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

3.1130 \(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=368 \[ \frac {6 c^2+27 i c d-56 d^2}{48 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}+\frac {d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 f (c-i d) (c+i d)^4 \sqrt {c+d \tan (e+f x)}}+\frac {\left (2 i c^3-12 c^2 d-33 i c d^2+58 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{9/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f (c-i d)^{3/2}}+\frac {-10 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}} \]

[Out]

-1/8*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^3/(c-I*d)^(3/2)/f+1/16*(2*I*c^3-12*c^2*d-33*I*c*d^2+58*

d^3)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^3/(c+I*d)^(9/2)/f+1/16*d*(2*c^3+9*I*c^2*d-17*c*d^2+60*I*d

^3)/a^3/(c-I*d)/(c+I*d)^4/f/(c+d*tan(f*x+e))^(1/2)-1/6/(I*c-d)/f/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^3+1

/24*(3*I*c-10*d)/a/(c+I*d)^2/f/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2+1/48*(6*c^2+27*I*c*d-56*d^2)/(I*c-d

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

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Rubi [A] time = 1.20, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3559, 3596, 3529, 3539, 3537, 63, 208} \[ \frac {d \left (9 i c^2 d+2 c^3-17 c d^2+60 i d^3\right )}{16 a^3 f (c-i d) (c+i d)^4 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\left (-12 c^2 d+2 i c^3-33 i c d^2+58 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{9/2}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f (c-i d)^{3/2}}+\frac {-10 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((-I/8)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*(c - I*d)^(3/2)*f) + (((2*I)*c^3 - 12*c^2*d - (3

3*I)*c*d^2 + 58*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*(c + I*d)^(9/2)*f) + (d*(2*c^3 +

(9*I)*c^2*d - 17*c*d^2 + (60*I)*d^3))/(16*a^3*(c - I*d)*(c + I*d)^4*f*Sqrt[c + d*Tan[e + f*x]]) - 1/(6*(I*c -

d)*f*(a + I*a*Tan[e + f*x])^3*Sqrt[c + d*Tan[e + f*x]]) + ((3*I)*c - 10*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e

+ f*x])^2*Sqrt[c + d*Tan[e + f*x]]) + (6*c^2 + (27*I)*c*d - 56*d^2)/(48*(I*c - d)^3*f*(a^3 + I*a^3*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 3559

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

p[(a*(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*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*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] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

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,

Rubi steps

\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {-\frac {1}{2} a (6 i c-13 d)-\frac {7}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx}{6 a^2 (i c-d)}\\ &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {-\frac {1}{2} a^2 \left (12 c^2+39 i c d-62 d^2\right )-\frac {5}{2} a^2 (3 c+10 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {\frac {3}{2} a^3 \left (4 i c^3-18 c^2 d-39 i c d^2+60 d^3\right )+\frac {3}{2} a^3 d \left (6 i c^2-27 c d-56 i d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac {d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {\frac {3}{2} a^3 \left (4 i c^4-18 c^3 d-33 i c^2 d^2+33 c d^3-56 i d^4\right )+\frac {3}{2} a^3 d \left (2 i c^3-9 c^2 d-17 i c d^2-60 d^3\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )}\\ &=\frac {d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^3 (c-i d)}+\frac {\left (2 c^3+12 i c^2 d-33 c d^2-58 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)^4}\\ &=\frac {d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}-\frac {\operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{16 a^3 (i c+d) f}-\frac {\left (2 i c^3-12 c^2 d-33 i c d^2+58 d^3\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)^4 f}\\ &=\frac {d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \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 )}{8 a^3 (c-i d) d f}-\frac {\left (2 c^3+12 i c^2 d-33 c d^2-58 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)^4 d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 (c-i d)^{3/2} f}+\frac {\left (2 i c^3-12 c^2 d-33 i c d^2+58 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{9/2} f}+\frac {d \left (2 c^3+9 i c^2 d-17 c d^2+60 i d^3\right )}{16 a^3 (c-i d) (c+i d)^4 f \sqrt {c+d \tan (e+f x)}}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3 \sqrt {c+d \tan (e+f x)}}+\frac {3 i c-10 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {6 c^2+27 i c d-56 d^2}{48 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}\\ \end {align*}

