3.1106 \(\int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=280 \[ \frac {\left (2 i c^3-4 c^2 d-i c d^2-2 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{5/2}}+\frac {c (-3 d+2 i c) \sqrt {c+d \tan (e+f x)}}{16 f (c+i d)^2 \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(-2 d+3 i c) \sqrt {c+d \tan (e+f x)}}{24 a f (c+i d) (a+i a \tan (e+f x))^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3} \]

[Out]

1/16*(2*I*c^3-4*c^2*d-I*c*d^2-2*d^3)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^3/(c+I*d)^(5/2)/f-1/8*I*a
rctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I*d)^(1/2)/a^3/f+1/6*I*(c+d*tan(f*x+e))^(1/2)/f/(a+I*a*tan(f*x
+e))^3+1/24*(3*I*c-2*d)*(c+d*tan(f*x+e))^(1/2)/a/(c+I*d)/f/(a+I*a*tan(f*x+e))^2+1/16*c*(2*I*c-3*d)*(c+d*tan(f*
x+e))^(1/2)/(c+I*d)^2/f/(a^3+I*a^3*tan(f*x+e))

________________________________________________________________________________________

Rubi [A]  time = 0.98, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3557, 3596, 3539, 3537, 63, 208} \[ \frac {\left (-4 c^2 d+2 i c^3-i c d^2-2 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{5/2}}+\frac {c (-3 d+2 i c) \sqrt {c+d \tan (e+f x)}}{16 f (c+i d)^2 \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(-2 d+3 i c) \sqrt {c+d \tan (e+f x)}}{24 a f (c+i d) (a+i a \tan (e+f x))^2}+\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((-I/8)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + (((2*I)*c^3 - 4*c^2*d - I*c*d
^2 - 2*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(5/2)*f) + ((I/6)*Sqrt[c + d*Ta
n[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c - 2*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*(c + I*d)*f*(a +
I*a*Tan[e + f*x])^2) + (c*((2*I)*c - 3*d)*Sqrt[c + d*Tan[e + f*x]])/(16*(c + I*d)^2*f*(a^3 + I*a^3*Tan[e + f*x
]))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3557

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> -Sim
p[(b*(a + b*Tan[e + f*x])^m*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*m), x] + Dist[1/(4*a^2*m), Int[((a + b*Tan[e + f*
x])^(m + 1)*Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x])/Sqrt[c + d*Tan[e + f*x]], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && IntegersQ[2
*m]

Rule 3596

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

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-a (6 c-i d)-5 a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx}{12 a^2}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {\int \frac {-3 a^2 \left (3 c d-i \left (4 c^2+2 d^2\right )\right )+3 a^2 (3 i c-2 d) d \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{48 a^4 (i c-d)}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\int \frac {3 a^3 \left (4 c^3+6 i c^2 d+c d^2+4 i d^3\right )+3 a^3 c (2 c+3 i d) d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{96 a^6 (c+i d)^2}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(c-i d) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^3}+\frac {\left (2 c^3+4 i c^2 d-c d^2+2 i d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{32 a^3 (c+i d)^2}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c+d) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{16 a^3 f}-\frac {\left (i \left (2 c^3+4 i c^2 d-c d^2+2 i d^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{32 a^3 (c+i d)^2 f}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(c-i d) \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{8 a^3 d f}-\frac {\left (2 c^3+4 i c^2 d-c d^2+2 i d^3\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{16 a^3 (c+i d)^2 d f}\\ &=-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}-\frac {\left (4 c^2 d-i \left (2 c^3-c d^2+2 i d^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{5/2} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 2.85, size = 329, normalized size = 1.18 \[ \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac {2 \cos (e+f x) (\sin (3 f x)+i \cos (3 f x)) \sqrt {c+d \tan (e+f x)} \left (i \left (9 c^2+14 i c d-2 d^2\right ) \sin (2 (e+f x))+\left (13 c^2+22 i c d-6 d^2\right ) \cos (2 (e+f x))+7 c^2+13 i c d-6 d^2\right )}{3 (c+i d)^2}+\frac {2 (\cos (3 e)+i \sin (3 e)) \left (\sqrt {-c+i d} \left (-2 i c^3+4 c^2 d+i c d^2+2 d^3\right ) \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )+2 (d+i c) (-c-i d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right )}{(-c-i d)^{5/2} \sqrt {-c+i d}}\right )}{32 f (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3,x]

