3.9.8 \(\int \frac {\sec ^3(c+d x) (B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [808]

Optimal. Leaf size=289 \[ \frac {(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}-\frac {a \left (2 a^4 b B-5 a^2 b^3 B+6 b^5 B-6 a^5 C+15 a^3 b^2 C-12 a b^4 C\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}-\frac {\left (a b B-3 a^2 C+2 b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}-\frac {a^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right ) \tan (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \]

[Out]

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

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Rubi [A]
time = 1.03, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {4157, 4114, 4175, 4167, 4083, 3855, 3916, 2738, 214} \begin {gather*} \frac {a (b B-a C) \tan (c+d x) \sec ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}-\frac {\left (-3 a^2 C+a b B+2 b^2 C\right ) \tan (c+d x)}{2 b^3 d \left (a^2-b^2\right )}-\frac {a^2 \left (-3 a^3 C+a^2 b B+6 a b^2 C-4 b^3 B\right ) \tan (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))}-\frac {a \left (-6 a^5 C+2 a^4 b B+15 a^3 b^2 C-5 a^2 b^3 B-12 a b^4 C+6 b^5 B\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*(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) - (a*(2*a^4*b*B - 5*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 15*a^3*b^2*C
- 12*a*b^4*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*B - 3*a^2*C + 2*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (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^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*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 4114

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

Rule 4157

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

Rubi steps

\begin {align*} \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx &=\int \frac {\sec ^4(c+d x) (B+C \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx\\ &=\frac {a (b B-a C) \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 a (b B-a C)-2 b (b B-a C) \sec (c+d x)-\left (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 {a (b B-a C) \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^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 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 (a b \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 C\right )+\left (a^2-b^2\right ) \left (a^2 b B-2 b^3 B-3 a^3 C+4 a b^2 C\right ) \sec (c+d x)-b \left (a^2-b^2\right ) \left (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 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 {a (b B-a C) \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^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 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 (a b^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 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 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 {a (b B-a C) \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^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 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 (a \left (2 a^4 b B-5 a^2 b^3 B+6 b^5 B-6 a^5 C+15 a^3 b^2 C-12 a b^4 C\right )\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 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 {a (b B-a C) \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^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 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 (a \left (2 a^4 b B-5 a^2 b^3 B+6 b^5 B-6 a^5 C+15 a^3 b^2 C-12 a b^4 C\right )\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 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 {a (b B-a C) \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^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 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 (a \left (2 a^4 b B-5 a^2 b^3 B+6 b^5 B-6 a^5 C+15 a^3 b^2 C-12 a b^4 C\right )\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 {a \left (2 a^4 b B-5 a^2 b^3 B+6 b^5 B-6 a^5 C+15 a^3 b^2 C-12 a 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 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 {a (b B-a C) \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^2 \left (a^2 b B-4 b^3 B-3 a^3 C+6 a b^2 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*}

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Mathematica [A]
time = 6.52, size = 418, normalized size = 1.45 \begin {gather*} \frac {a \left (2 a^4 b B-5 a^2 b^3 B+6 b^5 B-6 a^5 C+15 a^3 b^2 C-12 a b^4 C\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2} \left (-a^2+b^2\right )^2 d}+\frac {(-b B+3 a C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{b^4 d}+\frac {(b B-3 a C) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{b^4 d}+\frac {C \sin \left (\frac {1}{2} (c+d x)\right )}{b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {C \sin \left (\frac {1}{2} (c+d x)\right )}{b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {a^2 b B \sin (c+d x)-a^3 C \sin (c+d x)}{2 b^2 (-a+b) (a+b) d (b+a \cos (c+d x))^2}+\frac {-2 a^4 b B \sin (c+d x)+5 a^2 b^3 B \sin (c+d x)+4 a^5 C \sin (c+d x)-7 a^3 b^2 C \sin (c+d x)}{2 b^3 (-a+b)^2 (a+b)^2 d (b+a \cos (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.78, size = 380, normalized size = 1.31

