3.12.12 \(\int \frac {(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx\) [1112]

Optimal. Leaf size=274 \[ -\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {i c \left (2 c^2+3 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{3/2} f}+\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )} \]

[Out]

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

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Rubi [A]
time = 0.68, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3639, 3677, 3620, 3618, 65, 214} \begin {gather*} -\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 f (-d+i c) \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {i c \left (2 c^2+3 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{3/2}}-\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(4 d+3 i c) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/8*I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a^3*f) + ((I/16)*c*(2*c^2 + 3*d^2)
*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(a^3*(c + I*d)^(3/2)*f) + ((I*c - d)*Sqrt[c + d*Tan[e + f*x]
])/(6*f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c + 4*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*f*(a + I*a*Tan[e + f*x])^
2) - ((2*c^2 - I*c*d + 2*d^2)*Sqrt[c + d*Tan[e + f*x]])/(16*(I*c - d)*f*(a^3 + I*a^3*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 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 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 {(c+d \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^3} \, dx &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-\frac {1}{2} a \left (6 c^2-7 i c d+d^2\right )-\frac {1}{2} a (5 c-7 i d) d \tan (e+f x)}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx}{6 a^2}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}+\frac {\int \frac {\frac {3}{2} a^2 c (c+i d) (4 i c+5 d)+\frac {3}{2} a^2 (i c-d) (3 c-4 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{24 a^4 (i c-d)}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\int \frac {\frac {3}{2} a^3 (c+i d) \left (4 c^3-2 i c^2 d+5 c d^2-2 i d^3\right )+\frac {3}{2} a^3 (c+i d) d \left (2 c^2-i c d+2 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{48 a^6 (c+i d)^2}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(c-i d)^2 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{16 a^3}+\frac {\left (c \left (2 c^2+3 d^2\right )\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{32 a^3 (c+i d)}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\left (i (c-i 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^3 f}+\frac {\left (c \left (2 c^2+3 d^2\right )\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 (i c-d) f}\\ &=\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(c-i d)^2 \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 (c \left (2 c^2+3 d^2\right )\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 (c+i d) d f}\\ &=-\frac {i (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {i c \left (2 c^2+3 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{3/2} f}+\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c+4 d) \sqrt {c+d \tan (e+f x)}}{24 a f (a+i a \tan (e+f x))^2}-\frac {\left (2 c^2-i c d+2 d^2\right ) \sqrt {c+d \tan (e+f x)}}{16 (i c-d) f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}

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Mathematica [A]
time = 2.85, size = 311, normalized size = 1.14 \begin {gather*} \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac {2 i \left (c \sqrt {-c+i d} \left (2 c^2+3 d^2\right ) \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )+2 (-c-i d)^{3/2} (c-i d)^2 \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )\right ) (\cos (3 e)+i \sin (3 e))}{(-c-i d)^{3/2} \sqrt {-c+i d}}+\frac {2 \cos (e+f x) (i \cos (3 f x)+\sin (3 f x)) \left (7 c (c+i d)+\left (13 c^2+4 i c d+6 d^2\right ) \cos (2 (e+f x))+\left (9 i c^2+4 c d+10 i d^2\right ) \sin (2 (e+f x))\right ) \sqrt {c+d \tan (e+f x)}}{3 (c+i d)}\right )}{32 f (a+i a \tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(((2*I)*(c*Sqrt[-c + I*d]*(2*c^2 + 3*d^2)*ArcTan[Sqrt[c + d*Tan[e +
f*x]]/Sqrt[-c - I*d]] + 2*(-c - I*d)^(3/2)*(c - I*d)^2*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[3
*e] + I*Sin[3*e]))/((-c - I*d)^(3/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*f*x])*(7*c*(c + I
*d) + (13*c^2 + (4*I)*c*d + 6*d^2)*Cos[2*(e + f*x)] + ((9*I)*c^2 + 4*c*d + (10*I)*d^2)*Sin[2*(e + f*x)])*Sqrt[
c + d*Tan[e + f*x]])/(3*(c + I*d))))/(32*f*(a + I*a*Tan[e + f*x])^3)

