∫ 1_____________ (a+ia tan(e+fx))(c+d tan(e+fx))3∕2 dx

3.1128 \(\int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=205 \[ \frac {d (c-5 i d)}{2 a f (c-i d) (c+i d)^2 \sqrt {c+d \tan (e+f x)}}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a f (c-i d)^{3/2}}+\frac {(-4 d+i c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a f (c+i d)^{5/2}} \]

[Out]

-1/2*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a/(c-I*d)^(3/2)/f+1/2*(I*c-4*d)*arctanh((c+d*tan(f*x+e))^

(1/2)/(c+I*d)^(1/2))/a/(c+I*d)^(5/2)/f+1/2*(c-5*I*d)*d/a/(c-I*d)/(c+I*d)^2/f/(c+d*tan(f*x+e))^(1/2)-1/2/(I*c-d

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

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Rubi [A] time = 0.47, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3552, 3529, 3539, 3537, 63, 208} \[ \frac {d (c-5 i d)}{2 a f (c-i d) (c+i d)^2 \sqrt {c+d \tan (e+f x)}}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a f (c-i d)^{3/2}}+\frac {(-4 d+i c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a f (c+i d)^{5/2}} \]

Antiderivative was successfully veriﬁed.

[In]

Int[1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)),x]

[Out]

((-I/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*(c - I*d)^(3/2)*f) + ((I*c - 4*d)*ArcTanh[Sqrt[c +

d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*(c + I*d)^(5/2)*f) + ((c - (5*I)*d)*d)/(2*a*(c - I*d)*(c + I*d)^2*f*Sqrt

[c + d*Tan[e + f*x]]) - 1/(2*(I*c - d)*f*(a + I*a*Tan[e + f*x])*Sqrt[c + d*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 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((

b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +

f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c

- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

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 3552

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(a

*(c + d*Tan[e + f*x])^(n + 1))/(2*f*(b*c - a*d)*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c +

d*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]

&& NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0]

Rubi steps

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

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Mathematica [A] time = 3.24, size = 297, normalized size = 1.45 \[ \frac {\sec (e+f x) (\cos (f x)+i \sin (f x)) \left (\frac {2 \cos (e+f x) (\sin (f x)+i \cos (f x)) \sqrt {c+d \tan (e+f x)} \left (\left (c^2-i c d-4 d^2\right ) \cos (e+f x)+d (c-5 i d) \sin (e+f x)\right )}{(c-i d) (c+i d)^2 (c \cos (e+f x)+d \sin (e+f x))}-\frac {2 (\cos (e)+i \sin (e)) \left (i (-c-i d)^{5/2} \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )-i \sqrt {-c+i d} \left (c^2+3 i c d+4 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )\right )}{(-c-i d)^{5/2} (-c+i d)^{3/2}}\right )}{4 f (a+i a \tan (e+f x))} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])*(c + d*Tan[e + f*x])^(3/2)),x]

[Out]

(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((-2*((-I)*Sqrt[-c + I*d]*(c^2 + (3*I)*c*d + 4*d^2)*ArcTan[Sqrt[c + d*Ta

n[e + f*x]]/Sqrt[-c - I*d]] + I*(-c - I*d)^(5/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[e] + I*

Sin[e]))/((-c - I*d)^(5/2)*(-c + I*d)^(3/2)) + (2*Cos[e + f*x]*(I*Cos[f*x] + Sin[f*x])*((c^2 - I*c*d - 4*d^2)*

Cos[e + f*x] + (c - (5*I)*d)*d*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]])/((c - I*d)*(c + I*d)^2*(c*Cos[e + f*x]

+ d*Sin[e + f*x]))))/(4*f*(a + I*a*Tan[e + f*x]))

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fricas [B] time = 0.94, size = 1549, normalized size = 7.56 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((a*c^4 + 2*a*c^2*d^2 + a*d^4)*f*e^(4*I*f*x + 4*I*e) + (a*c^4 + 2*I*a*c^3*d + 2*I*a*c*d^3 - a*d^4)*f*e^(2*I*f

*x + 2*I*e))*sqrt(I/((-4*I*a^2*c^3 - 12*a^2*c^2*d + 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^2))*log((((4*I*a*c^2 + 8*a*c

*d - 4*I*a*d^2)*f*e^(2*I*f*x + 2*I*e) + (4*I*a*c^2 + 8*a*c*d - 4*I*a*d^2)*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(I/((-4*I*a^2*c^3 - 12*a^2*c^2*d + 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^