method	result	size
derivativedivides	\(\frac {-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-b B +3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (b B -3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}+\frac {2 a \left (\frac {\frac {\left (2 a^{2} b B -a \,b^{2} B -6 b^{3} B -4 a^{3} C +a^{2} b C +8 C \,b^{2} a \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 a^{2} b B +a \,b^{2} B -6 b^{3} B -4 a^{3} C -a^{2} b C +8 C \,b^{2} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (2 a^{4} b B -5 a^{2} b^{3} B +6 b^{5} B -6 a^{5} C +15 a^{3} b^{2} C -12 C a \,b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}}{d}\)	\(380\)
default	\(\frac {-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-b B +3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{b^{4}}-\frac {C}{b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (b B -3 a C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{4}}+\frac {2 a \left (\frac {\frac {\left (2 a^{2} b B -a \,b^{2} B -6 b^{3} B -4 a^{3} C +a^{2} b C +8 C \,b^{2} a \right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {b a \left (2 a^{2} b B +a \,b^{2} B -6 b^{3} B -4 a^{3} C -a^{2} b C +8 C \,b^{2} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}-\frac {\left (2 a^{4} b B -5 a^{2} b^{3} B +6 b^{5} B -6 a^{5} C +15 a^{3} b^{2} C -12 C a \,b^{4}\right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}}{d}\)	\(380\)
risch	\(\text {Expression too large to display}\)	\(1691\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^3*(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*a/b^4*((1/2*(2*B*a^2*b-B*a*b^2-6*B*b^3-4*C*a^3+C*a^2*b+8*C*a*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*(2*B*a^2*b+B*a*b^2-6*B*b^3-4*C*a^3-C*a^2*b+8*C*a*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*(2*B*a^4*b-5*B*a^2*b^3
+6*B*b^5-6*C*a^5+15*C*a^3*b^2-12*C*a*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))))

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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(sec(d*x+c)^3*(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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1026 vs. \(2 (277) = 554\).
time = 78.25, size = 2111, normalized size = 7.30 \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*(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 + 12*C*a^4*b^4 - 6*B*a^3*b^5)*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 + 12*C*a^3*b^5 - 6*B*a^2*b^6)*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 + 12*C*a^2*b^6 - 6*B*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*((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 + 13*C*a^4*b^5 - 5*B*a^3*b^6 - 2*C*a^2*b^7)*cos(d*x +
c)^2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 - 25*C*a^5*b^4 + 9*B*a^4*b^5 + 20*C*a^3*b^6 - 6*B*a^2*b^7 - 4*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 - 15*C*a^6*b^2 + 5*B*a^5*b^3 + 12*C*a^4*b^4 - 6*B*a^3*b^5)*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 + 12*C*a^3*b^5 - 6*B*a^2*b^6)*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 + 12*C*a^2*b^6 - 6*B*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))) - ((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 + 13*C*a^4*b^5 - 5*B*a^3*b^6 - 2*C*a^2*b^7)*cos(d*x + c)^
2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 - 25*C*a^5*b^4 + 9*B*a^4*b^5 + 20*C*a^3*b^6 - 6*B*a^2*b^7 - 4*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))]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (B + C \sec {\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (277) = 554\).
time = 0.57, size = 581, normalized size = 2.01 \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} + 12 \, C a^{2} b^{4} - 6 \, B a b^{5}\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} + 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} - 6 \, B a^{2} b^{4} \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 ) - 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 ) - 6 \, B a^{2} b^{4} \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*(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 + 12*C*a^2*b^4 - 6*B*a*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/
2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^4*b^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 + 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 - 6*B*a^2*b^4*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) - 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) - 6*B*a^2*b^4*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

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Mupad [B]
time = 17.50, size = 2500, normalized size = 8.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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)^5*(6*C*a^5 - 2*C*b^5 + 6*B*a^2*b^3 + B*a^3*b^2 + 4*C*a^2*b^3 - 12*C*a^3*b^2 - 2*B*a^4*b +
2*C*a*b^4 - 3*C*a^4*b))/((a*b^3 - b^4)*(a + b)^2) + (tan(c/2 + (d*x)/2)*(6*C*a^5 + 2*C*b^5 + 6*B*a^2*b^3 - B*
a^3*b^2 - 4*C*a^2*b^3 - 12*C*a^3*b^2 - 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))
- (2*tan(c/2 + (d*x)/2)^3*(6*C*a^6 - 2*C*b^6 + 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*B^2*b^12 + 72*C^2*a^12 - 8*B^2*a*b^11 - 72*C^2*a^11*b + 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*B*C*a*b^11 - 48*B*C*a^11*b + 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) + (((8*(4*B*b^18 - 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 - 12*B*a*b^17 - 12*C*a*b^17))/(a*b^15 + b^1
6 - 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*b - 3*C*a))/b^4)*1i)/b^4 + ((B*b - 3*C*a)*((8*tan(c/2 + (d*x)/2)*(4*B^2*b^12 + 72*C^2*a^12 - 8*B
^2*a*b^11 - 72*C^2*a^11*b + 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*B*C*a*b^11 - 48*B*C*a^11*b + 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) - (((8*(4*B*b^18 - 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 - 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*b - 3*C*a))/b^4)*1i)/b^4)/((16*
(108*C^3*a^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 - 7
02*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 - 108*B*C^2*a^11*b - 2
52*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 - 1
98*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*b^3 + 36*B^2*C*a^1
0*b^2))/(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) - ((B*b - 3*C
*a)*((8*tan(c/2 + (d*x)/2)*(4*B^2*b^12 + 72*C^2*a^12 - 8*B^2*a*b^11 - 72*C^2*a^11*b + 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*B*C*a*b^11 - 48*B*C*a^11*b + 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) + (((8*(4*B*b^18 - 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...

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