\(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx\) [1236]

Optimal result

Integrand size = 27, antiderivative size = 195 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=-\frac {i (a-i b)^2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3624, 3609, 3620, 3618, 65, 214} \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=\frac {2 \left (a^2 d+2 a b c-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}-\frac {i (a-i b)^2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \]

[In]

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

[Out]

((-I)*(a - I*b)^2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (I*(a + I*b)^2*(c + I*d
)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*a*b*c + a^2*d - b^2*d)*Sqrt[c + d*Tan[e + f
*x]])/f + (4*a*b*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*b^2*(c + d*Tan[e + f*x])^(5/2))/(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 3624

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \left (a^2-b^2+2 a b \tan (e+f x)\right ) (c+d \tan (e+f x))^{3/2} \, dx \\ & = \frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \sqrt {c+d \tan (e+f x)} \left (a^2 c-b^2 c-2 a b d+\left (2 a b c+a^2 d-b^2 d\right ) \tan (e+f x)\right ) \, dx \\ & = \frac {2 \left (2 a b c+a^2 d-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\int \frac {(a c-b c-a d-b d) (a c+b c+a d-b d)+2 (b c+a d) (a c-b d) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {2 \left (2 a b c+a^2 d-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {1}{2} \left ((a-i b)^2 (c-i d)^2\right ) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} \left ((a+i b)^2 (c+i d)^2\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {2 \left (2 a b c+a^2 d-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f}+\frac {\left (i (a-i b)^2 (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 )}{2 f}-\frac {\left (i (a+i b)^2 (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 )}{2 f} \\ & = \frac {2 \left (2 a b c+a^2 d-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f}-\frac {\left ((a-i b)^2 (c-i d)^2\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 )}{d f}-\frac {\left ((a+i b)^2 (c+i d)^2\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 )}{d f} \\ & = -\frac {i (a-i b)^2 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.92 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.04 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=\frac {\frac {6 b^2 (c+d \tan (e+f x))^{5/2}}{d}+5 i (a-i b)^2 \left (-3 (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c-3 i d+d \tan (e+f x))\right )-5 i (a+i b)^2 \left (-3 (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c+3 i d+d \tan (e+f x))\right )}{15 f} \]

[In]

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

[Out]

((6*b^2*(c + d*Tan[e + f*x])^(5/2))/d + (5*I)*(a - I*b)^2*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]
/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])) - (5*I)*(a + I*b)^2*(-3*(c + I*d)
^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f
*x])))/(15*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2505\) vs. \(2(165)=330\).

Time = 0.90 (sec) , antiderivative size = 2506, normalized size of antiderivative = 12.85

method	result	size
parts	\(\text {Expression too large to display}\)	\(2506\)
derivativedivides	\(\text {Expression too large to display}\)	\(2577\)
default	\(\text {Expression too large to display}\)	\(2577\)

[In]

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

[Out]

a^2*(2/f*d*(c+d*tan(f*x+e))^(1/2)-1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c+1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2-1/4/f*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*(c^2+d^2)^(1/2)+2*c)^(1/2)+2/f*d/
(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))*c-1/f*d/(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))*(c^2+d^2)^(1/2)+1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c-1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+1/
4/f*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*(c^2+d^2)^(1/
2)+2*c)^(1/2)-2/f*d/(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))*c+1/f*d/(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))*(c^2+d^2)^(1/2))+b^2*(2/5/d/f*(c+d*tan(f*x+e))^(5/2)
-2/f*d*(c+d*tan(f*x+e))^(1/2)+1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c-1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2+1/4/f*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*(c^2+d^2)^(1/2)+2*c)^(1/2)-2/f*d/(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))*c+1/f*d/(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))*(c^2+d^2)^(1/2)-1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*c+1/4/f/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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2-1/4/f*
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*(c^2+d^2)^(1/2)+2
*c)^(1/2)+2/f*d/(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))*c-1/f*d/(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))*(c^2+d^2)^(1/2))+2*a*b*(2/3/f*(c+d*tan(f*x+e))^(3/2)+2/f
*c*(c+d*tan(f*x+e))^(1/2)+1/4/f*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)-1/2/f*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-1/f/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arct
an((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))*(c^2+d^2)-1/f/(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))*(c^2+d^2)^(1/2)*c+2/f/(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))*c^2-1/4/f*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)+1/2/f*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*(c^2+d^2)^(1/2)+2*c)^(1/2)*c+1/f/(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))*(c^2+d^2)+1/f/(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))*(c^2+d^2)^(1/2)*c-2/f/(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))*c^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5471 vs. \(2 (157) = 314\).

