3.12.6 \(\int \frac {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx\) [1106]

Optimal. Leaf size=280 \[ -\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {\left (2 i c^3-4 c^2 d-i c d^2-2 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{5/2} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )} \]

[Out]

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/(c+I*d)^(5/2)/f-1/8*I*a
rctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I*d)^(1/2)/a^3/f+1/6*I*(c+d*tan(f*x+e))^(1/2)/f/(a+I*a*tan(f*x
+e))^3+1/24*(3*I*c-2*d)*(c+d*tan(f*x+e))^(1/2)/a/(c+I*d)/f/(a+I*a*tan(f*x+e))^2+1/16*c*(2*I*c-3*d)*(c+d*tan(f*
x+e))^(1/2)/(c+I*d)^2/f/(a^3+I*a^3*tan(f*x+e))

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Rubi [A]
time = 0.66, antiderivative size = 280, 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 = {3638, 3677, 3620, 3618, 65, 214} \begin {gather*} \frac {\left (2 i c^3-4 c^2 d-i c d^2-2 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 f (c+i d)^{5/2}}+\frac {c (-3 d+2 i c) \sqrt {c+d \tan (e+f x)}}{16 f (c+i d)^2 \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}+\frac {(-2 d+3 i c) \sqrt {c+d \tan (e+f x)}}{24 a f (c+i d) (a+i a \tan (e+f x))^2}+\frac {i \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[Sqrt[c + d*Tan[e + f*x]]/(a + I*a*Tan[e + f*x])^3,x]

[Out]

((-1/8*I)*Sqrt[c - I*d]*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*(c + I*d)^(5/2)*f) + ((I/6)*Sqrt[c + d*
Tan[e + f*x]])/(f*(a + I*a*Tan[e + f*x])^3) + (((3*I)*c - 2*d)*Sqrt[c + d*Tan[e + f*x]])/(24*a*(c + I*d)*f*(a
+ I*a*Tan[e + f*x])^2) + (c*((2*I)*c - 3*d)*Sqrt[c + d*Tan[e + f*x]])/(16*(c + I*d)^2*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 3638

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp
[(-b)*(a + b*Tan[e + f*x])^m*(Sqrt[c + d*Tan[e + f*x]]/(2*a*f*m)), x] + Dist[1/(4*a^2*m), Int[(a + b*Tan[e + f
*x])^(m + 1)*(Simp[2*a*c*m + b*d + a*d*(2*m + 1)*Tan[e + f*x], x]/Sqrt[c + d*Tan[e + f*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] && IntegersQ
[2*m]

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 {\sqrt {c+d \tan (e+f x)}}{(a+i a \tan (e+f x))^3} \, dx &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-a (6 c-i d)-5 a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 \sqrt {c+d \tan (e+f x)}} \, dx}{12 a^2}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {\int \frac {-3 a^2 \left (3 c d-i \left (4 c^2+2 d^2\right )\right )+3 a^2 (3 i c-2 d) d \tan (e+f x)}{(a+i a \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{48 a^4 (i c-d)}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\int \frac {3 a^3 \left (4 c^3+6 i c^2 d+c d^2+4 i d^3\right )+3 a^3 c (2 c+3 i d) d \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{96 a^6 (c+i d)^2}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(c-i d) \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 (c+i d)^2}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {(i c+d) \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 (i \left (2 c^3+4 i c^2 d-c d^2+2 i d^3\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 (c+i d)^2 f}\\ &=\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {(c-i d) \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 (c+i d)^2 d f}\\ &=-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{8 a^3 f}-\frac {\left (4 c^2 d-i \left (2 c^3-c d^2+2 i d^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{16 a^3 (c+i d)^{5/2} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{6 f (a+i a \tan (e+f x))^3}+\frac {(3 i c-2 d) \sqrt {c+d \tan (e+f x)}}{24 a (c+i d) f (a+i a \tan (e+f x))^2}+\frac {c (2 i c-3 d) \sqrt {c+d \tan (e+f x)}}{16 (c+i d)^2 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end {align*}

