\(\int \frac {\sec ^3(c+d x) (B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^2} \, dx\) [801]

Optimal result

Integrand size = 40, antiderivative size = 272 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (4 a b B-6 a^2 C-b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}+\frac {2 a^2 \left (2 a^2 b B-3 b^3 B-3 a^3 C+4 a b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^4 (a+b)^{3/2} d}+\frac {\left (2 a^2 b B-b^3 B-3 a^3 C+2 a b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (2 a b B-3 a^2 C+b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {4157, 4114, 4177, 4167, 4083, 3855, 3916, 2738, 214} \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (-6 a^2 C+4 a b B-b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}+\frac {a (b B-a C) \tan (c+d x) \sec ^2(c+d x)}{b d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\left (-3 a^2 C+2 a b B+b^2 C\right ) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left (a^2-b^2\right )}+\frac {2 a^2 \left (-3 a^3 C+2 a^2 b B+4 a b^2 C-3 b^3 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {\left (-3 a^3 C+2 a^2 b B+2 a b^2 C-b^3 B\right ) \tan (c+d x)}{b^3 d \left (a^2-b^2\right )} \]

[In]

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

[Out]

-1/2*((4*a*b*B - 6*a^2*C - b^2*C)*ArcTanh[Sin[c + d*x]])/(b^4*d) + (2*a^2*(2*a^2*b*B - 3*b^3*B - 3*a^3*C + 4*a
*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B
- b^3*B - 3*a^3*C + 2*a*b^2*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*b*B - 3*a^2*C + b^2*C)*Sec[c + d*x]*
Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^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 4177

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]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^4(c+d x) (B+C \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx \\ & = \frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {\sec ^2(c+d x) \left (2 a (b B-a C)-b (b B-a C) \sec (c+d x)-\left (2 a b B-3 a^2 C+b^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{b \left (a^2-b^2\right )} \\ & = -\frac {\left (2 a b B-3 a^2 C+b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-a \left (2 a b B-3 a^2 C+b^2 C\right )+b \left (2 a b B-a^2 C-b^2 C\right ) \sec (c+d x)+2 \left (2 a^2 b B-b^3 B-3 a^3 C+2 a b^2 C\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (2 a^2 b B-b^3 B-3 a^3 C+2 a b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (2 a b B-3 a^2 C+b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\int \frac {\sec (c+d x) \left (-a b \left (2 a b B-3 a^2 C+b^2 C\right )-\left (a^2-b^2\right ) \left (4 a b B-6 a^2 C-b^2 C\right ) \sec (c+d x)\right )}{a+b \sec (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )} \\ & = \frac {\left (2 a^2 b B-b^3 B-3 a^3 C+2 a b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (2 a b B-3 a^2 C+b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}-\frac {\left (4 a b B-6 a^2 C-b^2 C\right ) \int \sec (c+d x) \, dx}{2 b^4}+\frac {\left (a^2 \left (2 a^2 b B-3 b^3 B-3 a^3 C+4 a b^2 C\right )\right ) \int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{b^4 \left (a^2-b^2\right )} \\ & = -\frac {\left (4 a b B-6 a^2 C-b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}+\frac {\left (2 a^2 b B-b^3 B-3 a^3 C+2 a b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (2 a b B-3 a^2 C+b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (a^2 \left (2 a^2 b B-3 b^3 B-3 a^3 C+4 a b^2 C\right )\right ) \int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx}{b^5 \left (a^2-b^2\right )} \\ & = -\frac {\left (4 a b B-6 a^2 C-b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}+\frac {\left (2 a^2 b B-b^3 B-3 a^3 C+2 a b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (2 a b B-3 a^2 C+b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))}+\frac {\left (2 a^2 \left (2 a^2 b B-3 b^3 B-3 a^3 C+4 a b^2 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 ) d} \\ & = -\frac {\left (4 a b B-6 a^2 C-b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}+\frac {2 a^2 \left (2 a^2 b B-3 b^3 B-3 a^3 C+4 a b^2 C\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^4 (a+b)^{3/2} d}+\frac {\left (2 a^2 b B-b^3 B-3 a^3 C+2 a b^2 C\right ) \tan (c+d x)}{b^3 \left (a^2-b^2\right ) d}-\frac {\left (2 a b B-3 a^2 C+b^2 C\right ) \sec (c+d x) \tan (c+d x)}{2 b^2 \left (a^2-b^2\right ) d}+\frac {a (b B-a C) \sec ^2(c+d x) \tan (c+d x)}{b \left (a^2-b^2\right ) d (a+b \sec (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.97 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.61 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {2 a^2 \left (-2 a^2 b B+3 b^3 B+3 a^3 C-4 a b^2 C\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^4 \sqrt {a^2-b^2} \left (-a^2+b^2\right ) d}+\frac {\left (4 a b B-6 a^2 C-b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 b^4 d}+\frac {\left (-4 a b B+6 a^2 C+b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 b^4 d}+\frac {C}{4 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {C}{4 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {b B \sin \left (\frac {1}{2} (c+d x)\right )-2 a C \sin \left (\frac {1}{2} (c+d x)\right )}{b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {b B \sin \left (\frac {1}{2} (c+d x)\right )-2 a C \sin \left (\frac {1}{2} (c+d x)\right )}{b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {-a^3 b B \sin (c+d x)+a^4 C \sin (c+d x)}{b^3 (-a+b) (a+b) d (b+a \cos (c+d x))} \]