2)) + 2*(c - I*d)*e^(2*I*f*x + 2*I*e) + 2*c)*e^(-2*I*f*x - 2*I*e)) - ((a*c^4 + 2*a*c^2*d^2 + a*d^4)*f*e^(4*I*f

*x + 4*I*e) + (a*c^4 + 2*I*a*c^3*d + 2*I*a*c*d^3 - a*d^4)*f*e^(2*I*f*x + 2*I*e))*sqrt(I/((-4*I*a^2*c^3 - 12*a^

2*c^2*d + 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^2))*log((((-4*I*a*c^2 - 8*a*c*d + 4*I*a*d^2)*f*e^(2*I*f*x + 2*I*e) + (

-4*I*a*c^2 - 8*a*c*d + 4*I*a*d^2)*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(I/((-4*I*a^2*c^3 - 12*a^2*c^2*d + 12*I*a^2*c*d^2 + 4*a^2*d^3)*f^2)) + 2*(c - I*d)*e^(2*I*f*x + 2*I*e) +

2*c)*e^(-2*I*f*x - 2*I*e)) + ((a*c^4 + 2*a*c^2*d^2 + a*d^4)*f*e^(4*I*f*x + 4*I*e) + (a*c^4 + 2*I*a*c^3*d + 2*I

*a*c*d^3 - a*d^4)*f*e^(2*I*f*x + 2*I*e))*sqrt((I*c^2 - 8*c*d - 16*I*d^2)/((-4*I*a^2*c^5 + 20*a^2*c^4*d + 40*I*

a^2*c^3*d^2 - 40*a^2*c^2*d^3 - 20*I*a^2*c*d^4 + 4*a^2*d^5)*f^2))*log(-(c^2 + 5*I*c*d - 4*d^2 - ((2*I*a*c^3 - 6

*a*c^2*d - 6*I*a*c*d^2 + 2*a*d^3)*f*e^(2*I*f*x + 2*I*e) + (2*I*a*c^3 - 6*a*c^2*d - 6*I*a*c*d^2 + 2*a*d^3)*f)*s

qrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt((I*c^2 - 8*c*d - 16*I*d^2)/((-4*

I*a^2*c^5 + 20*a^2*c^4*d + 40*I*a^2*c^3*d^2 - 40*a^2*c^2*d^3 - 20*I*a^2*c*d^4 + 4*a^2*d^5)*f^2)) + (c^2 + 4*I*

c*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((2*I*a*c^3 - 6*a*c^2*d - 6*I*a*c*d^2 + 2*a*d^3)*f)) - ((a*c^4

+ 2*a*c^2*d^2 + a*d^4)*f*e^(4*I*f*x + 4*I*e) + (a*c^4 + 2*I*a*c^3*d + 2*I*a*c*d^3 - a*d^4)*f*e^(2*I*f*x + 2*I*

e))*sqrt((I*c^2 - 8*c*d - 16*I*d^2)/((-4*I*a^2*c^5 + 20*a^2*c^4*d + 40*I*a^2*c^3*d^2 - 40*a^2*c^2*d^3 - 20*I*a

^2*c*d^4 + 4*a^2*d^5)*f^2))*log(-(c^2 + 5*I*c*d - 4*d^2 - ((-2*I*a*c^3 + 6*a*c^2*d + 6*I*a*c*d^2 - 2*a*d^3)*f*

e^(2*I*f*x + 2*I*e) + (-2*I*a*c^3 + 6*a*c^2*d + 6*I*a*c*d^2 - 2*a*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((I*c^2 - 8*c*d - 16*I*d^2)/((-4*I*a^2*c^5 + 20*a^2*c^4*d + 40*I*a^2

*c^3*d^2 - 40*a^2*c^2*d^3 - 20*I*a^2*c*d^4 + 4*a^2*d^5)*f^2)) + (c^2 + 4*I*c*d)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f

*x - 2*I*e)/((2*I*a*c^3 - 6*a*c^2*d - 6*I*a*c*d^2 + 2*a*d^3)*f)) + (I*c^2 + I*d^2 + (I*c^2 + 2*c*d - 9*I*d^2)*

e^(4*I*f*x + 4*I*e) + (2*I*c^2 + 2*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)))/(4*(a*c^4 + 2*a*c^2*d^2 + a*d^4)*f*e^(4*I*f*x + 4*I*e) + (4*a*c^4 + 8*I*a*

c^3*d + 8*I*a*c*d^3 - 4*a*d^4)*f*e^(2*I*f*x + 2*I*e))