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((2*(Sqrt[-c + I*d]*((-2*I)*c^3 + 4*c^2*d + I*c*d^2 + 2*d^3)*ArcTan[
Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + 2*(-c - I*d)^(5/2)*(I*c + d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-
c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/((-c - I*d)^(5/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*
f*x])*(7*c^2 + (13*I)*c*d - 6*d^2 + (13*c^2 + (22*I)*c*d - 6*d^2)*Cos[2*(e + f*x)] + I*(9*c^2 + (14*I)*c*d - 2
*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d)^2)))/(32*f*(a + I*a*Tan[e + f*x])^3)

________________________________________________________________________________________

fricas [B]  time = 1.05, size = 1448, normalized size = 5.17 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/96*(3*(a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f*sqrt(-(c - I*d)/(a^6*f^2))*e^(6*I*f*x + 6*I*e)*log(1/8*((16*I*a^3*
f*e^(2*I*f*x + 2*I*e) + 16*I*a^3*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*
sqrt(-(c - I*d)/(a^6*f^2)) + 16*(c - I*d)*e^(2*I*f*x + 2*I*e) + 16*c)*e^(-2*I*f*x - 2*I*e)) - 3*(a^3*c^2 + 2*I
*a^3*c*d - a^3*d^2)*f*sqrt(-(c - I*d)/(a^6*f^2))*e^(6*I*f*x + 6*I*e)*log(1/8*((-16*I*a^3*f*e^(2*I*f*x + 2*I*e)
- 16*I*a^3*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(c - I*d)/(a^6*
f^2)) + 16*(c - I*d)*e^(2*I*f*x + 2*I*e) + 16*c)*e^(-2*I*f*x - 2*I*e)) + 24*(a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*
f*sqrt((4*I*c^6 - 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 + 4*c*d^5 - 4*I*d^6)/((-256*I*a^6*c^5 + 1280*a^6*c^4*
d + 2560*I*a^6*c^3*d^2 - 2560*a^6*c^2*d^3 - 1280*I*a^6*c*d^4 + 256*a^6*d^5)*f^2))*e^(6*I*f*x + 6*I*e)*log(-(2*
c^4 + 6*I*c^3*d - 5*c^2*d^2 + I*c*d^3 - 2*d^4 - ((16*I*a^3*c^3 - 48*a^3*c^2*d - 48*I*a^3*c*d^2 + 16*a^3*d^3)*f
*e^(2*I*f*x + 2*I*e) + (16*I*a^3*c^3 - 48*a^3*c^2*d - 48*I*a^3*c*d^2 + 16*a^3*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f
*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((4*I*c^6 - 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 + 4*c
*d^5 - 4*I*d^6)/((-256*I*a^6*c^5 + 1280*a^6*c^4*d + 2560*I*a^6*c^3*d^2 - 2560*a^6*c^2*d^3 - 1280*I*a^6*c*d^4 +
256*a^6*d^5)*f^2)) + (2*c^4 + 4*I*c^3*d - c^2*d^2 + 2*I*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((16
*I*a^3*c^3 - 48*a^3*c^2*d - 48*I*a^3*c*d^2 + 16*a^3*d^3)*f)) - 24*(a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f*sqrt((4*
I*c^6 - 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 + 4*c*d^5 - 4*I*d^6)/((-256*I*a^6*c^5 + 1280*a^6*c^4*d + 2560*I
*a^6*c^3*d^2 - 2560*a^6*c^2*d^3 - 1280*I*a^6*c*d^4 + 256*a^6*d^5)*f^2))*e^(6*I*f*x + 6*I*e)*log(-(2*c^4 + 6*I*
c^3*d - 5*c^2*d^2 + I*c*d^3 - 2*d^4 - ((-16*I*a^3*c^3 + 48*a^3*c^2*d + 48*I*a^3*c*d^2 - 16*a^3*d^3)*f*e^(2*I*f
*x + 2*I*e) + (-16*I*a^3*c^3 + 48*a^3*c^2*d + 48*I*a^3*c*d^2 - 16*a^3*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I
*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((4*I*c^6 - 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 + 4*c*d^5 - 4
*I*d^6)/((-256*I*a^6*c^5 + 1280*a^6*c^4*d + 2560*I*a^6*c^3*d^2 - 2560*a^6*c^2*d^3 - 1280*I*a^6*c*d^4 + 256*a^6
*d^5)*f^2)) + (2*c^4 + 4*I*c^3*d - c^2*d^2 + 2*I*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((16*I*a^3*c
^3 - 48*a^3*c^2*d - 48*I*a^3*c*d^2 + 16*a^3*d^3)*f)) - (-2*I*c^2 + 4*c*d + 2*I*d^2 + (-11*I*c^2 + 18*c*d + 4*I
*d^2)*e^(6*I*f*x + 6*I*e) + (-18*I*c^2 + 31*c*d + 10*I*d^2)*e^(4*I*f*x + 4*I*e) + (-9*I*c^2 + 17*c*d + 8*I*d^2
)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-6*I*f*x
- 6*I*e)/((a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f)