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Mathematica [B] time = 7.96, size = 989, normalized size = 2.69 \[ \frac {\sec ^3(e+f x) (\cos (3 e)+i \sin (3 e)) \left (\frac {2 \left (60 i d^4-17 c d^3+9 i c^2 d^2+2 c^3 d\right ) \left (\frac {\tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )}{2 \sqrt {-c-i d}}+\frac {\tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i d-c}}\right )}{2 \sqrt {i d-c}}\right ) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left (\tan ^2(e+f x)+1\right )}-\frac {i \left (4 c^4+18 i d c^3-33 d^2 c^2-33 i d^3 c-56 d^4\right ) \left (\frac {\tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )}{\sqrt {-c-i d}}-\frac {\tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i d-c}}\right )}{\sqrt {i d-c}}\right ) \sec (e+f x) (c+d \tan (e+f x))}{(c \cos (e+f x)+d \sin (e+f x)) \left (\tan ^2(e+f x)+1\right )}\right ) (\cos (f x)+i \sin (f x))^3}{32 (c-i d) (c+i d)^4 f (i \tan (e+f x) a+a)^3}+\frac {\sec ^3(e+f x) \sqrt {\sec (e+f x) (c \cos (e+f x)+d \sin (e+f x))} \left (\frac {\left (18 c^2+79 i d c-118 d^2\right ) \cos (2 f x) \left (\frac {1}{96} i \cos (e)-\frac {\sin (e)}{96}\right )}{(c+i d)^4}+\frac {(9 c+20 i d) \cos (4 f x) \left (\frac {1}{96} i \cos (e)+\frac {\sin (e)}{96}\right )}{(c+i d)^3}+\frac {\left (11 \cos (e) c^4+43 i d \cos (e) c^3+11 d \sin (e) c^3-46 d^2 \cos (e) c^2+43 i d^2 \sin (e) c^2+100 i d^3 \cos (e) c-46 d^3 \sin (e) c+192 d^4 \cos (e)+100 i d^4 \sin (e)\right ) \left (\frac {1}{96} \cos (3 e)+\frac {1}{96} i \sin (3 e)\right )}{(c-i d) (c+i d)^4 (-i c \cos (e)-i d \sin (e))}+\frac {\cos (6 f x) \left (\frac {1}{48} i \cos (3 e)+\frac {1}{48} \sin (3 e)\right )}{(c+i d)^2}+\frac {\left (18 c^2+79 i d c-118 d^2\right ) \left (\frac {\cos (e)}{96}+\frac {1}{96} i \sin (e)\right ) \sin (2 f x)}{(c+i d)^4}+\frac {(9 c+20 i d) \left (\frac {\cos (e)}{96}-\frac {1}{96} i \sin (e)\right ) \sin (4 f x)}{(c+i d)^3}+\frac {\left (\frac {1}{48} \cos (3 e)-\frac {1}{48} i \sin (3 e)\right ) \sin (6 f x)}{(c+i d)^2}+\frac {2 \left (\frac {1}{2} \cos (3 e-f x) d^5-\frac {1}{2} \cos (3 e+f x) d^5+\frac {1}{2} i \sin (3 e-f x) d^5-\frac {1}{2} i \sin (3 e+f x) d^5\right )}{(c-i d) (c+i d)^4 (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}\right ) (\cos (f x)+i \sin (f x))^3}{f (i \tan (e+f x) a+a)^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*Sqrt[Sec[e + f*x]*(c*Cos[e + f*x] + d*Sin[e + f*x])]*(((18*c^2 + (79

*I)*c*d - 118*d^2)*Cos[2*f*x]*((I/96)*Cos[e] - Sin[e]/96))/(c + I*d)^4 + ((9*c + (20*I)*d)*Cos[4*f*x]*((I/96)*

Cos[e] + Sin[e]/96))/(c + I*d)^3 + ((11*c^4*Cos[e] + (43*I)*c^3*d*Cos[e] - 46*c^2*d^2*Cos[e] + (100*I)*c*d^3*C

os[e] + 192*d^4*Cos[e] + 11*c^3*d*Sin[e] + (43*I)*c^2*d^2*Sin[e] - 46*c*d^3*Sin[e] + (100*I)*d^4*Sin[e])*(Cos[

3*e]/96 + (I/96)*Sin[3*e]))/((c - I*d)*(c + I*d)^4*((-I)*c*Cos[e] - I*d*Sin[e])) + (Cos[6*f*x]*((I/48)*Cos[3*e

] + Sin[3*e]/48))/(c + I*d)^2 + ((18*c^2 + (79*I)*c*d - 118*d^2)*(Cos[e]/96 + (I/96)*Sin[e])*Sin[2*f*x])/(c +

I*d)^4 + ((9*c + (20*I)*d)*(Cos[e]/96 - (I/96)*Sin[e])*Sin[4*f*x])/(c + I*d)^3 + ((Cos[3*e]/48 - (I/48)*Sin[3*

e])*Sin[6*f*x])/(c + I*d)^2 + (2*((d^5*Cos[3*e - f*x])/2 - (d^5*Cos[3*e + f*x])/2 + (I/2)*d^5*Sin[3*e - f*x] -