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giac [B] time = 1.17, size = 480, normalized size = 2.34 \[ \frac {4 \, {\left (i \, c - 4 \, d\right )} \arctan \left (-\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (2 \, a c^{2} f + 4 i \, a c d f - 2 \, a d^{2} f\right )} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {2 \, {\left ({\left (-i \, d \tan \left (f x + e\right ) - i \, c\right )} c d - 5 \, {\left (d \tan \left (f x + e\right ) + c\right )} d^{2} + 4 \, c d^{2} + 4 i \, d^{3}\right )}}{{\left (4 \, a c^{3} f + 4 i \, a c^{2} d f + 4 \, a c d^{2} f + 4 i \, a d^{3} f\right )} {\left (i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} - i \, \sqrt {d \tan \left (f x + e\right ) + c} c + \sqrt {d \tan \left (f x + e\right ) + c} d\right )}} + \frac {2 i \, \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{{\left (a c f - i \, a d f\right )} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(I*c - 4*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))))/((

2*a*c^2*f + 4*I*a*c*d*f - 2*a*d^2*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) - 2*((-I*

d*tan(f*x + e) - I*c)*c*d - 5*(d*tan(f*x + e) + c)*d^2 + 4*c*d^2 + 4*I*d^3)/((4*a*c^3*f + 4*I*a*c^2*d*f + 4*a*

c*d^2*f + 4*I*a*d^3*f)*(I*(d*tan(f*x + e) + c)^(3/2) - I*sqrt(d*tan(f*x + e) + c)*c + sqrt(d*tan(f*x + e) + c)

*d)) + 2*I*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*s

qrt(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*c*

f - I*a*d*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1))

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maple [B] time = 0.41, size = 580, normalized size = 2.83 \[ -\frac {d \sqrt {c +d \tan \left (f x +e \right )}\, c^{2}}{2 f a \left (i d +c \right )^{3} \left (i d -c \right ) \left (d \tan \left (f x +e \right )-i d \right )}-\frac {d^{3} \sqrt {c +d \tan \left (f x +e \right )}}{2 f a \left (i d +c \right )^{3} \left (i d -c \right ) \left (d \tan \left (f x +e \right )-i d \right )}+\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{3}}{2 f a \left (i d +c \right )^{3} \left (i d -c \right ) \sqrt {-i d -c}}+\frac {i d^{2} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c}{2 f a \left (i d +c \right )^{3} \left (i d -c \right ) \sqrt {-i d -c}}-\frac {2 d \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{2}}{f a \left (i d +c \right )^{3} \left (i d -c \right ) \sqrt {-i d -c}}-\frac {2 d^{3} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{f a \left (i d +c \right )^{3} \left (i d -c \right ) \sqrt {-i d -c}}-\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right ) c^{2}}{2 f a \left (i d -c \right )^{\frac {3}{2}} \left (i d +c \right )^{2}}+\frac {i d^{2} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{2 f a \left (i d -c \right )^{\frac {3}{2}} \left (i d +c \right )^{2}}+\frac {d \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right ) c}{f a \left (i d -c \right )^{\frac {3}{2}} \left (i d +c \right )^{2}}+\frac {2 i d^{2}}{f a \left (i d +c \right ) \left (i c -d \right ) \left (i c +d \right ) \sqrt {c +d \tan \left (f x +e \right )}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

rctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^2+1/2*I/f/a*d^2/(I*d-c)^(3/2)/(c+I*d)^2*arctan((c+d*tan(f*x+e))^

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

2/(c+I*d)/(I*c-d)/(I*c+d)/(c+d*tan(f*x+e))^(1/2)

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

[Out]

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

xponent.

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mupad [B] time = 15.81, size = 35674, normalized size = 174.02 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(10*a*d^7*f - ((-(240*c^2*d^7 - 240*d^9 - c*d^8*720i + c^3*d^6*80i + 160*c^4*d^5 - c^5*d^4*32i - a^2*c^6*f^

2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6

*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2

+ a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((

24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f

^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*

c^6*d^2*f^4)))^(1/2)*1i + a^2*d^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 +

a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^

4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2

))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 +

6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*

d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 2*a^2*c*d^5*f^2*(((1280*c^3*d^8 - 1200*c*d^10 +

208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f

^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2

+ 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8

*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)

/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) + 2*a^2*c^5*d

*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2

+ 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*

f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*

(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^

2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a

^4*c^6*d^2*f^4)))^(1/2) + a^2*c^2*d^4*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f

