\(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx\) [1118]

Optimal result

Integrand size = 30, antiderivative size = 285 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=-\frac {i (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {\left (2 i c^3+4 c^2 d-i c d^2+2 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 \sqrt {c+i d} f}+\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3} \]

[Out]

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

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3639, 3676, 3677, 3620, 3618, 65, 214} \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=\frac {\left (2 i c^3+4 c^2 d-i c d^2+2 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f \sqrt {c+i d}}-\frac {i (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d) (2 d+i c) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2} \]

[In]

Int[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((-1/8*I)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + (((2*I)*c^3 + 4*c^2*d - I
*c*d^2 + 2*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(16*a^3*Sqrt[c + I*d]*f) + ((c + I*d)*(I*c +
2*d)*Sqrt[c + d*Tan[e + f*x]])/(8*a*f*(a + I*a*Tan[e + f*x])^2) + (((2*I)*c^2 + 5*c*d - (4*I)*d^2)*Sqrt[c + d*
Tan[e + f*x]])/(16*f*(a^3 + I*a^3*Tan[e + f*x])) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(6*f*(a + I*a*Tan[e
+ f*x])^3)

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 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 3639

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

Rule 3676

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*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f
*m)), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[A*(a*c*m + b*d
*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]

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*} \text {integral}& = \frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\frac {3}{2} a \left (2 c^2-3 i c d+d^2\right )-\frac {3}{2} a (c-3 i d) d \tan (e+f x)\right )}{(a+i a \tan (e+f x))^2} \, dx}{6 a^2} \\ & = \frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac {\int \frac {\frac {3}{2} a^2 \left (4 c^3-9 i c^2 d-5 c d^2-2 i d^3\right )+\frac {3}{2} a^2 d \left (3 c^2-7 i c d-6 d^2\right ) \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{24 a^4} \\ & = \frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-\frac {3}{2} a^3 c (i c-d) \left (4 c^2-10 i c d-7 d^2\right )-\frac {3}{2} a^3 (i c-d) d \left (2 c^2-5 i c d-4 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{48 a^6 (i c-d)} \\ & = \frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}+\frac {(c-i d)^3 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^3}+\frac {\left (2 c^3-4 i c^2 d-c d^2-2 i d^3\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{32 a^3} \\ & = \frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {(i c+d)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{16 a^3 f}-\frac {\left (2 i c^3+4 c^2 d-i c d^2+2 d^3\right ) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{32 a^3 f} \\ & = \frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3}-\frac {(c-i d)^3 \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^3 d f}-\frac {\left (2 c^3-4 i c^2 d-c d^2-2 i d^3\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 )}{16 a^3 d f} \\ & = -\frac {i (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {\left (2 i c^3+4 c^2 d-i c d^2+2 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 \sqrt {c+i d} f}+\frac {(c+i d) (i c+2 d) \sqrt {c+d \tan (e+f x)}}{8 a f (a+i a \tan (e+f x))^2}+\frac {\left (2 i c^2+5 c d-4 i d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{6 f (a+i a \tan (e+f x))^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.45 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.36 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=-\frac {i \left (6 (c-i d)^{5/2} (c+i d) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )-3 \sqrt {c+i d} \left (2 c^3-4 i c^2 d-c d^2-2 i d^3\right ) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\frac {8 (c+i d) d (i c+3 d) \sqrt {c+d \tan (e+f x)}}{(-i+\tan (e+f x))^2}+\frac {2 (c+i d) \left (3 c^2-7 i c d-6 d^2\right ) \sqrt {c+d \tan (e+f x)}}{(-i+\tan (e+f x))^2}+\frac {3 \left (2 i c^3+3 c^2 d+i c d^2+4 d^3\right ) \sqrt {c+d \tan (e+f x)}}{-i+\tan (e+f x)}+\frac {8 i (c+i d) d (c+d \tan (e+f x))^{3/2}}{(-i+\tan (e+f x))^2}+\frac {8 i d (c+d \tan (e+f x))^{5/2}}{(-i+\tan (e+f x))^2}-\frac {8 i (c+d \tan (e+f x))^{7/2}}{(-i+\tan (e+f x))^3}\right )}{48 a^3 (c+i d) f} \]

