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3.1233 \(\int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=231 \[ -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {\sqrt {b} \left (-3 a^2 d+4 a b c+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 \sqrt {b c-a d}}-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2}+\frac {i \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2} \]

[Out]

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

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Rubi [A]  time = 0.78, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3568, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {\sqrt {b} \left (-3 a^2 d+4 a b c+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )^2 \sqrt {b c-a d}}-\frac {i \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2}+\frac {i \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*Sqrt[c + I*d]*ArcTan
h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - (Sqrt[b]*(4*a*b*c - 3*a^2*d + b^2*d)*ArcTanh[(Sqr
t[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)^2*Sqrt[b*c - a*d]*f) - (b*Sqrt[c + d*Tan[e + f*x
]])/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))

Rule 63

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^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 3539

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 3568

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

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
 /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3653

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

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\int \frac {\frac {1}{2} (-2 a c-b d)+(b c-a d) \tan (e+f x)+\frac {1}{2} b d \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{a^2+b^2}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {\int \frac {-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)}{\sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (b \left (4 a b c-3 a^2 d+b^2 d\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 \left (a^2+b^2\right )^2}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {(c-i d) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2}+\frac {(c+i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2}+\frac {\left (b \left (4 a b c-3 a^2 d+b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 \left (a^2+b^2\right )^2 f}\\ &=-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(i c-d) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 f}+\frac {(i c+d) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 f}+\frac {\left (b \left (4 a b c-3 a^2 d+b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{\left (a^2+b^2\right )^2 d f}\\ &=-\frac {\sqrt {b} \left (4 a b c-3 a^2 d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(c-i d) \operatorname {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 )}{(a-i b)^2 d f}-\frac {(c+i d) \operatorname {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 )}{(a+i b)^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 )}{(a-i b)^2 f}+\frac {i \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\sqrt {b} \left (4 a b c-3 a^2 d+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [A]  time = 2.03, size = 276, normalized size = 1.19 \[ -\frac {\frac {\sqrt {b} \sqrt {b c-a d} \left (-3 a^2 d+4 a b c+b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{a^2+b^2}+\frac {i \left ((a-i b)^2 \sqrt {c+i d} (a d-b c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+(a+i b)^2 \sqrt {c-i d} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )\right )}{a^2+b^2}+\frac {b^2 (c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)}-b d \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))^2,x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))^2,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.54, size = 1890, normalized size = 8.18 \[ \text {result too large to display} \]

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))^2,x)

[Out]

1/4/f/d/(a^2+b^2)^2*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)*b^2+1/4/f/d/(a^2+b^2)^2*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)*a^2*c+1/2/f/(a^2+b^2)^2*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)*a*
b-1/4/f/d/(a^2+b^2)^2*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)*b^2*c+1/f*d/(a^2+b^2)^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))*a^2-1/f*d/(a^2+b^2)^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))*b^2-
1/f*d/(a^2+b^2)^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))*a^2+1/f*d/(a^2+b^2)^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))*b^2+1/4/f/d/(a^2+b^2)^2*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)*a^2-1/4/f/d/(a^2+b^2)^2*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)*a^2-1/4/f/d/(a^2+b^2)^2*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)*b^2-1/
4/f/d/(a^2+b^2)^2*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)*a^2*c-1/2/f/(a^2+b^2)^2*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*ta
n(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b+2/f/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*ar
ctan((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)*a
*b-2/f/(a^2+b^2)^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))*a*b*c+1/4/f/d/(a^2+b^2)^2*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)*b^2*c-2/f/(a^2+b^2)^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))*(c^2+
d^2)^(1/2)*a*b+2/f/(a^2+b^2)^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))*a*b*c-1/f*d*b/(a^2+b^2)^2*(c+d*tan(f*x+e))^(1/2)/(tan(f*x+e)*b*d
+d*a)*a^2-1/f*d*b^3/(a^2+b^2)^2*(c+d*tan(f*x+e))^(1/2)/(tan(f*x+e)*b*d+d*a)-3/f*d*b/(a^2+b^2)^2/((a*d-b*c)*b)^
(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^2+4/f*b^2/(a^2+b^2)^2/((a*d-b*c)*b)^(1/2)*arctan(
(c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a*c+1/f*d*b^3/(a^2+b^2)^2/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*
x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
 more details)Is a*d-b*c positive or negative?