^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 24

0*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2

*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f

^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*

c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 4*a^2*c^3*d^3*f^2*(((1280*c^3*d^8 - 1200*c*d

^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6

*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d

^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a

^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^

4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) - a^2*c

^4*d^2*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^

6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^

2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2

*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*

c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4

+ 4*a^4*c^6*d^2*f^4)))^(1/2)*1i)/(512*(d^6 + c^2*d^4)*(a^2*d^6*f^2*1i - a^2*c^6*f^2*1i + 2*a^2*c*d^5*f^2 + 2*

a^2*c^5*d*f^2 + a^2*c^2*d^4*f^2*1i + 4*a^2*c^3*d^3*f^2 - a^2*c^4*d^2*f^2*1i)))^(1/2)*(104*a^3*c*d^9*f^3 - a^3*

d^10*f^3*24i + a^3*c^2*d^8*f^3*24i + 216*a^3*c^3*d^7*f^3 + a^3*c^4*d^6*f^3*120i + 120*a^3*c^5*d^5*f^3 + a^3*c^

6*d^4*f^3*72i + 8*a^3*c^7*d^3*f^3 - 2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*

(-(240*c^2*d^7 - 240*d^9 - c*d^8*720i + c^3*d^6*80i + 160*c^4*d^5 - c^5*d^4*32i - a^2*c^6*f^2*(((1280*c^3*d^8

- 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 +

4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4

*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d

^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^

2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/

2)*1i + a^2*d^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a

^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5

)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6

+ 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4

+ 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^

4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 2*a^2*c*d^5*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c

^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 -

1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f

^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4

+ 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^

4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) + 2*a^2*c^5*d*f^2*(((1280*c^3*d

^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^

2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2

+ 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^

3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 -

c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^

(1/2) + a^2*c^2*d^4*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 +

4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6

*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*

d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4

*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^

4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 4*a^2*c^3*d^3*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6

+ 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*

d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4

*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d

^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^

4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) - a^2*c^4*d^2*f^2*(((1280

*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*

d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^

8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7

+ 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16

*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f

^4)))^(1/2)*1i)/(512*(d^6 + c^2*d^4)*(a^2*d^6*f^2*1i - a^2*c^6*f^2*1i + 2*a^2*c*d^5*f^2 + 2*a^2*c^5*d*f^2 + a^

2*c^2*d^4*f^2*1i + 4*a^2*c^3*d^3*f^2 - a^2*c^4*d^2*f^2*1i)))^(1/2)*(a^2*c^2*d^7*f^2*64i - 32*a^2*c*d^8*f^2 - 3

2*a^2*c^3*d^6*f^2 + a^2*c^4*d^5*f^2*128i + 32*a^2*c^5*d^4*f^2 + a^2*c^6*d^3*f^2*64i + 32*a^2*c^7*d^2*f^2)) - 2

*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*(c*d^5*20i + 17*d^6 - 7*c^2*d^4 + c^3

*d^3*10i + 2*c^4*d^2))*(-(240*c^2*d^7 - 240*d^9 - c*d^8*720i + c^3*d^6*80i + 160*c^4*d^5 - c^5*d^4*32i - a^2*c

^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^

2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^

8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4

)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*

d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4

*a^4*c^6*d^2*f^4)))^(1/2)*1i + a^2*d^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*

f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 2

40*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^

2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*

f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4

*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 2*a^2*c*d^5*f^2*(((1280*c^3*d^8 - 1200*c*d^

10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*

d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^

6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^

4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4

*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) + 2*a^2*

c^5*d*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6

*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2

*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*

d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c

^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4

+ 4*a^4*c^6*d^2*f^4)))^(1/2) + a^2*c^2*d^4*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*

c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9

+ 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^

6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*

d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4

*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 4*a^2*c^3*d^3*f^2*(((1280*c^3*d^8 - 120

0*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^

2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*

c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1

i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6

+ c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) -

a^2*c^4*d^2*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c

^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i

)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 25

6*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4

*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^

4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i)/(512*(d^6 + c^2*d^4)*(a^2*d^6*f^2*1i - a^2*c^6*f^2*1i + 2*a^2*c*d^5*f^2

+ 2*a^2*c^5*d*f^2 + a^2*c^2*d^4*f^2*1i + 4*a^2*c^3*d^3*f^2 - a^2*c^4*d^2*f^2*1i)))^(1/2) + (a*c*d^6*f*39i)/2

