\(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx\) [125]

Optimal result

Integrand size = 35, antiderivative size = 209 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {(i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}-\frac {(B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right )}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )}{\left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]

[Out]

-(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f-(B-I*(A-C))*arctanh((c+d*tan(f*x+e)
)^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(5/2)/f-2*(2*c*(A-C)*d-B*(c^2-d^2))/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^(1/2)-2/3*(A
*d^2-B*c*d+C*c^2)/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(3/2)

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3709, 3610, 3620, 3618, 65, 214} \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {(i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}-\frac {(B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}}-\frac {2 \left (A d^2-B c d+c^2 C\right )}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )}{f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}} \]

[In]

Int[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(5/2),x]

[Out]

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

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 3610

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

Rule 3618

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

Rule 3620

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

Rule 3709

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.98 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.07 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=-\frac {2 C \left (c^2+d^2\right )+(B c+(-A+C) d) \left (i (c+i d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )\right )-3 B \left (i (c+i d) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )\right ) (c+d \tan (e+f x))}{3 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}} \]

[In]

Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(5/2),x]

[Out]

-1/3*(2*C*(c^2 + d^2) + (B*c + (-A + C)*d)*(I*(c + I*d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/
(c - I*d)] - (I*c + d)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)]) - 3*B*(I*(c + I*d)*Hy
pergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c +
d*Tan[e + f*x])/(c + I*d)])*(c + d*Tan[e + f*x]))/(d*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6787\) vs. \(2(184)=368\).

Time = 0.18 (sec) , antiderivative size = 6788, normalized size of antiderivative = 32.48

method	result	size
parts	\(\text {Expression too large to display}\)	\(6788\)
derivativedivides	\(\text {Expression too large to display}\)	\(20647\)
default	\(\text {Expression too large to display}\)	\(20647\)

[In]

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 13143 vs. \(2 (177) = 354\).

Time = 7.15 (sec) , antiderivative size = 13143, normalized size of antiderivative = 62.89 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(5/2),x)

[Out]

Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(c + d*tan(e + f*x))**(5/2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Timed out

Giac [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 35.92 (sec) , antiderivative size = 14163, normalized size of antiderivative = 67.77 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

int((A + B*tan(e + f*x) + C*tan(e + f*x)^2)/(c + d*tan(e + f*x))^(5/2),x)

[Out]