________________________________________________________________________________________

mupad [B]  time = 11.43, size = 28314, normalized size = 122.57 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12
- 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 +
63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24
*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 +
 4*a^6*b^2*f^4) - (((8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 -
 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d
^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2
 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b
^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f
^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + ((-b*(a*d - b*c))^(1/2)*((16*(c + d
*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40
*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2
 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*
a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c
^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2
 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^
6*b^2*f^4) + ((-b*(a*d - b*c))^(1/2)*((8*(32*b^15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480
*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f
^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 -
 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a
^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*
d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a
^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (8*(-b*(a*d - b*c))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(b^2*d - 3*
a^2*d + 4*a*b*c)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 1
60*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f
^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4
+ 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*
b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*
f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/((a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4
*a^6*b^2*f^4)*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d +
 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d +
 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(-b*(a*d - b*c))^(
1/2)*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*
f)))*(b^2*d - 3*a^2*d + 4*a*b*c)*1i)/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2
*d*f)) + ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^1
2 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11
+ 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 -
24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4
 + 4*a^6*b^2*f^4) + (((8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2
 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4
*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f
^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a
*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11
*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - ((-b*(a*d - b*c))^(1/2)*((16*(c +
 d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 +
40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f
^2 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 18
4*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2
*c^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f
^2 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*
a^6*b^2*f^4) - ((-b*(a*d - b*c))^(1/2)*((8*(32*b^15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 4
80*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9
*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4
 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64
*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*
c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4
*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + (8*(-b*(a*d - b*c))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(b^2*d -
3*a^2*d + 4*a*b*c)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 -
 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8
*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^
4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^
3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^
9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/((a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 +
 4*a^6*b^2*f^4)*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d
 + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d
 + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(-b*(a*d - b*c))
^(1/2)*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*
d*f)))*(b^2*d - 3*a^2*d + 4*a*b*c)*1i)/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b
^2*d*f)))/((16*(b^7*d^13 - 9*a^4*b^3*d^13 + 3*b^7*c^2*d^11 + 2*b^7*c^4*d^9 + 8*a*b^6*c^3*d^10 + 8*a*b^6*c^5*d^
8 + 24*a^3*b^4*c*d^12 - 22*a^2*b^5*c^2*d^11 - 22*a^2*b^5*c^4*d^9 + 24*a^3*b^4*c^3*d^10 - 9*a^4*b^3*c^2*d^11))/
(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(
e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d
^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a
^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^
8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4) - (((8*(20*b^11*c*d^11*f^2 - 52
*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20
*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c
^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9
*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a
^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4
*b^4*f^5 + 4*a^6*b^2*f^5) + ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^
13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^1
1*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d
^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 +
100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b
^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*
f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4) + ((-b*(a*d - b*c))^(1/2)*((8*(32*b^
15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*
a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^
3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*
f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64
*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6
*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (
8*(-b*(a*d - b*c))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(b^2*d - 3*a^2*d + 4*a*b*c)*(32*b^17*d^10*f^4 + 160*a^2*b^
15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 1
60*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^
2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8
*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a
^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^
9*f^4))/((a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)*(a^5*d*f - b^5*c*f - a^4*b*c*f +
a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f +
a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f +
a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(-b*(a*d - b*c))^(1/2)*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f -
