3.12.29 \(\int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx\) [1129]

Optimal. Leaf size=281 \[ -\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 (c-i d)^{3/2} f}+\frac {\left (2 i c^2-10 c d-23 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{7/2} f}+\frac {d \left (2 c^2+7 i c d+25 d^2\right )}{8 a^2 (c-i d) (c+i d)^3 f \sqrt {c+d \tan (e+f x)}}+\frac {2 i c-7 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \]

[Out]

-1/4*I*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a^2/(c-I*d)^(3/2)/f+1/8*(2*I*c^2-10*c*d-23*I*d^2)*arctanh
((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a^2/(c+I*d)^(7/2)/f+1/8*d*(2*c^2+7*I*c*d+25*d^2)/a^2/(c-I*d)/(c+I*d)^3/
f/(c+d*tan(f*x+e))^(1/2)+1/8*(2*I*c-7*d)/a^2/(c+I*d)^2/f/(c+d*tan(f*x+e))^(1/2)/(1+I*tan(f*x+e))-1/4/(I*c-d)/f
/(c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^2

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Rubi [A]
time = 0.54, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3640, 3677, 3610, 3620, 3618, 65, 214} \begin {gather*} \frac {d \left (2 c^2+7 i c d+25 d^2\right )}{8 a^2 f (c-i d) (c+i d)^3 \sqrt {c+d \tan (e+f x)}}+\frac {\left (2 i c^2-10 c d-23 i d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 f (c+i d)^{7/2}}+\frac {-7 d+2 i c}{8 a^2 f (c+i d)^2 (1+i \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 )}{4 a^2 f (c-i d)^{3/2}}-\frac {1}{4 f (-d+i c) (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/4*I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^2*(c - I*d)^(3/2)*f) + (((2*I)*c^2 - 10*c*d - (2
3*I)*d^2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(8*a^2*(c + I*d)^(7/2)*f) + (d*(2*c^2 + (7*I)*c*d +
25*d^2))/(8*a^2*(c - I*d)*(c + I*d)^3*f*Sqrt[c + d*Tan[e + f*x]]) + ((2*I)*c - 7*d)/(8*a^2*(c + I*d)^2*f*(1 +
I*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]) - 1/(4*(I*c - d)*f*(a + I*a*Tan[e + f*x])^2*Sqrt[c + d*Tan[e + f*x]
])

Rule 65

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 214

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

Rule 3610

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 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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 3640

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 3677

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

Rubi steps

\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2}} \, dx &=-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}-\frac {\int \frac {-\frac {1}{2} a (4 i c-9 d)-\frac {5}{2} i a d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx}{4 a^2 (i c-d)}\\ &=\frac {2 i c-7 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {1}{4 (i c-d) 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 (4 c^2+14 i c d-25 d^2\right )-\frac {3}{2} a^2 (2 c+7 i d) d \tan (e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{8 a^4 (c+i d)^2}\\ &=\frac {d \left (2 c^2+7 i c d+25 d^2\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 i c-7 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {1}{4 (i c-d) 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 (4 c^3+14 i c^2 d-19 c d^2+21 i d^3\right )-\frac {1}{2} a^2 d \left (2 c^2+7 i c d+25 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^4 (c+i d)^2 \left (c^2+d^2\right )}\\ &=\frac {d \left (2 c^2+7 i c d+25 d^2\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 i c-7 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{8 a^2 (c-i d)}+\frac {\left (2 c^2+10 i c d-23 d^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^2 (c+i d)^3}\\ &=\frac {d \left (2 c^2+7 i c d+25 d^2\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 i c-7 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{8 a^2 (i c+d) f}-\frac {\left (2 c^2+10 i c d-23 d^2\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{16 a^2 (i c-d)^3 f}\\ &=\frac {d \left (2 c^2+7 i c d+25 d^2\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 i c-7 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}-\frac {\text {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 )}{4 a^2 (c-i d) d f}-\frac {\left (2 c^2+10 i c d-23 d^2\right ) \text {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^2 (c+i d)^3 d f}\\ &=-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{4 a^2 (c-i d)^{3/2} f}-\frac {\left (10 c d-i \left (2 c^2-23 d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{8 a^2 (c+i d)^{7/2} f}+\frac {d \left (2 c^2+7 i c d+25 d^2\right )}{8 a^2 (c+i d)^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 i c-7 d}{8 a^2 (c+i d)^2 f (1+i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}-\frac {1}{4 (i c-d) f (a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}}\\ \end {align*}