Time = 1.13 (sec) , antiderivative size = 5471, normalized size of antiderivative = 28.06 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 25.70 (sec) , antiderivative size = 15671, normalized size of antiderivative = 80.36 \[ \int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*b^2*(c + d*tan(e + f*x))^(5/2))/(5*d*f) - atan(((((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f^2 -
2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*a*b*c^3*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*a^4
*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*
b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 +
b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2
*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2
+ 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b
^4*c^3*f^2 - 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*
b^3*c^2*d*f^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2
- 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^
2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^
2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*
a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 1
8*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3*d^3*
f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a^3*b*
c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*a^2*b
^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3 + 36*
a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3*f^
2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*
c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*
a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 +
3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 +
18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3*d^
3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a^3*
b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2)*1i - (((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f
^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*a*b*c^3*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((
8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48
*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*
d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^
6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^
4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^
2 + b^4*c^3*f^2 - 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 +
12*a*b^3*c^2*d*f^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3
*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b
^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 +
4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4
+ 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^
4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3
*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a
^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*
a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3
+ 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*(-(((8*a^4*c^3*f^2 + 8*b^4*c
^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a
*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6
+ 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d
^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*
d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b
^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12
*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2)*1i)/((((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*
d^3*f^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*a*b*c^3*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*
(-(((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2
- 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 +
a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 +
4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b
^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c
^3*f^2 + b^4*c^3*f^2 - 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f
^2 + 12*a*b^3*c^2*d*f^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^3*f^2 + 8*b^
4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 9
6*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*
d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^
2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c
^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*
a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 -
12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6
- 6*a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3
*d^3 + 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*(-(((8*a^4*c^3*f^2 + 8*
b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 +
96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^
8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*
c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4
*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 -
4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2
- 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) + (((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^
2*d^3*f^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*a*b*c^3*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2
)*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f
^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6
+ a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6
+ 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2
*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4
*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3
*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^3*f^2 + 8*
b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 +
96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^
8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*
c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4
*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 -
4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2
- 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d
^6 - 6*a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c
^3*d^3 + 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*(-(((8*a^4*c^3*f^2 +
8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2
+ 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 +
b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^
8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b
^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2
- 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f
^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) + (32*(a*b^5*d^8 + a^5*b*d^8 - a^6*c*d^7 + b^6*
c*d^7 + 2*a^3*b^3*d^8 - 2*a^6*c^3*d^5 - a^6*c^5*d^3 + 2*b^6*c^3*d^5 + b^6*c^5*d^3 + a*b^5*c^2*d^6 - a*b^5*c^4*
d^4 - a*b^5*c^6*d^2 + a^2*b^4*c*d^7 - a^4*b^2*c*d^7 + a^5*b*c^2*d^6 - a^5*b*c^4*d^4 - a^5*b*c^6*d^2 + 2*a^2*b^
4*c^3*d^5 + a^2*b^4*c^5*d^3 + 2*a^3*b^3*c^2*d^6 - 2*a^3*b^3*c^4*d^4 - 2*a^3*b^3*c^6*d^2 - 2*a^4*b^2*c^3*d^5 -
a^4*b^2*c^5*d^3))/f^3))*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^
2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^
2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b
^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^
2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2
*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d
^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2)*2i
- atan(((((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*a*
b*c^3*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2
+ 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b
*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*
b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^
8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2
+ 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*
f^2 + 3*a^4*c*d^2*f^2 + 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2
*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a
^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c
*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 +
4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2
+ 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*
a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2 + 3*
b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(
1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b^4
*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3 + 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a*b
^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*
a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*
c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 +
4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2
+ 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12
*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2 + 3
*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^
(1/2)*1i - (((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*
a*b*c^3*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f
^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3
*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^
4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*
b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^
2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^
3*f^2 + 3*a^4*c*d^2*f^2 + 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a
^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24
*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2
*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6
+ 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^
2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 1
2*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2 +
3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4))
^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b
^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3 + 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a
*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 2
4*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^
2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6
+ 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d
^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 +
12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2 +
3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4)
)^(1/2)*1i)/((((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 +
4*a*b*c^3*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3
*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a
^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*
a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 +
3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*
d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*
d^3*f^2 + 3*a^4*c*d^2*f^2 + 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18
*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 -
24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b
^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^
6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*
d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 +
12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2
+ 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4
))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6
*b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3 + 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16
*a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 -
24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*
b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c
^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4
*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4
+ 12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2
+ 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^
4))^(1/2) + (((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4
*a*b*c^3*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*
f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^
3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a
^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3
*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d
^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d
^3*f^2 + 3*a^4*c*d^2*f^2 + 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*
a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 2
4*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^
2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6
+ 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d
^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 +
12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2 +
3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4)
)^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*
b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3 + 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*
a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*((((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 -
24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b
^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^
6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*
d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 +
12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2
+ 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4
))^(1/2) + (32*(a*b^5*d^8 + a^5*b*d^8 - a^6*c*d^7 + b^6*c*d^7 + 2*a^3*b^3*d^8 - 2*a^6*c^3*d^5 - a^6*c^5*d^3 +
2*b^6*c^3*d^5 + b^6*c^5*d^3 + a*b^5*c^2*d^6 - a*b^5*c^4*d^4 - a*b^5*c^6*d^2 + a^2*b^4*c*d^7 - a^4*b^2*c*d^7 +
a^5*b*c^2*d^6 - a^5*b*c^4*d^4 - a^5*b*c^6*d^2 + 2*a^2*b^4*c^3*d^5 + a^2*b^4*c^5*d^3 + 2*a^3*b^3*c^2*d^6 - 2*a^
3*b^3*c^4*d^4 - 2*a^3*b^3*c^6*d^2 - 2*a^4*b^2*c^3*d^5 - a^4*b^2*c^5*d^3))/f^3))*((((8*a^4*c^3*f^2 + 8*b^4*c^3*
f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^
3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 +
4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4
+ 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4
+ 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) - a^4*c^3*f^2 - b^4*c^3*f^2 + 4*a*b^3*
d^3*f^2 - 4*a^3*b*d^3*f^2 + 3*a^4*c*d^2*f^2 + 3*b^4*c*d^2*f^2 + 6*a^2*b^2*c^3*f^2 - 12*a*b^3*c^2*d*f^2 + 12*a^
3*b*c^2*d*f^2 - 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2)*2i - ((4*b^2*c - 4*a*b*d)/(3*d*f) - (4*b^2*c)/(3*d*f))*(c
+ d*tan(e + f*x))^(3/2) - (c + d*tan(e + f*x))^(1/2)*(2*c*((4*b^2*c - 4*a*b*d)/(d*f) - (4*b^2*c)/(d*f)) - (2*
(a*d - b*c)^2)/(d*f) + (2*b^2*(c^2 + d^2))/(d*f))