3.1244 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{a+b \tan (e+f x)} \, dx\)

Optimal. Leaf size=195 \[ -\frac {2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} f \left (a^2+b^2\right )}+\frac {(c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)}-\frac {(c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)}+\frac {2 d^2 \sqrt {c+d \tan (e+f x)}}{b f} \]

[Out]

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

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Rubi [A]  time = 0.92, antiderivative size = 195, 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 = {3566, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac {2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} f \left (a^2+b^2\right )}+\frac {(c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)}-\frac {(c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)}+\frac {2 d^2 \sqrt {c+d \tan (e+f x)}}{b f} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x]),x]

[Out]

((c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((I*a + b)*f) - ((c + I*d)^(5/2)*ArcTanh[Sqr
t[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)*f) - (2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e +
f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)*f) + (2*d^2*Sqrt[c + d*Tan[e + f*x]])/(b*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 3566

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(m + n - 1)), x] + Dist[1/(d*(m + n -
1)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(
1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2,
x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
&& IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 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]

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

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Mathematica [A]  time = 0.58, size = 199, normalized size = 1.02 \[ \frac {2 \sqrt {b} d^2 \left (a^2+b^2\right ) \sqrt {c+d \tan (e+f x)}+b^{3/2} (b-i a) (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+b^{3/2} (b+i a) (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )-2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} f \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + b*Tan[e + f*x]),x]

[Out]

(b^(3/2)*((-I)*a + b)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + b^(3/2)*(I*a + b)*(c +
I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] - 2*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*
Tan[e + f*x]])/Sqrt[b*c - a*d]] + 2*Sqrt[b]*(a^2 + b^2)*d^2*Sqrt[c + d*Tan[e + f*x]])/(b^(3/2)*(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))^(5/2)/(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))^(5/2)/(a+b*tan(f*x+e)),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.44, size = 2802, normalized size = 14.37 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e)),x)

[Out]

2*d^2*(c+d*tan(f*x+e))^(1/2)/b/f-6/f*d*b/(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*c^2-2/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))*(c^2+d^2)^(1/2)*a*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+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^2+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))*(c^2+d^2)^(1/2)*a*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)*a*c^3-3/4/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-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+1/4/f*
d^2/(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)*b-1/4/f*d^2/(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-3/4/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)*b*c^2-1/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+1/2/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*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))*(c^2+d^2)^(1
/2)*b*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)*(c^2+d^2)^(1/2)*b*c-3/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*c-1/4/f/d/(a^2+b^2)*l
n((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^3+3/4/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+6/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*c+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))*(c^2+d^2)^(1/2)*b-2/f*d^3/b/(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^3-3/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^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)*(c^2+d^2)^(1/2)*a-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))*(c^2+d^2)^(1/2)*b+3/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*c+3/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^2-1/f*d^3/(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-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^3+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))*b*c^3+2/f*b^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))*c^3+3/4/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)*b*c^2+1/f*d^3/(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

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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))^(5/2)/(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 = 13.25, size = 27922, normalized size = 143.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*tan(e + f*x))^(5/2)/(a + b*tan(e + f*x)),x)

[Out]