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Mathematica [A]
time = 2.90, size = 329, normalized size = 1.18 \begin {gather*} \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac {2 \left (\sqrt {-c+i d} \left (-2 i c^3+4 c^2 d+i c d^2+2 d^3\right ) \text {ArcTan}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )+2 (-c-i d)^{5/2} (i c+d) \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)^{5/2} \sqrt {-c+i d}}+\frac {2 \cos (e+f x) (i \cos (3 f x)+\sin (3 f x)) \left (7 c^2+13 i c d-6 d^2+\left (13 c^2+22 i c d-6 d^2\right ) \cos (2 (e+f x))+i \left (9 c^2+14 i c d-2 d^2\right ) \sin (2 (e+f x))\right ) \sqrt {c+d \tan (e+f x)}}{3 (c+i d)^2}\right )}{32 f (a+i a \tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*((2*(Sqrt[-c + I*d]*((-2*I)*c^3 + 4*c^2*d + I*c*d^2 + 2*d^3)*ArcTan[
Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c - I*d]] + 2*(-c - I*d)^(5/2)*(I*c + d)*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-
c + I*d]])*(Cos[3*e] + I*Sin[3*e]))/((-c - I*d)^(5/2)*Sqrt[-c + I*d]) + (2*Cos[e + f*x]*(I*Cos[3*f*x] + Sin[3*
f*x])*(7*c^2 + (13*I)*c*d - 6*d^2 + (13*c^2 + (22*I)*c*d - 6*d^2)*Cos[2*(e + f*x)] + I*(9*c^2 + (14*I)*c*d - 2
*d^2)*Sin[2*(e + f*x)])*Sqrt[c + d*Tan[e + f*x]])/(3*(c + I*d)^2)))/(32*f*(a + I*a*Tan[e + f*x])^3)

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Maple [A]
time = 0.44, size = 382, normalized size = 1.36

