + 2
*d*x] + 14*a^2*A*b^5*Sin[c + 2*d*x] - 8*a^6*b*C*Sin[c + 2*d*x] + 2*a^4*b^3*C*Sin[c + 2*d*x] + 4*a^6*A*b*Sin[3*
c + 2*d*x] - 8*a^4*A*b^3*Sin[3*c + 2*d*x] + 4*a^2*A*b^5*Sin[3*c + 2*d*x] + a^7*A*Sin[2*c + 3*d*x] - 2*a^5*A*b^
2*Sin[2*c + 3*d*x] + a^3*A*b^4*Sin[2*c + 3*d*x] + a^7*A*Sin[4*c + 3*d*x] - 2*a^5*A*b^2*Sin[4*c + 3*d*x] + a^3*
A*b^4*Sin[4*c + 3*d*x]))/(a^2 - b^2)^2))/(4*a^4*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)

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Maple [A]
time = 0.36, size = 329, normalized size = 1.24 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(A+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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (247) = 494\).
time = 2.74, size = 1219, normalized size = 4.58 \begin {gather*} \left [-\frac {12 \, {\left (A a^{8} b - 3 \, A a^{6} b^{3} + 3 \, A a^{4} b^{5} - A a^{2} b^{7}\right )} d x \cos \left (d x + c\right )^{2} + 24 \, {\left (A a^{7} b^{2} - 3 \, A a^{5} b^{4} + 3 \, A a^{3} b^{6} - A a b^{8}\right )} d x \cos \left (d x + c\right ) + 12 \, {\left (A a^{6} b^{3} - 3 \, A a^{4} b^{5} + 3 \, A a^{2} b^{7} - A b^{9}\right )} d x - {\left (2 \, C a^{6} b^{2} + {\left (12 \, A + C\right )} a^{4} b^{4} - 15 \, A a^{2} b^{6} + 6 \, A b^{8} + {\left (2 \, C a^{8} + {\left (12 \, A + C\right )} a^{6} b^{2} - 15 \, A a^{4} b^{4} + 6 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{7} b + {\left (12 \, A + C\right )} a^{5} b^{3} - 15 \, A a^{3} b^{5} + 6 \, A a b^{7}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - 2 \, {\left ({\left (2 \, A - 3 \, C\right )} a^{7} b^{2} - {\left (13 \, A - 3 \, C\right )} a^{5} b^{4} + 17 \, A a^{3} b^{6} - 6 \, A a b^{8} + 2 \, {\left (A a^{9} - 3 \, A a^{7} b^{2} + 3 \, A a^{5} b^{4} - A a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, {\left (A - C\right )} a^{8} b - 5 \, {\left (4 \, A - C\right )} a^{6} b^{3} + {\left (25 \, A - C\right )} a^{4} b^{5} - 9 \, A a^{2} b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{12} - 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} - a^{6} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b - 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} - a^{5} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{10} b^{2} - 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} - a^{4} b^{8}\right )} d\right )}}, -\frac {6 \, {\left (A a^{8} b - 3 \, A a^{6} b^{3} + 3 \, A a^{4} b^{5} - A a^{2} b^{7}\right )} d x \cos \left (d x + c\right )^{2} + 12 \, {\left (A a^{7} b^{2} - 3 \, A a^{5} b^{4} + 3 \, A a^{3} b^{6} - A a b^{8}\right )} d x \cos \left (d x + c\right ) + 6 \, {\left (A a^{6} b^{3} - 3 \, A a^{4} b^{5} + 3 \, A a^{2} b^{7} - A b^{9}\right )} d x - {\left (2 \, C a^{6} b^{2} + {\left (12 \, A + C\right )} a^{4} b^{4} - 15 \, A a^{2} b^{6} + 6 \, A b^{8} + {\left (2 \, C a^{8} + {\left (12 \, A + C\right )} a^{6} b^{2} - 15 \, A