∫ (c+d tan(e+fx))3∕2 ____________________________

3.1238 \(\int \frac {(c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)} \, dx\)

Optimal. Leaf size=170 \[ -\frac {2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )}+\frac {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)}-\frac {(c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)} \]

[Out]

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

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

+b^2)/f/b^(1/2)

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Rubi [A] time = 0.52, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3573, 3539, 3537, 63, 208, 3634} \[ -\frac {2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )}+\frac {(c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)}-\frac {(c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*f)

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 3573

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1

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

f*x]], x], x] + Dist[(b*c - a*d)^2/(c^2 + d^2), Int[(1 + Tan[e + f*x]^2)/(Sqrt[a + b*Tan[e + f*x]]*(c + 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] && NeQ[c^2 + d^2

, 0]

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]

Rubi steps

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

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Mathematica [A] time = 0.39, size = 168, normalized size = 0.99 \[ \frac {\sqrt {b} (b-i a) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {b} (b+i a) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )-2 (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} f \left (a^2+b^2\right )} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)*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))^(3/2)/(a+b*tan(f*x+e)),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))^(3/2)/(a+b*tan(f*x+e)),x, algorithm="giac")

[Out]

Timed out

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maple [B] time = 0.42, size = 1964, normalized size = 11.55 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/f/d/(a^2+b^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*c-1/f/(a^2+b^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)*b*c-2/f*d/(a^2+b^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*c+1/f/(a^2+b^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*t

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

b^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*c^2+1/2/f/(a^2+b^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*c-1/f*d/(a^2+b^2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*ta

n(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+2/f*d/(a^2+b^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*c-1/f/(a^2+b^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*c^2-1/4/f*d/(a^2+b^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-1/2/f/(a^2+b^2)*ln((c+d*ta

n(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*

c+1/f*d/(a^2+b^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+1/4/f/(a^2+b^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-1/4/f/d/(a^2+b^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*c^2+1/f*d^2/(a^2+b^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-1/f*d^2/(a^2+b^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+1/4/f*d/(a^2+b^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+1/f/(a^2+b^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))*(c^2+d^2)^(1/2)*b*c-1/4/f/d/(a^2+b^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*c+2/f*d^2/(a^2

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

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mupad [B] time = 13.01, size = 19118, normalized size = 112.46 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(-((((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^

2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*

d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 -

2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d

^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^2*b^6*d^1

2*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^

10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 -

16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^

11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(-b*(a*d - b*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^9*d^10*f^

4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*

f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*

d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f + a^2*b*f))))/(b^3*f + a^2*b*f) - (32*(c + d*tan(e + f*x))^(1/2)*(

22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^

7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^

9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*

a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^

2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f))*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) - (32*(c + d

*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d

^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c

^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^

5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(b^3*f + a^2*b*f) - (((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^

6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^

4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11

*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b

^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2)

)/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^

8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^

4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*

b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(-b*(a*d - b

*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*

b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8

*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f + a^2*b*f))

))/(b^3*f + a^2*b*f) + (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*

f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^

5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2

- 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4

*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f))*(

-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c

^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7

*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 1

2*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2)*1i)/(b^3*f + a^2

*b*f))/((64*(a^2*b^2*d^18 + b^4*c^2*d^16 + 5*b^4*c^4*d^14 + 7*b^4*c^6*d^12 + 3*b^4*c^8*d^10 - 12*a*b^3*c^3*d^1

5 - 18*a*b^3*c^5*d^13 - 8*a*b^3*c^7*d^11 - 4*a^3*b*c^3*d^15 - 2*a^3*b*c^5*d^13 + 9*a^2*b^2*c^2*d^16 + 15*a^2*b

^2*c^4*d^14 + 7*a^2*b^2*c^6*d^12 - 2*a*b^3*c*d^17 - 2*a^3*b*c*d^17))/f^5 + (((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d

^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2

- 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*

b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*

f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c

^2*d^13*f^2))/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12

*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^

5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10

*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*

(-b*(a*d - b*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f

^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2

*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f

+ a^2*b*f))))/(b^3*f + a^2*b*f) - (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a

^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 1

2*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3

*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 +

66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f +

a^2*b*f))*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^

16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 -

8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*

c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^

3*f + a^2*b*f) + (((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^

3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2

*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^

12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^

2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + ((((-b*(a*d - b*c)^3)^(1/2)*((32*(4*a^

2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b

^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d

^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3

*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(-b*(a*d - b*c)^3)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^

9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7

*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a

^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/(f^4*(b^3*f + a^2*b*f))))/(b^3*f + a^2*b*f) + (32*(c + d*tan(e + f*x)

)^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^

2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b

^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9

*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*

c*d^12*f^2))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f))*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f) + (