method	result	size
derivativedivides	\(\frac {2 d^{4} \left (\frac {\frac {-\frac {c d \left (5 i c d +2 c^{2}-3 d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {2 d \left (12 i c^{3} d -8 i c \,d^{3}+3 c^{4}-16 c^{2} d^{2}+d^{4}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {d \left (11 i c^{4} d -29 i c^{2} d^{3}+4 i d^{5}+2 c^{5}-25 c^{3} d^{2}+17 c \,d^{4}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (2 i c^{4}-5 i c^{2} d^{2}-2 i d^{4}-6 c^{3} d -c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}}{16 d^{4}}-\frac {i \sqrt {i d -c}\, \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 d^{4}}\right )}{f \,a^{3}}\)	\(382\)
default	\(\frac {2 d^{4} \left (\frac {\frac {-\frac {c d \left (5 i c d +2 c^{2}-3 d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {2 d \left (12 i c^{3} d -8 i c \,d^{3}+3 c^{4}-16 c^{2} d^{2}+d^{4}\right ) \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {d \left (11 i c^{4} d -29 i c^{2} d^{3}+4 i d^{5}+2 c^{5}-25 c^{3} d^{2}+17 c \,d^{4}\right ) \sqrt {c +d \tan \left (f x +e \right )}}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{\left (-d \tan \left (f x +e \right )+i d \right )^{3}}-\frac {\left (2 i c^{4}-5 i c^{2} d^{2}-2 i d^{4}-6 c^{3} d -c \,d^{3}\right ) \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{2 \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right ) \sqrt {-i d -c}}}{16 d^{4}}-\frac {i \sqrt {i d -c}\, \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{16 d^{4}}\right )}{f \,a^{3}}\)	\(382\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f/a^3*d^4*(1/16/d^4*((-1/2*c*d*(5*I*c*d+2*c^2-3*d^2)/(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*(-16*c^2*d^2+d^4+12*I*c^3*d-8*I*c*d^3+3*c^4)/(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*(1
1*I*c^4*d-29*I*c^2*d^3+4*I*d^5+2*c^5-25*c^3*d^2+17*c*d^4)/(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*(-6*c^3*d-c*d^3+2*I*c^4-5*I*c^2*d^2-2*I*d^4)/(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)^(1/2)/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))^(1/2)/(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. 1478 vs. \(2 (230) = 460\).
time = 1.80, size = 1478, normalized size = 5.28 \begin {gather*} -\frac {{\left (6 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f \sqrt {-\frac {c - i \, d}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-2 \, {\left ({\left (i \, a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, 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 - i \, d}{a^{6} f^{2}}} - {\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - c\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) - 6 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f \sqrt {-\frac {c - i \, d}{a^{6} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (-2 \, {\left ({\left (-i \, a^{3} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, 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 - i \, d}{a^{6} f^{2}}} - {\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - c\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}\right ) + 3 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f \sqrt {-\frac {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}}{{\left (i \, a^{6} c^{5} - 5 \, a^{6} c^{4} d - 10 i \, a^{6} c^{3} d^{2} + 10 \, a^{6} c^{2} d^{3} + 5 i \, a^{6} c d^{4} - a^{6} d^{5}\right )} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (2 \, c^{4} + 6 i \, c^{3} d - 5 \, c^{2} d^{2} + i \, c d^{3} - 2 \, d^{4} + {\left ({\left (i \, a^{3} c^{3} - 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} + a^{3} d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, a^{3} c^{3} - 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} + a^{3} d^{3}\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} - 16 \, c^{5} d - 20 i \, c^{4} d^{2} - 15 i \, c^{2} d^{4} + 4 \, c d^{5} - 4 i \, d^{6}}{{\left (i \, a^{6} c^{5} - 5 \, a^{6} c^{4} d - 10 i \, a^{6} c^{3} d^{2} + 10 \, a^{6} c^{2} d^{3} + 5 i \, a^{6} c d^{4} - a^{6} d^{5}\right )} f^{2}}} + {\left (2 \, c^{4} + 4 i \, c^{3} d - c^{2} d^{2} + 2 i \, c d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{16 \, {\left (-i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} f}\right ) - 3 \, {\left (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f \sqrt {-\frac {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}}{{\left (i \, a^{6} c^{5} - 5 \, a^{6} c^{4} d - 10 i \, a^{6} c^{3} d^{2} + 10 \, a^{6} c^{2} d^{3} + 5 i \, a^{6} c d^{4} - a^{6} d^{5}\right )} f^{2}}} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (2 \, c^{4} + 6 i \, c^{3} d - 5 \, c^{2} d^{2} + i \, c d^{3} - 2 \, d^{4} + {\left ({\left (-i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} - a^{3} d^{3}\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} - 