________________________________________________________________________________________

giac [B]  time = 0.85, size = 633, normalized size = 2.26 \[ \frac {2 \, {\left (2 \, c^{3} + 4 i \, c^{2} d - c d^{2} + 2 i \, d^{3}\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (8 i \, a^{3} c^{2} f - 16 \, a^{3} c d f - 8 i \, a^{3} d^{2} f\right )} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {6 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{2} d - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{3} d + 6 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{4} d + 9 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d^{2} - 36 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d^{2} + 27 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d^{2} + 28 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{3} - 48 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{3} + 4 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{4} - 39 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{4} + 12 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{5}}{{\left (48 \, a^{3} c^{2} f + 96 i \, a^{3} c d f - 48 \, a^{3} d^{2} f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{3}} - \frac {{\left (-i \, c - d\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{2 \, a^{3} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

2*(2*c^3 + 4*I*c^2*d - c*d^2 + 2*I*d^3)*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x
+ e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c
+ 8*sqrt(c^2 + d^2))))/((8*I*a^3*c^2*f - 16*a^3*c*d*f - 8*I*a^3*d^2*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c
- sqrt(c^2 + d^2)) + 1)) + (6*(d*tan(f*x + e) + c)^(5/2)*c^2*d - 12*(d*tan(f*x + e) + c)^(3/2)*c^3*d + 6*sqrt(
d*tan(f*x + e) + c)*c^4*d + 9*I*(d*tan(f*x + e) + c)^(5/2)*c*d^2 - 36*I*(d*tan(f*x + e) + c)^(3/2)*c^2*d^2 + 2
7*I*sqrt(d*tan(f*x + e) + c)*c^3*d^2 + 28*(d*tan(f*x + e) + c)^(3/2)*c*d^3 - 48*sqrt(d*tan(f*x + e) + c)*c^2*d
^3 + 4*I*(d*tan(f*x + e) + c)^(3/2)*d^4 - 39*I*sqrt(d*tan(f*x + e) + c)*c*d^4 + 12*sqrt(d*tan(f*x + e) + c)*d^
5)/((48*a^3*c^2*f + 96*I*a^3*c*d*f - 48*a^3*d^2*f)*(d*tan(f*x + e) - I*d)^3) - 1/2*(-I*c - d)*arctan(4*(sqrt(d
*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) - I*sqrt(-8
*c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/(a^3*sqrt(-8*c + 8*sqrt(c^2 + d^2
))*f*(-I*d/(c - sqrt(c^2 + d^2)) + 1))

________________________________________________________________________________________

maple [B]  time = 0.44, size = 1315, normalized size = 4.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x)

[Out]

2/3*I/f/a^3*d^4/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2)*c-I/f/a^3*d^2/(d*tan
(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2)*c^3+1/8/f/a^3*d/(d*tan(f*x+e)-I*d)^3*c^3/(
3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)-3/16/f/a^3*d^3/(d*tan(f*x+e)-I*d)^3*c/(3*I*c^2*d-I*d^3+c^3
-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)+5/16*I/f/a^3*d^2/(d*tan(f*x+e)-I*d)^3*c^2/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*
tan(f*x+e))^(5/2)-1/8*I/f/a^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d
-c)^(1/2))*c^4-1/4/f/a^3*d/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2)*c^4+4/3/f
/a^3*d^3/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2)*c^2-1/12/f/a^3*d^5/(d*tan(f
*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2)+1/4*I/f/a^3*d^6/(d*tan(f*x+e)-I*d)^3/(3*I*c^
2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)+5/16*I/f/a^3*d^2/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*ar
ctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^2+1/8*I/f/a^3*d^4/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*a
rctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))+1/8/f/a^3*d/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c
+d*tan(f*x+e))^(1/2)*c^5-25/16/f/a^3*d^3/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(
1/2)*c^3+17/16/f/a^3*d^5/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)*c+11/16*I/f
/a^3*d^2/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(1/2)*c^4-1/8*I/f/a^3*(I*d-c)^(1/
2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))-29/16*I/f/a^3*d^4/(d*tan(f*x+e)-I*d)^3/(3*I*c^2*d-I*d^3+c^3-3*
c*d^2)*(c+d*tan(f*x+e))^(1/2)*c^2+3/8/f/a^3*d/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x
+e))^(1/2)/(-I*d-c)^(1/2))*c^3+1/16/f/a^3*d^3/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x
+e))^(1/2)/(-I*d-c)^(1/2))*c