(2*d^2*(c + d*tan(e + f*x))^(1/2))/(b*f) - atan(((((((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13
*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^
5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^
4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b
^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) - (32*(c + d*t
an(e + f*x))^(1/2)*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1
i + 2*a*b*f^2)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 2
4*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^
4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1
i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(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)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2
*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2
+ 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*
a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^
3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d
^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a
^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d
^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5
*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d
^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522
*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b
^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4
*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13
*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^
6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2
))/(b*f^5))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(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^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 +
2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b
^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*
d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d
^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^
9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*
d^19))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i +
2*a*b*f^2)))^(1/2)*1i - (((((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^
4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c
^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*
f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*
b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) + (32*(c + d*tan(e + f*x))^(1/2)*(-(c*
d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(
16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*
a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4
+ 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c
^3*d^2*10i)/(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)*(16*a^7*b*d^15*f
^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b
^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f
^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 17
0*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5
*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*
d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a
^7*b*c^2*d^13*f^2))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i -
b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d
^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^1
0*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 -
207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^
3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*
c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^
2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b
^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5))*(-(c*d^4*5i
+ 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(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^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c
^2*d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c
^5*d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^1
5 - 12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^
10 + 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*
d^18 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4))*(-(c*d^4
*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*1i)/
((((((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4
+ 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6
*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9
*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*
b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) - (32*(c + d*tan(e + f*x))^(1/2)*(-(c*d^4*5i + 5*c^4*d + c^5*1
i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^
2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*
a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4
+ 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(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)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 -
8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^
8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*
f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 -
150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^
5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6
*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f
^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)
))^(1/2) + (32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15
*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f
^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2
+ a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a
^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3
*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f
^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 104*a^5*
b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5
- 10*c^2*d^3 - c^3*d^2*10i)/(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^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^1