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

(f*(a + I*a*Tan[e + f*x])^3) + (Sec[e + f*x]^3*(Cos[3*e] + I*Sin[3*e])*(Cos[f*x] + I*Sin[f*x])^3*(((-I)*(4*c^4

+ (18*I)*c^3*d - 33*c^2*d^2 - (33*I)*c*d^3 - 56*d^4)*(ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]]/Sqrt[-c

- I*d] - ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]]/Sqrt[-c + I*d])*Sec[e + f*x]*(c + d*Tan[e + f*x]))/(

(c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e + f*x]^2)) + (2*(2*c^3*d + (9*I)*c^2*d^2 - 17*c*d^3 + (60*I)*d^4)

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

+ I*d]]/(2*Sqrt[-c + I*d]))*Sec[e + f*x]*(c + d*Tan[e + f*x]))/((c*Cos[e + f*x] + d*Sin[e + f*x])*(1 + Tan[e

+ f*x]^2))))/(32*(c - I*d)*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])^3)

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fricas [B] time = 5.17, size = 2645, normalized size = 7.19 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I*a^3*c*d^5 - 24*a^3*

d^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*d^4 - 96*I*a^3*c*d^5

+ 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt(I/((-64*I*a^6*c^3 - 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a^6*d^3)*f

^2))*log((((16*I*a^3*c^2 + 32*a^3*c*d - 16*I*a^3*d^2)*f*e^(2*I*f*x + 2*I*e) + (16*I*a^3*c^2 + 32*a^3*c*d - 16*

I*a^3*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/((-64*I*a^6*c^3

- 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a^6*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)) - ((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I*a^3*c*d^5

- 24*a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*d^4 - 96*I*

a^3*c*d^5 + 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt(I/((-64*I*a^6*c^3 - 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a

^6*d^3)*f^2))*log((((-16*I*a^3*c^2 - 32*a^3*c*d + 16*I*a^3*d^2)*f*e^(2*I*f*x + 2*I*e) + (-16*I*a^3*c^2 - 32*a^

3*c*d + 16*I*a^3*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/((-6

4*I*a^6*c^3 - 192*a^6*c^2*d + 192*I*a^6*c*d^2 + 64*a^6*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)) - ((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I

*a^3*c*d^5 - 24*a^3*d^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*

d^4 - 96*I*a^3*c*d^5 + 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt((4*I*c^6 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*

d^3 + 2481*I*c^2*d^4 - 3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504

*a^6*c^6*d^3 - 32256*I*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c

*d^8 + 256*a^6*d^9)*f^2))*log((2*c^4 + 14*I*c^3*d - 45*c^2*d^2 - 91*I*c*d^3 + 58*d^4 + ((16*I*a^3*c^5 - 80*a^3

*c^4*d - 160*I*a^3*c^3*d^2 + 160*a^3*c^2*d^3 + 80*I*a^3*c*d^4 - 16*a^3*d^5)*f*e^(2*I*f*x + 2*I*e) + (16*I*a^3*

c^5 - 80*a^3*c^4*d - 160*I*a^3*c^3*d^2 + 160*a^3*c^2*d^3 + 80*I*a^3*c*d^4 - 16*a^3*d^5)*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 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*d^3

+ 2481*I*c^2*d^4 - 3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504*a^

6*c^6*d^3 - 32256*I*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c*d^

8 + 256*a^6*d^9)*f^2)) + (2*c^4 + 12*I*c^3*d - 33*c^2*d^2 - 58*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^5 + 80*a^3*c^4*d + 160*I*a^3*c^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*f)) +

((24*a^3*c^6 + 48*I*a^3*c^5*d + 24*a^3*c^4*d^2 + 96*I*a^3*c^3*d^3 - 24*a^3*c^2*d^4 + 48*I*a^3*c*d^5 - 24*a^3*d

^6)*f*e^(8*I*f*x + 8*I*e) + (24*a^3*c^6 + 96*I*a^3*c^5*d - 120*a^3*c^4*d^2 - 120*a^3*c^2*d^4 - 96*I*a^3*c*d^5

+ 24*a^3*d^6)*f*e^(6*I*f*x + 6*I*e))*sqrt((4*I*c^6 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*d^3 + 2481*I*c^2*d^4

- 3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504*a^6*c^6*d^3 - 32256*

I*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c*d^8 + 256*a^6*d^9)*f

^2))*log((2*c^4 + 14*I*c^3*d - 45*c^2*d^2 - 91*I*c*d^3 + 58*d^4 + ((-16*I*a^3*c^5 + 80*a^3*c^4*d + 160*I*a^3*c

^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*f*e^(2*I*f*x + 2*I*e) + (-16*I*a^3*c^5 + 80*a^3*c^4*d