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

mupad [B]  time = 10.25, size = 35270, normalized size = 125.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^(1/2)/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

log(58*a^3*c^2*d^11*f - (((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*
d^4*8i - a^6*c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^
4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4
+ (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*
f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 +
(49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6
*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/
1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i - a^6*d^4*f^2*((((25*c^3*d^12)/4
- 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^
2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10
*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*
f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^
11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4)
- (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64
+ (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4
))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i + a^6*c^2*d^2*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c
^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^
6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*
f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (
7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^1
2*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (
65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a
^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24
i - 16*a^6*c*d^3*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^
4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4
+ (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*
f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 +
(49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6
*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/
1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2) + 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/
4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*
c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 +
10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^
2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (
c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4
) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/
64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f
^4))*(256*d^6 + 256*c^2*d^4))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2*1i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2
- 4*a^6*c^3*d*f^2 - a^6*c^2*d^2*f^2*6i)))^(1/2)*(7168*a^9*c*d^11*f^3 - a^9*d^12*f^3*4096i + 32*(c + d*tan(e +
f*x))^(1/2)*((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*d^4*8i - a^6*
c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f
^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^
11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c
^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/
512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4
*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^
8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d
^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i - a^6*d^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 +
(41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6
*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*
c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*(
(((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*
1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 -
(69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)
/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 +
256*c^2*d^4))^(1/2)*4i + a^6*c^2*d^2*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 1
0*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2
) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8
*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256
+ (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4
*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/
1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 +
4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24i - 16*a^6*c*
d^3*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f
^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^
11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c
^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/
512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4
*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^
8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d
^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2) + 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/4 - 5*c*d^14
+ (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 +
6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 +
2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4
*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128
)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256
- (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^
4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6
+ 256*c^2*d^4))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2*1i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2 - 4*a^6*c^3*d
*f^2 - a^6*c^2*d^2*f^2*6i)))^(1/2)*(2048*a^6*c*d^6*f^2 - a^6*c^2*d^5*f^2*8192i - 12288*a^6*c^3*d^4*f^2 + a^6*c
^4*d^3*f^2*8192i + 2048*a^6*c^5*d^2*f^2)*(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*
a^6*c^2*d^2*f^2) - a^9*c^2*d^10*f^3*9216i + 19456*a^9*c^3*d^9*f^3 - a^9*c^4*d^8*f^3*7168i + 19456*a^9*c^5*d^7*
f^3 - a^9*c^6*d^6*f^3*3072i + 9216*a^9*c^7*d^5*f^3 - a^9*c^8*d^4*f^3*1024i + 2048*a^9*c^9*d^3*f^3) + 32*(c + d
*tan(e + f*x))^(1/2)*(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*a^6*c^2*d^2*f^2)*(c*
d^7*4i - 8*d^8 - 19*c^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*24i + 8*c^6*d^2))*((c*d^10*20i + 15*c^2*d^9 +
c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*d^4*8i - a^6*c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 +
(41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 +
6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2
*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*
((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)
*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256
- (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4
)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6
+ 256*c^2*d^4))^(1/2)*4i - a^6*d^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c
^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) -
(((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^
2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 +
(7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^
12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/102
4 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*
a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i + a^6*c^2*d^2*f
^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 +
a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2
+ (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^
4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 +
(5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4
+ 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8
)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^