[In]

Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((-1/48*I)*(6*(c - I*d)^(5/2)*(c + I*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] - 3*Sqrt[c + I*d]*(2*c
^3 - (4*I)*c^2*d - c*d^2 - (2*I)*d^3)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + (8*(c + I*d)*d*(I*c +
3*d)*Sqrt[c + d*Tan[e + f*x]])/(-I + Tan[e + f*x])^2 + (2*(c + I*d)*(3*c^2 - (7*I)*c*d - 6*d^2)*Sqrt[c + d*Tan
[e + f*x]])/(-I + Tan[e + f*x])^2 + (3*((2*I)*c^3 + 3*c^2*d + I*c*d^2 + 4*d^3)*Sqrt[c + d*Tan[e + f*x]])/(-I +
Tan[e + f*x]) + ((8*I)*(c + I*d)*d*(c + d*Tan[e + f*x])^(3/2))/(-I + Tan[e + f*x])^2 + ((8*I)*d*(c + d*Tan[e
+ f*x])^(5/2))/(-I + Tan[e + f*x])^2 - ((8*I)*(c + d*Tan[e + f*x])^(7/2))/(-I + Tan[e + f*x])^3))/(a^3*(c + I*
d)*f)

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.59

method	result	size
derivativedivides	\(\frac {2 d^{4} \left (-\frac {i \left (i d -c \right )^{\frac {5}{2}} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 d^{4}}+\frac {\frac {-\frac {d \left (i c^{4} d +i c^{2} d^{3}+4 i d^{5}+2 c^{5}+5 c^{3} d^{2}+7 c \,d^{4}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {2 d \left (6 i c^{5} d +20 i c^{3} d^{3}-2 i c \,d^{5}+3 c^{6}+5 c^{4} d^{2}-15 c^{2} d^{4}-d^{6}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {d c \left (7 i c^{5} d +10 i c^{3} d^{3}-13 i c \,d^{5}+2 c^{6}-5 c^{4} d^{2}-20 c^{2} d^{4}+3 d^{6}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (2 i c^{6}+5 i c^{4} d^{2}+5 i c^{2} d^{4}-2 i d^{6}-2 c^{5} d -5 c^{3} d^{3}-7 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}}{16 d^{4}}\right )}{f \,a^{3}}\)	\(452\)
default	\(\frac {2 d^{4} \left (-\frac {i \left (i d -c \right )^{\frac {5}{2}} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 d^{4}}+\frac {\frac {-\frac {d \left (i c^{4} d +i c^{2} d^{3}+4 i d^{5}+2 c^{5}+5 c^{3} d^{2}+7 c \,d^{4}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {2 d \left (6 i c^{5} d +20 i c^{3} d^{3}-2 i c \,d^{5}+3 c^{6}+5 c^{4} d^{2}-15 c^{2} d^{4}-d^{6}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {d c \left (7 i c^{5} d +10 i c^{3} d^{3}-13 i c \,d^{5}+2 c^{6}-5 c^{4} d^{2}-20 c^{2} d^{4}+3 d^{6}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (2 i c^{6}+5 i c^{4} d^{2}+5 i c^{2} d^{4}-2 i d^{6}-2 c^{5} d -5 c^{3} d^{3}-7 c \,d^{5}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}}{16 d^{4}}\right )}{f \,a^{3}}\)	\(452\)