- (17*a*c^2*d^5*f)/2 + (a*c^3*d^4*f*3i)/2 - (a*c^4*d^3*f)/2)*(-(240*c^2*d^7 - 240*d^9 - c*d^8*720i + c^3*d^6*8

0i + 160*c^4*d^5 - c^5*d^4*32i - a^2*c^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^

8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 +

240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*

d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^

6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a

^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + a^2*d^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10

+ 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^

2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*

f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*

c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d

^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 2*a^2

*c*d^5*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^

6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^

2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2

*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*

c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4

+ 4*a^4*c^6*d^2*f^4)))^(1/2) + 2*a^2*c^5*d*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2

*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^

9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c

^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2

*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 +

4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) + a^2*c^2*d^4*f^2*(((1280*c^3*d^8 - 1200*c*

d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^

6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*

d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(

a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c

^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 4

*a^2*c^3*d^3*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*

c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1

i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 2

56*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 +

4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d

^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) - a^2*c^4*d^2*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4

)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*

c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4

*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a

^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*

f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i)/(512*(d^6 + c^2*d^4)*(a^2*d^6*f^2

*1i - a^2*c^6*f^2*1i + 2*a^2*c*d^5*f^2 + 2*a^2*c^5*d*f^2 + a^2*c^2*d^4*f^2*1i + 4*a^2*c^3*d^3*f^2 - a^2*c^4*d^

2*f^2*1i)))^(1/2) - log(10*a*d^7*f - (-(d^11*240i - 1200*c*d^10 - c^2*d^9*1920i + 1280*c^3*d^8 + c^4*d^7*240i

+ 208*c^5*d^6 + c^6*d^5*96i + 32*c^7*d^4 + a^2*c^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*

d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 +

c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 +

56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1

/2) + a^2*d^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^

5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(

a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*

f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^2*d^6*f^2*((2280704*c^2

*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d

^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8

*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*

a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 6*a^2*c^4*d^4*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600

576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800

*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*

d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d

^2*f^4))^(1/2) + 4*a^2*c^6*d^2*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033

344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 5

7600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4

+ 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2))/(512*a^2*d^14*f

^2 + 2560*a^2*c^2*d^12*f^2 + 5120*a^2*c^4*d^10*f^2 + 5120*a^2*c^6*d^8*f^2 + 2560*a^2*c^8*d^6*f^2 + 512*a^2*c^1

0*d^4*f^2))^(1/2)*((-(d^11*240i - 1200*c*d^10 - c^2*d^9*1920i + 1280*c^3*d^8 + c^4*d^7*240i + 208*c^5*d^6 + c^

6*d^5*96i + 32*c^7*d^4 + a^2*c^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 70

33344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i +

57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^

4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + a^2*d^8*f^2*

((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i +

1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4

*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*

d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^2*d^6*f^2*((2280704*c^2*d^20 - 73984*d^22

- c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706

560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4

+ 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 +

8*a^4*c^14*d^2*f^4))^(1/2) + 6*a^2*c^4*d^4*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*50

93376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^

13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*

c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4

*a^2*c^6*d^2*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5

*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a

^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f

^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2))/(512*a^2*d^14*f^2 + 2560*a^2*c^2*

d^12*f^2 + 5120*a^2*c^4*d^10*f^2 + 5120*a^2*c^6*d^8*f^2 + 2560*a^2*c^8*d^6*f^2 + 512*a^2*c^10*d^4*f^2))^(1/2)*

(104*a^3*c*d^9*f^3 - a^3*d^10*f^3*24i + a^3*c^2*d^8*f^3*24i + 216*a^3*c^3*d^7*f^3 + a^3*c^4*d^6*f^3*120i + 120

*a^3*c^5*d^5*f^3 + a^3*c^6*d^4*f^3*72i + 8*a^3*c^7*d^3*f^3 + 2*(-(d^11*240i - 1200*c*d^10 - c^2*d^9*1920i + 12

80*c^3*d^8 + c^4*d^7*240i + 208*c^5*d^6 + c^6*d^5*96i + 32*c^7*d^4 + a^2*c^8*f^2*((2280704*c^2*d^20 - 73984*d^

22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*7

06560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^

4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4

+ 8*a^4*c^14*d^2*f^4))^(1/2) + a^2*d^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*509337

6i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*3

8400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*

d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2

*c^2*d^6*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^1

7*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c

^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 +

56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 6*a^2*c^4*d^4*f^2*((2280704*c^2*d^20

- 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 -

c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*

c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c

^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^6*d^2*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i