[In]

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

[Out]

(-2*a^2*(-2*a^2*b*B + 3*b^3*B + 3*a^3*C - 4*a*b^2*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)*d) + ((4*a*b*B - 6*a^2*C - b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*
b^4*d) + ((-4*a*b*B + 6*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*b^4*d) + C/(4*b^2*d*(Cos[(
c + d*x)/2] - Sin[(c + d*x)/2])^2) - C/(4*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (b*B*Sin[(c + d*x)/
2] - 2*a*C*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (b*B*Sin[(c + d*x)/2] - 2*a*C*Sin
[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (-(a^3*b*B*Sin[c + d*x]) + a^4*C*Sin[c + d*x])/
(b^3*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x]))

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {\frac {C}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 B b -4 C a -C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 B a b -6 C \,a^{2}-C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {2 a^{2} \left (\frac {a b \left (B b -C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 B \,a^{2} b -3 B \,b^{3}-3 a^{3} C +4 C a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}-\frac {C}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 B b -4 C a -C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 B a b +6 C \,a^{2}+C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}}{d}\)	\(330\)
default	\(\frac {\frac {C}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 B b -4 C a -C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (4 B a b -6 C \,a^{2}-C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{4}}-\frac {2 a^{2} \left (\frac {a b \left (B b -C a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {\left (2 B \,a^{2} b -3 B \,b^{3}-3 a^{3} C +4 C a \,b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4}}-\frac {C}{2 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 B b -4 C a -C b}{2 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-4 B a b +6 C \,a^{2}+C \,b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{4}}}{d}\)	\(330\)
risch	\(\text {Expression too large to display}\)	\(1276\)

[In]

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

[Out]

1/d*(1/2*C/b^2/(tan(1/2*d*x+1/2*c)-1)^2-1/2*(2*B*b-4*C*a-C*b)/b^3/(tan(1/2*d*x+1/2*c)-1)+1/2*(4*B*a*b-6*C*a^2-
C*b^2)/b^4*ln(tan(1/2*d*x+1/2*c)-1)-2*a^2/b^4*(a*b*(B*b-C*a)/(a^2-b^2)*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^
2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)-(2*B*a^2*b-3*B*b^3-3*C*a^3+4*C*a*b^2)/(a+b)/(a-b)/((a+b)*(a-b))^(1/2)*arctanh(
(a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))-1/2*C/b^2/(tan(1/2*d*x+1/2*c)+1)^2-1/2*(2*B*b-4*C*a-C*b)/b^3/(t
an(1/2*d*x+1/2*c)+1)+1/2/b^4*(-4*B*a*b+6*C*a^2+C*b^2)*ln(tan(1/2*d*x+1/2*c)+1))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 643 vs. \(2 (260) = 520\).