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Mathematica [A]
time = 4.92, size = 388, normalized size = 1.38 \begin {gather*} \frac {\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left (\frac {2 \left (\sqrt {-c+i d} \left (-2 i c^3+8 c^2 d+13 i c d^2+23 d^3\right ) \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )-2 i (-c-i d)^{7/2} \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right ) (\cos (2 e)+i \sin (2 e))}{(-c-i d)^{7/2} (-c+i d)^{3/2}}+\frac {(\cos (2 f x)-i \sin (2 f x)) \left (\left (12 i c^3-23 c^2 d+26 i c d^2+23 d^3\right ) \cos (e+f x)+\left (4 i c^3-5 c^2 d+18 i c d^2+41 d^3\right ) \cos (3 (e+f x))-4 \left (2 c^3+3 i c^2 d+16 c d^2-43 i d^3\right ) \cos ^2(e+f x) \sin (e+f x)\right ) \sqrt {c+d \tan (e+f x)}}{2 (c-i d) (c+i d)^3 (c \cos (e+f x)+d \sin (e+f x))}\right )}{16 f (a+i a \tan (e+f x))^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*((2*(Sqrt[-c + I*d]*((-2*I)*c^3 + 8*c^2*d + (13*I)*c*d^2 + 23*d^3)*A
rcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] - (2*I)*(-c - I*d)^(7/2)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-
c + I*d]])*(Cos[2*e] + I*Sin[2*e]))/((-c - I*d)^(7/2)*(-c + I*d)^(3/2)) + ((Cos[2*f*x] - I*Sin[2*f*x])*(((12*I
)*c^3 - 23*c^2*d + (26*I)*c*d^2 + 23*d^3)*Cos[e + f*x] + ((4*I)*c^3 - 5*c^2*d + (18*I)*c*d^2 + 41*d^3)*Cos[3*(
e + f*x)] - 4*(2*c^3 + (3*I)*c^2*d + 16*c*d^2 - (43*I)*d^3)*Cos[e + f*x]^2*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*
x]])/(2*(c - I*d)*(c + I*d)^3*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(16*f*(a + I*a*Tan[e + f*x])^2)

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Maple [A]
time = 0.32, size = 399, normalized size = 1.42