4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24i - 16*a^6*c*d^3*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 +
(41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 +
6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2
*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*
((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)
*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256
- (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4
)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6
+ 256*c^2*d^4))^(1/2) + 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10
*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2)
- (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*
f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256
+ (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*
a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1
024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 +
4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2))/(2048*(d^6 + c^
2*d^4)*(a^6*c^4*f^2*1i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2 - 4*a^6*c^3*d*f^2 - a^6*c^2*d^2*f^2*6i)))^(1/2) + a^
3*c^3*d^10*f*82i + 12*a^3*c^4*d^9*f + a^3*c^5*d^8*f*108i - 38*a^3*c^6*d^7*f + a^3*c^7*d^6*f*58i - 32*a^3*c^8*d
^5*f + a^3*c^9*d^4*f*12i - 8*a^3*c^10*d^3*f - a^3*c*d^12*f*12i)*((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c
^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*d^4*8i - a^6*c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 +
(85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 +
4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6
*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/51
2 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4
+ a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/102
4 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^
4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/
2)*4i - a^6*d^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4
)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4
+ (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f
^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (
49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*
a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1
024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i + a^6*c^2*d^2*f^2*((((25*c^3*d^12)
/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6
*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 +
10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d
^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 +
(c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^
4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)
/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*
f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24i - 16*a^6*c*d^3*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 +
(85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 +
4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6
*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/51
2 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4
+ a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/102
4 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^
4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/
2) + 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d
^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/
4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6
*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 +
(49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)
/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4
+ 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2
*1i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2 - 4*a^6*c^3*d*f^2 - a^6*c^2*d^2*f^2*6i)))^(1/2) + log(58*a^3*c^2*d^11*f
- (((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*d^4*8i + a^6*c^4*f^2*
((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6
*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 +
(93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f
^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5
*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4
*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1
024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 +
4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i + a^6*d^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*
d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4
*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5
)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^
3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^1
2*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2
*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a
^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2
*d^4))^(1/2)*4i - a^6*c^2*d^2*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^
6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((6
5*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4
*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^
5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^
2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (
11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*
c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24i + 16*a^6*c*d^3*f^2*
((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6
*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 +
(93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f
^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5
*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4
*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1
024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 +
4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2) - 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^
5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c
^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d
^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*
c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a
^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c
^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/
(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c
^2*d^4))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2*1i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2 - 4*a^6*c^3*d*f^2 - a
^6*c^2*d^2*f^2*6i)))^(1/2)*(7168*a^9*c*d^11*f^3 - a^9*d^12*f^3*4096i + 32*(c + d*tan(e + f*x))^(1/2)*((c*d^10*
20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*d^4*8i + a^6*c^4*f^2*((((25*c^3*d^
12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*
a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/
4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^
6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128
+ (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2
*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d
^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d
^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i + a^6*d^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85
*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*
a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^
8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 -
(7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a
^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 -
(65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 +
a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*
4i - a^6*c^2*d^2*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^