Time = 11.64 (sec) , antiderivative size = 1343, normalized size of antiderivative = 4.94 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*(2*((3*C*a^6 - 2*B*a^5*b - 4*C*a^4*b^2 + 3*B*a^3*b^3)*cos(d*x + c)^3 + (3*C*a^5*b - 2*B*a^4*b^2 - 4*C*a^3
*b^3 + 3*B*a^2*b^4)*cos(d*x + c)^2)*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2
*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b
^2)) + ((6*C*a^7 - 4*B*a^6*b - 11*C*a^5*b^2 + 8*B*a^4*b^3 + 4*C*a^3*b^4 - 4*B*a^2*b^5 + C*a*b^6)*cos(d*x + c)^
3 + (6*C*a^6*b - 4*B*a^5*b^2 - 11*C*a^4*b^3 + 8*B*a^3*b^4 + 4*C*a^2*b^5 - 4*B*a*b^6 + C*b^7)*cos(d*x + c)^2)*l
og(sin(d*x + c) + 1) - ((6*C*a^7 - 4*B*a^6*b - 11*C*a^5*b^2 + 8*B*a^4*b^3 + 4*C*a^3*b^4 - 4*B*a^2*b^5 + C*a*b^
6)*cos(d*x + c)^3 + (6*C*a^6*b - 4*B*a^5*b^2 - 11*C*a^4*b^3 + 8*B*a^3*b^4 + 4*C*a^2*b^5 - 4*B*a*b^6 + C*b^7)*c
os(d*x + c)^2)*log(-sin(d*x + c) + 1) + 2*(C*a^4*b^3 - 2*C*a^2*b^5 + C*b^7 - 2*(3*C*a^6*b - 2*B*a^5*b^2 - 5*C*
a^4*b^3 + 3*B*a^3*b^4 + 2*C*a^2*b^5 - B*a*b^6)*cos(d*x + c)^2 - (3*C*a^5*b^2 - 2*B*a^4*b^3 - 6*C*a^3*b^4 + 4*B
*a^2*b^5 + 3*C*a*b^6 - 2*B*b^7)*cos(d*x + c))*sin(d*x + c))/((a^5*b^4 - 2*a^3*b^6 + a*b^8)*d*cos(d*x + c)^3 +
(a^4*b^5 - 2*a^2*b^7 + b^9)*d*cos(d*x + c)^2), -1/4*(4*((3*C*a^6 - 2*B*a^5*b - 4*C*a^4*b^2 + 3*B*a^3*b^3)*cos(
d*x + c)^3 + (3*C*a^5*b - 2*B*a^4*b^2 - 4*C*a^3*b^3 + 3*B*a^2*b^4)*cos(d*x + c)^2)*sqrt(-a^2 + b^2)*arctan(-sq
rt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - ((6*C*a^7 - 4*B*a^6*b - 11*C*a^5*b^2 + 8*B*a
^4*b^3 + 4*C*a^3*b^4 - 4*B*a^2*b^5 + C*a*b^6)*cos(d*x + c)^3 + (6*C*a^6*b - 4*B*a^5*b^2 - 11*C*a^4*b^3 + 8*B*a
^3*b^4 + 4*C*a^2*b^5 - 4*B*a*b^6 + C*b^7)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) + ((6*C*a^7 - 4*B*a^6*b - 11*C
*a^5*b^2 + 8*B*a^4*b^3 + 4*C*a^3*b^4 - 4*B*a^2*b^5 + C*a*b^6)*cos(d*x + c)^3 + (6*C*a^6*b - 4*B*a^5*b^2 - 11*C
*a^4*b^3 + 8*B*a^3*b^4 + 4*C*a^2*b^5 - 4*B*a*b^6 + C*b^7)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(C*a^4*b^
3 - 2*C*a^2*b^5 + C*b^7 - 2*(3*C*a^6*b - 2*B*a^5*b^2 - 5*C*a^4*b^3 + 3*B*a^3*b^4 + 2*C*a^2*b^5 - B*a*b^6)*cos(
d*x + c)^2 - (3*C*a^5*b^2 - 2*B*a^4*b^3 - 6*C*a^3*b^4 + 4*B*a^2*b^5 + 3*C*a*b^6 - 2*B*b^7)*cos(d*x + c))*sin(d
*x + c))/((a^5*b^4 - 2*a^3*b^6 + a*b^8)*d*cos(d*x + c)^3 + (a^4*b^5 - 2*a^2*b^7 + b^9)*d*cos(d*x + c)^2)]