method	result	size
derivativedivides	\(\frac {2 d^{3} \left (-\frac {i \left (\frac {-\frac {d \left (2 i c^{4}-7 i c^{2} d^{2}-9 i d^{4}-11 c^{3} d -11 c \,d^{3}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2 \left (2 i c d +c^{2}-d^{2}\right )}+\frac {d \left (2 i c^{5}-22 i c^{3} d^{2}-24 i c \,d^{4}-15 c^{4} d -4 c^{2} d^{3}+11 d^{5}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{4 i c d +2 c^{2}-2 d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (12 i c^{4} d -11 i c^{2} d^{3}-23 i d^{5}+2 c^{5}-31 c^{3} d^{2}-33 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}\right )}{8 d^{3} \left (i d +c \right )^{3} \left (i d -c \right )}-\frac {1}{\left (i c -d \right ) \left (i c +d \right ) \left (i d +c \right )^{2} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {i \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 \left (i d -c \right )^{\frac {3}{2}} \left (i d +c \right )^{3} d^{3}}\right )}{f \,a^{2}}\)	\(399\)
default	\(\frac {2 d^{3} \left (-\frac {i \left (\frac {-\frac {d \left (2 i c^{4}-7 i c^{2} d^{2}-9 i d^{4}-11 c^{3} d -11 c \,d^{3}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{2 \left (2 i c d +c^{2}-d^{2}\right )}+\frac {d \left (2 i c^{5}-22 i c^{3} d^{2}-24 i c \,d^{4}-15 c^{4} d -4 c^{2} d^{3}+11 d^{5}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{4 i c d +2 c^{2}-2 d^{2}}}{\left (-d \tan \left (f x +e \right )+i d \right )^{2}}-\frac {\left (12 i c^{4} d -11 i c^{2} d^{3}-23 i d^{5}+2 c^{5}-31 c^{3} d^{2}-33 c \,d^{4}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (2 i c d +c^{2}-d^{2}\right ) \sqrt {-i d -c}}\right )}{8 d^{3} \left (i d +c \right )^{3} \left (i d -c \right )}-\frac {1}{\left (i c -d \right ) \left (i c +d \right ) \left (i d +c \right )^{2} \sqrt {c +d \tan \left (f x +e \right )}}-\frac {i \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{8 \left (i d -c \right )^{\frac {3}{2}} \left (i d +c \right )^{3} d^{3}}\right )}{f \,a^{2}}\)	\(399\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/a^2*d^3*(-1/8*I/d^3/(c+I*d)^3/(I*d-c)*((-1/2*d*(2*I*c^4-7*I*c^2*d^2-9*I*d^4-11*c^3*d-11*c*d^3)/(2*I*c*d+c^
2-d^2)*(c+d*tan(f*x+e))^(3/2)+1/2*d*(2*I*c^5-22*I*c^3*d^2-24*I*c*d^4-15*c^4*d-4*c^2*d^3+11*d^5)/(2*I*c*d+c^2-d
^2)*(c+d*tan(f*x+e))^(1/2))/(-d*tan(f*x+e)+I*d)^2-1/2*(-31*c^3*d^2-33*c*d^4+12*I*c^4*d-11*I*c^2*d^3-23*I*d^5+2
*c^5)/(2*I*c*d+c^2-d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2)))-1/(I*c-d)/(I*c+d)/(c+I*d
)^2/(c+d*tan(f*x+e))^(1/2)-1/8*I/(I*d-c)^(3/2)/(c+I*d)^3*(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/d^3*arctan((c+d*tan(f*x
+e))^(1/2)/(I*d-c)^(1/2)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^2/(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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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2298 vs. \(2 (230) = 460\).
time = 2.27, size = 2298, normalized size = 8.18 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(8*((-I*a^2*c^5 + a^2*c^4*d - 2*I*a^2*c^3*d^2 + 2*a^2*c^2*d^3 - I*a^2*c*d^4 + a^2*d^5)*f*e^(6*I*f*x + 6*I
*e) + (-I*a^2*c^5 + 3*a^2*c^4*d + 2*I*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + 3*I*a^2*c*d^4 - a^2*d^5)*f*e^(4*I*f*x + 4*