6 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*
d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 -
30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17
- 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d
^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i +
d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (((((32*(4*b^9*c*d^12*f^4
- 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a
^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c
^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f
^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*
c*d^12*f^4))/(b*f^5) + (32*(c + d*tan(e + f*x))^(1/2)*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*
d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6
*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6
*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*
f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(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)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*
c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7
*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f
^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 +
2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6
*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^1
4*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d
+ c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (32*(12*a^6*b*d^
18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 -
48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f
^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 6
40*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^
4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*
c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2
- 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c
^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2*10i
)/(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^6*d^20 - 2*a^6*d^20 + 3
0*a^6*c^2*d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^
6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 -
12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a
^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 6
00*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14
+ 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^3*d^2
*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2) + (64*(a^5*d^23 - a^3*b^2*d^23 - 6*a^5*c^4*d^19 - 8*a^5
*c^6*d^17 - 3*a^5*c^8*d^15 + b^5*c^3*d^20 + 6*b^5*c^5*d^18 + 18*b^5*c^7*d^16 + 28*b^5*c^9*d^14 + 21*b^5*c^11*d
^12 + 6*b^5*c^13*d^10 - 3*a*b^4*c^2*d^21 - 18*a*b^4*c^4*d^19 - 62*a*b^4*c^6*d^17 - 108*a*b^4*c^8*d^15 - 87*a*b
^4*c^10*d^13 - 26*a*b^4*c^12*d^11 + 3*a^2*b^3*c*d^22 + 8*a^4*b*c^3*d^20 + 42*a^4*b*c^5*d^18 + 48*a^4*b*c^7*d^1
6 + 17*a^4*b*c^9*d^14 + 17*a^2*b^3*c^3*d^20 + 78*a^2*b^3*c^5*d^18 + 162*a^2*b^3*c^7*d^16 + 143*a^2*b^3*c^9*d^1
4 + 45*a^2*b^3*c^11*d^12 - 3*a^3*b^2*c^2*d^21 - 42*a^3*b^2*c^4*d^19 - 118*a^3*b^2*c^6*d^17 - 117*a^3*b^2*c^8*d
^15 - 39*a^3*b^2*c^10*d^13 - 3*a^4*b*c*d^22))/(b*f^5)))*(-(c*d^4*5i + 5*c^4*d + c^5*1i + d^5 - 10*c^2*d^3 - c^
3*d^2*10i)/(4*(a^2*f^2*1i - b^2*f^2*1i + 2*a*b*f^2)))^(1/2)*2i - atan(((((((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^1
3*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^
10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 +
12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8
*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/
(b*f^5) - (32*(c + d*tan(e + f*x))^(1/2)*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(
a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6
*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 +
8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*(-(5*c*d^4 +
c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(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)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*
d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*
c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^
11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2
+ 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114
*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^1
4*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^
3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 +
a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^
7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^
2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 37
2*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*
b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c
^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 +
58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d
^14*f^2))/(b*f^5))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(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^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16
+ 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3
*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^
3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8
*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*
c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*
c*d^19))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b
*f^2*2i)))^(1/2)*1i - (((((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 +
16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*
d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4
- 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7
*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) + (32*(c + d*tan(e + f*x))^(1/2)*(-(5*c*d
^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^10*
d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*
c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^
5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d