4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4
+ (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*
f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 +
(49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6
*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/
1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24i + 16*a^6*c*d^3*f^2*((((25*c^3*d^
12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*
a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/
4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^
6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128
+ (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2
*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d
^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d
^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2) - 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (
85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 +
4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*
c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512
- (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 +
a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024
- (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4
+ a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2
))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2*1i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2 - 4*a^6*c^3*d*f^2 - a^6*c^2*d^2*f^
2*6i)))^(1/2)*(2048*a^6*c*d^6*f^2 - a^6*c^2*d^5*f^2*8192i - 12288*a^6*c^3*d^4*f^2 + a^6*c^4*d^3*f^2*8192i + 20
48*a^6*c^5*d^2*f^2)*(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*a^6*c^2*d^2*f^2) - a^
9*c^2*d^10*f^3*9216i + 19456*a^9*c^3*d^9*f^3 - a^9*c^4*d^8*f^3*7168i + 19456*a^9*c^5*d^7*f^3 - a^9*c^6*d^6*f^3
*3072i + 9216*a^9*c^7*d^5*f^3 - a^9*c^8*d^4*f^3*1024i + 2048*a^9*c^9*d^3*f^3) + 32*(c + d*tan(e + f*x))^(1/2)*
(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*a^6*c^2*d^2*f^2)*(c*d^7*4i - 8*d^8 - 19*c
^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*24i + 8*c^6*d^2))*((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4
*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*d^4*8i + a^6*c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (8
5*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4
*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c
^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512
- (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 +
a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024
- (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4
+ a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)
*4i + a^6*d^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/
(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 +
(55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2
+ 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49
*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^
12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/102
4 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*
a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i - a^6*c^2*d^2*f^2*((((25*c^3*d^12)/4
- 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c
^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 1
0*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2
*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c
^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4)
- (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/6
4 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^
4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24i + 16*a^6*c*d^3*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (8
5*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4
*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c
^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512
- (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 +
a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024
- (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4
+ a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)
- 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4
)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4
+ (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f
^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (
49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*
a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1
024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2*1
i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2 - 4*a^6*c^3*d*f^2 - a^6*c^2*d^2*f^2*6i)))^(1/2) + a^3*c^3*d^10*f*82i + 12
*a^3*c^4*d^9*f + a^3*c^5*d^8*f*108i - 38*a^3*c^6*d^7*f + a^3*c^7*d^6*f*58i - 32*a^3*c^8*d^5*f + a^3*c^9*d^4*f*
12i - 8*a^3*c^10*d^3*f - a^3*c*d^12*f*12i)*((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i +
24*c^6*d^5 - c^7*d^4*8i + a^6*c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c
^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) -
(((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^
2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 +
(7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^
12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/102
4 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*
a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i + a^6*d^4*f^2*(
(((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*
d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (
93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^
2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*
c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*
a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/10
24 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 +
4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i - a^6*c^2*d^2*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c
^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*
c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*
d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25
*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(
a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*
c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)
/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*
c^2*d^4))^(1/2)*24i + 16*a^6*c*d^3*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c
^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) -
(((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^
2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 +
(7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^
12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/102
4 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*
a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2) - 16*a^6*c^3*d*f^2
*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^
6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 +
(93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*
f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (
5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 +
4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/
1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4
+ 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2*1i + a^6*d^4*f^2*1i
+ 4*a^6*c*d^3*f^2 - 4*a^6*c^3*d*f^2 - a^6*c^2*d^2*f^2*6i)))^(1/2) - log((((20*c*d^14 + c^2*d^13*65i - 25*c^3*d
^12 + c^4*d^11*110i - 82*c^5*d^10 + c^6*d^9*93i - 85*c^7*d^8 + c^8*d^7*40i - 40*c^9*d^6 + c^10*d^5*8i - 8*c^11
*d^4 + 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i