+ c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*

d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*

f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^

4))^(1/2))/(512*a^2*d^14*f^2 + 2560*a^2*c^2*d^12*f^2 + 5120*a^2*c^4*d^10*f^2 + 5120*a^2*c^6*d^8*f^2 + 2560*a^2

*c^8*d^6*f^2 + 512*a^2*c^10*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^

2*2i)*(a^2*c^2*d^7*f^2*64i - 32*a^2*c*d^8*f^2 - 32*a^2*c^3*d^6*f^2 + a^2*c^4*d^5*f^2*128i + 32*a^2*c^5*d^4*f^2

+ a^2*c^6*d^3*f^2*64i + 32*a^2*c^7*d^2*f^2)) + 2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*

c*d*f^2*2i)*(c*d^5*20i + 17*d^6 - 7*c^2*d^4 + c^3*d^3*10i + 2*c^4*d^2)) + (a*c*d^6*f*39i)/2 - (17*a*c^2*d^5*f)

/2 + (a*c^3*d^4*f*3i)/2 - (a*c^4*d^3*f)/2)*(-(d^11*240i - 1200*c*d^10 - c^2*d^9*1920i + 1280*c^3*d^8 + c^4*d^7

*240i + 208*c^5*d^6 + c^6*d^5*96i + 32*c^7*d^4 + a^2*c^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i

+ c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*

d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*

f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^

4))^(1/2) + a^2*d^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^1

8 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d

^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^

8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^2*d^6*f^2*((22807

04*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000

*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f

^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4

+ 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 6*a^2*c^4*d^4*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^

21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i +

684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^

4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*

c^14*d^2*f^4))^(1/2) + 4*a^2*c^6*d^2*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i

- 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*3840

0i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^1

0*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2))/(512*a^2*

d^14*f^2 + 2560*a^2*c^2*d^12*f^2 + 5120*a^2*c^4*d^10*f^2 + 5120*a^2*c^6*d^8*f^2 + 2560*a^2*c^8*d^6*f^2 + 512*a

^2*c^10*d^4*f^2))^(1/2) - ((d^2*2i)/(a*f*(c^2 + d^2)) + (d*(c - d*5i)*(c + d*tan(e + f*x))*1i)/(2*a*f*(c*1i -

d)*(c^2 + d^2)))/((c + d*1i)*(c + d*tan(e + f*x))^(1/2) - (c + d*tan(e + f*x))^(3/2)) - log(10*a*d^7*f - ((120

0*c*d^10 - d^11*240i + c^2*d^9*1920i - 1280*c^3*d^8 - c^4*d^7*240i - 208*c^5*d^6 - c^6*d^5*96i - 32*c^7*d^4 +

a^2*c^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17

*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^

16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 +

56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + a^2*d^8*f^2*((2280704*c^2*d^20 - 7398

4*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^

15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^1

4*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4

*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^2*d^6*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d

^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 +

c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 5

6*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/

2) + 6*a^2*c^4*d^4*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18

- c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^

12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8

*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^6*d^2*f^2*((228070

4*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*

c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^

4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4

+ 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2))/(512*a^2*d^14*f^2 + 2560*a^2*c^2*d^12*f^2 + 5120*a^2*c^4*d

^10*f^2 + 5120*a^2*c^6*d^8*f^2 + 2560*a^2*c^8*d^6*f^2 + 512*a^2*c^10*d^4*f^2))^(1/2)*(((1200*c*d^10 - d^11*240

i + c^2*d^9*1920i - 1280*c^3*d^8 - c^4*d^7*240i - 208*c^5*d^6 - c^6*d^5*96i - 32*c^7*d^4 + a^2*c^8*f^2*((22807

04*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000

*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f

^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4

+ 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + a^2*d^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600

576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800

*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*

d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d

^2*f^4))^(1/2) + 4*a^2*c^2*d^6*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033

344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 5

7600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4

+ 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 6*a^2*c^4*d^4*

f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840

i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 +

a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c

^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^6*d^2*f^2*((2280704*c^2*d^20 - 73984*

d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15

*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*

f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f

^4 + 8*a^4*c^14*d^2*f^4))^(1/2))/(512*a^2*d^14*f^2 + 2560*a^2*c^2*d^12*f^2 + 5120*a^2*c^4*d^10*f^2 + 5120*a^2*

c^6*d^8*f^2 + 2560*a^2*c^8*d^6*f^2 + 512*a^2*c^10*d^4*f^2))^(1/2)*(104*a^3*c*d^9*f^3 - a^3*d^10*f^3*24i + a^3*

c^2*d^8*f^3*24i + 216*a^3*c^3*d^7*f^3 + a^3*c^4*d^6*f^3*120i + 120*a^3*c^5*d^5*f^3 + a^3*c^6*d^4*f^3*72i + 8*a