I*e))*sqrt(-1/16*I/((I*a^4*c^3 + 3*a^4*c^2*d - 3*I*a^4*c*d^2 - a^4*d^3)*f^2))*log(-2*(4*((I*a^2*c^2 + 2*a^2*c*
d - I*a^2*d^2)*f*e^(2*I*f*x + 2*I*e) + (I*a^2*c^2 + 2*a^2*c*d - I*a^2*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(-1/16*I/((I*a^4*c^3 + 3*a^4*c^2*d - 3*I*a^4*c*d^2 - a^4*d^3)*f^
2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + 8*((I*a^2*c^5 - a^2*c^4*d + 2*I*a^2*c^3*d^2 -
2*a^2*c^2*d^3 + I*a^2*c*d^4 - a^2*d^5)*f*e^(6*I*f*x + 6*I*e) + (I*a^2*c^5 - 3*a^2*c^4*d - 2*I*a^2*c^3*d^2 - 2
*a^2*c^2*d^3 - 3*I*a^2*c*d^4 + a^2*d^5)*f*e^(4*I*f*x + 4*I*e))*sqrt(-1/16*I/((I*a^4*c^3 + 3*a^4*c^2*d - 3*I*a^
4*c*d^2 - a^4*d^3)*f^2))*log(-2*(4*((-I*a^2*c^2 - 2*a^2*c*d + I*a^2*d^2)*f*e^(2*I*f*x + 2*I*e) + (-I*a^2*c^2 -
2*a^2*c*d + I*a^2*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(-1/1
6*I/((I*a^4*c^3 + 3*a^4*c^2*d - 3*I*a^4*c*d^2 - a^4*d^3)*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*
x - 2*I*e)) + ((I*a^2*c^5 - a^2*c^4*d + 2*I*a^2*c^3*d^2 - 2*a^2*c^2*d^3 + I*a^2*c*d^4 - a^2*d^5)*f*e^(6*I*f*x
+ 6*I*e) + (I*a^2*c^5 - 3*a^2*c^4*d - 2*I*a^2*c^3*d^2 - 2*a^2*c^2*d^3 - 3*I*a^2*c*d^4 + a^2*d^5)*f*e^(4*I*f*x
+ 4*I*e))*sqrt(-(-4*I*c^4 + 40*c^3*d + 192*I*c^2*d^2 - 460*c*d^3 - 529*I*d^4)/((-I*a^4*c^7 + 7*a^4*c^6*d + 21*
I*a^4*c^5*d^2 - 35*a^4*c^4*d^3 - 35*I*a^4*c^3*d^4 + 21*a^4*c^2*d^5 + 7*I*a^4*c*d^6 - a^4*d^7)*f^2))*log(1/8*(2
*I*c^3 - 12*c^2*d - 33*I*c*d^2 + 23*d^3 + ((a^2*c^4 + 4*I*a^2*c^3*d - 6*a^2*c^2*d^2 - 4*I*a^2*c*d^3 + a^2*d^4)
*f*e^(2*I*f*x + 2*I*e) + (a^2*c^4 + 4*I*a^2*c^3*d - 6*a^2*c^2*d^2 - 4*I*a^2*c*d^3 + a^2*d^4)*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^4 + 40*c^3*d + 192*I*c^2*d^2 - 460*c
*d^3 - 529*I*d^4)/((-I*a^4*c^7 + 7*a^4*c^6*d + 21*I*a^4*c^5*d^2 - 35*a^4*c^4*d^3 - 35*I*a^4*c^3*d^4 + 21*a^4*c
^2*d^5 + 7*I*a^4*c*d^6 - a^4*d^7)*f^2)) + (2*I*c^3 - 10*c^2*d - 23*I*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x -
2*I*e)/((a^2*c^4 + 4*I*a^2*c^3*d - 6*a^2*c^2*d^2 - 4*I*a^2*c*d^3 + a^2*d^4)*f)) + ((-I*a^2*c^5 + a^2*c^4*d -
2*I*a^2*c^3*d^2 + 2*a^2*c^2*d^3 - I*a^2*c*d^4 + a^2*d^5)*f*e^(6*I*f*x + 6*I*e) + (-I*a^2*c^5 + 3*a^2*c^4*d + 2
*I*a^2*c^3*d^2 + 2*a^2*c^2*d^3 + 3*I*a^2*c*d^4 - a^2*d^5)*f*e^(4*I*f*x + 4*I*e))*sqrt(-(-4*I*c^4 + 40*c^3*d +
192*I*c^2*d^2 - 460*c*d^3 - 529*I*d^4)/((-I*a^4*c^7 + 7*a^4*c^6*d + 21*I*a^4*c^5*d^2 - 35*a^4*c^4*d^3 - 35*I*a
^4*c^3*d^4 + 21*a^4*c^2*d^5 + 7*I*a^4*c*d^6 - a^4*d^7)*f^2))*log(1/8*(2*I*c^3 - 12*c^2*d - 33*I*c*d^2 + 23*d^3
- ((a^2*c^4 + 4*I*a^2*c^3*d - 6*a^2*c^2*d^2 - 4*I*a^2*c*d^3 + a^2*d^4)*f*e^(2*I*f*x + 2*I*e) + (a^2*c^4 + 4*I
*a^2*c^3*d - 6*a^2*c^2*d^2 - 4*I*a^2*c*d^3 + a^2*d^4)*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^4 + 40*c^3*d + 192*I*c^2*d^2 - 460*c*d^3 - 529*I*d^4)/((-I*a^4*c^7 + 7*a^4*
c^6*d + 21*I*a^4*c^5*d^2 - 35*a^4*c^4*d^3 - 35*I*a^4*c^3*d^4 + 21*a^4*c^2*d^5 + 7*I*a^4*c*d^6 - a^4*d^7)*f^2))
+ (2*I*c^3 - 10*c^2*d - 23*I*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((a^2*c^4 + 4*I*a^2*c^3*d - 6*a
^2*c^2*d^2 - 4*I*a^2*c*d^3 + a^2*d^4)*f)) - 2*(c^3 + I*c^2*d + c*d^2 + I*d^3 + (3*c^3 + 4*I*c^2*d + 17*c*d^2 -
