3.1.92 \(\int (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x)) \, dx\) [92]

Optimal. Leaf size=224 \[ -\frac {(i a+b) (A-i B-C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(i a-b) (A+i B-C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 (A b+a B-b C) \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (2 b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f} \]

[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)^(1/2)/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)^(1/2)/f+2*(A*b+B*a-C*b)*(c+d*tan(f*x+e))^(1/2)/f-2/15*(-5*B*b*d-5*C
*a*d+2*C*b*c)*(c+d*tan(f*x+e))^(3/2)/d^2/f+2/5*b*C*tan(f*x+e)*(c+d*tan(f*x+e))^(3/2)/d/f

________________________________________________________________________________________

Rubi [A]
time = 0.42, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3718, 3711, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (a B+A b-b C) \sqrt {c+d \tan (e+f x)}}{f}-\frac {(b+i a) \sqrt {c-i d} (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(-b+i a) \sqrt {c+i d} (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}-\frac {2 (-5 a C d-5 b B d+2 b c C) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((I*a + b)*(A - I*B - C)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f) + ((I*a - b)*(A +
I*B - C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(A*b + a*B - b*C)*Sqrt[c + d*T
an[e + f*x]])/f - (2*(2*b*c*C - 5*b*B*d - 5*a*C*d)*(c + d*Tan[e + f*x])^(3/2))/(15*d^2*f) + (2*b*C*Tan[e + f*x
]*(c + d*Tan[e + f*x])^(3/2))/(5*d*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 3609

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

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 3718

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

Rubi steps

\begin {align*} \int (a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (\frac {1}{2} (2 b c C-5 a A d)-\frac {5}{2} (A b+a B-b C) d \tan (e+f x)+\frac {1}{2} (2 b c C-5 b B d-5 a C d) \tan ^2(e+f x)\right ) \, dx}{5 d}\\ &=-\frac {2 (2 b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac {2 \int \sqrt {c+d \tan (e+f x)} \left (\frac {5}{2} (b B-a (A-C)) d-\frac {5}{2} (A b+a B-b C) d \tan (e+f x)\right ) \, dx}{5 d}\\ &=\frac {2 (A b+a B-b C) \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (2 b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac {2 \int \frac {\frac {5}{2} d (b B c+b (A-C) d-a (A c-c C-B d))-\frac {5}{2} 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}{5 d}\\ &=\frac {2 (A b+a B-b C) \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (2 b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}+\frac {1}{2} ((a-i b) (A-i B-C) (c-i d)) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} ((a+i b) (A+i B-C) (c+i d)) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 (A b+a B-b C) \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (2 b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac {((i a-b) (A+i B-C) (c+i d)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}+\frac {((a-i b) (A-i B-C) (i c+d)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 (A b+a B-b C) \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (2 b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}-\frac {((a+i b) (A+i B-C) (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 )}{d f}+\frac {((i a+b) (A-i B-C) (i c+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 )}{d f}\\ &=-\frac {(i a+b) (A-i B-C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {(i a-b) (A+i B-C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 (A b+a B-b C) \sqrt {c+d \tan (e+f x)}}{f}-\frac {2 (2 b c C-5 b B d-5 a C d) (c+d \tan (e+f x))^{3/2}}{15 d^2 f}+\frac {2 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}}{5 d f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.32, size = 220, normalized size = 0.98 \begin {gather*} \frac {\frac {2 (-2 b c C+5 b B d+5 a C d) (c+d \tan (e+f x))^{3/2}}{d}+6 b C \tan (e+f x) (c+d \tan (e+f x))^{3/2}+15 (i a+b) (A-i B-C) d \left (-\sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)}\right )+15 (-i a+b) (A+i B-C) d \left (-\sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\sqrt {c+d \tan (e+f x)}\right )}{15 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(-2*b*c*C + 5*b*B*d + 5*a*C*d)*(c + d*Tan[e + f*x])^(3/2))/d + 6*b*C*Tan[e + f*x]*(c + d*Tan[e + f*x])^(3/
2) + 15*(I*a + b)*(A - I*B - C)*d*(-(Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]) + Sqrt[c +
d*Tan[e + f*x]]) + 15*((-I)*a + b)*(A + I*B - C)*d*(-(Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c +
I*d]]) + Sqrt[c + d*Tan[e + f*x]]))/(15*d*f)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1397\) vs. \(2(194)=388\).
time = 0.48, size = 1398, normalized size = 6.24 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x,method=_RETURNVERBOSE)