^2)/(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)*(16*a^7*b*d^15*f^2 - 14*a*b^7
*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f
^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b
^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3
*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11
*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 11
4*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13
*f^2))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f
^2*2i)))^(1/2) + (32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^
3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10
*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^
10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2
+ 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a
^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*
d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 10
4*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5))*(-(5*c*d^4 + c^4*d*5i + c^5 +
d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(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^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^
16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7
*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 -
30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^1
7 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*
d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d
^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*1i)/((((((32*(4*b^9*c*d^12*f^4 -
4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^
2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^
3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^
4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c
*d^12*f^4))/(b*f^5) - (32*(c + d*tan(e + f*x))^(1/2)*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c
^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*
f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c
^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*
(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(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)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2
+ 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 +
100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^
3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*
c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^
12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a
^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5
*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(12*a^6*b*d^18*f^2 + 8*a^7*c
*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14
*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5
*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d
^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f
^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 1
92*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7
*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60
*a^6*b*c^4*d^14*f^2))/(b*f^5))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(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^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a
^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c
^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 -
180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450
*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 -
40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12
+ 12*a^5*b*c*d^19))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b
^2*f^2 + a*b*f^2*2i)))^(1/2) + (((((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d
^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3
*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2
*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 1
6*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) + (32*(c + d*tan(e + f*x))^(1/2)
*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*
(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40
*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^
4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i -
10*c^3*d^2)/(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)*(16*a^7*b*d^15*f^2 -
14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c
^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 -
12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^
4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3
*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11
*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b
*c^2*d^13*f^2))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i + c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^
2 + a*b*f^2*2i)))^(1/2) + (32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 +
8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3
*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b
^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7
*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*
f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a
*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17
*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5))*(-(5*c*d^4 + c^4*d*5i
+ c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(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^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b
^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*
a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c
^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^
3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4
*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4))*(-(5*c*d^4 + c^4*d*5i
+ c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (64*(a^5*d^23 - a^3*b
^2*d^23 - 6*a^5*c^4*d^19 - 8*a^5*c^6*d^17 - 3*a^5*c^8*d^15 + b^5*c^3*d^20 + 6*b^5*c^5*d^18 + 18*b^5*c^7*d^16 +