42*I*d^3)*e^(6*I*f*x + 6*I*e) + (7*c^3 + 13*I*c^2*d + 21*c*d^2 - 33*I*d^3)*e^(4*I*f*x + 4*I*e) + 5*(c^3 + 2*I
*c^2*d + c*d^2 + 2*I*d^3)*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)))/((I*a^2*c^5 - a^2*c^4*d + 2*I*a^2*c^3*d^2 - 2*a^2*c^2*d^3 + I*a^2*c*d^4 - a^2*d^5)*f*e^(6*I*f*x +
6*I*e) + (I*a^2*c^5 - 3*a^2*c^4*d - 2*I*a^2*c^3*d^2 - 2*a^2*c^2*d^3 - 3*I*a^2*c*d^4 + a^2*d^5)*f*e^(4*I*f*x +
4*I*e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {1}{c \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - 2 i c \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )} - c \sqrt {c + d \tan {\left (e + f x \right )}} + d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{3}{\left (e + f x \right )} - 2 i d \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )} - d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}\, dx}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (230) = 460\).
time = 0.94, size = 588, normalized size = 2.09 \begin {gather*} \frac {2 \, d^{3}}{{\left (a^{2} c^{4} f + 2 i \, a^{2} c^{3} d f + 2 i \, a^{2} c d^{3} f - a^{2} d^{4} f\right )} \sqrt {d \tan \left (f x + e\right ) + c}} - \frac {{\left (2 i \, c^{2} - 10 \, c d - 23 i \, d^{2}\right )} \arctan \left (\frac {2 \, {\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 {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{4 \, {\left (a^{2} c^{3} f + 3 i \, a^{2} c^{2} d f - 3 \, a^{2} c d^{2} f - i \, a^{2} d^{3} f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {2 \, \arctan \left (\frac {2 \, {\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 {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} - i \, \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}}}\right )}{-4 \, {\left (-i \, a^{2} c f - a^{2} d f\right )} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d - 2 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d + 9 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{2} - 13 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{2} + 11 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{3}}{8 \, {\left (a^{2} c^{3} f + 3 i \, a^{2} c^{2} d f - 3 \, a^{2} c d^{2} f - i \, a^{2} d^{3} f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d^3/((a^2*c^4*f + 2*I*a^2*c^3*d*f + 2*I*a^2*c*d^3*f - a^2*d^4*f)*sqrt(d*tan(f*x + e) + c)) - 1/4*(2*I*c^2 -
10*c*d - 23*I*d^2)*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-2
*c + 2*sqrt(c^2 + d^2)) + I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2)))
)/((a^2*c^3*f + 3*I*a^2*c^2*d*f - 3*a^2*c*d^2*f - I*a^2*d^3*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c
^2 + d^2)) + 1)) - 2*arctan(2*(sqrt(d*tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(
-2*c + 2*sqrt(c^2 + d^2)) - I*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-2*c + 2*sqrt(c^2 + d^2)
)))/((4*I*a^2*c*f + 4*a^2*d*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) + 1/8*(2*(d*ta
n(f*x + e) + c)^(3/2)*c*d - 2*sqrt(d*tan(f*x + e) + c)*c^2*d + 9*I*(d*tan(f*x + e) + c)^(3/2)*d^2 - 13*I*sqrt(
d*tan(f*x + e) + c)*c*d^2 + 11*sqrt(d*tan(f*x + e) + c)*d^3)/((a^2*c^3*f + 3*I*a^2*c^2*d*f - 3*a^2*c*d^2*f - I
*a^2*d^3*f)*(d*tan(f*x + e) - I*d)^2)