 b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f -
 b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)) + ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e
 + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^
8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^
3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8
*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4) + (((8*(20*b^11*c*d^11*f^2 - 52*
a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*
b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^
2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*
f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^
4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*
b^4*f^5 + 4*a^6*b^2*f^5) - ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^1
3*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11
*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^
8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 1
00*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^
11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f
^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4) - ((-b*(a*d - b*c))^(1/2)*((8*(32*b^1
5*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a
^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3
*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f
^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*
a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*
c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + (8
*(-b*(a*d - b*c))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(b^2*d - 3*a^2*d + 4*a*b*c)*(32*b^17*d^10*f^4 + 160*a^2*b^1
5*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 16
0*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2
*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*
f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^
7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9
*f^4))/((a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)*(a^5*d*f - b^5*c*f - a^4*b*c*f + a
*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a
*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f - b^5*c*f - a^4*b*c*f + a
*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(-b*(a*d - b*c))^(1/2)*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f -
b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f)))*(b^2*d - 3*a^2*d + 4*a*b*c))/(2*(a^5*d*f -
b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f))))*(-b*(a*d - b*c))^(1/2)*(b^2*d - 3*a^2*d +
4*a*b*c)*1i)/(a^5*d*f - b^5*c*f - a^4*b*c*f + a*b^4*d*f - 2*a^2*b^3*c*f + 2*a^3*b^2*d*f) - atan(((((((8*(32*b^
15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*
a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^
3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*
f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64
*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6
*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (
16*(c + d*tan(e + f*x))^(1/2)*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f
^2*6i)))^(1/2)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160
*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4
 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 +
48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^
14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^
4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^
6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) + (
16*(c + d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11
*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^
3*d^8*f^2 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f
^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a
^11*b^2*c^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c
*d^10*f^2 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f
^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^
(1/2) + (8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4
*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*
a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b
^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*
f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^
5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b
^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 +
 17*a^4*b^5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*
a^5*b^4*c*d^11 + 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4
*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4
+ 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b
^2*f^2*6i)))^(1/2)*1i - (((((8*(32*b^15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d
^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a
^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b
^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^
2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 +
 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5
 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + (16*(c + d*tan(e + f*x))^(1/2)*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i -
4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d
^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^
14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2
*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f
^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*
b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 +
 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^
3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 2
0*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 -
20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*
b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d
^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 +
48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 +
b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f
^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) + (8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^
2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^
9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2
+ 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b
^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11
*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5))*(-(c*1i + d
)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) + (16*(c + d*tan(e + f*x))
^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*
b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^
3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11)
)/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1
i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2)*1i)/((((((8*(32*b^15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 -
 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*
f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4
+ 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*
a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c
*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4