+ 160*I*a^3*c^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*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 - 48*c^5*d - 276*I*c^4*d^2 + 1024*c^3*d^3 + 2481*I*c^2*d^4 -

3828*c*d^5 - 3364*I*d^6)/((-256*I*a^6*c^9 + 2304*a^6*c^8*d + 9216*I*a^6*c^7*d^2 - 21504*a^6*c^6*d^3 - 32256*I

*a^6*c^5*d^4 + 32256*a^6*c^4*d^5 + 21504*I*a^6*c^3*d^6 - 9216*a^6*c^2*d^7 - 2304*I*a^6*c*d^8 + 256*a^6*d^9)*f^

2)) + (2*c^4 + 12*I*c^3*d - 33*c^2*d^2 - 58*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^5

+ 80*a^3*c^4*d + 160*I*a^3*c^3*d^2 - 160*a^3*c^2*d^3 - 80*I*a^3*c*d^4 + 16*a^3*d^5)*f)) - (-2*I*c^4 + 4*c^3*d

+ 4*c*d^3 + 2*I*d^4 + (-11*I*c^4 + 32*c^3*d + 3*I*c^2*d^2 + 146*c*d^3 - 292*I*d^4)*e^(8*I*f*x + 8*I*e) + (-29

*I*c^4 + 97*c^3*d + 67*I*c^2*d^2 + 211*c*d^3 - 210*I*d^4)*e^(6*I*f*x + 6*I*e) + (-27*I*c^4 + 90*c^3*d + 71*I*c

^2*d^2 + 90*c*d^3 + 98*I*d^4)*e^(4*I*f*x + 4*I*e) + (-11*I*c^4 + 29*c^3*d + 7*I*c^2*d^2 + 29*c*d^3 + 18*I*d^4)

*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)))/((96*a^3*c^6

+ 192*I*a^3*c^5*d + 96*a^3*c^4*d^2 + 384*I*a^3*c^3*d^3 - 96*a^3*c^2*d^4 + 192*I*a^3*c*d^5 - 96*a^3*d^6)*f*e^(8

*I*f*x + 8*I*e) + (96*a^3*c^6 + 384*I*a^3*c^5*d - 480*a^3*c^4*d^2 - 480*a^3*c^2*d^4 - 384*I*a^3*c*d^5 + 96*a^3

*d^6)*f*e^(6*I*f*x + 6*I*e))

________________________________________________________________________________________

giac [B] time = 2.65, size = 787, normalized size = 2.14 \[ \frac {2 \, d^{4}}{{\left (-i \, a^{3} c^{5} f + 3 \, a^{3} c^{4} d f + 2 i \, a^{3} c^{3} d^{2} f + 2 \, a^{3} c^{2} d^{3} f + 3 i \, a^{3} c d^{4} f - a^{3} d^{5} f\right )} \sqrt {d \tan \left (f x + e\right ) + c}} - \frac {4 \, {\left (2 i \, c^{3} - 12 \, c^{2} d - 33 i \, c d^{2} + 58 \, 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 (16 \, a^{3} c^{4} f + 64 i \, a^{3} c^{3} d f - 96 \, a^{3} c^{2} d^{2} f - 64 i \, a^{3} c d^{3} f + 16 \, a^{3} d^{4} f\right )} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {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 )}{2 \, {\left (a^{3} c f - i \, a^{3} d 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 (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 + 33 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d^{2} - 84 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d^{2} + 51 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d^{2} - 84 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{3} + 268 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{3} - 204 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{3} + 196 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{4} - 279 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{4} + 120 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{5}\right )}}{{\left (-96 i \, a^{3} c^{4} f + 384 \, a^{3} c^{3} d f + 576 i \, a^{3} c^{2} d^{2} f - 384 \, a^{3} c d^{3} f - 96 i \, a^{3} d^{4} f\right )} {\left (-i \, d \tan \left (f x + e\right ) - d\right )}^{3}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

(d*tan(f*x + e) + c)) - 4*(2*I*c^3 - 12*c^2*d - 33*I*c*d^2 + 58*d^3)*arctan(4*(sqrt(d*tan(f*x + e) + c)*c - sq

rt(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))))/((16*a^3*c^4*f + 64*I*a^3*c^3*d*f - 96*a^3*c^2*d^2*f - 64*

I*a^3*c*d^3*f + 16*a^3*d^4*f)*sqrt(-8*c + 8*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 1/2*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*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*c*f - I*a^3*d*f)*sq

rt(-8*c + 8*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) + 2*(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 + 33*I*(d*tan(f*x + e) + c)^(5/2)*c*d^2 - 84