28*b^5*c^9*d^14 + 21*b^5*c^11*d^12 + 6*b^5*c^13*d^10 - 3*a*b^4*c^2*d^21 - 18*a*b^4*c^4*d^19 - 62*a*b^4*c^6*d^
17 - 108*a*b^4*c^8*d^15 - 87*a*b^4*c^10*d^13 - 26*a*b^4*c^12*d^11 + 3*a^2*b^3*c*d^22 + 8*a^4*b*c^3*d^20 + 42*a
^4*b*c^5*d^18 + 48*a^4*b*c^7*d^16 + 17*a^4*b*c^9*d^14 + 17*a^2*b^3*c^3*d^20 + 78*a^2*b^3*c^5*d^18 + 162*a^2*b^
3*c^7*d^16 + 143*a^2*b^3*c^9*d^14 + 45*a^2*b^3*c^11*d^12 - 3*a^3*b^2*c^2*d^21 - 42*a^3*b^2*c^4*d^19 - 118*a^3*
b^2*c^6*d^17 - 117*a^3*b^2*c^8*d^15 - 39*a^3*b^2*c^10*d^13 - 3*a^4*b*c*d^22))/(b*f^5)))*(-(5*c*d^4 + c^4*d*5i
+ c^5 + d^5*1i - c^2*d^3*10i - 10*c^3*d^2)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*2i + (atan((((-b^3*(a*d
- b*c)^5)^(1/2)*((32*(c + d*tan(e + f*x))^(1/2)*(b^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 +
2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b
^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*
d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d
^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^
9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*
d^19))/(b*f^4) - ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 1
5*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*
b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*
d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f
^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 3
48*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^
6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2
+ 90*a^3*b^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5) +
((-b^3*(a*d - b*c)^5)^(1/2)*((32*(c + d*tan(e + f*x))^(1/2)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d
^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10
*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a
^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b
^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*
d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 -
44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f^4) + (((3
2*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^
9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^
9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 +
8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d
^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) - (32*(-b^3*(a*d - b*c)^5)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^10*
d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*
c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^
5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b^4*f^5*(a^2 + b^2)))*(-b^3*(a*d - b*c)^5)^(1/2))/(b^3*f*(a^2 + b^2))
))/(b^3*f*(a^2 + b^2))))/(b^3*f*(a^2 + b^2)))*1i)/(b^3*f*(a^2 + b^2)) + ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(c +
d*tan(e + f*x))^(1/2)*(b^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*
d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*
d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 -
12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10
+ 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^1
8 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4) + ((-b^3*(a*
d - b*c)^5)^(1/2)*((32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*
c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^
10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*
d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f
^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3
*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^
3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 -
104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5) - ((-b^3*(a*d - b*c)^5)^(1/2)
*((32*(c + d*tan(e + f*x))^(1/2)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2
+ 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 +
100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a
^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4
*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d
^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*
a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f^4) - (((32*(4*b^9*c*d^12*f^4 - 4*a*b
^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*
c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10
*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24
*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*
f^4))/(b*f^5) + (32*(-b^3*(a*d - b*c)^5)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*
f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2
*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3
*c*d^9*f^4))/(b^4*f^5*(a^2 + b^2)))*(-b^3*(a*d - b*c)^5)^(1/2))/(b^3*f*(a^2 + b^2))))/(b^3*f*(a^2 + b^2))))/(b
^3*f*(a^2 + b^2)))*1i)/(b^3*f*(a^2 + b^2)))/((64*(a^5*d^23 - a^3*b^2*d^23 - 6*a^5*c^4*d^19 - 8*a^5*c^6*d^17 -
3*a^5*c^8*d^15 + b^5*c^3*d^20 + 6*b^5*c^5*d^18 + 18*b^5*c^7*d^16 + 28*b^5*c^9*d^14 + 21*b^5*c^11*d^12 + 6*b^5*
c^13*d^10 - 3*a*b^4*c^2*d^21 - 18*a*b^4*c^4*d^19 - 62*a*b^4*c^6*d^17 - 108*a*b^4*c^8*d^15 - 87*a*b^4*c^10*d^13
- 26*a*b^4*c^12*d^11 + 3*a^2*b^3*c*d^22 + 8*a^4*b*c^3*d^20 + 42*a^4*b*c^5*d^18 + 48*a^4*b*c^7*d^16 + 17*a^4*b
*c^9*d^14 + 17*a^2*b^3*c^3*d^20 + 78*a^2*b^3*c^5*d^18 + 162*a^2*b^3*c^7*d^16 + 143*a^2*b^3*c^9*d^14 + 45*a^2*b
^3*c^11*d^12 - 3*a^3*b^2*c^2*d^21 - 42*a^3*b^2*c^4*d^19 - 118*a^3*b^2*c^6*d^17 - 117*a^3*b^2*c^8*d^15 - 39*a^3
*b^2*c^10*d^13 - 3*a^4*b*c*d^22))/(b*f^5) - ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(c + d*tan(e + f*x))^(1/2)*(b^6*d
^20 - 2*a^6*d^20 + 30*a^6*c^2*d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^16 + 18*
b^6*c^6*d^14 + 45*b^6*c^8*d^12 - 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*d^13 +
180*a*b^5*c^9*d^11 - 12*a*b^5*c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 - 30*a^2*
b^4*c^4*d^16 + 450*a^2*b^4*c^6*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17 - 600*