Sympy [F]

\[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\left (B + C \sec {\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.41 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=-\frac {\frac {4 \, {\left (3 \, C a^{5} - 2 \, B a^{4} b - 4 \, C a^{3} b^{2} + 3 \, B a^{2} b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {4 \, {\left (C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a^{2} b^{3} - b^{5}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )}} - \frac {{\left (6 \, C a^{2} - 4 \, B a b + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{4}} + \frac {{\left (6 \, C a^{2} - 4 \, B a b + C b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{4}} - \frac {2 \, {\left (4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} b^{3}}}{2 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 28.30 (sec) , antiderivative size = 6677, normalized size of antiderivative = 24.55 \[ \int \frac {\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

- ((tan(c/2 + (d*x)/2)^5*(6*C*a^4 - 2*B*b^4 + C*b^4 + 2*B*a^2*b^2 - 5*C*a^2*b^2 + 2*B*a*b^3 - 4*B*a^3*b + 3*C*
a*b^3 - 3*C*a^3*b))/((a*b^3 - b^4)*(a + b)) + (2*tan(c/2 + (d*x)/2)^3*(C*b^4 - 6*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b
^3 + 4*B*a^3*b))/(b*(a*b^2 - b^3)*(a + b)) + (tan(c/2 + (d*x)/2)*(2*B*b^4 + 6*C*a^4 + C*b^4 - 2*B*a^2*b^2 - 5*
C*a^2*b^2 + 2*B*a*b^3 - 4*B*a^3*b - 3*C*a*b^3 + 3*C*a^3*b))/(b^3*(a + b)*(a - b)))/(d*(a + b - tan(c/2 + (d*x)
/2)^2*(3*a + b) - tan(c/2 + (d*x)/2)^6*(a - b) + tan(c/2 + (d*x)/2)^4*(3*a - b))) - (atan(((((8*tan(c/2 + (d*x
)/2)*(72*C^2*a^10 + C^2*b^10 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b^8 - 32*B^2*a^3*b^7 + 20*B^2*a^4*b^6 +
64*B^2*a^5*b^5 - 64*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*
a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 - 8*B*C*a*b^9 - 96*B*C*a^9*b + 1
6*B*C*a^2*b^8 - 40*B*C*a^3*b^7 + 64*B*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B*C*a^6*b^4 + 176*B*C*a^7*b^3 + 96*B*C*
a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (((8*(2*C*b^15 + 12*B*a^2*b^13 + 12*B*a^3*b^12 - 20*B*a^4*b^11 -
4*B*a^5*b^10 + 8*B*a^6*b^9 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*
C*a^7*b^8 - 8*B*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (4*tan(c/2 + (d*x)/2)*(6*C*a^2 + C*b^2 - 4*B*a
*b)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 -
a^3*b^6)))*(6*C*a^2 + C*b^2 - 4*B*a*b))/(2*b^4))*(6*C*a^2 + C*b^2 - 4*B*a*b)*1i)/(2*b^4) + (((8*tan(c/2 + (d*
x)/2)*(72*C^2*a^10 + C^2*b^10 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b^8 - 32*B^2*a^3*b^7 + 20*B^2*a^4*b^6
+ 64*B^2*a^5*b^5 - 64*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2
*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 - 8*B*C*a*b^9 - 96*B*C*a^9*b +