*I*(d*tan(f*x + e) + c)^(3/2)*c^2*d^2 + 51*I*sqrt(d*tan(f*x + e) + c)*c^3*d^2 - 84*(d*tan(f*x + e) + c)^(5/2)*

d^3 + 268*(d*tan(f*x + e) + c)^(3/2)*c*d^3 - 204*sqrt(d*tan(f*x + e) + c)*c^2*d^3 + 196*I*(d*tan(f*x + e) + c)

^(3/2)*d^4 - 279*I*sqrt(d*tan(f*x + e) + c)*c*d^4 + 120*sqrt(d*tan(f*x + e) + c)*d^5)/((-96*I*a^3*c^4*f + 384*

a^3*c^3*d*f + 576*I*a^3*c^2*d^2*f - 384*a^3*c*d^3*f - 96*I*a^3*d^4*f)*(-I*d*tan(f*x + e) - d)^3)

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maple [B] time = 0.46, size = 3053, normalized size = 8.30 \[ \text {output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

3/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^2+5/2/f/a^3*d^9/(c+I*d)^4/(I*d-c)/(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)-7/4/f/a^3*d^7/(c+I*d)^4/(I*d-c)/(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))^(5/2)-29/8/f/a^3*d^7/(c+I*d)^4/(I*d-c)/(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))-2*I/f/a^3*d^4/(I*c+d)/(I*c-d)/(c+I*d)^3/

(c+d*tan(f*x+e))^(1/2)+49/12*I/f/a^3*d^8/(c+I*d)^4/(I*d-c)/(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)+73/16*I/f/a^3*d^6/(c+I*d)^4/(I*d-c)/(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^3-173/16*I/f/a^3*d^8/(c+I*d)^4/(I*d-c)/(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-59/4*I/f/a^3*d^4/(c+I*d)^4/(I*d-c)/(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^4-155/12*I/f/a^3*d^6/(c+I*d)^4/(I*d-c)/(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^2-1/2/f/a^3*d^3/(I*d-c)^(3/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d

-c)^(1/2))*c-1/8*I/f/a^3*d^4/(I*d-c)^(3/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))-1/8*I/f/a^3/

(I*d-c)^(3/2)/(c+I*d)^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^4+67/16*I/f/a^3*d^6/(c+I*d)^4/(I*d-c)/(

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))^(5/2)*c-109/12/f/a^3*d^3/(c+I*d)^4/(I*d-c)/

(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^5+45/8*I/f/a^3*d^4/(c+I*d)^4/(I*d-

c)/(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+149/16*I/f/a

^3*d^6/(c+I*d)^4/(I*d-c)/(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+39/8/f/a^3*d^5/(c+I*d)^4/(I*d-c)/(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^2+53/12/f/a^3*d^5/(c+I*d)^4/(I*d-c)/(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+55/4/f/a^3*d^7/(c+I*d)^4/(I*d-c)/(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-1/8/f/a^3*d/(c+I*d)^4/(I*d-c)/(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^8+51/8/f/a^3*d^3/(c+I*d)^4/(I*d-c)/(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^6-95/8/f/a^3*d^5/(c+I*d)^4/(I*d-c)/(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^4-127/8/f/a^3*d^7/(c+I*d)^4/(I*d-c)/(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^2-1/f/a^3*d/(c+I*d)^4/(I*d-c)/(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^6-1/8/f/a^3*d/(c+I*d)^4/(I*d-c)/(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))^(5/2)*c^6-15/16*I/f/a^3*d^2/(c+I*d)^4/(I*d-c)/(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))^(5/2)*c^5+13/4*I/f/a^3*d^4/(c+I*d)^4/(I*d-c)/(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))^(5/2)*c^3+9/4*I/f/a^3*d^2/(c+I*d)^4/(I*d-c)/(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^6+1/8*I/f/a^3/(c+I*d)^4/(I*d-c)/(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^7-57/16*I/f/a^3*d^2/(c+I*d)^4/(I*d-c)/(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^5+15/2/f/a^3*d^3/(c+I*d)^4/

(I*d-c)/(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+25/8/f/

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

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

a^3*d/(c+I*d)^4/(I*d-c)/(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^7-21/16*I/

f/a^3*d^2/(c+I*d)^4/(I*d-c)/(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^7+225/

16*I/f/a^3*d^4/(c+I*d)^4/(I*d-c)/(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

________________________________________________________________________________________

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))^3/(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.