))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (16*(c + d*tan(e + f*x))^(1/2)*(-(c*1
i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2)*(32*b^17*d^10*f^4 + 1
60*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^1
0*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^
4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^
5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^
4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15
*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2
*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(68*a*
b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*
b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2 + 204*a^3*b^10*c^2*d^9*f^2
 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^
8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c^2*d^9*f^2 + 116*a*b^12*c^2
*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2 - 296*a^8*b^5*c*d^10*f^2 -
 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4
*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) + (8*(20*b^11*c*d^11*f^2 - 52*
a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*
b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^
2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*
f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^
4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*
b^4*f^5 + 4*a^6*b^2*f^5))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6
i)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 - 9*a^6*b^3*d^12 +
3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63*a^2*b^7*c^2*d^10
 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 +
9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-
(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) + (((((8*(32*b^15
*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^
12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*
d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^
4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a
*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c
*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + (16
*(c + d*tan(e + f*x))^(1/2)*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2
*6i)))^(1/2)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a
^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 +
 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48
*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14
*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4
+ 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*
b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) - (16
*(c + d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f
^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*
d^8*f^2 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2
 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^1
1*b^2*c^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d
^10*f^2 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4
 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1
/2) + (8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d
^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^
4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4
*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^
2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5
+ b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3
*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 1
7*a^4*b^5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^
5*b^4*c*d^11 + 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b
^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 +
6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2
*f^2*6i)))^(1/2) + (16*(b^7*d^13 - 9*a^4*b^3*d^13 + 3*b^7*c^2*d^11 + 2*b^7*c^4*d^9 + 8*a*b^6*c^3*d^10 + 8*a*b^
6*c^5*d^8 + 24*a^3*b^4*c*d^12 - 22*a^2*b^5*c^2*d^11 - 22*a^2*b^5*c^4*d^9 + 24*a^3*b^4*c^3*d^10 - 9*a^4*b^3*c^2
*d^11))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5)))*(-(c*1i + d)/(4*(a^4*f^2*1i + b^
4*f^2*1i - 4*a*b^3*f^2 + 4*a^3*b*f^2 - a^2*b^2*f^2*6i)))^(1/2)*2i - atan(((((((8*(32*b^15*d^11*f^4 + 96*a^2*b^
13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^12*b^3*d^11*f^4 + 32*
b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*d^8*f^4 - 320*a^6*b^9
*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^4 - 288*a^10*b^5*c^2*
d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a*b^14*c^3*d^8*f^4 + 3
20*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c*d^10*f^4 + 64*a^11*b
^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (16*(c + d*tan(e + f*x))
^(1/2)*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2)*(32*b^17*d^10
*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^4 - 288*a^10
*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*c^2*d^8*f^4
+ 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48
*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 336*a^5*b^12*
c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4*c*d^9*f^4 +
 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c + d*1i)/(4*
(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(68
*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a
^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2 + 204*a^3*b^10*c^2*d^9*
f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68
*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c^2*d^9*f^2 + 116*a*b^12*
c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2 - 296*a^8*b^5*c*d^10*f^
2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c + d*1i)
/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) + (8*(20*b^11*c*d^11*f^2 - 52*a*
b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^
11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*
d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^
2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*
b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^
4*f^5 + 4*a^6*b^2*f^5))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1
/2) - (16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c
^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63*a^2*b^7*c^2*d^10 - 12*a
^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b
^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c + d*
1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2)*1i - (((((8*(32*b^15*d^11*f^4
 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64*a^12*b^3*d^
11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c^3*d^8*f^4 -
 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8*f^4 - 288*a
^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 64*a*b^14*c^3
*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^6*c*d^10*f^4
 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + (16*(c + d*t
an(e + f*x))^(1/2)*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2)*(
32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^
4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*
c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*
d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 3