16*B*C*a^2*b^8 - 40*B*C*a^3*b^7 + 64*B*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B*C*a^6*b^4 + 176*B*C*a^7*b^3 + 96*B*C
*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (((8*(2*C*b^15 + 12*B*a^2*b^13 + 12*B*a^3*b^12 - 20*B*a^4*b^11
- 4*B*a^5*b^10 + 8*B*a^6*b^9 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12
*C*a^7*b^8 - 8*B*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (4*tan(c/2 + (d*x)/2)*(6*C*a^2 + C*b^2 - 4*B*
a*b)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7
- a^3*b^6)))*(6*C*a^2 + C*b^2 - 4*B*a*b))/(2*b^4))*(6*C*a^2 + C*b^2 - 4*B*a*b)*1i)/(2*b^4))/((16*(108*C^3*a^11
- 54*C^3*a^10*b - 48*B^3*a^4*b^7 - 24*B^3*a^5*b^6 + 80*B^3*a^6*b^5 + 16*B^3*a^7*b^4 - 32*B^3*a^8*b^3 + 4*C^3*
a^3*b^8 - 4*C^3*a^4*b^7 + 41*C^3*a^5*b^6 - 9*C^3*a^6*b^5 + 63*C^3*a^7*b^4 + 81*C^3*a^8*b^3 - 216*C^3*a^9*b^2 -
216*B*C^2*a^10*b - 3*B*C^2*a^2*b^9 + 3*B*C^2*a^3*b^8 - 63*B*C^2*a^4*b^7 + 15*B*C^2*a^5*b^6 - 186*B*C^2*a^6*b^
5 - 162*B*C^2*a^7*b^4 + 468*B*C^2*a^8*b^3 + 108*B*C^2*a^9*b^2 + 24*B^2*C*a^3*b^8 - 6*B^2*C*a^4*b^7 + 168*B^2*C
*a^5*b^6 + 108*B^2*C*a^6*b^5 - 336*B^2*C*a^7*b^4 - 72*B^2*C*a^8*b^3 + 144*B^2*C*a^9*b^2))/(a*b^11 + b^12 - a^2
*b^10 - a^3*b^9) - (((8*tan(c/2 + (d*x)/2)*(72*C^2*a^10 + C^2*b^10 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b
^8 - 32*B^2*a^3*b^7 + 20*B^2*a^4*b^6 + 64*B^2*a^5*b^5 - 64*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*
C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^
8*b^2 - 8*B*C*a*b^9 - 96*B*C*a^9*b + 16*B*C*a^2*b^8 - 40*B*C*a^3*b^7 + 64*B*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B
*C*a^6*b^4 + 176*B*C*a^7*b^3 + 96*B*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (((8*(2*C*b^15 + 12*B*a^2*
b^13 + 12*B*a^3*b^12 - 20*B*a^4*b^11 - 4*B*a^5*b^10 + 8*B*a^6*b^9 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^
11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*B*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (4*tan(c
/2 + (d*x)/2)*(6*C*a^2 + C*b^2 - 4*B*a*b)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a
^6*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*(6*C*a^2 + C*b^2 - 4*B*a*b))/(2*b^4))*(6*C*a^2 + C*b^2 - 4*B
*a*b))/(2*b^4) + (((8*tan(c/2 + (d*x)/2)*(72*C^2*a^10 + C^2*b^10 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b^8
- 32*B^2*a^3*b^7 + 20*B^2*a^4*b^6 + 64*B^2*a^5*b^5 - 64*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*C^
2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*
b^2 - 8*B*C*a*b^9 - 96*B*C*a^9*b + 16*B*C*a^2*b^8 - 40*B*C*a^3*b^7 + 64*B*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B*C
*a^6*b^4 + 176*B*C*a^7*b^3 + 96*B*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (((8*(2*C*b^15 + 12*B*a^2*b^
13 + 12*B*a^3*b^12 - 20*B*a^4*b^11 - 4*B*a^5*b^10 + 8*B*a^6*b^9 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11