a^{4} b^{4} + 6 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, C a^{7} b + {\left (12 \, A + C\right )} a^{5} b^{3} - 15 \, A a^{3} b^{5} + 6 \, A a b^{7}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left ({\left (2 \, A - 3 \, C\right )} a^{7} b^{2} - {\left (13 \, A - 3 \, C\right )} a^{5} b^{4} + 17 \, A a^{3} b^{6} - 6 \, A a b^{8} + 2 \, {\left (A a^{9} - 3 \, A a^{7} b^{2} + 3 \, A a^{5} b^{4} - A a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (4 \, {\left (A - C\right )} a^{8} b - 5 \, {\left (4 \, A - C\right )} a^{6} b^{3} + {\left (25 \, A - C\right )} a^{4} b^{5} - 9 \, A a^{2} b^{7}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{12} - 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} - a^{6} b^{6}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{11} b - 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} - a^{5} b^{7}\right )} d \cos \left (d x + c\right ) + {\left (a^{10} b^{2} - 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} - a^{4} b^{8}\right )} d\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(12*(A*a^8*b - 3*A*a^6*b^3 + 3*A*a^4*b^5 - A*a^2*b^7)*d*x*cos(d*x + c)^2 + 24*(A*a^7*b^2 - 3*A*a^5*b^4 +
3*A*a^3*b^6 - A*a*b^8)*d*x*cos(d*x + c) + 12*(A*a^6*b^3 - 3*A*a^4*b^5 + 3*A*a^2*b^7 - A*b^9)*d*x - (2*C*a^6*b
^2 + (12*A + C)*a^4*b^4 - 15*A*a^2*b^6 + 6*A*b^8 + (2*C*a^8 + (12*A + C)*a^6*b^2 - 15*A*a^4*b^4 + 6*A*a^2*b^6)
*cos(d*x + c)^2 + 2*(2*C*a^7*b + (12*A + C)*a^5*b^3 - 15*A*a^3*b^5 + 6*A*a*b^7)*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*((2*A - 3*C)*a^7*b^2 - (13*A - 3*C)*a^5*b^4
+ 17*A*a^3*b^6 - 6*A*a*b^8 + 2*(A*a^9 - 3*A*a^7*b^2 + 3*A*a^5*b^4 - A*a^3*b^6)*cos(d*x + c)^2 + (4*(A - C)*a^
8*b - 5*(4*A - C)*a^6*b^3 + (25*A - C)*a^4*b^5 - 9*A*a^2*b^7)*cos(d*x + c))*sin(d*x + c))/((a^12 - 3*a^10*b^2
+ 3*a^8*b^4 - a^6*b^6)*d*cos(d*x + c)^2 + 2*(a^11*b - 3*a^9*b^3 + 3*a^7*b^5 - a^5*b^7)*d*cos(d*x + c) + (a^10*
b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*d), -1/2*(6*(A*a^8*b - 3*A*a^6*b^3 + 3*A*a^4*b^5 - A*a^2*b^7)*d*x*cos(d
*x + c)^2 + 12*(A*a^7*b^2 - 3*A*a^5*b^4 + 3*A*a^3*b^6 - A*a*b^8)*d*x*cos(d*x + c) + 6*(A*a^6*b^3 - 3*A*a^4*b^5
+ 3*A*a^2*b^7 - A*b^9)*d*x - (2*C*a^6*b^2 + (12*A + C)*a^4*b^4 - 15*A*a^2*b^6 + 6*A*b^8 + (2*C*a^8 + (12*A +
C)*a^6*b^2 - 15*A*a^4*b^4 + 6*A*a^2*b^6)*cos(d*x + c)^2 + 2*(2*C*a^7*b + (12*A + C)*a^5*b^3 - 15*A*a^3*b^5 + 6
*A*a*b^7)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x +
c))) - ((2*A - 3*C)*a^7*b^2 - (13*A - 3*C)*a^5*b^4 + 17*A*a^3*b^6 - 6*A*a*b^8 + 2*(A*a^9 - 3*A*a^7*b^2 + 3*A*a
^5*b^4 - A*a^3*b^6)*cos(d*x + c)^2 + (4*(A - C)*a^8*b - 5*(4*A - C)*a^6*b^3 + (25*A - C)*a^4*b^5 - 9*A*a^2*b^7
)*cos(d*x + c))*sin(d*x + c))/((a^12 - 3*a^10*b^2 + 3*a^8*b^4 - a^6*b^6)*d*cos(d*x + c)^2 + 2*(a^11*b - 3*a^9*
b^3 + 3*a^7*b^5 - a^5*b^7)*d*cos(d*x + c) + (a^10*b^2 - 3*a^8*b^4 + 3*a^6*b^6 - a^4*b^8)*d)]

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