(log(96*A^3*c^3*d^13*f^2 - ((((((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d
^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(c^10*f^4 + d^10*
f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(((((320*A^4*c^2*d^8*f^4 - 16*A^
4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2
*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^
8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*
d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*
f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5))/4 - 32*A*d^21*f^4 - 160*A*c^2*d^19*f^4 - 128*A*c^4*d^17*f^4 + 896*A
*c^6*d^15*f^4 + 3136*A*c^8*d^13*f^4 + 4928*A*c^10*d^11*f^4 + 4480*A*c^12*d^9*f^4 + 2432*A*c^14*d^7*f^4 + 736*A
*c^16*d^5*f^4 + 96*A*c^18*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(320*A^2*c^4*d^14*f^3 - 16*A^2*d^18*f^3 + 1
024*A^2*c^6*d^12*f^3 + 1440*A^2*c^8*d^10*f^3 + 1024*A^2*c^10*d^8*f^3 + 320*A^2*c^12*d^6*f^3 - 16*A^2*c^16*d^2*
f^3))*(((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2
*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10
*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 + 240*A^3*c^5*d^11*f^2 + 320*A^3*c^7*d^9*f^2 + 240*A^
3*c^9*d^7*f^2 + 96*A^3*c^11*d^5*f^2 + 16*A^3*c^13*d^3*f^2 + 16*A^3*c*d^15*f^2)*(((320*A^4*c^2*d^8*f^4 - 16*A^4
*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*
c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8
*d^2*f^4))^(1/2))/4 + (log(96*A^3*c^3*d^13*f^2 - ((((-((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d
^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) + 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4
*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(((-((320
*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2)
+ 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4
+ 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*
c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*
d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5))/4 - 32*A*d^21*f^4 - 160*A*c^2*d^19*f^4 - 12
8*A*c^4*d^17*f^4 + 896*A*c^6*d^15*f^4 + 3136*A*c^8*d^13*f^4 + 4928*A*c^10*d^11*f^4 + 4480*A*c^12*d^9*f^4 + 243
2*A*c^14*d^7*f^4 + 736*A*c^16*d^5*f^4 + 96*A*c^18*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(320*A^2*c^4*d^14*f
^3 - 16*A^2*d^18*f^3 + 1024*A^2*c^6*d^12*f^3 + 1440*A^2*c^8*d^10*f^3 + 1024*A^2*c^10*d^8*f^3 + 320*A^2*c^12*d^
6*f^3 - 16*A^2*c^16*d^2*f^3))*(-((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*
d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) + 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(c^10*f^4 + d^10
*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 + 240*A^3*c^5*d^11*f^2 + 320
*A^3*c^7*d^9*f^2 + 240*A^3*c^9*d^7*f^2 + 96*A^3*c^11*d^5*f^2 + 16*A^3*c^13*d^3*f^2 + 16*A^3*c*d^15*f^2)*(-((32
0*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2)
+ 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^
4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 - log(96*A^3*c^3*d^13*f^2 - ((((320*A^4*c^2*d^8*f^4 - 16*A^4*d^1
0*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*
d^2*f^2 - 20*A^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 +
80*c^8*d^2*f^4))^(1/2)*(896*A*c^6*d^15*f^4 - (((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 +
1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(1
6*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(c + d*
tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14
*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 +
64*c^21*d^2*f^5) - 160*A*c^2*d^19*f^4 - 128*A*c^4*d^17*f^4 - 32*A*d^21*f^4 + 3136*A*c^8*d^13*f^4 + 4928*A*c^10
*d^11*f^4 + 4480*A*c^12*d^9*f^4 + 2432*A*c^14*d^7*f^4 + 736*A*c^16*d^5*f^4 + 96*A*c^18*d^3*f^4) + (c + d*tan(e
+ f*x))^(1/2)*(320*A^2*c^4*d^14*f^3 - 16*A^2*d^18*f^3 + 1024*A^2*c^6*d^12*f^3 + 1440*A^2*c^8*d^10*f^3 + 1024*