+ 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*B*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (4*tan(c/2
+ (d*x)/2)*(6*C*a^2 + C*b^2 - 4*B*a*b)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6
*b^8))/(b^4*(a*b^8 + b^9 - a^2*b^7 - a^3*b^6)))*(6*C*a^2 + C*b^2 - 4*B*a*b))/(2*b^4))*(6*C*a^2 + C*b^2 - 4*B*a
*b))/(2*b^4)))*(6*C*a^2 + C*b^2 - 4*B*a*b)*1i)/(b^4*d) - (a^2*atan(((a^2*((8*tan(c/2 + (d*x)/2)*(72*C^2*a^10 +
C^2*b^10 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b^8 - 32*B^2*a^3*b^7 + 20*B^2*a^4*b^6 + 64*B^2*a^5*b^5 - 6
4*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^
5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 - 8*B*C*a*b^9 - 96*B*C*a^9*b + 16*B*C*a^2*b^8 - 40*
B*C*a^3*b^7 + 64*B*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B*C*a^6*b^4 + 176*B*C*a^7*b^3 + 96*B*C*a^8*b^2))/(a*b^8 +
b^9 - a^2*b^7 - a^3*b^6) + (a^2*((8*(2*C*b^15 + 12*B*a^2*b^13 + 12*B*a^3*b^12 - 20*B*a^4*b^11 - 4*B*a^5*b^10 +
8*B*a^6*b^9 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*B
*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (8*a^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^
3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^
8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*((a + b)^3*(a - b)^3)^(1/2)*
(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*((a + b)^3*(a - b)^3)^(
1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2)*1i)/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4) + (a^2*((8*tan(c
/2 + (d*x)/2)*(72*C^2*a^10 + C^2*b^10 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b^8 - 32*B^2*a^3*b^7 + 20*B^2*
a^4*b^6 + 64*B^2*a^5*b^5 - 64*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7
+ 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 - 8*B*C*a*b^9 - 96*B*C*
a^9*b + 16*B*C*a^2*b^8 - 40*B*C*a^3*b^7 + 64*B*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B*C*a^6*b^4 + 176*B*C*a^7*b^3
+ 96*B*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (a^2*((8*(2*C*b^15 + 12*B*a^2*b^13 + 12*B*a^3*b^12 - 20
*B*a^4*b^11 - 4*B*a^5*b^10 + 8*B*a^6*b^9 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*
a^6*b^9 - 12*C*a^7*b^8 - 8*B*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*a^2*tan(c/2 + (d*x)/2)*((a + b
)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4
*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*
((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b
^4))*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2)*1i)/(b^10 - 3*a^2*b^8 + 3*a^4*b^6
- a^6*b^4))/((16*(108*C^3*a^11 - 54*C^3*a^10*b - 48*B^3*a^4*b^7 - 24*B^3*a^5*b^6 + 80*B^3*a^6*b^5 + 16*B^3*a^
7*b^4 - 32*B^3*a^8*b^3 + 4*C^3*a^3*b^8 - 4*C^3*a^4*b^7 + 41*C^3*a^5*b^6 - 9*C^3*a^6*b^5 + 63*C^3*a^7*b^4 + 81*
C^3*a^8*b^3 - 216*C^3*a^9*b^2 - 216*B*C^2*a^10*b - 3*B*C^2*a^2*b^9 + 3*B*C^2*a^3*b^8 - 63*B*C^2*a^4*b^7 + 15*B