________________________________________________________________________________________

mupad [B] time = 13.88, size = 106340, normalized size = 288.97 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(6960*a^3*d^15*f - ((-(c*d^12*6300i + 3360*d^13 - 945*c^2*d^11 + c^3*d^10*1365i + 315*c^4*d^9 + c^5*d^8*693

i + 672*c^6*d^7 - c^7*d^6*288i - 72*c^8*d^5 + c^9*d^4*8i - a^6*c^10*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16

)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a

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

*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089

*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 +

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

a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (8914

5*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d

^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a^6

*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^

16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c

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

56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*4i + a^6*d^10*f^2*(4*(((29973*c^2*d

^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64

- (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d

^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*

c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)

*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a

^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4)

+ (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 3

4*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*

c^6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*

d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6

- 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 +

70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*4i + 24*a^6*c*

d^9*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^

8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4

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

c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7

)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56

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

*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^

11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a

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

*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339

*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^

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

)^2)^(1/2) + 24*a^6*c^9*d*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^

10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*

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

c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c

^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*

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

8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^

6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a

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

6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2

+ 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^

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

+ 8*a^6*c^14*d^2*f^2))^2)^(1/2) - a^6*c^2*d^8*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^1

2)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d

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

10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c

^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^

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

8*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675

*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2

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

*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^1

4 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^

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

+ 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*52i - 32*a^6*c^3*d^7*f^2*(4*(((29973*c^2*d^14)/1024 - (8

41*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/

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

^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512

- (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^1

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

4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19

- (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6

*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 +

70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005

*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)

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

8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2) - a^6*c^4*d^6*f^2*(4*(((2997

3*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*

d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^1

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

(11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^

5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4

+ 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^

2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9

)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 +

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

6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c

^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^1

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

12*a^6*c^5*d^5*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 +

(423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 +

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

4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512

+ (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^

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

4*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 +

729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14

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

f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*

d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*

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

14*d^2*f^2))^2)^(1/2) + a^6*c^6*d^4*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (

987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8

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

+ 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/12

8 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14

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

*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 -

(18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16

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

^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c

^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c

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

12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*56i - 32*a^6*c^7*d^3*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/25

6 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*

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

*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d

^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^1

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

*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^

2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*

1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a^6*c^8

*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/

4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*

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

a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2) + a^6*c^8*d^2*f^2*(4*(((29973*c^2*d^14)

/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (

c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*

f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*

d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)

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

c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + ((

(840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^

12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*

d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18

- (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2

*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*

a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*52i)/(2048*(d^6 +

c^2*d^4)*(a^6*c^10*f^2*1i - a^6*d^10*f^2*1i - 6*a^6*c*d^9*f^2 - 6*a^6*c^9*d*f^2 + a^6*c^2*d^8*f^2*13i + 8*a^6*

c^3*d^7*f^2 + a^6*c^4*d^6*f^2*14i + 28*a^6*c^5*d^5*f^2 - a^6*c^6*d^4*f^2*14i + 8*a^6*c^7*d^3*f^2 - a^6*c^8*d^2

*f^2*13i)))^(1/2)*(267264*a^9*c*d^17*f^3 - a^9*d^18*f^3*57344i + a^9*c^2*d^16*f^3*211968i + 765952*a^9*c^3*d^1

5*f^3 + a^9*c^4*d^14*f^3*1349632i + 396288*a^9*c^5*d^13*f^3 + a^9*c^6*d^12*f^3*2135040i - 727040*a^9*c^7*d^11*

f^3 + a^9*c^8*d^10*f^3*1290240i - 906240*a^9*c^9*d^9*f^3 + a^9*c^10*d^8*f^3*173056i - 270336*a^9*c^11*d^7*f^3

- a^9*c^12*d^6*f^3*59392i + 13312*a^9*c^13*d^5*f^3 + a^9*c^14*d^4*f^3*3072i + 2048*a^9*c^15*d^3*f^3 + 32*(c +

d*tan(e + f*x))^(1/2)*(-(c*d^12*6300i + 3360*d^13 - 945*c^2*d^11 + c^3*d^10*1365i + 315*c^4*d^9 + c^5*d^8*693i

+ 672*c^6*d^7 - c^7*d^6*288i - 72*c^8*d^5 + c^9*d^4*8i - a^6*c^10*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)

/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^

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

d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*

c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 +

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

^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145

*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^

5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a^6*

c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^1

6)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^

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

56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*4i + a^6*d^10*f^2*(4*(((29973*c^2*d^

14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64

- (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^

10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c

^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*

1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^

12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) +

(((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34

*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c

^6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d

^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6

- 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 +

70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*4i + 24*a^6*c*d

^9*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8

)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*

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

^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)

/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*

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

(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^1

1 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^

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

c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*

c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2

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

^2)^(1/2) + 24*a^6*c^9*d*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^1

0)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d

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

^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^

7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a

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

8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6

*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^

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

*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2

+ 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2

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

+ 8*a^6*c^14*d^2*f^2))^2)^(1/2) - a^6*c^2*d^8*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12