[In]

int((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

2/f/a^3*d^4*(-1/16*I*(I*d-c)^(5/2)/d^4*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))+1/16/d^4*((-1/2*d*(I*c^4*d
+I*c^2*d^3+4*I*d^5+2*c^5+5*c^3*d^2+7*c*d^4)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(5/2)+2/3*d*(5*c^4*
d^2-15*c^2*d^4-d^6+6*I*c^5*d+20*I*c^3*d^3-2*I*c*d^5+3*c^6)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*(c+d*tan(f*x+e))^(3/2
)-1/2*d*c*(-5*c^4*d^2-20*c^2*d^4+3*d^6+7*I*c^5*d+10*I*c^3*d^3-13*I*c*d^5+2*c^6)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)*
(c+d*tan(f*x+e))^(1/2))/(-d*tan(f*x+e)+I*d)^3-1/2*(-2*c^5*d-5*c^3*d^3-7*c*d^5+2*I*c^6+5*I*c^4*d^2+5*I*c^2*d^4-
2*I*d^6)/(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))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1252 vs. \(2 (225) = 450\).

Time = 0.53 (sec) , antiderivative size = 1252, normalized size of antiderivative = 4.39 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/192*(6*a^3*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^6*f^2))*e^(6*I*f*x + 6
*I*e)*log(2*(c^3 - 2*I*c^2*d - c*d^2 + (I*a^3*f*e^(2*I*f*x + 2*I*e) + I*a^3*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*
I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^
5)/(a^6*f^2)) + (c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(c^2 - 2*I*c*d -
d^2)) - 6*a^3*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^6*f^2))*e^(6*I*f*x +
6*I*e)*log(2*(c^3 - 2*I*c^2*d - c*d^2 + (-I*a^3*f*e^(2*I*f*x + 2*I*e) - I*a^3*f)*sqrt(((c - I*d)*e^(2*I*f*x +
2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I
*d^5)/(a^6*f^2)) + (c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(c^2 - 2*I*c*
d - d^2)) + 3*a^3*f*sqrt(-(4*I*c^6 + 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 - 4*c*d^5 - 4*I*d^6)/((I*a^6*c - a
^6*d)*f^2))*e^(6*I*f*x + 6*I*e)*log(-1/16*(2*c^4 - 2*I*c^3*d + 3*c^2*d^2 - 3*I*c*d^3 + 2*d^4 - ((I*a^3*c - a^3
*d)*f*e^(2*I*f*x + 2*I*e) + (I*a^3*c - a^3*d)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x +
2*I*e) + 1))*sqrt(-(4*I*c^6 + 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 - 4*c*d^5 - 4*I*d^6)/((I*a^6*c - a^6*d)*f
^2)) + (2*c^4 - 4*I*c^3*d - c^2*d^2 - 2*I*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((I*a^3*c - a^3*d)*
f)) - 3*a^3*f*sqrt(-(4*I*c^6 + 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 - 4*c*d^5 - 4*I*d^6)/((I*a^6*c - a^6*d)*
f^2))*e^(6*I*f*x + 6*I*e)*log(-1/16*(2*c^4 - 2*I*c^3*d + 3*c^2*d^2 - 3*I*c*d^3 + 2*d^4 - ((-I*a^3*c + a^3*d)*f
*e^(2*I*f*x + 2*I*e) + (-I*a^3*c + a^3*d)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*
e) + 1))*sqrt(-(4*I*c^6 + 16*c^5*d - 20*I*c^4*d^2 - 15*I*c^2*d^4 - 4*c*d^5 - 4*I*d^6)/((I*a^6*c - a^6*d)*f^2))
+ (2*c^4 - 4*I*c^3*d - c^2*d^2 - 2*I*c*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((I*a^3*c - a^3*d)*f))
+ 2*(2*I*c^2 - 4*c*d - 2*I*d^2 + (11*I*c^2 + 18*c*d - 4*I*d^2)*e^(6*I*f*x + 6*I*e) + (18*I*c^2 + 17*c*d + 2*I*
d^2)*e^(4*I*f*x + 4*I*e) + (9*I*c^2 - 5*c*d + 4*I*d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e
) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-6*I*f*x - 6*I*e)/(a^3*f)

Sympy [F]

\[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=\frac {i \left (\int \frac {c^{2} \sqrt {c + d \tan {\left (e + f x \right )}}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \frac {d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx + \int \frac {2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx\right )}{a^{3}} \]

[In]

integrate((c+d*tan(f*x+e))**(5/2)/(a+I*a*tan(f*x+e))**3,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 591 vs. \(2 (225) = 450\).