A^2*c^10*d^8*f^3 + 320*A^2*c^12*d^6*f^3 - 16*A^2*c^16*d^2*f^3))*(((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 176
0*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 2
0*A^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*
f^4))^(1/2) + 240*A^3*c^5*d^11*f^2 + 320*A^3*c^7*d^9*f^2 + 240*A^3*c^9*d^7*f^2 + 96*A^3*c^11*d^5*f^2 + 16*A^3*
c^13*d^3*f^2 + 16*A^3*c*d^15*f^2)*(((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c
^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) - 4*A^2*c^5*f^2 + 40*A^2*c^3*d^2*f^2 - 20*A^2*c*d^4*f^2)/(16*c^10*f^4
+ 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) - log(96*A^3*c^3*d
^13*f^2 - ((-((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c
^8*d^2*f^4)^(1/2) + 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2
*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(896*A*c^6*d^15*f^4 - (-((320*A^4*c^2*d^
8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) + 4*A^2*c^5
*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 +
160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c
^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d
^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 160*A*c^2*d^19*f^4 - 128*A*c^4*d^17*f^4 - 3
2*A*d^21*f^4 + 3136*A*c^8*d^13*f^4 + 4928*A*c^10*d^11*f^4 + 4480*A*c^12*d^9*f^4 + 2432*A*c^14*d^7*f^4 + 736*A*
c^16*d^5*f^4 + 96*A*c^18*d^3*f^4) + (c + d*tan(e + f*x))^(1/2)*(320*A^2*c^4*d^14*f^3 - 16*A^2*d^18*f^3 + 1024*
A^2*c^6*d^12*f^3 + 1440*A^2*c^8*d^10*f^3 + 1024*A^2*c^10*d^8*f^3 + 320*A^2*c^12*d^6*f^3 - 16*A^2*c^16*d^2*f^3)
)*(-((320*A^4*c^2*d^8*f^4 - 16*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^
4)^(1/2) + 4*A^2*c^5*f^2 - 40*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4
+ 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) + 240*A^3*c^5*d^11*f^2 + 320*A^3*c^7*d^9*f^2 + 24
0*A^3*c^9*d^7*f^2 + 96*A^3*c^11*d^5*f^2 + 16*A^3*c^13*d^3*f^2 + 16*A^3*c*d^15*f^2)*(-((320*A^4*c^2*d^8*f^4 - 1
6*A^4*d^10*f^4 - 1760*A^4*c^4*d^6*f^4 + 1600*A^4*c^6*d^4*f^4 - 400*A^4*c^8*d^2*f^4)^(1/2) + 4*A^2*c^5*f^2 - 40
*A^2*c^3*d^2*f^2 + 20*A^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d
^4*f^4 + 80*c^8*d^2*f^4))^(1/2) + (log(((((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600
*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) - 4*C^2*c^5*f^2 + 40*C^2*c^3*d^2*f^2 - 20*C^2*c*d^4*f^2)/(c^10*f
^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^
(1/2)*(320*C^2*c^4*d^14*f^3 - 16*C^2*d^18*f^3 + 1024*C^2*c^6*d^12*f^3 + 1440*C^2*c^8*d^10*f^3 + 1024*C^2*c^10*
d^8*f^3 + 320*C^2*c^12*d^6*f^3 - 16*C^2*c^16*d^2*f^3) + ((((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c
^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) - 4*C^2*c^5*f^2 + 40*C^2*c^3*d^2*f^2 - 20*C^2*c
*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(896*
C*c^6*d^15*f^4 - ((((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400
*C^4*c^8*d^2*f^4)^(1/2) - 4*C^2*c^5*f^2 + 40*C^2*c^3*d^2*f^2 - 20*C^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*
d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 +
640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^
13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5))/4 - 160*C*c^2*d^19*
f^4 - 128*C*c^4*d^17*f^4 - 32*C*d^21*f^4 + 3136*C*c^8*d^13*f^4 + 4928*C*c^10*d^11*f^4 + 4480*C*c^12*d^9*f^4 +
2432*C*c^14*d^7*f^4 + 736*C*c^16*d^5*f^4 + 96*C*c^18*d^3*f^4))/4))/4 - 96*C^3*c^3*d^13*f^2 - 240*C^3*c^5*d^11*
f^2 - 320*C^3*c^7*d^9*f^2 - 240*C^3*c^9*d^7*f^2 - 96*C^3*c^11*d^5*f^2 - 16*C^3*c^13*d^3*f^2 - 16*C^3*c*d^15*f^
2)*(((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^
4)^(1/2) - 4*C^2*c^5*f^2 + 40*C^2*c^3*d^2*f^2 - 20*C^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^
4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 + (log(((-((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760