+ 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d
^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c
^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 +
128*a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*
c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^
11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*
c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 +
448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^
2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9
*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4
+ 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8
*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 24*a^6*c^4*d^4*f^2*(-(c
*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^
23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25
*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^
6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(
1/2) + 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*
2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c
^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a
^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f
^4 + 128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 20480*a^6*c^4*d^10*f^2 + 204
80*a^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2)*(a^9*d^12*f^3*4096i - 7168*a^9*c*d^
11*f^3 + a^9*c^2*d^10*f^3*9216i - 19456*a^9*c^3*d^9*f^3 + a^9*c^4*d^8*f^3*7168i - 19456*a^9*c^5*d^7*f^3 + a^9*
c^6*d^6*f^3*3072i - 9216*a^9*c^7*d^5*f^3 + a^9*c^8*d^4*f^3*1024i - 2048*a^9*c^9*d^3*f^3 + 32*((20*c*d^14 + c^2
*d^13*65i - 25*c^3*d^12 + c^4*d^11*110i - 82*c^5*d^10 + c^6*d^9*93i - 85*c^7*d^8 + c^8*d^7*40i - 40*c^9*d^6 +
c^10*d^5*8i - 8*c^11*d^4 + 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d
^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^
19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d
^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*
a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c
^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i
+ 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*
d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896
*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^29*448i
+ 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i +
1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16
)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4
+ 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 24*
a^6*c^4*d^4*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 11
79*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 +
c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^
12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a
^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*
c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^
11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*
c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 +
448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 20480*a^
6*c^4*d^10*f^2 + 20480*a^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2)*(c + d*tan(e +
f*x))^(1/2)*(2048*a^6*c*d^6*f^2 - a^6*c^2*d^5*f^2*8192i - 12288*a^6*c^3*d^4*f^2 + a^6*c^4*d^3*f^2*8192i + 2048
*a^6*c^5*d^2*f^2)*(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*a^6*c^2*d^2*f^2)) + 32*
(c + d*tan(e + f*x))^(1/2)*(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*a^6*c^2*d^2*f^
2)*(c*d^7*4i - 8*d^8 - 19*c^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*24i + 8*c^6*d^2))*((20*c*d^14 + c^2*d^1
3*65i - 25*c^3*d^12 + c^4*d^11*110i - 82*c^5*d^10 + c^6*d^9*93i - 85*c^7*d^8 + c^8*d^7*40i - 40*c^9*d^6 + c^10
*d^5*8i - 8*c^11*d^4 + 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26
- c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*3
52i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*
f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12
*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d
^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 92
6*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16
*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^1
2*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^29*448i + 6
4*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 189
0*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(1
6*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1
120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 24*a^6*
c^4*d^4*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c
^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^1
3*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f
^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*
c^14*d^2*f^4))^(1/2) + 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*
d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d
^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*
d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448
*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 20480*a^6*c^
4*d^10*f^2 + 20480*a^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2) + 58*a^3*c^2*d^11*f
+ a^3*c^3*d^10*f*82i + 12*a^3*c^4*d^9*f + a^3*c^5*d^8*f*108i - 38*a^3*c^6*d^7*f + a^3*c^7*d^6*f*58i - 32*a^3*
c^8*d^5*f + a^3*c^9*d^4*f*12i - 8*a^3*c^10*d^3*f - a^3*c*d^12*f*12i)*((20*c*d^14 + c^2*d^13*65i - 25*c^3*d^12
+ c^4*d^11*110i - 82*c^5*d^10 + c^6*d^9*93i - 85*c^7*d^8 + c^8*d^7*40i - 40*c^9*d^6 + c^10*d^5*8i - 8*c^11*d^4
+ 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1
179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18
+ c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d
^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*
a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*
d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d
^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*
d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448
*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^
28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^2
1*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16
*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4
+ 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 24*a^6*c^4*d^4*f^2*(-(c*d^2
9*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1
608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^1
4*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^
10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2)
+ 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462
i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*
d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*
c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 +
128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 20480*a^6*c^4*d^10*f^2 + 20480*a
^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2) - log(((-(25*c^3*d^12 - c^2*d^13*65i -
20*c*d^14 - c^4*d^11*110i + 82*c^5*d^10 - c^6*d^9*93i + 85*c^7*d^8 - c^8*d^7*40i + 40*c^9*d^6 - c^10*d^5*8i +
8*c^11*d^4 + 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25
*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*
c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*
a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*
f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i
- 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^2
0 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128
*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6
*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^29*448i + 64*d^30 - 1
184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22