)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^

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

0*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^

5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^1

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

*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*

c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2

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

c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14

- (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2

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

28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*52i - 32*a^6*c^3*d^7*f^2*(4*(((29973*c^2*d^14)/1024 - (84

1*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/2

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

12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 -

(4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16

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

+ 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 -

(89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*

c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 +

70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*

c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/

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

*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2) - a^6*c^4*d^6*f^2*(4*(((29973

*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d

^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12

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

11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5

)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4

+ 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2

*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)

/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 5

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

615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^

13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10

*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*56i - 11

2*a^6*c^5*d^5*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (

423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 2

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

+ 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512

+ (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^1

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

*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 +

729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*

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

^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d

^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c

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

4*d^2*f^2))^2)^(1/2) + a^6*c^6*d^4*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (9

87*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*

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

28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128

+ (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*

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

d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 -

(18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*

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

2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^

7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^

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

2*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*56i - 32*a^6*c^7*d^3*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256

- (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c

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

f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^

15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12

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

c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2

*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1

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

d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4

+ 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f

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

^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2) + a^6*c^8*d^2*f^2*(4*(((29973*c^2*d^14)/

1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c

^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f

^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d

^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/

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

^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((

840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^1

2*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d

^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18

- (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*

c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a

^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*52i)/(2048*(d^6 + c

^2*d^4)*(a^6*c^10*f^2*1i - a^6*d^10*f^2*1i - 6*a^6*c*d^9*f^2 - 6*a^6*c^9*d*f^2 + a^6*c^2*d^8*f^2*13i + 8*a^6*c

^3*d^7*f^2 + a^6*c^4*d^6*f^2*14i + 28*a^6*c^5*d^5*f^2 - a^6*c^6*d^4*f^2*14i + 8*a^6*c^7*d^3*f^2 - a^6*c^8*d^2*

f^2*13i)))^(1/2)*(a^6*d^6*f^2 - a^6*c^6*f^2 + a^6*c*d^5*f^2*6i + a^6*c^5*d*f^2*6i - 15*a^6*c^2*d^4*f^2 - a^6*c

^3*d^3*f^2*20i + 15*a^6*c^4*d^2*f^2)*(2048*a^6*c*d^12*f^2 - a^6*c^2*d^11*f^2*12288i - 26624*a^6*c^3*d^10*f^2 +

a^6*c^4*d^9*f^2*16384i - 28672*a^6*c^5*d^8*f^2 + a^6*c^6*d^7*f^2*57344i + 28672*a^6*c^7*d^6*f^2 + a^6*c^8*d^5

*f^2*16384i + 26624*a^6*c^9*d^4*f^2 - a^6*c^10*d^3*f^2*12288i - 2048*a^6*c^11*d^2*f^2)) - 32*(c + d*tan(e + f*

x))^(1/2)*(a^6*d^6*f^2 - a^6*c^6*f^2 + a^6*c*d^5*f^2*6i + a^6*c^5*d*f^2*6i - 15*a^6*c^2*d^4*f^2 - a^6*c^3*d^3*

f^2*20i + 15*a^6*c^4*d^2*f^2)*(c*d^9*2868i + 3368*d^10 + 1699*c^2*d^8 + c^3*d^7*114i + 989*c^4*d^6 - c^5*d^5*7

44i - 296*c^6*d^4 + c^7*d^3*72i + 8*c^8*d^2))*(-(c*d^12*6300i + 3360*d^13 - 945*c^2*d^11 + c^3*d^10*1365i + 31

5*c^4*d^9 + c^5*d^8*693i + 672*c^6*d^7 - c^7*d^6*288i - 72*c^8*d^5 + c^9*d^4*8i - a^6*c^10*f^2*(4*(((29973*c^2

*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/

64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6

*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((1186

1*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/12

8)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70

*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4

) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 +

34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^

6*c^6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*

c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d

^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2

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

0*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)

/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d

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

14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/

128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a

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

256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11

+ (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6

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

^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c

^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2

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

2)^(1/2)*4i + 24*a^6*c*d^9*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d

^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2

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

*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*

c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28

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

+ 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c

^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*

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

^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/

2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f

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

2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2) + 24*a^6*c^9*d*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d

^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12

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

c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171

*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*

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

28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 456

75*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f

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

^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d

^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*

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

2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2) - a^6*c^2*d^8*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^

16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/

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

^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (40

89*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4

+ a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 5

6*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89

145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14

*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 70*a

^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*

d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6

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

+ 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*52i - 32*a^6*c^3*d^7*f^2*(4*(((29