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Mupad [B]
time = 15.10, size = 2500, normalized size = 8.90 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(575*a^2*d^11*f - ((-(c*d^10*1155i + 525*d^11 - 315*c^2*d^9 + c^3*d^8*175i - 140*c^4*d^7 + c^5*d^6*168i + 5
6*c^6*d^5 - c^7*d^4*8i - a^4*c^8*f^2*((((525*d^15 - 8085*c^2*d^13 + 6195*c^4*d^11 + 609*c^6*d^9 + 228*c^8*d^7
+ 24*c^10*d^5)*1i)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^6*d^6*f^2
+ 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2) + (10115*c^3*d^12 - 3255*c*d^14 - 973*c^5*d^10 + 57*c^7*d^8 + 8*c^
9*d^6 + 8*c^11*d^4)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^6*d^6*f^
2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2))^2 - 4*((((207*c*d^11)/8 - (21*c^3*d^9)/16 + (21*c^5*d^7)/16 + (3
*c^7*d^5)/8)*1i)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^6*d^6*f^4 +
15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4) - ((763*c^2*d^10)/32 - (529*d^12)/64 + (315*c^4*d^8)/64 + (7*c^6*d^6
)/8 - (c^8*d^4)/16)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^6*d^6*f^
4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*1i - a^4*d^8*f^2*((((525*d^15 - 8
085*c^2*d^13 + 6195*c^4*d^11 + 609*c^6*d^9 + 228*c^8*d^7 + 24*c^10*d^5)*1i)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*a
^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^6*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2) + (10115*
c^3*d^12 - 3255*c*d^14 - 973*c^5*d^10 + 57*c^7*d^8 + 8*c^9*d^6 + 8*c^11*d^4)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*
a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^6*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2))^2 - 4*(
(((207*c*d^11)/8 - (21*c^3*d^9)/16 + (21*c^5*d^7)/16 + (3*c^7*d^5)/8)*1i)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*a^8
*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^6*d^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4) - ((763*c^2
*d^10)/32 - (529*d^12)/64 + (315*c^4*d^8)/64 + (7*c^6*d^6)/8 - (c^8*d^4)/16)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*
a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^6*d^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4))*(256*d^
6 + 256*c^2*d^4))^(1/2)*1i + a^4*c^2*d^6*f^2*((((525*d^15 - 8085*c^2*d^13 + 6195*c^4*d^11 + 609*c^6*d^9 + 228*
c^8*d^7 + 24*c^10*d^5)*1i)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^6
*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2) + (10115*c^3*d^12 - 3255*c*d^14 - 973*c^5*d^10 + 57*c^7*d^
8 + 8*c^9*d^6 + 8*c^11*d^4)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^
6*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2))^2 - 4*((((207*c*d^11)/8 - (21*c^3*d^9)/16 + (21*c^5*d^7)
/16 + (3*c^7*d^5)/8)*1i)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^6*d
^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4) - ((763*c^2*d^10)/32 - (529*d^12)/64 + (315*c^4*d^8)/64 + (7
*c^6*d^6)/8 - (c^8*d^4)/16)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^
6*d^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i - 4*a^4*c^3*d^5*f^2*((
((525*d^15 - 8085*c^2*d^13 + 6195*c^4*d^11 + 609*c^6*d^9 + 228*c^8*d^7 + 24*c^10*d^5)*1i)/(a^4*c^12*f^2 + a^4*
d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^6*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*
f^2) + (10115*c^3*d^12 - 3255*c*d^14 - 973*c^5*d^10 + 57*c^7*d^8 + 8*c^9*d^6 + 8*c^11*d^4)/(a^4*c^12*f^2 + a^4
*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 + 20*a^4*c^6*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2
*f^2))^2 - 4*((((207*c*d^11)/8 - (21*c^3*d^9)/16 + (21*c^5*d^7)/16 + (3*c^7*d^5)/8)*1i)/(a^8*c^12*f^4 + a^8*d^
12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^6*d^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^
4) - ((763*c^2*d^10)/32 - (529*d^12)/64 + (315*c^4*d^8)/64 + (7*c^6*d^6)/8 - (c^8*d^4)/16)/(a^8*c^12*f^4 + a^8
*d^12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 20*a^8*c^6*d^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2
*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2) + a^4*c^4*d^4*f^2*((((525*d^15 - 8085*c^2*d^13 + 6195*c^4*d^11 + 609*c^6
*d^9 + 228*c^8*d^7 + 24*c^10*d^5)*1i)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2 +
20*a^4*c^6*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2) + (10115*c^3*d^12 - 3255*c*d^14 - 973*c^5*d^10
+ 57*c^7*d^8 + 8*c^9*d^6 + 8*c^11*d^4)/(a^4*c^12*f^2 + a^4*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*a^4*c^4*d^8*f^2
+ 20*a^4*c^6*d^6*f^2 + 15*a^4*c^8*d^4*f^2 + 6*a^4*c^10*d^2*f^2))^2 - 4*((((207*c*d^11)/8 - (21*c^3*d^9)/16 + (
21*c^5*d^7)/16 + (3*c^7*d^5)/8)*1i)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4 + 2
0*a^8*c^6*d^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4) - ((763*c^2*d^10)/32 - (529*d^12)/64 + (315*c^4*d
^8)/64 + (7*c^6*d^6)/8 - (c^8*d^4)/16)/(a^8*c^12*f^4 + a^8*d^12*f^4 + 6*a^8*c^2*d^10*f^4 + 15*a^8*c^4*d^8*f^4
+ 20*a^8*c^6*d^6*f^4 + 15*a^8*c^8*d^4*f^4 + 6*a^8*c^10*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*10i + 4*a^4*c^
5*d^3*f^2*((((525*d^15 - 8085*c^2*d^13 + 6195*c^4*d^11 + 609*c^6*d^9 + 228*c^8*d^7 + 24*c^10*d^5)*1i)/(a^4*c^1
2*f^2 + a^4*d^12*f^2 + 6*a^4*c^2*d^10*f^2 + 15*...

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