^3*c^7*d^3*f^3 + 2*((1200*c*d^10 - d^11*240i + c^2*d^9*1920i - 1280*c^3*d^8 - c^4*d^7*240i - 208*c^5*d^6 - c^6

*d^5*96i - 32*c^7*d^4 + a^2*c^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 703

3344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i +

57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4

+ 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + a^2*d^8*f^2*(

(2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1

856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*

d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d

^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^2*d^6*f^2*((2280704*c^2*d^20 - 73984*d^22

- c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*7065

60i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 +

28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 +

8*a^4*c^14*d^2*f^4))^(1/2) + 6*a^2*c^4*d^4*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*509

3376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^1

3*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c

^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*

a^2*c^6*d^2*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*

d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^

4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^

4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2))/(512*a^2*d^14*f^2 + 2560*a^2*c^2*d

^12*f^2 + 5120*a^2*c^4*d^10*f^2 + 5120*a^2*c^6*d^8*f^2 + 2560*a^2*c^8*d^6*f^2 + 512*a^2*c^10*d^4*f^2))^(1/2)*(

c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*(a^2*c^2*d^7*f^2*64i - 32*a^2*c*d^8*f^2

- 32*a^2*c^3*d^6*f^2 + a^2*c^4*d^5*f^2*128i + 32*a^2*c^5*d^4*f^2 + a^2*c^6*d^3*f^2*64i + 32*a^2*c^7*d^2*f^2))

+ 2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*(c*d^5*20i + 17*d^6 - 7*c^2*d^4 +

c^3*d^3*10i + 2*c^4*d^2)) + (a*c*d^6*f*39i)/2 - (17*a*c^2*d^5*f)/2 + (a*c^3*d^4*f*3i)/2 - (a*c^4*d^3*f)/2)*((

1200*c*d^10 - d^11*240i + c^2*d^9*1920i - 1280*c^3*d^8 - c^4*d^7*240i - 208*c^5*d^6 - c^6*d^5*96i - 32*c^7*d^4

+ a^2*c^8*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d

^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4

*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4

+ 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + a^2*d^8*f^2*((2280704*c^2*d^20 - 7

3984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7

*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*

d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*

d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^2*d^6*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^

3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14

+ c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4

+ 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^

(1/2) + 6*a^2*c^4*d^4*f^2*((2280704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d

^18 - c^5*d^17*5667840i + 1856000*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10

*d^12)/(a^4*c^16*f^4 + a^4*d^16*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*

c^8*d^8*f^4 + 56*a^4*c^10*d^6*f^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2) + 4*a^2*c^6*d^2*f^2*((228

0704*c^2*d^20 - 73984*d^22 - c*d^21*600576i + c^3*d^19*5093376i - 7033344*c^4*d^18 - c^5*d^17*5667840i + 18560

00*c^6*d^16 - c^7*d^15*706560i + 684800*c^8*d^14 + c^9*d^13*38400i + 57600*c^10*d^12)/(a^4*c^16*f^4 + a^4*d^16

*f^4 + 8*a^4*c^2*d^14*f^4 + 28*a^4*c^4*d^12*f^4 + 56*a^4*c^6*d^10*f^4 + 70*a^4*c^8*d^8*f^4 + 56*a^4*c^10*d^6*f

^4 + 28*a^4*c^12*d^4*f^4 + 8*a^4*c^14*d^2*f^4))^(1/2))/(512*a^2*d^14*f^2 + 2560*a^2*c^2*d^12*f^2 + 5120*a^2*c^

4*d^10*f^2 + 5120*a^2*c^6*d^8*f^2 + 2560*a^2*c^8*d^6*f^2 + 512*a^2*c^10*d^4*f^2))^(1/2) + log(10*a*d^7*f - (((

c*d^8*720i + 240*d^9 - 240*c^2*d^7 - c^3*d^6*80i - 160*c^4*d^5 + c^5*d^4*32i - a^2*c^6*f^2*(((1280*c^3*d^8 - 1

200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*

a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^

2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)

*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d

^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*

1i + a^2*d^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*

c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1

i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 2

56*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 +

4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d

^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 2*a^2*c*d^5*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*

d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 19

20*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2

+ 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 +

4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d

^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) + 2*a^2*c^5*d*f^2*(((1280*c^3*d^8