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Maple [A]
time = 0.42, size = 362, normalized size = 1.32

method	result	size
derivativedivides	\(\frac {2 d^{4} \left (\frac {\frac {-\frac {d \left (3 i c^{3} d +5 i c \,d^{3}+2 c^{4}+2 c^{2} d^{2}-2 d^{4}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {2 \left (3 c^{2}+5 d^{2}\right ) d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d \left (9 i c^{5} d -6 i c^{3} d^{3}-7 i c \,d^{5}+2 c^{6}-14 c^{4} d^{2}-6 c^{2} d^{4}+2 d^{6}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {c \left (2 i c^{4}+i c^{2} d^{2}-3 i d^{4}-4 c^{3} d -6 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}}{16 d^{4}}+\frac {i \left (i d -c \right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 d^{4}}\right )}{f \,a^{3}}\)	\(362\)
default	\(\frac {2 d^{4} \left (\frac {\frac {-\frac {d \left (3 i c^{3} d +5 i c \,d^{3}+2 c^{4}+2 c^{2} d^{2}-2 d^{4}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {2 \left (3 c^{2}+5 d^{2}\right ) d \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d \left (9 i c^{5} d -6 i c^{3} d^{3}-7 i c \,d^{5}+2 c^{6}-14 c^{4} d^{2}-6 c^{2} d^{4}+2 d^{6}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {c \left (2 i c^{4}+i c^{2} d^{2}-3 i d^{4}-4 c^{3} d -6 c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}}{16 d^{4}}+\frac {i \left (i d -c \right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 d^{4}}\right )}{f \,a^{3}}\)	\(362\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/a^3*d^4*(1/16/d^4*((-1/2*d*(3*I*c^3*d+5*I*c*d^3+2*c^4+2*c^2*d^2-2*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*(3*c^2+5*d^2)*d*(c+d*tan(f*x+e))^(3/2)-1/2*d*(9*I*c^5*d-6*I*c^3*d^3-7*I*c*d^5+2*c^6-14*c
^4*d^2-6*c^2*d^4+2*d^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*c*(2*I
*c^4+I*c^2*d^2-3*I*d^4-4*c^3*d-6*c*d^3)/(3*I*c^2*d-I*d^3+c^3-3*c*d^2)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(
1/2)/(-I*d-c)^(1/2)))+1/16*I*(I*d-c)^(3/2)/d^4*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((c+d*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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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. 1268 vs. \(2 (226) = 452\).
time = 1.81, size = 1268, normalized size = 4.63 \begin {gather*} \frac {{\left (6 \, {\left (i \, a^{3} c - a^{3} d\right )} f \sqrt {-\frac {c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {2 \, {\left (-i \, c^{2} - c d + {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}}{a^{6} f^{2}}} + {\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{i \, c + d}\right ) + 6 \, {\left (-i \, a^{3} c + a^{3} d\right )} f \sqrt {-\frac {c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {2 \, {\left (-i \, c^{2} - c d - {\left (a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3} f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}}{a^{6} f^{2}}} + {\left (-i \, c^{2} - 2 \, c d + i \, d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{i \, c + d}\right ) + 3 \, {\left (-i \, a^{3} c + a^{3} d\right )} f \sqrt {-\frac {-4 i \, c^{6} - 12 i \, c^{4} d^{2} - 9 i \, c^{2} d^{4}}{{\left (-i \, a^{6} c^{3} + 3 \, a^{6} c^{2} d + 3 i \, a^{6} c d^{2} - a^{6} d^{3}\right )} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {{\left (-2 i \, c^{4} + 2 \, c^{3} d - 3 i \, c^{2} d^{2} + 3 \, c d^{3} + {\left ({\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {-4 i \, c^{6} - 12 i \, c^{4} d^{2} - 9 i \, c^{2} d^{4}}{{\left (-i \, a^{6} c^{3} + 3 \, a^{6} c^{2} d + 3 i \, a^{6} c d^{2} - a^{6} d^{3}\right )} f^{2}}} + {\left (-2 i \, c^{4} - 3 i \, c^{2} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{16 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f}\right ) + 3 \, {\left (i \, a^{3} c - a^{3} d\right )} f \sqrt {-\frac {-4 i \, c^{6} - 12 i \, c^{4} d^{2} - 9 i \, c^{2} d^{4}}{{\left (-i \, a^{6} c^{3} + 3 \, a^{6} c^{2} d + 3 i \, a^{6} c d^{2} - a^{6} d^{3}\right )} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-\frac {{\left (-2 i \, c^{4} + 2 \, c^{3} d - 3 i \, c^{2} d^{2} + 3 \, c d^{3} - {\left ({\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {-4 i \, c^{6} - 12 i \, c^{4} d^{2} - 9 i \, c^{2} d^{4}}{{\left (-i \, a^{6} c^{3} + 3 \, a^{6} c^{2} d + 3 i \, a^{6} c d^{2} - a^{6} d^{3}\right )} f^{2}}} + {\left (-2 i \, c^{4} - 3 i \, c^{2} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{16 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f}\right ) - 2 \, {\left (2 \, c^{2} + 4 i \, c d - 2 \, d^{2} + {\left (11 \, c^{2} + 8 \, d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (18 \, c^{2} + 7 i \, c d + 8 \, d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (9 \, c^{2} + 11 i \, c d - 2 \, d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{192 \, {\left (i \, a^{3} c - a^{3} d\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(6*(I*a^3*c - a^3*d)*f*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a^6*f^2))*e^(6*I*f*x + 6*I*e)*log(-2*(
-I*c^2 - c*d + (a^3*f*e^(2*I*f*x + 2*I*e) + 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^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a^6*f^2)) + (-I*c^2 - 2*c*d + I*d^2)*e^(2*I*f*x + 2*
I*e))*e^(-2*I*f*x - 2*I*e)/(I*c + d)) + 6*(-I*a^3*c + a^3*d)*f*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a^6*
f^2))*e^(6*I*f*x + 6*I*e)*log(-2*(-I*c^2 - c*d - (a^3*f*e^(2*I*f*x + 2*I*e) + 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^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a^6*f^2)) + (-I*c^
2 - 2*c*d + I*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(I*c + d)) + 3*(-I*a^3*c + a^3*d)*f*sqrt(-(-4*I*c
^6 - 12*I*c^4*d^2 - 9*I*c^2*d^4)/((-I*a^6*c^3 + 3*a^6*c^2*d + 3*I*a^6*c*d^2 - a^6*d^3)*f^2))*e^(6*I*f*x + 6*I*
e)*log(-1/16*(-2*I*c^4 + 2*c^3*d - 3*I*c^2*d^2 + 3*c*d^3 + ((a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f*e^(2*I*f*x + 2
*I*e) + (a^3*c^2 + 2*I*a^3*c*d - 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(-(-4*I*c^6 - 12*I*c^4*d^2 - 9*I*c^2*d^4)/((-I*a^6*c^3 + 3*a^6*c^2*d + 3*I*a^6*c*d^2 - a^6*d^3)*f
^2)) + (-2*I*c^4 - 3*I*c^2*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/((a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f
)) + 3*(I*a^3*c - a^3*d)*f*sqrt(-(-4*I*c^6 - 12*I*c^4*d^2 - 9*I*c^2*d^4)/((-I*a^6*c^3 + 3*a^6*c^2*d + 3*I*a^6*
c*d^2 - a^6*d^3)*f^2))*e^(6*I*f*x + 6*I*e)*log(-1/16*(-2*I*c^4 + 2*c^3*d - 3*I*c^2*d^2 + 3*c*d^3 - ((a^3*c^2 +
2*I*a^3*c*d - a^3*d^2)*f*e^(2*I*f*x + 2*I*e) + (a^3*c^2 + 2*I*a^3*c*d - 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(-(-4*I*c^6 - 12*I*c^4*d^2 - 9*I*c^2*d^4)/((-I*a^6*c^3 +
3*a^6*c^2*d + 3*I*a^6*c*d^2 - a^6*d^3)*f^2)) + (-2*I*c^4 - 3*I*c^2*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I
*e)/((a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f)) - 2*(2*c^2 + 4*I*c*d - 2*d^2 + (11*c^2 + 8*d^2)*e^(6*I*f*x + 6*I*e)
+ (18*c^2 + 7*I*c*d + 8*d^2)*e^(4*I*f*x + 4*I*e) + (9*c^2 + 11*I*c*d - 2*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)/((I*a^3*c - a^3*d)*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {i \left (\int \frac {c \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 \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*(Integral(c*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) + Inte
gral(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