32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*

b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 +

2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a

^3*b^2*c^5*d^11))/f^4)*(-b*(a*d - b*c)^3)^(1/2))/(b^3*f + a^2*b*f)))*(-b*(a*d - b*c)^3)^(1/2)*2i)/(b^3*f + a^2

*b*f) - atan(((((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^1

2*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*

c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^

2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4

*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + (((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*

f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4

*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4

+ 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d

^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1

i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24

*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4

+ 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3

)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*

b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^

5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^

10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b

^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c

^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2))*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4

*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b

^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4

*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12

+ 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/

(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*1i - (((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^

2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*

f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*

b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11

*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + (((

32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 +

28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b

^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4

- 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f*x))^(1/2)*((c*d^2*3i - 3*c^2*d -

c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4

*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*

a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/

f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (32*(c + d*tan(e

+ f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d

^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28

*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c

^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^

4*b^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2))

*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (32*(c + d*tan(e + f*

x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^

4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 1

2*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^

4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*1i)/((64*(a^2*b^2*d^1

8 + b^4*c^2*d^16 + 5*b^4*c^4*d^14 + 7*b^4*c^6*d^12 + 3*b^4*c^8*d^10 - 12*a*b^3*c^3*d^15 - 18*a*b^3*c^5*d^13 -

8*a*b^3*c^7*d^11 - 4*a^3*b*c^3*d^15 - 2*a^3*b*c^5*d^13 + 9*a^2*b^2*c^2*d^16 + 15*a^2*b^2*c^4*d^14 + 7*a^2*b^2*

c^6*d^12 - 2*a*b^3*c*d^17 - 2*a^3*b*c*d^17))/f^5 + (((32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 -

15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2

- 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3

*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^

2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5 + (((32*

(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*

a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*

c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 3

2*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((c*d^2*3i - 3*c^2*d - c^3

*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^

5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6

*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4

)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f

*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10

*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^

3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*

d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b

^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2))*((

c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) - (32*(c + d*tan(e + f*x))

^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c

^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a

^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*

((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (((32*(a*b^5*d^15*f^2

+ 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6*

c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f^

2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b^

5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2

+ 4*a^5*b*c^2*d^13*f^2))/f^5 + (((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^

2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 +

20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*

c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f

*x))^(1/2)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^9*d^10*

f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^

8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*

c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*

b*f^2)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 -

14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*

f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*

a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f

^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i

- b^2*f^2*1i + 2*a*b*f^2)))^(1/2))*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^

2)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5

*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4

*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*

c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*

f^2)))^(1/2)))*((c*d^2*3i - 3*c^2*d - c^3*1i + d^3)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*2i - atan

(((((((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8

*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 1

2*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d

^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((3*c*d^2 - c^2*

d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a

^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 -

8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4)

)/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f

*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10

*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^

3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*

d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b

^3*c*d^12*f^2))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(a

*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^1

0*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b

^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10

*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b

^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f

^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8

*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12

*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*

b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*

2i)))^(1/2)*1i - (((((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 +

12*b^8*c^4*d^8*f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c

^2*d^10*f^4 + 12*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 -

16*a*b^7*c^3*d^9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f*x))^(1/2)*(

(3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*

d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5

*c^2*d^8*f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7

*b^2*c*d^9*f^4))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(

c + d*tan(e + f*x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 +

28*b^7*c^3*d^10*f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d

^11*f^2 - 28*a^3*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 +

2*a^5*b^2*c^4*d^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12

*f^2 + 54*a^4*b^3*c*d^12*f^2))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))

^(1/2) + (32*(a*b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 +

21*b^6*c^5*d^10*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8

*f^2 + 59*a^3*b^3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a

^4*b^2*c^5*d^10*f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14

*f^2 - 30*a^4*b^2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 -

b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b

^5*c^4*d^12 - 8*b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*

b^2*c*d^15 - 12*a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6

*d^10 + 48*a^3*b^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2

*f^2 + a*b*f^2*2i)))^(1/2)*1i)/((64*(a^2*b^2*d^18 + b^4*c^2*d^16 + 5*b^4*c^4*d^14 + 7*b^4*c^6*d^12 + 3*b^4*c^8

*d^10 - 12*a*b^3*c^3*d^15 - 18*a*b^3*c^5*d^13 - 8*a*b^3*c^7*d^11 - 4*a^3*b*c^3*d^15 - 2*a^3*b*c^5*d^13 + 9*a^2