16 \, c^{5} d - 20 i \, c^{4} d^{2} - 15 i \, c^{2} d^{4} + 4 \, c d^{5} - 4 i \, d^{6}}{{\left (i \, a^{6} c^{5} - 5 \, a^{6} c^{4} d - 10 i \, a^{6} c^{3} d^{2} + 10 \, a^{6} c^{2} d^{3} + 5 i \, a^{6} c d^{4} - a^{6} d^{5}\right )} f^{2}}} + {\left (2 \, c^{4} + 4 i \, c^{3} d - c^{2} d^{2} + 2 i \, c d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{16 \, {\left (-i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} f}\right ) + 2 \, {\left (-2 i \, c^{2} + 4 \, c d + 2 i \, d^{2} + {\left (-11 i \, c^{2} + 18 \, c d + 4 i \, d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-18 i \, c^{2} + 31 \, c d + 10 i \, d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-9 i \, c^{2} + 17 \, c d + 8 i \, 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 (a^{3} c^{2} + 2 i \, a^{3} c d - a^{3} d^{2}\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(6*(a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*f*sqrt(-(c - I*d)/(a^6*f^2))*e^(6*I*f*x + 6*I*e)*log(-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 - I*d)/(a^6*f^2)) - (c - I*d)*e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) - 6*(a^3*c^2 + 2*I*a^3*c*d -
a^3*d^2)*f*sqrt(-(c - I*d)/(a^6*f^2))*e^(6*I*f*x + 6*I*e)*log(-2*((-I*a^3*f*e^(2*I*f*x + 2*I*e) - I*a^3*f)*sqr
t(((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 - I*d)/(a^6*f^2)) - (c - I*d)*
e^(2*I*f*x + 2*I*e) - c)*e^(-2*I*f*x - 2*I*e)) + 3*(a^3*c^2 + 2*I*a^3*c*d - a^3*d^2)*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^5 - 5*a^6*c^4*d - 10*I*a^6*c^3*d^2 + 10*a^6*c^
2*d^3 + 5*I*a^6*c*d^4 - a^6*d^5)*f^2))*e^(6*I*f*x + 6*I*e)*log(1/16*(2*c^4 + 6*I*c^3*d - 5*c^2*d^2 + I*c*d^3 -
2*d^4 + ((I*a^3*c^3 - 3*a^3*c^2*d - 3*I*a^3*c*d^2 + a^3*d^3)*f*e^(2*I*f*x + 2*I*e) + (I*a^3*c^3 - 3*a^3*c^2*d
- 3*I*a^3*c*d^2 + a^3*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(
-(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^5 - 5*a^6*c^4*d - 10*I*a^6*c
^3*d^2 + 10*a^6*c^2*d^3 + 5*I*a^6*c*d^4 - a^6*d^5)*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^3 + 3*a^3*c^2*d + 3*I*a^3*c*d^2 - a^3*d^3)*f)) - 3*(a^3*c^2 + 2*I*
a^3*c*d - a^3*d^2)*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^5
- 5*a^6*c^4*d - 10*I*a^6*c^3*d^2 + 10*a^6*c^2*d^3 + 5*I*a^6*c*d^4 - a^6*d^5)*f^2))*e^(6*I*f*x + 6*I*e)*log(1/1
6*(2*c^4 + 6*I*c^3*d - 5*c^2*d^2 + I*c*d^3 - 2*d^4 + ((-I*a^3*c^3 + 3*a^3*c^2*d + 3*I*a^3*c*d^2 - a^3*d^3)*f*e
^(2*I*f*x + 2*I*e) + (-I*a^3*c^3 + 3*a^3*c^2*d + 3*I*a^3*c*d^2 - a^3*d^3)*f)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*
e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(-(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^5 - 5*a^6*c^4*d - 10*I*a^6*c^3*d^2 + 10*a^6*c^2*d^3 + 5*I*a^6*c*d^4 - a^6*d^5)*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^3 + 3*a^3*c^2*d + 3*
I*a^3*c*d^2 - a^3*d^3)*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 + 31*c*d + 10*I*d^2)*e^(4*I*f*x + 4*I*e) + (-9*I*c^2 + 17*c*d + 8*I*d^2)*e^(2*I*f*x + 2*I*e))*s
qrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-6*I*f*x - 6*I*e)/((a^3*c^2 + 2*I
*a^3*c*d - a^3*d^2)*f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {i \int \frac {\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}{a^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*Integral(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)/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. 632 vs. \(2 (230) = 460\).
time = 0.69, size = 632, normalized size = 2.26 \begin {gather*} \frac {{\left (2 \, c^{3} + 4 i \, c^{2} d - c d^{2} + 2 i \, 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 )}{-8 \, {\left (-i \, a^{3} c^{2} f + 2 \, a^{3} c d f + i \, a^{3} d^{2} 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 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c^{2} d - 12 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{3} d + 6 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{4} d + 9 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} c d^{2} - 36 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c^{2} d^{2} + 27 i \, \sqrt {d \tan \left (f x + e\right ) + c} c^{3} d^{2} + 28 \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} c d^{3} - 48 \, \sqrt {d \tan \left (f x + e\right ) + c} c^{2} d^{3} + 4 i \, {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} d^{4} - 39 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{4} + 12 \, \sqrt {d \tan \left (f x + e\right ) + c} d^{5}}{48 \, {\left (a^{3} c^{2} f + 2 i \, a^{3} c d f - a^{3} d^{2} f\right )} {\left (d \tan \left (f x + e\right ) - i \, d\right )}^{3}} - \frac {{\left (-i \, c - d\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))^(1/2)/(a+I*a*tan(f*x+e))^3,x, algorithm="giac")