- c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^1
6*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c
^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 24*a^6*c^4*d^4*f^
2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 -
c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130
i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a
^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f
^4))^(1/2) + 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5
*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i +
149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 +
448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12
*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 20480*a^6*c^4*d^10*f^2
+ 20480*a^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2)*(a^9*d^12*f^3*4096i - 7168*a^
9*c*d^11*f^3 + a^9*c^2*d^10*f^3*9216i - 19456*a^9*c^3*d^9*f^3 + a^9*c^4*d^8*f^3*7168i - 19456*a^9*c^5*d^7*f^3
+ a^9*c^6*d^6*f^3*3072i - 9216*a^9*c^7*d^5*f^3 + a^9*c^8*d^4*f^3*1024i - 2048*a^9*c^9*d^3*f^3 + 32*(-(25*c^3*d
^12 - c^2*d^13*65i - 20*c*d^14 - c^4*d^11*110i + 82*c^5*d^10 - c^6*d^9*93i + 85*c^7*d^8 - c^8*d^7*40i + 40*c^9
*d^6 - c^10*d^5*8i + 8*c^11*d^4 + 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 69
5*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 +
c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^1
2*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4
+ 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d
^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^
21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 1
6*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^
4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^
29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*
1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^
14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d
^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2
) + 24*a^6*c^4*d^4*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*246
2i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12
*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12
*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4
+ 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i
- 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^
20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 12
8*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^
6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 2
0480*a^6*c^4*d^10*f^2 + 20480*a^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2)*(c + d*t
an(e + f*x))^(1/2)*(2048*a^6*c*d^6*f^2 - a^6*c^2*d^5*f^2*8192i - 12288*a^6*c^3*d^4*f^2 + a^6*c^4*d^3*f^2*8192i
+ 2048*a^6*c^5*d^2*f^2)*(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*a^6*c^2*d^2*f^2)
) + 32*(c + d*tan(e + f*x))^(1/2)*(a^6*c^4*f^2 + a^6*d^4*f^2 + a^6*c*d^3*f^2*4i - a^6*c^3*d*f^2*4i - 6*a^6*c^2
*d^2*f^2)*(c*d^7*4i - 8*d^8 - 19*c^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*24i + 8*c^6*d^2))*(-(25*c^3*d^12
- c^2*d^13*65i - 20*c*d^14 - c^4*d^11*110i + 82*c^5*d^10 - c^6*d^9*93i + 85*c^7*d^8 - c^8*d^7*40i + 40*c^9*d^
6 - c^10*d^5*8i + 8*c^11*d^4 + 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c
^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^1
1*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c
^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 +
448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28
- c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*
20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a
^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 +
896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^29*
448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*160
8i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*
d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10
*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) +
24*a^6*c^4*d^4*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i
+ 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^
18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^
4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 1
28*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i -
695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20
+ c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a
^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f
^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 2048
0*a^6*c^4*d^10*f^2 + 20480*a^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2) + 58*a^3*c^
2*d^11*f + a^3*c^3*d^10*f*82i + 12*a^3*c^4*d^9*f + a^3*c^5*d^8*f*108i - 38*a^3*c^6*d^7*f + a^3*c^7*d^6*f*58i -
32*a^3*c^8*d^5*f + a^3*c^9*d^4*f*12i - 8*a^3*c^10*d^3*f - a^3*c*d^12*f*12i)*(-(25*c^3*d^12 - c^2*d^13*65i - 2
0*c*d^14 - c^4*d^11*110i + 82*c^5*d^10 - c^6*d^9*93i + 85*c^7*d^8 - c^8*d^7*40i + 40*c^9*d^6 - c^10*d^5*8i + 8
*c^11*d^4 + 4*a^6*c^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*
2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c
^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a
^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f
^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 4*a^6*d^8*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i -
695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20
+ c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*
a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*
f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 16*a^6*c^2*d^6*f^2*(-(c*d^29*448i + 64*d^30 - 11
84*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22
- c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16
*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^
8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2) + 24*a^6*c^4*d^4*f^2
*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*d^25*2462i + 1179*c^6*d^24 - c
^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i + 149*c^12*d^18 + c^13*d^17*130i
- 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 + 448*a^12*c^4*d^12*f^4 + 896*a^
12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*d^4*f^4 + 128*a^12*c^14*d^2*f^
4))^(1/2) + 16*a^6*c^6*d^2*f^2*(-(c*d^29*448i + 64*d^30 - 1184*c^2*d^28 - c^3*d^27*1160i - 695*c^4*d^26 - c^5*
d^25*2462i + 1179*c^6*d^24 - c^7*d^23*1608i + 1890*c^8*d^22 - c^9*d^21*20i + 926*c^10*d^20 + c^11*d^19*352i +
149*c^12*d^18 + c^13*d^17*130i - 25*c^14*d^16)/(16*a^12*c^16*f^4 + 16*a^12*d^16*f^4 + 128*a^12*c^2*d^14*f^4 +
448*a^12*c^4*d^12*f^4 + 896*a^12*c^6*d^10*f^4 + 1120*a^12*c^8*d^8*f^4 + 896*a^12*c^10*d^6*f^4 + 448*a^12*c^12*
d^4*f^4 + 128*a^12*c^14*d^2*f^4))^(1/2))/(2048*a^6*d^14*f^2 + 10240*a^6*c^2*d^12*f^2 + 20480*a^6*c^4*d^10*f^2
+ 20480*a^6*c^6*d^8*f^2 + 10240*a^6*c^8*d^6*f^2 + 2048*a^6*c^10*d^4*f^2))^(1/2) + (((c + d*tan(e + f*x))^(1/2)
*(25*c*d^2 - c^2*d*10i + d^3*20i)*1i)/(80*a^3*f) + ((c + d*tan(e + f*x))^(3/2)*(24*c*d^2 - c^2*d*12i + d^3*4i)
)/(48*a^3*f*(c*1i - d)) - (d*(c*d*3i + 2*c^2)*(c + d*tan(e + f*x))^(5/2)*1i)/(16*a^3*f*(2*c*d - c^2*1i + d^2*1
i)))/((c + d*tan(e + f*x))*(c*d*6i + 3*c^2 - 3*d^2) + (c + d*tan(e + f*x))^3 + 3*c*d^2 - c^2*d*3i - (3*c + d*3
i)*(c + d*tan(e + f*x))^2 - c^3 + d^3*1i)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {i \int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e))**3,x)

[Out]

I*Integral(sqrt(c + d*tan(e + f*x))/(tan(e + f*x)**3 - 3*I*tan(e + f*x)**2 - 3*tan(e + f*x) + I), x)/a**3

________________________________________________________________________________________