*b^2*c^2*d^16 + 15*a^2*b^2*c^4*d^14 + 7*a^2*b^2*c^6*d^12 - 2*a*b^3*c*d^17 - 2*a^3*b*c*d^17))/f^5 + (((((32*(4*

a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*f^4 + 28*a^2

*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12*a^4*b^4*c^4

*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^9*f^4 - 32*a

^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*((3*c*d^2 - c^2*d*3i - c^3 +

d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f

^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*c^2

*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*((3*c*

d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(2

2*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*f^2 - 18*b^7

*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3*b^4*c^4*d^9

*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d^9*f^2 + 8*a

^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^3*c*d^12*f^2

))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(a*b^5*d^15*f^2

+ 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10*f^2 - 3*b^6

*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^3*c^2*d^13*f

^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*f^2 - 61*a*b

^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^2*c*d^14*f^2

+ 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2

) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*b^5*c^6*d^10

+ 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*a^4*b*c^2*d^

14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b^2*c^3*d^13

- 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) +

(((((32*(4*a^2*b^6*d^12*f^4 + 8*a^4*b^4*d^12*f^4 + 4*a^6*b^2*d^12*f^4 + 12*b^8*c^2*d^10*f^4 + 12*b^8*c^4*d^8*

f^4 + 28*a^2*b^6*c^2*d^10*f^4 + 24*a^2*b^6*c^4*d^8*f^4 - 32*a^3*b^5*c^3*d^9*f^4 + 20*a^4*b^4*c^2*d^10*f^4 + 12

*a^4*b^4*c^4*d^8*f^4 - 16*a^5*b^3*c^3*d^9*f^4 + 4*a^6*b^2*c^2*d^10*f^4 - 16*a*b^7*c*d^11*f^4 - 16*a*b^7*c^3*d^

9*f^4 - 32*a^3*b^5*c*d^11*f^4 - 16*a^5*b^3*c*d^11*f^4))/f^5 + (32*(c + d*tan(e + f*x))^(1/2)*((3*c*d^2 - c^2*d

*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^

4*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8

*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))

/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(c + d*tan(e + f*

x))^(1/2)*(22*b^7*c*d^12*f^2 - 14*a*b^6*d^13*f^2 + 4*a^3*b^4*d^13*f^2 - 14*a^5*b^2*d^13*f^2 + 28*b^7*c^3*d^10*

f^2 - 18*b^7*c^5*d^8*f^2 + 24*a^2*b^5*c^3*d^10*f^2 + 12*a^2*b^5*c^5*d^8*f^2 - 88*a^3*b^4*c^2*d^11*f^2 - 28*a^3

*b^4*c^4*d^9*f^2 + 60*a^4*b^3*c^3*d^10*f^2 - 2*a^4*b^3*c^5*d^8*f^2 - 44*a^5*b^2*c^2*d^11*f^2 + 2*a^5*b^2*c^4*d

^9*f^2 + 8*a^6*b*c*d^12*f^2 + 20*a*b^6*c^2*d^11*f^2 + 66*a*b^6*c^4*d^9*f^2 - 28*a^2*b^5*c*d^12*f^2 + 54*a^4*b^

3*c*d^12*f^2))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(a*

b^5*d^15*f^2 + 4*a^5*b*d^15*f^2 - b^6*c*d^14*f^2 - 15*a^3*b^3*d^15*f^2 + 23*b^6*c^3*d^12*f^2 + 21*b^6*c^5*d^10

*f^2 - 3*b^6*c^7*d^8*f^2 - 29*a^2*b^4*c^3*d^12*f^2 - 81*a^2*b^4*c^5*d^10*f^2 + a^2*b^4*c^7*d^8*f^2 + 59*a^3*b^

3*c^2*d^13*f^2 + 75*a^3*b^3*c^4*d^11*f^2 + a^3*b^3*c^6*d^9*f^2 - 32*a^4*b^2*c^3*d^12*f^2 - 2*a^4*b^2*c^5*d^10*

f^2 - 61*a*b^5*c^2*d^13*f^2 - 25*a*b^5*c^4*d^11*f^2 + 37*a*b^5*c^6*d^9*f^2 + 53*a^2*b^4*c*d^14*f^2 - 30*a^4*b^

2*c*d^14*f^2 + 4*a^5*b*c^2*d^13*f^2))/f^5)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^

2*2i)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^16 + 2*a^4*b*d^16 + 4*b^5*c^2*d^14 + 8*b^5*c^4*d^12 - 8*

b^5*c^6*d^10 + 3*b^5*c^8*d^8 - 8*a*b^4*c^3*d^13 + 48*a*b^4*c^5*d^11 - 8*a*b^4*c^7*d^9 - 8*a^3*b^2*c*d^15 - 12*

a^4*b*c^2*d^14 + 2*a^4*b*c^4*d^12 + 12*a^2*b^3*c^2*d^14 - 72*a^2*b^3*c^4*d^12 + 12*a^2*b^3*c^6*d^10 + 48*a^3*b

^2*c^3*d^13 - 8*a^3*b^2*c^5*d^11))/f^4)*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2

i)))^(1/2)))*((3*c*d^2 - c^2*d*3i - c^3 + d^3*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*2i

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{a + b \tan {\left (e + f x \right )}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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