Time = 0.97 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.07 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=-\frac {1}{48} \, d^{4} {\left (\frac {6 \, {\left (2 i \, c^{3} + 4 \, c^{2} d - i \, c d^{2} + 2 \, d^{3}\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 )}{a^{3} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d^{4} f {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {12 \, {\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\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 )}{a^{3} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} d^{4} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} - \frac {6 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{2} - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{3} + 6 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{4} - 15 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d + 12 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d + 3 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{2} - 20 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{2} + 12 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{2} + 4 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{3} + 9 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{3}}{{\left (d \tan \left (f x + e\right ) - i \, d\right )}^{3} a^{3} d^{3} f}\right )} \]

[In]

integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

-1/48*d^4*(6*(2*I*c^3 + 4*c^2*d - I*c*d^2 + 2*d^3)*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^3*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d^4*f*(I*d/(c - sqrt(c^2 + d^2)) + 1)) +
12*(-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*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^3*sqrt(-2*c + 2*sqrt(c^2 + d^2))*d^4*f*(-I*d/(c - sqrt(c^2 + d^2)) + 1)) - (6*(d*ta
n(f*x + e) + c)^(5/2)*c^2 - 12*(d*tan(f*x + e) + c)^(3/2)*c^3 + 6*sqrt(d*tan(f*x + e) + c)*c^4 - 15*I*(d*tan(f
*x + e) + c)^(5/2)*c*d + 12*I*(d*tan(f*x + e) + c)^(3/2)*c^2*d + 3*I*sqrt(d*tan(f*x + e) + c)*c^3*d - 12*(d*ta
n(f*x + e) + c)^(5/2)*d^2 - 20*(d*tan(f*x + e) + c)^(3/2)*c*d^2 + 12*sqrt(d*tan(f*x + e) + c)*c^2*d^2 + 4*I*(d
*tan(f*x + e) + c)^(3/2)*d^3 + 9*I*sqrt(d*tan(f*x + e) + c)*c*d^3)/((d*tan(f*x + e) - I*d)^3*a^3*d^3*f))

Mupad [B] (verification not implemented)

Time = 10.41 (sec) , antiderivative size = 9472, normalized size of antiderivative = 33.24 \[ \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^3} \, dx=\text {Too large to display} \]

[In]

int((c + d*tan(e + f*x))^(5/2)/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