[Out]

2/f/d^2*(1/5*C*b*(c+d*tan(f*x+e))^(5/2)+1/3*B*b*d*(c+d*tan(f*x+e))^(3/2)+1/3*C*a*d*(c+d*tan(f*x+e))^(3/2)-1/3*
C*b*c*(c+d*tan(f*x+e))^(3/2)+A*b*d^2*(c+d*tan(f*x+e))^(1/2)+B*a*d^2*(c+d*tan(f*x+e))^(1/2)-C*b*d^2*(c+d*tan(f*
x+e))^(1/2)+d^2*(1/4/d*(1/2*(-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*a*c-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b-B*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*a*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a-C*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(-2*A*(c^2+d^2)^(1/2)*b*d-2*B*(c^2+d^2)^(1/2)*a*d+2*C*(c^2+d^2)
^(1/2)*b*d-1/2*(-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-A*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a-C*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(
1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/4/d*(-1
/2*(-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-A*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*b*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d-B*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)
-c-(c^2+d^2)^(1/2))+2*(2*A*(c^2+d^2)^(1/2)*b*d+2*B*(c^2+d^2)^(1/2)*a*d-2*C*(c^2+d^2)^(1/2)*b*d+1/2*(-A*(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b
*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d-B*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*b*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+C*(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*b*d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^
(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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((c+d*tan(f*x+e))^(1/2)*(a+b*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(1/2)*(a+b*tan(f*x+e))*(A+B*tan(f*x+e)+C*tan(f*x+e)**2),x)

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Mupad [B]
time = 60.11, size = 2500, normalized size = 11.16 \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))*(c + d*tan(e + f*x))^(1/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2),x)

[Out]