973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^1

0*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a

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

(((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*

d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f

^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*

c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d

^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2

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

(6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6

*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d

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

*c^4*d^6*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c

^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^1

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

a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^

9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4

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

f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c

^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 +

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

8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 +

(339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^

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

*f^2))^2)^(1/2)*56i - 112*a^6*c^5*d^5*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 +

(987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 +

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

4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/

128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^

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

12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15

- (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^

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

*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223

*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6

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

c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2) + a^6*c^6*d^4*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (

5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*

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

+ 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/

256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^1

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

*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^1

7)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(

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

f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 2

8917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 +

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

^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2))^2)^(1/2)*56i - 32*a^6*c^7*d^3*f^2*(4*(((29973*c^2*d^1

4)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 -

(c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^1

0*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^

3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1

i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*c^6*d^10*f^4 + 70*a^1

2*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) +

(((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*

c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^

6*d^10*f^2 + 70*a^6*c^8*d^8*f^2 + 56*a^6*c^10*d^6*f^2 + 28*a^6*c^12*d^4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^

18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 -

2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56*a^6*c^6*d^10*f^2 + 7

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

2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024 + (423*c^8*d^8)/102

4 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*

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

^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/512 + (c^9*d^7)/128

- (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*d^12*f^4 + 56*a^12*

c^6*d^10*f^4 + 70*a^12*c^8*d^8*f^4 + 56*a^12*c^10*d^6*f^4 + 28*a^12*c^12*d^4*f^4 + 8*a^12*c^14*d^2*f^4))*(256*

d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2 + 729*c^8*d^11 + (

879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4

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

d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^9*d^10 + (339*c^11*

d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^6*c^4*d^12*f^2 + 56

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

1/2)*52i)/(2048*(d^6 + c^2*d^4)*(a^6*c^10*f^2*1i - a^6*d^10*f^2*1i - 6*a^6*c*d^9*f^2 - 6*a^6*c^9*d*f^2 + a^6*c

^2*d^8*f^2*13i + 8*a^6*c^3*d^7*f^2 + a^6*c^4*d^6*f^2*14i + 28*a^6*c^5*d^5*f^2 - a^6*c^6*d^4*f^2*14i + 8*a^6*c^

7*d^3*f^2 - a^6*c^8*d^2*f^2*13i)))^(1/2) - 91746*a^3*c^2*d^13*f - a^3*c^3*d^12*f*106018i + 59298*a^3*c^4*d^11*

f + a^3*c^5*d^10*f*9918i + 1722*a^3*c^6*d^9*f - a^3*c^7*d^8*f*1458i + 558*a^3*c^8*d^7*f - a^3*c^9*d^6*f*402i +

32*a^3*c^10*d^5*f - a^3*c^11*d^4*f*36i - 8*a^3*c^12*d^3*f + a^3*c*d^14*f*39772i)*(-(c*d^12*6300i + 3360*d^13

- 945*c^2*d^11 + c^3*d^10*1365i + 315*c^4*d^9 + c^5*d^8*693i + 672*c^6*d^7 - c^7*d^6*288i - 72*c^8*d^5 + c^9*d

^4*8i - a^6*c^10*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 + (987*c^6*d^10)/1024

+ (423*c^8*d^8)/1024 + (3*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4

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

f^4 + 8*a^12*c^14*d^2*f^4) + (((11861*c^3*d^13)/512 - (4089*c*d^15)/256 + (171*c^5*d^11)/128 + (261*c^7*d^9)/5

12 + (c^9*d^7)/128 - (3*c^11*d^5)/128)*1i)/(a^12*c^16*f^4 + a^12*d^16*f^4 + 8*a^12*c^2*d^14*f^4 + 28*a^12*c^4*

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

^14*d^2*f^4))*(256*d^6 + 256*c^2*d^4) + (((840*d^19 - (89145*c^2*d^17)/4 + 45675*c^4*d^15 - (18123*c^6*d^13)/2

+ 729*c^8*d^11 + (879*c^10*d^9)/4 + 34*c^12*d^7 + 6*c^14*d^5)*1i)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^

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

4*f^2 + 8*a^6*c^14*d^2*f^2) - (6615*c*d^18 - (166005*c^3*d^16)/4 + 28917*c^5*d^14 - (2223*c^7*d^12)/2 + 344*c^

9*d^10 + (339*c^11*d^8)/4 - 6*c^13*d^6 - 2*c^15*d^4)/(a^6*c^16*f^2 + a^6*d^16*f^2 + 8*a^6*c^2*d^14*f^2 + 28*a^

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

c^14*d^2*f^2))^2)^(1/2)*4i + a^6*d^10*f^2*(4*(((29973*c^2*d^14)/1024 - (841*d^16)/256 - (5247*c^4*d^12)/1024 +