[Out]

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

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Giac [A]
time = 0.54, size = 491, normalized size = 1.85 \begin {gather*} \frac {\frac {{\left (2 \, C a^{6} + 12 \, A a^{4} b^{2} + C a^{4} b^{2} - 15 \, A a^{2} b^{4} + 6 \, 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^{8} - 2 \, a^{6} b^{2} + a^{4} b^{4}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {3 \, {\left (d x + c\right )} A b}{a^{4}} + \frac {4 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, A b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\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 {2 \, A \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 )} a^{3}}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 14.15, size = 2500, normalized size = 9.40 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(c/2 + (d*x)/2)*(2*A*a^5 + 6*A*b^5 - 12*A*a^2*b^3 - 4*A*a^3*b^2 + C*a^3*b^2 + 3*A*a*b^4 + 2*A*a^4*b - 4*C
*a^4*b))/((a + b)*(a^5 - 2*a^4*b + a^3*b^2)) - (tan(c/2 + (d*x)/2)^5*(2*A*a^5 - 6*A*b^5 + 12*A*a^2*b^3 - 4*A*a
^3*b^2 + C*a^3*b^2 + 3*A*a*b^4 - 2*A*a^4*b + 4*C*a^4*b))/((a^3*b - a^4)*(a + b)^2) + (2*tan(c/2 + (d*x)/2)^3*(
2*A*a^6 - 6*A*b^6 + 13*A*a^2*b^4 - 6*A*a^4*b^2 + 3*C*a^4*b^2))/(a*(a^2*b - a^3)*(a + b)^2*(a - b)))/(d*(2*a*b
+ tan(c/2 + (d*x)/2)^2*(2*a*b - a^2 + 3*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 + a^2 - 3*b^2))) - (6*A*b*atan(((3*A*b*((8*tan(c/2 + (d*x)/2)*(72*A^2*b^12 + 4*C^2*a^12 -
72*A^2*a*b^11 - 288*A^2*a^2*b^10 + 288*A^2*a^3*b^9 + 441*A^2*a^4*b^8 - 432*A^2*a^5*b^7 - 288*A^2*a^6*b^6 + 288
*A^2*a^7*b^5 + 36*A^2*a^8*b^4 - 72*A^2*a^9*b^3 + 36*A^2*a^10*b^2 + C^2*a^8*b^4 + 4*C^2*a^10*b^2 + 12*A*C*a^4*b
^8 - 6*A*C*a^6*b^6 - 36*A*C*a^8*b^4 + 48*A*C*a^10*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9
*b^4 - 3*a^10*b^3 - 3*a^11*b^2) + (A*b*((8*(4*C*a^18 + 12*A*a^8*b^10 - 6*A*a^9*b^9 - 54*A*a^10*b^8 + 24*A*a^11
*b^7 + 96*A*a^12*b^6 - 42*A*a^13*b^5 - 78*A*a^14*b^4 + 36*A*a^15*b^3 + 24*A*a^16*b^2 - 2*C*a^11*b^7 + 2*C*a^12
*b^6 + 6*C*a^15*b^3 - 6*C*a^16*b^2 - 12*A*a^17*b - 4*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b
^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) - (A*b*tan(c/2 + (d*x)/2)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a
^10*b^8 - 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2)*24i)/(a^4*(a^12*b
+ a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2)))*3i)/a^4))/a^4 + (3*A*b*((8*tan
(c/2 + (d*x)/2)*(72*A^2*b^12 + 4*C^2*a^12 - 72*A^2*a*b^11 - 288*A^2*a^2*b^10 + 288*A^2*a^3*b^9 + 441*A^2*a^4*b
^8 - 432*A^2*a^5*b^7 - 288*A^2*a^6*b^6 + 288*A^2*a^7*b^5 + 36*A^2*a^8*b^4 - 72*A^2*a^9*b^3 + 36*A^2*a^10*b^2 +
C^2*a^8*b^4 + 4*C^2*a^10*b^2 + 12*A*C*a^4*b^8 - 6*A*C*a^6*b^6 - 36*A*C*a^8*b^4 + 48*A*C*a^10*b^2))/(a^12*b +
a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (A*b*((8*(4*C*a^18 + 12*A*a^8*b^