*C^2*a^5*b^6 - 186*B*C^2*a^6*b^5 - 162*B*C^2*a^7*b^4 + 468*B*C^2*a^8*b^3 + 108*B*C^2*a^9*b^2 + 24*B^2*C*a^3*b^
8 - 6*B^2*C*a^4*b^7 + 168*B^2*C*a^5*b^6 + 108*B^2*C*a^6*b^5 - 336*B^2*C*a^7*b^4 - 72*B^2*C*a^8*b^3 + 144*B^2*C
*a^9*b^2))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) + (a^2*((8*tan(c/2 + (d*x)/2)*(72*C^2*a^10 + C^2*b^10 - 2*C^2*
a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b^8 - 32*B^2*a^3*b^7 + 20*B^2*a^4*b^6 + 64*B^2*a^5*b^5 - 64*B^2*a^6*b^4 - 32
*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26*C^2*a^5*b^5 + 17*C^2*a^6
*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 - 8*B*C*a*b^9 - 96*B*C*a^9*b + 16*B*C*a^2*b^8 - 40*B*C*a^3*b^7 + 64*B
*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B*C*a^6*b^4 + 176*B*C*a^7*b^3 + 96*B*C*a^8*b^2))/(a*b^8 + b^9 - a^2*b^7 - a^
3*b^6) + (a^2*((8*(2*C*b^15 + 12*B*a^2*b^13 + 12*B*a^3*b^12 - 20*B*a^4*b^11 - 4*B*a^5*b^10 + 8*B*a^6*b^9 + 6*C
*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^8 - 8*B*a*b^14))/(a*b^11
+ b^12 - a^2*b^10 - a^3*b^9) + (8*a^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*
a^2*b - 4*C*a*b^2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8*a^6*b^8))/((a*b^8 + b^9
- a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3
- 2*B*a^2*b - 4*C*a*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*((a + b)^3*(a - b)^3)^(1/2)*(3*B*b^3 + 3*
C*a^3 - 2*B*a^2*b - 4*C*a*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4) - (a^2*((8*tan(c/2 + (d*x)/2)*(72*C^2
*a^10 + C^2*b^10 - 2*C^2*a*b^9 - 72*C^2*a^9*b + 16*B^2*a^2*b^8 - 32*B^2*a^3*b^7 + 20*B^2*a^4*b^6 + 64*B^2*a^5*
b^5 - 64*B^2*a^6*b^4 - 32*B^2*a^7*b^3 + 32*B^2*a^8*b^2 + 11*C^2*a^2*b^8 - 20*C^2*a^3*b^7 + 23*C^2*a^4*b^6 - 26
*C^2*a^5*b^5 + 17*C^2*a^6*b^4 + 120*C^2*a^7*b^3 - 120*C^2*a^8*b^2 - 8*B*C*a*b^9 - 96*B*C*a^9*b + 16*B*C*a^2*b^
8 - 40*B*C*a^3*b^7 + 64*B*C*a^4*b^6 - 40*B*C*a^5*b^5 - 176*B*C*a^6*b^4 + 176*B*C*a^7*b^3 + 96*B*C*a^8*b^2))/(a
*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (a^2*((8*(2*C*b^15 + 12*B*a^2*b^13 + 12*B*a^3*b^12 - 20*B*a^4*b^11 - 4*B*a^5
*b^10 + 8*B*a^6*b^9 + 6*C*a^2*b^13 - 16*C*a^3*b^12 - 14*C*a^4*b^11 + 28*C*a^5*b^10 + 6*C*a^6*b^9 - 12*C*a^7*b^
8 - 8*B*a*b^14))/(a*b^11 + b^12 - a^2*b^10 - a^3*b^9) - (8*a^2*tan(c/2 + (d*x)/2)*((a + b)^3*(a - b)^3)^(1/2)*
(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2)*(8*a*b^13 - 8*a^2*b^12 - 16*a^3*b^11 + 16*a^4*b^10 + 8*a^5*b^9 - 8
*a^6*b^8))/((a*b^8 + b^9 - a^2*b^7 - a^3*b^6)*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*((a + b)^3*(a - b)^3)
^(1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))*((a + b)^3*(a -
b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2))/(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4)))*((a + b)^3
*(a - b)^3)^(1/2)*(3*B*b^3 + 3*C*a^3 - 2*B*a^2*b - 4*C*a*b^2)*2i)/(d*(b^10 - 3*a^2*b^8 + 3*a^4*b^6 - a^6*b^4))