*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^2*c^3*d^2*f^2 + 20
*C^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)
*((c + d*tan(e + f*x))^(1/2)*(320*C^2*c^4*d^14*f^3 - 16*C^2*d^18*f^3 + 1024*C^2*c^6*d^12*f^3 + 1440*C^2*c^8*d^
10*f^3 + 1024*C^2*c^10*d^8*f^3 + 320*C^2*c^12*d^6*f^3 - 16*C^2*c^16*d^2*f^3) + ((-((320*C^4*c^2*d^8*f^4 - 16*C
^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^
2*c^3*d^2*f^2 + 20*C^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c
^8*d^2*f^4))^(1/2)*(896*C*c^6*d^15*f^4 - ((-((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1
600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^2*c^3*d^2*f^2 + 20*C^2*c*d^4*f^2)/(c^1
0*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x)
)^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128
*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*
f^5))/4 - 160*C*c^2*d^19*f^4 - 128*C*c^4*d^17*f^4 - 32*C*d^21*f^4 + 3136*C*c^8*d^13*f^4 + 4928*C*c^10*d^11*f^4
+ 4480*C*c^12*d^9*f^4 + 2432*C*c^14*d^7*f^4 + 736*C*c^16*d^5*f^4 + 96*C*c^18*d^3*f^4))/4))/4 - 96*C^3*c^3*d^1
3*f^2 - 240*C^3*c^5*d^11*f^2 - 320*C^3*c^7*d^9*f^2 - 240*C^3*c^9*d^7*f^2 - 96*C^3*c^11*d^5*f^2 - 16*C^3*c^13*d
^3*f^2 - 16*C^3*c*d^15*f^2)*(-((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^
4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^2*c^3*d^2*f^2 + 20*C^2*c*d^4*f^2)/(c^10*f^4 + d^10*f
^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 - log(- (((320*C^4*c^2*d^8*f^4
- 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) - 4*C^2*c^5*f^2
+ 40*C^2*c^3*d^2*f^2 - 20*C^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c
^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(320*C^2*c^4*d^14*f^3 - 16*C^2*d^18*f^3 + 1024
*C^2*c^6*d^12*f^3 + 1440*C^2*c^8*d^10*f^3 + 1024*C^2*c^10*d^8*f^3 + 320*C^2*c^12*d^6*f^3 - 16*C^2*c^16*d^2*f^3
) - (((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f
^4)^(1/2) - 4*C^2*c^5*f^2 + 40*C^2*c^3*d^2*f^2 - 20*C^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4
+ 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*((((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760
*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) - 4*C^2*c^5*f^2 + 40*C^2*c^3*d^2*f^2 - 20
*C^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f
^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^
5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 6
40*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*C*d^21*f^4 - 160*C*c^2*d^19*f^4 - 128*C*c^4*d^17*f^4 + 896*C*c^6*d^15*
f^4 + 3136*C*c^8*d^13*f^4 + 4928*C*c^10*d^11*f^4 + 4480*C*c^12*d^9*f^4 + 2432*C*c^14*d^7*f^4 + 736*C*c^16*d^5*
f^4 + 96*C*c^18*d^3*f^4)) - 96*C^3*c^3*d^13*f^2 - 240*C^3*c^5*d^11*f^2 - 320*C^3*c^7*d^9*f^2 - 240*C^3*c^9*d^7
*f^2 - 96*C^3*c^11*d^5*f^2 - 16*C^3*c^13*d^3*f^2 - 16*C^3*c*d^15*f^2)*(((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4
- 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) - 4*C^2*c^5*f^2 + 40*C^2*c^3*d^2*f
^2 - 20*C^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^
8*d^2*f^4))^(1/2) - log(- (-((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*
f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^2*c^3*d^2*f^2 + 20*C^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^
10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/
2)*(320*C^2*c^4*d^14*f^3 - 16*C^2*d^18*f^3 + 1024*C^2*c^6*d^12*f^3 + 1440*C^2*c^8*d^10*f^3 + 1024*C^2*c^10*d^8
*f^3 + 320*C^2*c^12*d^6*f^3 - 16*C^2*c^16*d^2*f^3) - (-((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*
d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^2*c^3*d^2*f^2 + 20*C^2*c*d^
4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2
)*((-((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c^8*d^2*f