- atan((((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(
c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8
*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*
c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25
*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024
- d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/2
56)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d
^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^
4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*
c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)
*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1
024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^
2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*
d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2
*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5
)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (28
5*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c
^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/
2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i - ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^
6*c^3*d^3*f^2) + 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4
*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*
d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((
5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/12
8)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)
/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*
d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 -
(35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2
))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512
+ (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024
+ (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*
f^2*(d^6 + c^2*d^4)))^(1/2) - 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*
80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^
6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((
55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*
c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) -
((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6
)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i)/(((2*a^3*f*(1536*a^6*c*d^
5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*
c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10
- (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f
^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/5
12 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/102
4 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^
6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40
i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4
+ 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512
- (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)
/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d
^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*
d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i
- 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 -
18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6
+ 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/12
8 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1
024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4
)))^(1/2) + ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) + 65536*a^12*c*d^2*f
^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i
+ 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 +
10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 -
(25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/
1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^
4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c
^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^
7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((
((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/
128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^
8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) - 32*a^
6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*
c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6
*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6
*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 -
(285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (2
25*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))
^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 4*a^3*f*(c*d^10*30i + 8*d^11 - 25*c^2*d^9 + c^3*d^8*55i - 153*
c^4*d^7 - c^5*d^6*165i + 96*c^6*d^5 + c^7*d^4*30i - 4*c^8*d^3)))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4
*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 - 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*
d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((
5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/12
8)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)
/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*2i - atan
((((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(c + d*
tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d
^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^
7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d
^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^1
6/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a
^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i
- 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^
6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15
)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(
a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 +
(11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^2*(c +
d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*
(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*
c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/
(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*
d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^1
2)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2
048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i - ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*
d^3*f^2) + 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*4
0i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(
a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^
15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)
/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024
+ (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 +
c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c
^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 -
4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55
*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155
*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d
^6 + c^2*d^4)))^(1/2) - 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i +
80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^
6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2
*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^
13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c
^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 -
(c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*1i)/(((2*a^3*f*(1536*a^6*c*d^5*f^2
+ a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) - 65536*a^12*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(20*c*d^10
+ c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35
*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2
- 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (
55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (1
55*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*
(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*
c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c
^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*
c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024
- d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/25
6)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 32*a^6*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*20
i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^2))*(-(20*c*d^10 + c^2*d^9*55i - 35*
c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5
*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*
c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5
*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 -
(665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1
/2) + ((2*a^3*f*(1536*a^6*c*d^5*f^2 + a^6*c^2*d^4*f^2*2560i - 1024*a^6*c^3*d^3*f^2) + 65536*a^12*c*d^2*f^4*(c
+ d*tan(e + f*x))^(1/2)*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c
^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^
4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c
^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 -
d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256
)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7
*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)
/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*
d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1
i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/102
4 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) - 32*a^6*f^2*
(c + d*tan(e + f*x))^(1/2)*(c*d^7*20i + 8*d^8 - 45*c^2*d^6 - c^3*d^5*80i + 80*c^4*d^4 + c^5*d^3*40i - 8*c^6*d^
2))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*40i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(
((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*
1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*
c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4
*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024 + (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2)
)/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2) + 4*a^3*f*(c*d^10*30i + 8*d^11 - 25*c^2*d^9 + c^3*d^8*55i - 153*c^4*d^
7 - c^5*d^6*165i + 96*c^6*d^5 + c^7*d^4*30i - 4*c^8*d^3)))*(-(20*c*d^10 + c^2*d^9*55i - 35*c^3*d^8 + c^4*d^7*4
0i - 72*c^5*d^6 - c^6*d^5*40i + 8*c^7*d^4 + 4*a^6*f^2*(((5*c*d^10 - (35*c^3*d^8)/4 - 18*c^5*d^6 + 2*c^7*d^4)/(
a^6*f^2) + (((55*c^2*d^9)/4 + 10*c^4*d^7 - 10*c^6*d^5)*1i)/(a^6*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*((((5*c*d^
15)/256 + (35*c^3*d^13)/512 - (25*c^5*d^11)/128 - (285*c^7*d^9)/512 + (55*c^9*d^7)/128 - (5*c^11*d^5)/128)*1i)
/(a^12*f^4) - ((21*c^2*d^14)/1024 - d^16/256 + (225*c^4*d^12)/1024 + (155*c^6*d^10)/1024 - (665*c^8*d^8)/1024
+ (11*c^10*d^6)/64 - (c^12*d^4)/256)/(a^12*f^4)))^(1/2))/(2048*a^6*f^2*(d^6 + c^2*d^4)))^(1/2)*2i - (((c + d*t
an(e + f*x))^(3/2)*(10*c*d^3 + 6*c^3*d - d^4*2i - c^2*d^2*6i))/(24*a^3*f) - ((c + d*tan(e + f*x))^(1/2)*(c*d^4
*15i + 10*c^4*d + 20*c^2*d^3 + c^3*d^2*5i))/(80*a^3*f) + (d*(c + d*tan(e + f*x))^(5/2)*(5*c*d + c^2*2i - d^2*4
i)*1i)/(16*a^3*f))/((c + d*tan(e + f*x))*(c*d*6i + 3*c^2 - 3*d^2) + (c + d*tan(e + f*x))^3 + 3*c*d^2 - c^2*d*3
i - (3*c + d*3i)*(c + d*tan(e + f*x))^2 - c^3 + d^3*1i)