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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. 624 vs. \(2 (226) = 452\).
time = 0.75, size = 624, normalized size = 2.28 \begin {gather*} -\frac {{\left (2 \, c^{3} + 3 \, c 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 )}{-8 \, {\left (i \, a^{3} c f - a^{3} 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 {-6 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{2} d + 12 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{3} d - 6 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{4} d - 3 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d^{2} - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d^{2} + 15 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d^{2} - 6 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} d^{3} + 20 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{3} + 6 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{3} - 20 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{4} + 9 \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{4} + 6 i \, \sqrt {d \tan \left (f x + e\right ) + c} d^{5}}{-48 \, {\left (i \, a^{3} c f - a^{3} d f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{3}} - \frac {{\left (-i \, c^{2} - 2 \, c d + 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 \, a^{3} \sqrt {-2 \, c + 2 \, \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*c^3 + 3*c*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)))
)/((-8*I*a^3*c*f + 8*a^3*d*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + (-6*I*(d*tan(f
*x + e) + c)^(5/2)*c^2*d + 12*I*(d*tan(f*x + e) + c)^(3/2)*c^3*d - 6*I*sqrt(d*tan(f*x + e) + c)*c^4*d - 3*(d*t
an(f*x + e) + c)^(5/2)*c*d^2 - 12*(d*tan(f*x + e) + c)^(3/2)*c^2*d^2 + 15*sqrt(d*tan(f*x + e) + c)*c^3*d^2 - 6
*I*(d*tan(f*x + e) + c)^(5/2)*d^3 + 20*I*(d*tan(f*x + e) + c)^(3/2)*c*d^3 + 6*I*sqrt(d*tan(f*x + e) + c)*c^2*d
^3 - 20*(d*tan(f*x + e) + c)^(3/2)*d^4 + 9*sqrt(d*tan(f*x + e) + c)*c*d^4 + 6*I*sqrt(d*tan(f*x + e) + c)*d^5)/
((-48*I*a^3*c*f + 48*a^3*d*f)*(d*tan(f*x + e) - I*d)^3) - 1/4*(-I*c^2 - 2*c*d + 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^3*sqrt(-2*c + 2*sqrt(c^2 + d^2))*f*(-
I*d/(c - sqrt(c^2 + d^2)) + 1))