^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^2*c^3*d^2*f^2 + 20*C^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4
+ 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*
c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d
^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 32*C*d^21*f^4 - 160*C*
c^2*d^19*f^4 - 128*C*c^4*d^17*f^4 + 896*C*c^6*d^15*f^4 + 3136*C*c^8*d^13*f^4 + 4928*C*c^10*d^11*f^4 + 4480*C*c
^12*d^9*f^4 + 2432*C*c^14*d^7*f^4 + 736*C*c^16*d^5*f^4 + 96*C*c^18*d^3*f^4)) - 96*C^3*c^3*d^13*f^2 - 240*C^3*c
^5*d^11*f^2 - 320*C^3*c^7*d^9*f^2 - 240*C^3*c^9*d^7*f^2 - 96*C^3*c^11*d^5*f^2 - 16*C^3*c^13*d^3*f^2 - 16*C^3*c
*d^15*f^2)*(-((320*C^4*c^2*d^8*f^4 - 16*C^4*d^10*f^4 - 1760*C^4*c^4*d^6*f^4 + 1600*C^4*c^6*d^4*f^4 - 400*C^4*c
^8*d^2*f^4)^(1/2) + 4*C^2*c^5*f^2 - 40*C^2*c^3*d^2*f^2 + 20*C^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2
*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) + (log(8*B^3*d^16*f^2 - ((((320*B^4*c^2*
d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) + 4*B^2*c
^5*f^2 - 40*B^2*c^3*d^2*f^2 + 20*B^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6
*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(320*B^2*c^4*d^14*f^3 - 16*B^2*d^18*f^3 + 1024*B^
2*c^6*d^12*f^3 + 1440*B^2*c^8*d^10*f^3 + 1024*B^2*c^10*d^8*f^3 + 320*B^2*c^12*d^6*f^3 - 16*B^2*c^16*d^2*f^3) +
((((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4
)^(1/2) + 4*B^2*c^5*f^2 - 40*B^2*c^3*d^2*f^2 + 20*B^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4
*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(((((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^
6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) + 4*B^2*c^5*f^2 - 40*B^2*c^3*d^2*f^2 + 20*B^2*c*d^4*
f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(c + d*tan
(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^
5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*
c^21*d^2*f^5))/4 + 96*B*c*d^20*f^4 + 736*B*c^3*d^18*f^4 + 2432*B*c^5*d^16*f^4 + 4480*B*c^7*d^14*f^4 + 4928*B*c
^9*d^12*f^4 + 3136*B*c^11*d^10*f^4 + 896*B*c^13*d^8*f^4 - 128*B*c^15*d^6*f^4 - 160*B*c^17*d^4*f^4 - 32*B*c^19*
d^2*f^4))/4))/4 + 40*B^3*c^2*d^14*f^2 + 72*B^3*c^4*d^12*f^2 + 40*B^3*c^6*d^10*f^2 - 40*B^3*c^8*d^8*f^2 - 72*B^
3*c^10*d^6*f^2 - 40*B^3*c^12*d^4*f^2 - 8*B^3*c^14*d^2*f^2)*(((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4
*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) + 4*B^2*c^5*f^2 - 40*B^2*c^3*d^2*f^2 + 20*B^2
*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4
+ (log(8*B^3*d^16*f^2 - ((-((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f
^4 - 400*B^4*c^8*d^2*f^4)^(1/2) - 4*B^2*c^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4
+ 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(320*B^2
*c^4*d^14*f^3 - 16*B^2*d^18*f^3 + 1024*B^2*c^6*d^12*f^3 + 1440*B^2*c^8*d^10*f^3 + 1024*B^2*c^10*d^8*f^3 + 320*
B^2*c^12*d^6*f^3 - 16*B^2*c^16*d^2*f^3) + ((-((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 +
1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) - 4*B^2*c^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(c^
10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(((-((320*B^4*c^2*
d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) - 4*B^2*c
^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6
*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*
f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 +
2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5))/4 + 96*B*c*d^20*f^4 + 736*B*c^3*d^18*f^4 + 2432*B*c^
5*d^16*f^4 + 4480*B*c^7*d^14*f^4 + 4928*B*c^9*d^12*f^4 + 3136*B*c^11*d^10*f^4 + 896*B*c^13*d^8*f^4 - 128*B*c^1
5*d^6*f^4 - 160*B*c^17*d^4*f^4 - 32*B*c^19*d^2*f^4))/4))/4 + 40*B^3*c^2*d^14*f^2 + 72*B^3*c^4*d^12*f^2 + 40*B^