36*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4
*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(c
 + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) - (16*(c + d*tan(e + f*x
))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40*a^7*b^6*d^1
1*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2 + 204*a^3*b
^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*a^7*b^6*c^2*
d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c^2*d^9*f^2 +
 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2 - 296*a^8*b
^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*
(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) + (8*(20*b^11*c*d^11
*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12
*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b^7*c^3*d^9*f^2 - 168*
a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^
3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 - 256*a^2*b^9*c*d^11*f
^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^
5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b
^2*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 - 9*a^6*b^3*d^
12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63*a^2*b^7*c^2
*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d
^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4
))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2)*1i)/((((((8*(32*b
^15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11*f^4 - 64
*a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a^5*b^10*c
^3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^6*c^3*d^8
*f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^10*f^4 + 6
4*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 320*a^9*b^
6*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) -
(16*(c + d*tan(e + f*x))^(1/2)*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^
2)))^(1/2)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4 - 160*a^8
*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^8*f^4 + 2
72*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f^4 + 48*a
^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a^3*b^14*c
*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d^9*f^4 +
112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^
2*f^4))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) + (16*(c + d
*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11*f^2 + 40
*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2
 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f^2 + 184*
a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a^11*b^2*c
^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c*d^10*f^2
 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^
6*b^2*f^4))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) + (8*(20
*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*
a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b^7*c^3*d^
9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^
2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 - 256*a^2*
b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 +
4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*
4i - 6*a^2*b^2*f^2)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 -
9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63
*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a
^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4
*a^6*b^2*f^4))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) + (((
((8*(32*b^15*d^11*f^4 + 96*a^2*b^13*d^11*f^4 - 320*a^6*b^9*d^11*f^4 - 480*a^8*b^7*d^11*f^4 - 288*a^10*b^5*d^11
*f^4 - 64*a^12*b^3*d^11*f^4 + 32*b^15*c^2*d^9*f^4 + 96*a^2*b^13*c^2*d^9*f^4 + 320*a^3*b^12*c^3*d^8*f^4 + 640*a
^5*b^10*c^3*d^8*f^4 - 320*a^6*b^9*c^2*d^9*f^4 + 640*a^7*b^8*c^3*d^8*f^4 - 480*a^8*b^7*c^2*d^9*f^4 + 320*a^9*b^
6*c^3*d^8*f^4 - 288*a^10*b^5*c^2*d^9*f^4 + 64*a^11*b^4*c^3*d^8*f^4 - 64*a^12*b^3*c^2*d^9*f^4 + 64*a*b^14*c*d^1
0*f^4 + 64*a*b^14*c^3*d^8*f^4 + 320*a^3*b^12*c*d^10*f^4 + 640*a^5*b^10*c*d^10*f^4 + 640*a^7*b^8*c*d^10*f^4 + 3
20*a^9*b^6*c*d^10*f^4 + 64*a^11*b^4*c*d^10*f^4))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^
2*f^5) + (16*(c + d*tan(e + f*x))^(1/2)*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a
^2*b^2*f^2)))^(1/2)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 + 160*a^6*b^11*d^10*f^4
- 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^10*f^4 + 48*b^17*c^2*d^
8*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 + 400*a^8*b^9*c^2*d^8*f
^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*b^16*c*d^9*f^4 + 112*a
^3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*f^4 + 336*a^11*b^6*c*d
^9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 +
 4*a^6*b^2*f^4))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) - (
16*(c + d*tan(e + f*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 - 88*a^5*b^8*d^11
*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^
3*d^8*f^2 + 204*a^3*b^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^6*b^7*c^3*d^8*f
^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + 100*a^9*b^4*c^2*d^9*f^2 + 4*a^10*b^3*c^3*d^8*f^2 - 4*a
^11*b^2*c^2*d^9*f^2 + 116*a*b^12*c^2*d^9*f^2 + 104*a^2*b^11*c*d^10*f^2 + 48*a^4*b^9*c*d^10*f^2 - 304*a^6*b^7*c
*d^10*f^2 - 296*a^8*b^5*c*d^10*f^2 - 56*a^10*b^3*c*d^10*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f
^4 + 4*a^6*b^2*f^4))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2)
 + (8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8*d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12
*f^2 - 4*a^9*b^2*d^12*f^2 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 + 72*a^4*b
^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^
2*d^10*f^2 + 4*a^8*b^3*c^3*d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4*d^8*f^2 -
 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b
^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a
^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 - 3*a^2*b^7*d^12 + 17*a^4*b^
5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2*b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*
d^11 + 63*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5*c^2*d^10 + 18*a^4*b^5*c^4*d
^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^
4*f^4 + 4*a^6*b^2*f^4))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1
/2) + (16*(b^7*d^13 - 9*a^4*b^3*d^13 + 3*b^7*c^2*d^11 + 2*b^7*c^4*d^9 + 8*a*b^6*c^3*d^10 + 8*a*b^6*c^5*d^8 + 2
4*a^3*b^4*c*d^12 - 22*a^2*b^5*c^2*d^11 - 22*a^2*b^5*c^4*d^9 + 24*a^3*b^4*c^3*d^10 - 9*a^4*b^3*c^2*d^11))/(a^8*
f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5)))*(-(c + d*1i)/(4*(a^4*f^2 + b^4*f^2 - a*b^3*f^
2*4i + a^3*b*f^2*4i - 6*a^2*b^2*f^2)))^(1/2)*2i - (b*d*(c + d*tan(e + f*x))^(1/2))/((a^2 + b^2)*(b*f*(c + d*ta
n(e + f*x)) + a*d*f - b*c*f))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \]

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))**2,x)

[Out]

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

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