3.2.18 \(\int \frac {(a+b \tan (e+f x)) (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^{3/2}} \, dx\) [118]

Optimal. Leaf size=201 \[ -\frac {(i a+b) (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {(i a-b) (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}+\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{d^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b C \sqrt {c+d \tan (e+f x)}}{d^2 f} \]

[Out]

-(I*a+b)*(A-I*B-C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(3/2)/f+(I*a-b)*(A+I*B-C)*arctanh((c+
d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(3/2)/f+2*(-a*d+b*c)*(A*d^2-B*c*d+C*c^2)/d^2/(c^2+d^2)/f/(c+d*tan(f
*x+e))^(1/2)+2*b*C*(c+d*tan(f*x+e))^(1/2)/d^2/f

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Rubi [A]
time = 0.37, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3716, 3711, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (b c-a d) \left (A d^2-B c d+c^2 C\right )}{d^2 f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {(b+i a) (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}}+\frac {(-b+i a) (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}}+\frac {2 b C \sqrt {c+d \tan (e+f x)}}{d^2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

-(((I*a + b)*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(3/2)*f)) + ((I*a - b)*
(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) + (2*(b*c - a*d)*(c^2*C - B
*c*d + A*d^2))/(d^2*(c^2 + d^2)*f*Sqrt[c + d*Tan[e + f*x]]) + (2*b*C*Sqrt[c + d*Tan[e + f*x]])/(d^2*f)

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 3711

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])
^m*Simp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0]
&&  !LeQ[m, -1]

Rule 3716

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

Rubi steps

\begin {align*} \int \frac {(a+b \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{3/2}} \, dx &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{d^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {a d (A c-c C+B d)+b \left (c^2 C-B c d+A d^2\right )+d (A b c+a B c-b c C-a A d+b B d+a C d) \tan (e+f x)+b C \left (c^2+d^2\right ) \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )}\\ &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{d^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b C \sqrt {c+d \tan (e+f x)}}{d^2 f}+\frac {\int \frac {d (a (A c-c C+B d)-b (B c-(A-C) d))+d (A b c+a B c-b c C-a A d+b B d+a C d) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )}\\ &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{d^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b C \sqrt {c+d \tan (e+f x)}}{d^2 f}+\frac {((a-i b) (A-i B-C)) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c-i d)}+\frac {((a+i b) (A+i B-C)) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (c+i d)}\\ &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{d^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b C \sqrt {c+d \tan (e+f x)}}{d^2 f}+\frac {(i (a-i b) (A-i B-C)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (c-i d) f}-\frac {((i a-b) (A+i B-C)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (c+i d) f}\\ &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{d^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b C \sqrt {c+d \tan (e+f x)}}{d^2 f}-\frac {((a-i b) (A-i B-C)) \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 )}{(c-i d) d f}-\frac {((a+i b) (A+i B-C)) \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 )}{(c+i d) d f}\\ &=-\frac {(i a+b) (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {(i a-b) (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}+\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{d^2 \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}+\frac {2 b C \sqrt {c+d \tan (e+f x)}}{d^2 f}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.71, size = 290, normalized size = 1.44 \begin {gather*} \frac {(A b+a B-b C) \left (-\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {i \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )-\frac {2 (-2 b c C+b B d+2 a C d)}{d \sqrt {c+d \tan (e+f x)}}+\frac {(A b c+a B c-b c C-a A d+b B d+a C d) \left ((-i c+d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )+(i c+d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )}{\left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {2 C (a+b \tan (e+f x))}{\sqrt {c+d \tan (e+f x)}}}{d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Tan[e + f*x])*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

((A*b + a*B - b*C)*(((-I)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (I*ArcTanh[Sqrt[c +
d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d]) - (2*(-2*b*c*C + b*B*d + 2*a*C*d))/(d*Sqrt[c + d*Tan[e + f*x]]
) + ((A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + a*C*d)*(((-I)*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[
e + f*x])/(c - I*d)] + (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)]))/((c^2 + d^2
)*Sqrt[c + d*Tan[e + f*x]]) + (2*C*(a + b*Tan[e + f*x]))/Sqrt[c + d*Tan[e + f*x]])/(d*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(6236\) vs. \(2(177)=354\).
time = 0.48, size = 6237, normalized size = 31.03

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(6237\)
default	\(\text {Expression too large to display}\)	\(6237\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(3/2),x)

[Out]

Integral((a + b*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(c + d*tan(e + f*x))**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 41.07, size = 2500, normalized size = 12.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(3/2),x)