3*c^6*d^10*f^2 - 40*B^3*c^8*d^8*f^2 - 72*B^3*c^10*d^6*f^2 - 40*B^3*c^12*d^4*f^2 - 8*B^3*c^14*d^2*f^2)*(-((320*
B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) -
4*B^2*c^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4
+ 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 - log((((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*
f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) + 4*B^2*c^5*f^2 - 40*B^2*c^3*d^2*f^2 + 20*B^2*c*d^4*f^
2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*((
c + d*tan(e + f*x))^(1/2)*(320*B^2*c^4*d^14*f^3 - 16*B^2*d^18*f^3 + 1024*B^2*c^6*d^12*f^3 + 1440*B^2*c^8*d^10*
f^3 + 1024*B^2*c^10*d^8*f^3 + 320*B^2*c^12*d^6*f^3 - 16*B^2*c^16*d^2*f^3) - (((320*B^4*c^2*d^8*f^4 - 16*B^4*d^
10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) + 4*B^2*c^5*f^2 - 40*B^2*c^3
*d^2*f^2 + 20*B^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 +
80*c^8*d^2*f^4))^(1/2)*(96*B*c*d^20*f^4 - (((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1
600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) + 4*B^2*c^5*f^2 - 40*B^2*c^3*d^2*f^2 + 20*B^2*c*d^4*f^2)/(16*
c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(c + d*ta
n(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f
^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64
*c^21*d^2*f^5) + 736*B*c^3*d^18*f^4 + 2432*B*c^5*d^16*f^4 + 4480*B*c^7*d^14*f^4 + 4928*B*c^9*d^12*f^4 + 3136*B
*c^11*d^10*f^4 + 896*B*c^13*d^8*f^4 - 128*B*c^15*d^6*f^4 - 160*B*c^17*d^4*f^4 - 32*B*c^19*d^2*f^4)) + 8*B^3*d^
16*f^2 + 40*B^3*c^2*d^14*f^2 + 72*B^3*c^4*d^12*f^2 + 40*B^3*c^6*d^10*f^2 - 40*B^3*c^8*d^8*f^2 - 72*B^3*c^10*d^
6*f^2 - 40*B^3*c^12*d^4*f^2 - 8*B^3*c^14*d^2*f^2)*(((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*
f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) + 4*B^2*c^5*f^2 - 40*B^2*c^3*d^2*f^2 + 20*B^2*c*d^4*f^
2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) -
log((-((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*
f^4)^(1/2) - 4*B^2*c^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^
4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(320*B^2*c^4*d^14*f
^3 - 16*B^2*d^18*f^3 + 1024*B^2*c^6*d^12*f^3 + 1440*B^2*c^8*d^10*f^3 + 1024*B^2*c^10*d^8*f^3 + 320*B^2*c^12*d^
6*f^3 - 16*B^2*c^16*d^2*f^3) - (-((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6
*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^(1/2) - 4*B^2*c^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(16*c^10*f^4 +
16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(96*B*c*d^20*f^4 - (
-((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d^2*f^4)^
(1/2) - 4*B^2*c^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 1
60*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*
d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*
f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) + 736*B*c^3*d^18*f^4 + 2432*
B*c^5*d^16*f^4 + 4480*B*c^7*d^14*f^4 + 4928*B*c^9*d^12*f^4 + 3136*B*c^11*d^10*f^4 + 896*B*c^13*d^8*f^4 - 128*B
*c^15*d^6*f^4 - 160*B*c^17*d^4*f^4 - 32*B*c^19*d^2*f^4)) + 8*B^3*d^16*f^2 + 40*B^3*c^2*d^14*f^2 + 72*B^3*c^4*d
^12*f^2 + 40*B^3*c^6*d^10*f^2 - 40*B^3*c^8*d^8*f^2 - 72*B^3*c^10*d^6*f^2 - 40*B^3*c^12*d^4*f^2 - 8*B^3*c^14*d^
2*f^2)*(-((320*B^4*c^2*d^8*f^4 - 16*B^4*d^10*f^4 - 1760*B^4*c^4*d^6*f^4 + 1600*B^4*c^6*d^4*f^4 - 400*B^4*c^8*d
^2*f^4)^(1/2) - 4*B^2*c^5*f^2 + 40*B^2*c^3*d^2*f^2 - 20*B^2*c*d^4*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8
*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2) + ((2*B*c)/(3*(c^2 + d^2)) + (2*B*(c^2 - d^2
)*(c + d*tan(e + f*x)))/(c^2 + d^2)^2)/(f*(c + d*tan(e + f*x))^(3/2)) - ((2*A*d)/(3*(c^2 + d^2)) + (4*A*c*d*(c
+ d*tan(e + f*x)))/(c^2 + d^2)^2)/(f*(c + d*tan(e + f*x))^(3/2)) - ((2*C*c^2)/(3*(c^2 + d^2)) - (4*C*c*d^2*(c
+ d*tan(e + f*x)))/(c^2 + d^2)^2)/(d*f*(c + d*tan(e + f*x))^(3/2))