[Out]

(2*c^3 + 4*I*c^2*d - c*d^2 + 2*I*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))))/((8*I*a^3*c^2*f - 16*a^3*c*d*f - 8*I*a^3*d^2*f)*sqrt(-2*c + 2*sqrt(c^2 + d^2))*(I*d/(c -
sqrt(c^2 + d^2)) + 1)) + 1/48*(6*(d*tan(f*x + e) + c)^(5/2)*c^2*d - 12*(d*tan(f*x + e) + c)^(3/2)*c^3*d + 6*sq
rt(d*tan(f*x + e) + c)*c^4*d + 9*I*(d*tan(f*x + e) + c)^(5/2)*c*d^2 - 36*I*(d*tan(f*x + e) + c)^(3/2)*c^2*d^2
+ 27*I*sqrt(d*tan(f*x + e) + c)*c^3*d^2 + 28*(d*tan(f*x + e) + c)^(3/2)*c*d^3 - 48*sqrt(d*tan(f*x + e) + c)*c^
2*d^3 + 4*I*(d*tan(f*x + e) + c)^(3/2)*d^4 - 39*I*sqrt(d*tan(f*x + e) + c)*c*d^4 + 12*sqrt(d*tan(f*x + e) + c)
*d^5)/((a^3*c^2*f + 2*I*a^3*c*d*f - a^3*d^2*f)*(d*tan(f*x + e) - I*d)^3) - 1/4*(-I*c - d)*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 = 10.25, size = 2500, normalized size = 8.93 \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))^(1/2)/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

log(58*a^3*c^2*d^11*f - (((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*
d^4*8i - a^6*c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^
4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4
+ (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*
f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 +
(49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6
*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/
1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i - a^6*d^4*f^2*((((25*c^3*d^12)/4
- 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^
2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10
*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*
f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^
11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4)
- (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64
+ (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4
))*(256*d^6 + 256*c^2*d^4))^(1/2)*4i + a^6*c^2*d^2*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c
^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^
6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*
f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (
7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^1
2*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (
65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a
^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2)*24
i - 16*a^6*c*d^3*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^
4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4
+ (55*c^4*d^11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*
f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 +
(49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6
*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/
1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 +
6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4))*(256*d^6 + 256*c^2*d^4))^(1/2) + 16*a^6*c^3*d*f^2*((((25*c^3*d^12)/
4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*
c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^11)/2 + (93*c^6*d^9)/4 +
10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^
2*f^2))^2 + 4*((((25*c^3*d^13)/512 - (7*c*d^15)/256 + (7*c^5*d^11)/64 + (49*c^7*d^9)/512 + (5*c^9*d^7)/128 + (
c^11*d^5)/128)*1i)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f^4
) - (d^16/256 - (69*c^2*d^14)/1024 - (65*c^4*d^12)/1024 - (11*c^6*d^10)/1024 + (25*c^8*d^8)/1024 + (c^10*d^6)/
64 + (c^12*d^4)/256)/(a^12*c^8*f^4 + a^12*d^8*f^4 + 4*a^12*c^2*d^6*f^4 + 6*a^12*c^4*d^4*f^4 + 4*a^12*c^6*d^2*f
^4))*(256*d^6 + 256*c^2*d^4))^(1/2))/(2048*(d^6 + c^2*d^4)*(a^6*c^4*f^2*1i + a^6*d^4*f^2*1i + 4*a^6*c*d^3*f^2
- 4*a^6*c^3*d*f^2 - a^6*c^2*d^2*f^2*6i)))^(1/2)*(7168*a^9*c*d^11*f^3 - a^9*d^12*f^3*4096i + 32*(c + d*tan(e +
f*x))^(1/2)*((c*d^10*20i + 15*c^2*d^9 + c^3*d^8*35i + 40*c^4*d^7 + c^5*d^6*8i + 24*c^6*d^5 - c^7*d^4*8i - a^6*
c^4*f^2*((((25*c^3*d^12)/4 - 5*c*d^14 + (41*c^5*d^10)/2 + (85*c^7*d^8)/4 + 10*c^9*d^6 + 2*c^11*d^4)/(a^6*c^8*f
^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2) - (((65*c^2*d^13)/4 + (55*c^4*d^
11)/2 + (93*c^6*d^9)/4 + 10*c^8*d^7 + 2*c^10*d^5)*1i)/(a^6*c^8*f^2 + a^6*d^8*f^2 + 4*a^6*c^2*d^6*f^2 + 6*a^6*c
^4*d^4*f^2 + 4*a^6*c^6*d^2*f^2))^2 + 4*((((25*c...

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