[Out]

atan((((c + d*tan(e + f*x))^(1/2)*(16*A^2*a^2*d^10*f^3 - 16*B^2*a^2*d^10*f^3 + 16*C^2*a^2*d^10*f^3 + 32*A^2*a^
2*c^2*d^8*f^3 - 32*A^2*a^2*c^6*d^4*f^3 - 16*A^2*a^2*c^8*d^2*f^3 - 32*B^2*a^2*c^2*d^8*f^3 + 32*B^2*a^2*c^6*d^4*
f^3 + 16*B^2*a^2*c^8*d^2*f^3 + 32*C^2*a^2*c^2*d^8*f^3 - 32*C^2*a^2*c^6*d^4*f^3 - 16*C^2*a^2*c^8*d^2*f^3 - 32*A
*C*a^2*d^10*f^3 - 64*A*B*a^2*c*d^9*f^3 + 64*B*C*a^2*c*d^9*f^3 - 192*A*B*a^2*c^3*d^7*f^3 - 192*A*B*a^2*c^5*d^5*
f^3 - 64*A*B*a^2*c^7*d^3*f^3 - 64*A*C*a^2*c^2*d^8*f^3 + 64*A*C*a^2*c^6*d^4*f^3 + 32*A*C*a^2*c^8*d^2*f^3 + 192*
B*C*a^2*c^3*d^7*f^3 + 192*B*C*a^2*c^5*d^5*f^3 + 64*B*C*a^2*c^7*d^3*f^3) - ((((8*A^2*a^2*c^3*f^2 - 8*B^2*a^2*c^
3*f^2 + 8*C^2*a^2*c^3*f^2 - 16*A*B*a^2*d^3*f^2 - 16*A*C*a^2*c^3*f^2 + 16*B*C*a^2*d^3*f^2 - 24*A^2*a^2*c*d^2*f^
2 + 24*B^2*a^2*c*d^2*f^2 - 24*C^2*a^2*c*d^2*f^2 + 48*A*B*a^2*c^2*d*f^2 + 48*A*C*a^2*c*d^2*f^2 - 48*B*C*a^2*c^2
*d*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(A^4*a^4 + B^4*a^4 + C^4*a^4 - 4*A*C
^3*a^4 - 4*A^3*C*a^4 + 2*A^2*B^2*a^4 + 6*A^2*C^2*a^4 + 2*B^2*C^2*a^4 - 4*A*B^2*C*a^4))^(1/2) - 4*A^2*a^2*c^3*f
^2 + 4*B^2*a^2*c^3*f^2 - 4*C^2*a^2*c^3*f^2 + 8*A*B*a^2*d^3*f^2 + 8*A*C*a^2*c^3*f^2 - 8*B*C*a^2*d^3*f^2 + 12*A^
2*a^2*c*d^2*f^2 - 12*B^2*a^2*c*d^2*f^2 + 12*C^2*a^2*c*d^2*f^2 - 24*A*B*a^2*c^2*d*f^2 - 24*A*C*a^2*c*d^2*f^2 +
24*B*C*a^2*c^2*d*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1
/2)*((((8*A^2*a^2*c^3*f^2 - 8*B^2*a^2*c^3*f^2 + 8*C^2*a^2*c^3*f^2 - 16*A*B*a^2*d^3*f^2 - 16*A*C*a^2*c^3*f^2 +
16*B*C*a^2*d^3*f^2 - 24*A^2*a^2*c*d^2*f^2 + 24*B^2*a^2*c*d^2*f^2 - 24*C^2*a^2*c*d^2*f^2 + 48*A*B*a^2*c^2*d*f^2
+ 48*A*C*a^2*c*d^2*f^2 - 48*B*C*a^2*c^2*d*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f
^4)*(A^4*a^4 + B^4*a^4 + C^4*a^4 - 4*A*C^3*a^4 - 4*A^3*C*a^4 + 2*A^2*B^2*a^4 + 6*A^2*C^2*a^4 + 2*B^2*C^2*a^4 -
4*A*B^2*C*a^4))^(1/2) - 4*A^2*a^2*c^3*f^2 + 4*B^2*a^2*c^3*f^2 - 4*C^2*a^2*c^3*f^2 + 8*A*B*a^2*d^3*f^2 + 8*A*C
*a^2*c^3*f^2 - 8*B*C*a^2*d^3*f^2 + 12*A^2*a^2*c*d^2*f^2 - 12*B^2*a^2*c*d^2*f^2 + 12*C^2*a^2*c*d^2*f^2 - 24*A*B
*a^2*c^2*d*f^2 - 24*A*C*a^2*c*d^2*f^2 + 24*B*C*a^2*c^2*d*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d
^2*f^4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c
^11*d^2*f^5) - 32*B*a*d^12*f^4 - 256*A*a*c^3*d^9*f^4 - 384*A*a*c^5*d^7*f^4 - 256*A*a*c^7*d^5*f^4 - 64*A*a*c^9*
d^3*f^4 - 96*B*a*c^2*d^10*f^4 - 64*B*a*c^4*d^8*f^4 + 64*B*a*c^6*d^6*f^4 + 96*B*a*c^8*d^4*f^4 + 32*B*a*c^10*d^2
*f^4 + 256*C*a*c^3*d^9*f^4 + 384*C*a*c^5*d^7*f^4 + 256*C*a*c^7*d^5*f^4 + 64*C*a*c^9*d^3*f^4 - 64*A*a*c*d^11*f^