((2*B*a*d - 4*C*a*c)/(d*f) + (4*C*a*c)/(d*f))*(c + d*tan(e + f*x))^(1/2) + ((2*B*b*d - 6*C*b*c)/(3*d^2*f) + (4
*C*b*c)/(3*d^2*f))*(c + d*tan(e + f*x))^(3/2) + (c + d*tan(e + f*x))^(1/2)*(2*c*((2*B*b*d - 6*C*b*c)/(d^2*f) +
(4*C*b*c)/(d^2*f)) + (2*A*b*d^2 + 6*C*b*c^2 - 4*B*b*c*d)/(d^2*f) - (2*C*b*(d^4*f + c^2*d^2*f))/(d^4*f^2)) - a
tan(((((8*(4*A*b*d^4*f^2 - 4*C*b*d^4*f^2 + 4*A*b*c^2*d^2*f^2 - 4*C*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e
+ f*x))^(1/2)*((A^2*b^2*c)/(4*f^2) - (4*A*C^3*b^4*d^2*f^4 - B^4*b^4*d^2*f^4 - C^4*b^4*d^2*f^4 - A^4*b^4*d^2*f
^4 + 4*A^3*C*b^4*d^2*f^4 - 4*A^2*B^2*b^4*c^2*f^4 + 2*A^2*B^2*b^4*d^2*f^4 - 6*A^2*C^2*b^4*d^2*f^4 - 4*B^2*C^2*b
^4*c^2*f^4 + 2*B^2*C^2*b^4*d^2*f^4 + 4*A*B^3*b^4*c*d*f^4 - 4*A^3*B*b^4*c*d*f^4 + 4*B*C^3*b^4*c*d*f^4 - 4*B^3*C
*b^4*c*d*f^4 + 8*A*B^2*C*b^4*c^2*f^4 - 4*A*B^2*C*b^4*d^2*f^4 - 12*A*B*C^2*b^4*c*d*f^4 + 12*A^2*B*C*b^4*c*d*f^4
)^(1/2)/(4*f^4) - (B^2*b^2*c)/(4*f^2) + (C^2*b^2*c)/(4*f^2) - (A*B*b^2*d)/(2*f^2) - (A*C*b^2*c)/(2*f^2) + (B*C
*b^2*d)/(2*f^2))^(1/2))*((A^2*b^2*c)/(4*f^2) - (4*A*C^3*b^4*d^2*f^4 - B^4*b^4*d^2*f^4 - C^4*b^4*d^2*f^4 - A^4*
b^4*d^2*f^4 + 4*A^3*C*b^4*d^2*f^4 - 4*A^2*B^2*b^4*c^2*f^4 + 2*A^2*B^2*b^4*d^2*f^4 - 6*A^2*C^2*b^4*d^2*f^4 - 4*
B^2*C^2*b^4*c^2*f^4 + 2*B^2*C^2*b^4*d^2*f^4 + 4*A*B^3*b^4*c*d*f^4 - 4*A^3*B*b^4*c*d*f^4 + 4*B*C^3*b^4*c*d*f^4
- 4*B^3*C*b^4*c*d*f^4 + 8*A*B^2*C*b^4*c^2*f^4 - 4*A*B^2*C*b^4*d^2*f^4 - 12*A*B*C^2*b^4*c*d*f^4 + 12*A^2*B*C*b^
4*c*d*f^4)^(1/2)/(4*f^4) - (B^2*b^2*c)/(4*f^2) + (C^2*b^2*c)/(4*f^2) - (A*B*b^2*d)/(2*f^2) - (A*C*b^2*c)/(2*f^
2) + (B*C*b^2*d)/(2*f^2))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(A^2*b^2*d^4 - B^2*b^2*d^4 + C^2*b^2*d^4 - A^
2*b^2*c^2*d^2 + B^2*b^2*c^2*d^2 - C^2*b^2*c^2*d^2 - 2*A*C*b^2*d^4 + 2*A*C*b^2*c^2*d^2 + 4*A*B*b^2*c*d^3 - 4*B*
C*b^2*c*d^3))/f^2)*((A^2*b^2*c)/(4*f^2) - (4*A*C^3*b^4*d^2*f^4 - B^4*b^4*d^2*f^4 - C^4*b^4*d^2*f^4 - A^4*b^4*d
^2*f^4 + 4*A^3*C*b^4*d^2*f^4 - 4*A^2*B^2*b^4*c^2*f^4 + 2*A^2*B^2*b^4*d^2*f^4 - 6*A^2*C^2*b^4*d^2*f^4 - 4*B^2*C
^2*b^4*c^2*f^4 + 2*B^2*C^2*b^4*d^2*f^4 + 4*A*B^3*b^4*c*d*f^4 - 4*A^3*B*b^4*c*d*f^4 + 4*B*C^3*b^4*c*d*f^4 - 4*B
^3*C*b^4*c*d*f^4 + 8*A*B^2*C*b^4*c^2*f^4 - 4*A*B^2*C*b^4*d^2*f^4 - 12*A*B*C^2*b^4*c*d*f^4 + 12*A^2*B*C*b^4*c*d
*f^4)^(1/2)/(4*f^4) - (B^2*b^2*c)/(4*f^2) + (C^2*b^2*c)/(4*f^2) - (A*B*b^2*d)/(2*f^2) - (A*C*b^2*c)/(2*f^2) +
(B*C*b^2*d)/(2*f^2))^(1/2)*1i - (((8*(4*A*b*d^4*f^2 - 4*C*b*d^4*f^2 + 4*A*b*c^2*d^2*f^2 - 4*C*b*c^2*d^2*f^2))/
f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((A^2*b^2*c)/(4*f^2) - (4*A*C^3*b^4*d^2*f^4 - B^4*b^4*d^2*f^4 - C^4*