a^3*b^3*c^5*d^15 + 600*a^3*b^3*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d^16 - 4
50*a^4*b^2*c^6*d^14 + 30*a^4*b^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4) - ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(12*a
^6*b*d^18*f^2 + 8*a^7*c*d^17*f^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^1
6*f^2 - 48*b^7*c^4*d^14*f^2 + 30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2
*d^16*f^2 + 558*a^2*b^5*c^4*d^14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*
f^2 - 640*a^3*b^4*c^3*d^15*f^2 - 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 +
372*a^4*b^3*c^2*d^16*f^2 + 42*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a
^5*b^2*c^3*d^15*f^2 + 192*a^5*b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d
^13*f^2 - 312*a*b^6*c^7*d^11*f^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*
a^6*b*c^2*d^16*f^2 - 60*a^6*b*c^4*d^14*f^2))/(b*f^5) + ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(c + d*tan(e + f*x))^(
1/2)*(16*a^7*b*d^15*f^2 - 14*a*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^
5*b^3*d^15*f^2 - 10*b^8*c^3*d^12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 +
36*a^2*b^6*c^5*d^10*f^2 - 12*a^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3
*b^5*c^6*d^9*f^2 - 170*a^4*b^4*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c
^2*d^13*f^2 + 190*a^5*b^3*c^4*d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13
*f^2 + 110*a*b^7*c^4*d^11*f^2 - 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*
b^2*c*d^14*f^2 + 48*a^7*b*c^2*d^13*f^2))/(b*f^4) + (((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13
*f^4 - 4*a^5*b^4*d^13*f^4 + 16*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^
5*d^8*f^4 - 48*a^3*b^6*c^2*d^11*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^
4 - 24*a^5*b^4*c^2*d^11*f^4 - 20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b
^8*c^4*d^9*f^4 + 16*a^2*b^7*c*d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) - (32*(-b^3*(a
*d - b*c)^5)^(1/2)*(c + d*tan(e + f*x))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 -
16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8
*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b^4*f^5*(a^2 +
b^2)))*(-b^3*(a*d - b*c)^5)^(1/2))/(b^3*f*(a^2 + b^2))))/(b^3*f*(a^2 + b^2))))/(b^3*f*(a^2 + b^2))))/(b^3*f*(
a^2 + b^2)) + ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(c + d*tan(e + f*x))^(1/2)*(b^6*d^20 - 2*a^6*d^20 + 30*a^6*c^2*
d^18 - 30*a^6*c^4*d^16 + 2*a^6*c^6*d^14 + 6*b^6*c^2*d^18 + 15*b^6*c^4*d^16 + 18*b^6*c^6*d^14 + 45*b^6*c^8*d^12
- 24*b^6*c^10*d^10 + 3*b^6*c^12*d^8 + 12*a*b^5*c^5*d^15 - 180*a*b^5*c^7*d^13 + 180*a*b^5*c^9*d^11 - 12*a*b^5*
c^11*d^9 - 180*a^5*b*c^3*d^17 + 180*a^5*b*c^5*d^15 - 12*a^5*b*c^7*d^13 - 30*a^2*b^4*c^4*d^16 + 450*a^2*b^4*c^6
*d^14 - 450*a^2*b^4*c^8*d^12 + 30*a^2*b^4*c^10*d^10 + 40*a^3*b^3*c^3*d^17 - 600*a^3*b^3*c^5*d^15 + 600*a^3*b^3
*c^7*d^13 - 40*a^3*b^3*c^9*d^11 - 30*a^4*b^2*c^2*d^18 + 450*a^4*b^2*c^4*d^16 - 450*a^4*b^2*c^6*d^14 + 30*a^4*b
^2*c^8*d^12 + 12*a^5*b*c*d^19))/(b*f^4) + ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(12*a^6*b*d^18*f^2 + 8*a^7*c*d^17*f
^2 + a^2*b^5*d^18*f^2 - 15*a^4*b^3*d^18*f^2 + 8*a^7*c^3*d^15*f^2 - 3*b^7*c^2*d^16*f^2 - 48*b^7*c^4*d^14*f^2 +
30*b^7*c^6*d^12*f^2 + 72*b^7*c^8*d^10*f^2 - 3*b^7*c^10*d^8*f^2 - 171*a^2*b^5*c^2*d^16*f^2 + 558*a^2*b^5*c^4*d^
14*f^2 + 522*a^2*b^5*c^6*d^12*f^2 - 207*a^2*b^5*c^8*d^10*f^2 + a^2*b^5*c^10*d^8*f^2 - 640*a^3*b^4*c^3*d^15*f^2
- 372*a^3*b^4*c^5*d^13*f^2 + 360*a^3*b^4*c^7*d^11*f^2 + 2*a^3*b^4*c^9*d^9*f^2 + 372*a^4*b^3*c^2*d^16*f^2 + 42
*a^4*b^3*c^4*d^14*f^2 - 348*a^4*b^3*c^6*d^12*f^2 - 3*a^4*b^3*c^8*d^10*f^2 + 88*a^5*b^2*c^3*d^15*f^2 + 192*a^5*
b^2*c^5*d^13*f^2 + 2*a*b^6*c*d^17*f^2 + 144*a*b^6*c^3*d^15*f^2 - 228*a*b^6*c^5*d^13*f^2 - 312*a*b^6*c^7*d^11*f
^2 + 58*a*b^6*c^9*d^9*f^2 + 90*a^3*b^4*c*d^17*f^2 - 104*a^5*b^2*c*d^17*f^2 - 48*a^6*b*c^2*d^16*f^2 - 60*a^6*b*
c^4*d^14*f^2))/(b*f^5) - ((-b^3*(a*d - b*c)^5)^(1/2)*((32*(c + d*tan(e + f*x))^(1/2)*(16*a^7*b*d^15*f^2 - 14*a
*b^7*d^15*f^2 - 8*a^8*c*d^14*f^2 + 38*b^8*c*d^14*f^2 + 4*a^3*b^5*d^15*f^2 + 2*a^5*b^3*d^15*f^2 - 10*b^8*c^3*d^
12*f^2 - 102*b^8*c^5*d^10*f^2 + 18*b^8*c^7*d^8*f^2 + 100*a^2*b^6*c^3*d^12*f^2 + 36*a^2*b^6*c^5*d^10*f^2 - 12*a
^2*b^6*c^7*d^8*f^2 - 60*a^3*b^5*c^2*d^13*f^2 + 140*a^3*b^5*c^4*d^11*f^2 + 44*a^3*b^5*c^6*d^9*f^2 - 170*a^4*b^4
*c^3*d^12*f^2 - 150*a^4*b^4*c^5*d^10*f^2 + 2*a^4*b^4*c^7*d^8*f^2 + 162*a^5*b^3*c^2*d^13*f^2 + 190*a^5*b^3*c^4*
d^11*f^2 - 2*a^5*b^3*c^6*d^9*f^2 - 120*a^6*b^2*c^3*d^12*f^2 + 114*a*b^7*c^2*d^13*f^2 + 110*a*b^7*c^4*d^11*f^2
- 114*a*b^7*c^6*d^9*f^2 - 44*a^2*b^6*c*d^14*f^2 - 2*a^4*b^4*c*d^14*f^2 - 88*a^6*b^2*c*d^14*f^2 + 48*a^7*b*c^2*
d^13*f^2))/(b*f^4) - (((32*(4*b^9*c*d^12*f^4 - 4*a*b^8*d^13*f^4 - 8*a^3*b^6*d^13*f^4 - 4*a^5*b^4*d^13*f^4 + 16
*b^9*c^3*d^10*f^4 + 12*b^9*c^5*d^8*f^4 + 40*a^2*b^7*c^3*d^10*f^4 + 24*a^2*b^7*c^5*d^8*f^4 - 48*a^3*b^6*c^2*d^1
1*f^4 - 40*a^3*b^6*c^4*d^9*f^4 + 32*a^4*b^5*c^3*d^10*f^4 + 12*a^4*b^5*c^5*d^8*f^4 - 24*a^5*b^4*c^2*d^11*f^4 -
20*a^5*b^4*c^4*d^9*f^4 + 8*a^6*b^3*c^3*d^10*f^4 - 24*a*b^8*c^2*d^11*f^4 - 20*a*b^8*c^4*d^9*f^4 + 16*a^2*b^7*c*
d^12*f^4 + 20*a^4*b^5*c*d^12*f^4 + 8*a^6*b^3*c*d^12*f^4))/(b*f^5) + (32*(-b^3*(a*d - b*c)^5)^(1/2)*(c + d*tan(
e + f*x))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*
c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*
a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b^4*f^5*(a^2 + b^2)))*(-b^3*(a*d - b*c)^5)^(
1/2))/(b^3*f*(a^2 + b^2))))/(b^3*f*(a^2 + b^2))))/(b^3*f*(a^2 + b^2))))/(b^3*f*(a^2 + b^2))))*(-b^3*(a*d - b*c
)^5)^(1/2)*2i)/(b^3*f*(a^2 + b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{a + b \tan {\left (e + f x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(5/2)/(a+b*tan(f*x+e)),x)

[Out]

Integral((c + d*tan(e + f*x))**(5/2)/(a + b*tan(e + f*x)), x)

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