4 + 64*C*a*c*d^11*f^4))*((((8*A^2*a^2*c^3*f^2 - 8*B^2*a^2*c^3*f^2 + 8*C^2*a^2*c^3*f^2 - 16*A*B*a^2*d^3*f^2 - 1
6*A*C*a^2*c^3*f^2 + 16*B*C*a^2*d^3*f^2 - 24*A^2*a^2*c*d^2*f^2 + 24*B^2*a^2*c*d^2*f^2 - 24*C^2*a^2*c*d^2*f^2 +
48*A*B*a^2*c^2*d*f^2 + 48*A*C*a^2*c*d^2*f^2 - 48*B*C*a^2*c^2*d*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^
4*f^4 + 48*c^4*d^2*f^4)*(A^4*a^4 + B^4*a^4 + C^4*a^4 - 4*A*C^3*a^4 - 4*A^3*C*a^4 + 2*A^2*B^2*a^4 + 6*A^2*C^2*a
^4 + 2*B^2*C^2*a^4 - 4*A*B^2*C*a^4))^(1/2) - 4*A^2*a^2*c^3*f^2 + 4*B^2*a^2*c^3*f^2 - 4*C^2*a^2*c^3*f^2 + 8*A*B
*a^2*d^3*f^2 + 8*A*C*a^2*c^3*f^2 - 8*B*C*a^2*d^3*f^2 + 12*A^2*a^2*c*d^2*f^2 - 12*B^2*a^2*c*d^2*f^2 + 12*C^2*a^
2*c*d^2*f^2 - 24*A*B*a^2*c^2*d*f^2 - 24*A*C*a^2*c*d^2*f^2 + 24*B*C*a^2*c^2*d*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c
^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*1i + ((c + d*tan(e + f*x))^(1/2)*(16*A^2*a^2*d^10*f^3 - 16*B^2*a^2*d^10*f^
3 + 16*C^2*a^2*d^10*f^3 + 32*A^2*a^2*c^2*d^8*f^3 - 32*A^2*a^2*c^6*d^4*f^3 - 16*A^2*a^2*c^8*d^2*f^3 - 32*B^2*a^
2*c^2*d^8*f^3 + 32*B^2*a^2*c^6*d^4*f^3 + 16*B^2*a^2*c^8*d^2*f^3 + 32*C^2*a^2*c^2*d^8*f^3 - 32*C^2*a^2*c^6*d^4*
f^3 - 16*C^2*a^2*c^8*d^2*f^3 - 32*A*C*a^2*d^10*f^3 - 64*A*B*a^2*c*d^9*f^3 + 64*B*C*a^2*c*d^9*f^3 - 192*A*B*a^2
*c^3*d^7*f^3 - 192*A*B*a^2*c^5*d^5*f^3 - 64*A*B*a^2*c^7*d^3*f^3 - 64*A*C*a^2*c^2*d^8*f^3 + 64*A*C*a^2*c^6*d^4*
f^3 + 32*A*C*a^2*c^8*d^2*f^3 + 192*B*C*a^2*c^3*d^7*f^3 + 192*B*C*a^2*c^5*d^5*f^3 + 64*B*C*a^2*c^7*d^3*f^3) - (
(((8*A^2*a^2*c^3*f^2 - 8*B^2*a^2*c^3*f^2 + 8*C^2*a^2*c^3*f^2 - 16*A*B*a^2*d^3*f^2 - 16*A*C*a^2*c^3*f^2 + 16*B*
C*a^2*d^3*f^2 - 24*A^2*a^2*c*d^2*f^2 + 24*B^2*a^2*c*d^2*f^2 - 24*C^2*a^2*c*d^2*f^2 + 48*A*B*a^2*c^2*d*f^2 + 48
*A*C*a^2*c*d^2*f^2 - 48*B*C*a^2*c^2*d*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(
A^4*a^4 + B^4*a^4 + C^4*a^4 - 4*A*C^3*a^4 - 4*A^3*C*a^4 + 2*A^2*B^2*a^4 + 6*A^2*C^2*a^4 + 2*B^2*C^2*a^4 - 4*A*
B^2*C*a^4))^(1/2) - 4*A^2*a^2*c^3*f^2 + 4*B^2*a^2*c^3*f^2 - 4*C^2*a^2*c^3*f^2 + 8*A*B*a^2*d^3*f^2 + 8*A*C*a^2*
c^3*f^2 - 8*B*C*a^2*d^3*f^2 + 12*A^2*a^2*c*d^2*f^2 - 12*B^2*a^2*c*d^2*f^2 + 12*C^2*a^2*c*d^2*f^2 - 24*A*B*a^2*
c^2*d*f^2 - 24*A*C*a^2*c*d^2*f^2 + 24*B*C*a^2*c^2*d*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^
4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*((((8*A^2*a^2*c^3*f^2 - 8*B^2*a^2*c^3*f^2 + 8*C^2*a^2*c^3*f^2 - 16*A*B*
a^2*d^3*f^2 - 16*A*C*a^2*c^3*f^2 + 16*B*C*a^2*d...

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