10 - 6*A*a^9*b^9 - 54*A*a^10*b^8 + 24*A*a^11*b^7 + 96*A*a^12*b^6 - 42*A*a^13*b^5 - 78*A*a^14*b^4 + 36*A*a^15*b
^3 + 24*A*a^16*b^2 - 2*C*a^11*b^7 + 2*C*a^12*b^6 + 6*C*a^15*b^3 - 6*C*a^16*b^2 - 12*A*a^17*b - 4*C*a^17*b))/(a
^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) + (A*b*tan(c/2 + (d*x)/
2)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 -
32*a^15*b^3 - 8*a^16*b^2)*24i)/(a^4*(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 -
3*a^11*b^2)))*3i)/a^4))/a^4)/((16*(108*A^3*b^12 - 54*A^3*a*b^11 - 486*A^3*a^2*b^10 + 243*A^3*a^3*b^9 + 864*A^3
*a^4*b^8 - 378*A^3*a^5*b^7 - 702*A^3*a^6*b^6 + 216*A^3*a^7*b^5 + 216*A^3*a^8*b^4 + 12*A*C^2*a^11*b + 3*A*C^2*a
^7*b^5 + 12*A*C^2*a^9*b^3 + 18*A^2*C*a^3*b^9 + 18*A^2*C*a^4*b^8 - 18*A^2*C*a^5*b^7 - 54*A^2*C*a^7*b^5 - 54*A^2
*C*a^8*b^4 + 108*A^2*C*a^9*b^3 + 36*A^2*C*a^10*b^2))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12
*b^4 - 3*a^13*b^3 - 3*a^14*b^2) + (A*b*((8*tan(c/2 + (d*x)/2)*(72*A^2*b^12 + 4*C^2*a^12 - 72*A^2*a*b^11 - 288*
A^2*a^2*b^10 + 288*A^2*a^3*b^9 + 441*A^2*a^4*b^8 - 432*A^2*a^5*b^7 - 288*A^2*a^6*b^6 + 288*A^2*a^7*b^5 + 36*A^
2*a^8*b^4 - 72*A^2*a^9*b^3 + 36*A^2*a^10*b^2 + C^2*a^8*b^4 + 4*C^2*a^10*b^2 + 12*A*C*a^4*b^8 - 6*A*C*a^6*b^6 -
36*A*C*a^8*b^4 + 48*A*C*a^10*b^2))/(a^12*b + a^13 - a^6*b^7 - a^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 -
3*a^11*b^2) + (A*b*((8*(4*C*a^18 + 12*A*a^8*b^10 - 6*A*a^9*b^9 - 54*A*a^10*b^8 + 24*A*a^11*b^7 + 96*A*a^12*b^6
- 42*A*a^13*b^5 - 78*A*a^14*b^4 + 36*A*a^15*b^3 + 24*A*a^16*b^2 - 2*C*a^11*b^7 + 2*C*a^12*b^6 + 6*C*a^15*b^3
- 6*C*a^16*b^2 - 12*A*a^17*b - 4*C*a^17*b))/(a^15*b + a^16 - a^9*b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*
a^13*b^3 - 3*a^14*b^2) - (A*b*tan(c/2 + (d*x)/2)*(8*a^17*b - 8*a^8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^
7 - 48*a^12*b^6 + 48*a^13*b^5 + 32*a^14*b^4 - 32*a^15*b^3 - 8*a^16*b^2)*24i)/(a^4*(a^12*b + a^13 - a^6*b^7 - a
^7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2)))*3i)/a^4)*3i)/a^4 - (A*b*((8*tan(c/2 + (d*x)/2)*(72
*A^2*b^12 + 4*C^2*a^12 - 72*A^2*a*b^11 - 288*A^2*a^2*b^10 + 288*A^2*a^3*b^9 + 441*A^2*a^4*b^8 - 432*A^2*a^5*b^
7 - 288*A^2*a^6*b^6 + 288*A^2*a^7*b^5 + 36*A^2*a^8*b^4 - 72*A^2*a^9*b^3 + 36*A^2*a^10*b^2 + C^2*a^8*b^4 + 4*C^
2*a^10*b^2 + 12*A*C*a^4*b^8 - 6*A*C*a^6*b^6 - 36*A*C*a^8*b^4 + 48*A*C*a^10*b^2))/(a^12*b + a^13 - a^6*b^7 - a^
7*b^6 + 3*a^8*b^5 + 3*a^9*b^4 - 3*a^10*b^3 - 3*a^11*b^2) - (A*b*((8*(4*C*a^18 + 12*A*a^8*b^10 - 6*A*a^9*b^9 -
54*A*a^10*b^8 + 24*A*a^11*b^7 + 96*A*a^12*b^6 - 42*A*a^13*b^5 - 78*A*a^14*b^4 + 36*A*a^15*b^3 + 24*A*a^16*b^2
- 2*C*a^11*b^7 + 2*C*a^12*b^6 + 6*C*a^15*b^3 - 6*C*a^16*b^2 - 12*A*a^17*b - 4*C*a^17*b))/(a^15*b + a^16 - a^9*
b^7 - a^10*b^6 + 3*a^11*b^5 + 3*a^12*b^4 - 3*a^13*b^3 - 3*a^14*b^2) + (A*b*tan(c/2 + (d*x)/2)*(8*a^17*b - 8*a^
8*b^10 + 8*a^9*b^9 + 32*a^10*b^8 - 32*a^11*b^7 ...

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