- 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 +

4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4

*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d

^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^

2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/

2) + a^2*c^2*d^4*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*

a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^

5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6

+ 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^

4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c

^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + 4*a^2*c^3*d^3*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 3

2*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^1

1 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^

4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*

f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 +

a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) - a^2*c^4*d^2*f^2*(((1280*c^

3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4

*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f

^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6

*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^

8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)

))^(1/2)*1i)/(512*(d^6 + c^2*d^4)*(a^2*d^6*f^2*1i - a^2*c^6*f^2*1i + 2*a^2*c*d^5*f^2 + 2*a^2*c^5*d*f^2 + a^2*c

^2*d^4*f^2*1i + 4*a^2*c^3*d^3*f^2 - a^2*c^4*d^2*f^2*1i)))^(1/2)*(104*a^3*c*d^9*f^3 - a^3*d^10*f^3*24i + a^3*c^

2*d^8*f^3*24i + 216*a^3*c^3*d^7*f^3 + a^3*c^4*d^6*f^3*120i + 120*a^3*c^5*d^5*f^3 + a^3*c^6*d^4*f^3*72i + 8*a^3

*c^7*d^3*f^3 - 2*(c + d*tan(e + f*x))^(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*((c*d^8*720i + 240*d^

9 - 240*c^2*d^7 - c^3*d^6*80i - 160*c^4*d^5 + c^5*d^4*32i - a^2*c^6*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^

5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) +

((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a

^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 +

a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*

c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i + a^2*d^6*f^2*((

(1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2

*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a

^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c

*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)

+ (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*

d^2*f^4)))^(1/2)*1i + 2*a^2*c*d^5*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 +

a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^

4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2

))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 +

6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*

d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) + 2*a^2*c^5*d*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208

*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2)

+ ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 +

6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^

4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a

^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) + a^2*c^2*d^4*f^

2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6

*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2

+ a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((

24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f

^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*

c^6*d^2*f^4)))^(1/2)*1i + 4*a^2*c^3*d^3*f^2*(((1280*c^3*d^8 - 1200*c*d^10 + 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8

*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2) + ((240*d^11 - 1920*c^2*d^9 +

240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d

^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6

*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^

4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2) - a^2*c^4*d^2*f^2*(((1280*c^3*d^8 - 1200*c*d^10

+ 208*c^5*d^6 + 32*c^7*d^4)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^

2*f^2) + ((240*d^11 - 1920*c^2*d^9 + 240*c^4*d^7 + 96*c^6*d^5)*1i)/(a^2*c^8*f^2 + a^2*d^8*f^2 + 4*a^2*c^2*d^6*

f^2 + 6*a^2*c^4*d^4*f^2 + 4*a^2*c^6*d^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((24*c*d^7 + 6*c^3*d^5)*1i)/(a^4*

c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4) + (16*d^8 - c^2*d^6 + c^4*d

^4)/(a^4*c^8*f^4 + a^4*d^8*f^4 + 4*a^4*c^2*d^6*f^4 + 6*a^4*c^4*d^4*f^4 + 4*a^4*c^6*d^2*f^4)))^(1/2)*1i)/(512*(

d^6 + c^2*d^4)*(a^2*d^6*f^2*1i - a^2*c^6*f^2*1i + 2*a^2*c*d^5*f^2 + 2*a^2*c^5*d*f^2 + a^2*c^2*d^4*f^2*1i + 4*a

^2*c^3*d^3*f^2 - a^2*c^4*d^2*f^2*1i)))^(1/2)*(a^2*c^2*d^7*f^2*64i - 32*a^2*c*d^8*f^2 - 32*a^2*c^3*d^6*f^2 + a^

2*c^4*d^5*f^2*128i + 32*a^2*c^5*d^4*f^2 + a^2*c^6*d^3*f^2*64i + 32*a^2*c^7*d^2*f^2)) - 2*(c + d*tan(e + f*x))^

(1/2)*(a^2*d^2*f^2 - a^2*c^2*f^2 + a^2*c*d*f^2*2i)*(c*d^5*20i + 17*d^6 - 7*c^2*d^4 + c^3*d^3*10i + 2*c^4*d^2))

*((c*d^8*720i + 240*d^9 - 240*c^2*d^7 - c^3*d^6*80i - 160*c^4*d^5 + c^5*d^4*32i - a^2*c^6*f^2*(((1280*c^3*d^8