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Mupad [B]
time = 9.21, size = 2500, normalized size = 9.12 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((c + d*tan(e + f*x))^(1/2)*(5*c*d^3 + 10*c^3*d + d^4*10i + c^2*d^2*15i))/(80*a^3*f) - (d*(c + d*tan(e + f*x)
)^(5/2)*(c*d + c^2*2i + d^2*2i)*1i)/(16*(a^3*c*f + a^3*d*f*1i)) + (d*(c^2*6i + d^2*10i)*(c + d*tan(e + f*x))^(
3/2)*1i)/(24*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
*3i - (3*c + d*3i)*(c + d*tan(e + f*x))^2 - c^3 + d^3*1i) - log((((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*
d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^
26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 -
c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 6
4*a^12*c^6*d^2*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^2
2 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4
+ 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 8*a^6*c^2*d^2*f^
2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^
19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 +
96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d
^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2)*(2048*a^9*d^9*f^3 + a^9*c*d^8*f^3*5120i - 32*((12*c*d^12 - d^13*4i - c^2
*d^11*9i + 59*c^3*d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((21
6*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i
- 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^
12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*
176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)
/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2)
- 8*a^6*c^2*d^2*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209
*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*
a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^
2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(a^6*c^2*d^3*f^2*4096i - 20
48*a^6*c*d^4*f^2 + 2048*a^6*c^3*d^2*f^2)*(a^6*d^2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*2i) + a^9*c^3*d^6*f^3*8192i
- 4096*a^9*c^4*d^5*f^3 + a^9*c^5*d^4*f^3*3072i - 2048*a^9*c^6*d^3*f^3) + 32*(c + d*tan(e + f*x))^(1/2)*(a^6*d^
2*f^2 - a^6*c^2*f^2 + a^6*c*d*f^2*2i)*(c*d^7*8i + 4*d^8 - 5*c^2*d^6 + c^3*d^5*16i - 16*c^4*d^4 + c^5*d^3*8i -
8*c^6*d^2))*((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d
^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*
d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^1
2*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*
d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111
*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4
*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 8*a^6*c^2*d^2*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*17
6i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(
16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2))/(
2048*a^6*d^10*f^2 + 6144*a^6*c^2*d^8*f^2 + 6144*a^6*c^4*d^6*f^2 + 2048*a^6*c^6*d^4*f^2))^(1/2) - a^3*c^2*d^10*
f*42i + 28*a^3*c^3*d^9*f - a^3*c^4*d^8*f*64i + 108*a^3*c^5*d^7*f + a^3*c^6*d^6*f*30i + 44*a^3*c^7*d^5*f + a^3*
c^8*d^4*f*36i - 8*a^3*c^9*d^3*f - 12*a^3*c*d^11*f)*((12*c*d^12 - d^13*4i - c^2*d^11*9i + 59*c^3*d^10 + c^4*d^9
*39i + 51*c^5*d^8 + c^6*d^7*64i + c^8*d^5*24i - 8*c^9*d^4 - 4*a^6*c^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96
i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*330i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i +
9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2
*f^4))^(1/2) - 4*a^6*d^4*f^2*((216*c^2*d^24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111*c^4*d^22 + c^5*d^21*3
30i - 209*c^6*d^20 + c^7*d^19*36i - 111*c^8*d^18 - c^9*d^17*54i + 9*c^10*d^16)/(16*a^12*c^8*f^4 + 16*a^12*d^8*
f^4 + 64*a^12*c^2*d^6*f^4 + 96*a^12*c^4*d^4*f^4 + 64*a^12*c^6*d^2*f^4))^(1/2) - 8*a^6*c^2*d^2*f^2*((216*c^2*d^
24 - 16*d^26 - c*d^25*96i + c^3*d^23*176i + 111...

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