b^4*d^2*f^4 - A^4*b^4*d^2*f^4 + 4*A^3*C*b^4*d^2*f^4 - 4*A^2*B^2*b^4*c^2*f^4 + 2*A^2*B^2*b^4*d^2*f^4 - 6*A^2*C^
2*b^4*d^2*f^4 - 4*B^2*C^2*b^4*c^2*f^4 + 2*B^2*C^2*b^4*d^2*f^4 + 4*A*B^3*b^4*c*d*f^4 - 4*A^3*B*b^4*c*d*f^4 + 4*
B*C^3*b^4*c*d*f^4 - 4*B^3*C*b^4*c*d*f^4 + 8*A*B^2*C*b^4*c^2*f^4 - 4*A*B^2*C*b^4*d^2*f^4 - 12*A*B*C^2*b^4*c*d*f
^4 + 12*A^2*B*C*b^4*c*d*f^4)^(1/2)/(4*f^4) - (B^2*b^2*c)/(4*f^2) + (C^2*b^2*c)/(4*f^2) - (A*B*b^2*d)/(2*f^2) -
(A*C*b^2*c)/(2*f^2) + (B*C*b^2*d)/(2*f^2))^(1/2))*((A^2*b^2*c)/(4*f^2) - (4*A*C^3*b^4*d^2*f^4 - B^4*b^4*d^2*f
^4 - C^4*b^4*d^2*f^4 - A^4*b^4*d^2*f^4 + 4*A^3*C*b^4*d^2*f^4 - 4*A^2*B^2*b^4*c^2*f^4 + 2*A^2*B^2*b^4*d^2*f^4 -
6*A^2*C^2*b^4*d^2*f^4 - 4*B^2*C^2*b^4*c^2*f^4 + 2*B^2*C^2*b^4*d^2*f^4 + 4*A*B^3*b^4*c*d*f^4 - 4*A^3*B*b^4*c*d
*f^4 + 4*B*C^3*b^4*c*d*f^4 - 4*B^3*C*b^4*c*d*f^4 + 8*A*B^2*C*b^4*c^2*f^4 - 4*A*B^2*C*b^4*d^2*f^4 - 12*A*B*C^2*
b^4*c*d*f^4 + 12*A^2*B*C*b^4*c*d*f^4)^(1/2)/(4*f^4) - (B^2*b^2*c)/(4*f^2) + (C^2*b^2*c)/(4*f^2) - (A*B*b^2*d)/
(2*f^2) - (A*C*b^2*c)/(2*f^2) + (B*C*b^2*d)/(2*f^2))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(A^2*b^2*d^4 - B^2
*b^2*d^4 + C^2*b^2*d^4 - A^2*b^2*c^2*d^2 + B^2*b^2*c^2*d^2 - C^2*b^2*c^2*d^2 - 2*A*C*b^2*d^4 + 2*A*C*b^2*c^2*d
^2 + 4*A*B*b^2*c*d^3 - 4*B*C*b^2*c*d^3))/f^2)*((A^2*b^2*c)/(4*f^2) - (4*A*C^3*b^4*d^2*f^4 - B^4*b^4*d^2*f^4 -
C^4*b^4*d^2*f^4 - A^4*b^4*d^2*f^4 + 4*A^3*C*b^4*d^2*f^4 - 4*A^2*B^2*b^4*c^2*f^4 + 2*A^2*B^2*b^4*d^2*f^4 - 6*A^
2*C^2*b^4*d^2*f^4 - 4*B^2*C^2*b^4*c^2*f^4 + 2*B^2*C^2*b^4*d^2*f^4 + 4*A*B^3*b^4*c*d*f^4 - 4*A^3*B*b^4*c*d*f^4
+ 4*B*C^3*b^4*c*d*f^4 - 4*B^3*C*b^4*c*d*f^4 + 8*A*B^2*C*b^4*c^2*f^4 - 4*A*B^2*C*b^4*d^2*f^4 - 12*A*B*C^2*b^4*c
*d*f^4 + 12*A^2*B*C*b^4*c*d*f^4)^(1/2)/(4*f^4) - (B^2*b^2*c)/(4*f^2) + (C^2*b^2*c)/(4*f^2) - (A*B*b^2*d)/(2*f^
2) - (A*C*b^2*c)/(2*f^2) + (B*C*b^2*d)/(2*f^2))^(1/2)*1i)/((16*(B^3*b^3*d^5 - A^3*b^3*c^3*d^2 + B^3*b^3*c^2*d^
3 + C^3*b^3*c^3*d^2 + A^2*B*b^3*d^5 + B*C^2*b^3*d^5 - A^3*b^3*c*d^4 + C^3*b^3*c*d^4 - A*B^2*b^3*c*d^4 - 3*A*C^
2*b^3*c*d^4 + 3*A^2*C*b^3*c*d^4 + B^2*C*b^3*c*d^4 - A*B^2*b^3*c^3*d^2 + A^2*B*b^3*c^2*d^3 - 3*A*C^2*b^3*c^3*d^
2 + 3*A^2*C*b^3*c^3*d^2 + B*C^2*b^3*c^2*d^3 + B^2*C*b^3*c^3*d^2 - 2*A*B*C*b^3*d^5 - 2*A*B*C*b^3*c^2*d^3))/f^3
+ (((8*(4*A*b*d^4*f^2 - 4*C*b*d^4*f^2 + 4*A*b*c^2*d^2*f^2 - 4*C*b*c^2*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e +
f*x))^(1/2)*((A^2*b^2*c)/(4*f^2) - (4*A*C^3*b^4*d^2*f^4 - B^4*b^4*d^2*f^4 - C^4*b^4*d^2*f^4 - A^4*b^4*d^2*f^4
+ 4*A^3*C*b^4*d^2*f^4 - 4*A^2*B^2*b^4*c^2*f^4 +...

________________________________________________________________________________________