3.1254 \(\int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=216 \[ -\frac {2 b \left (a d (2 b c-a d)-b^2 \left (2 c^2+d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{d^2 f \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {(-b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}}+\frac {(b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.50, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3565, 3630, 3539, 3537, 63, 208} \[ -\frac {2 b \left (a d (2 b c-a d)-b^2 \left (2 c^2+d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{d^2 f \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {(-b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}}+\frac {(b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

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

Rule 208

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

Rule 3537

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

Rule 3539

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

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3630

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

Rubi steps

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

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Mathematica [C]  time = 1.86, size = 287, normalized size = 1.33 \[ \frac {-i b \left (3 a^2-b^2\right ) \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )+\frac {\left (a^3 (-d)+3 a^2 b c+3 a b^2 d-b^3 c\right ) \left ((d-i c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )+(d+i c) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )}{\left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {4 b^2 (b c-2 a d)}{d \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 (a+b \tan (e+f x))}{\sqrt {c+d \tan (e+f x)}}}{d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*b*(3*a^2 - b^2)*(ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/Sqrt[c - I*d] - ArcTanh[Sqrt[c + d*Tan[
e + f*x]]/Sqrt[c + I*d]]/Sqrt[c + I*d]) + (4*b^2*(b*c - 2*a*d))/(d*Sqrt[c + d*Tan[e + f*x]]) + ((3*a^2*b*c - b
^3*c - a^3*d + 3*a*b^2*d)*(((-I)*c + d)*Hypergeometric2F1[-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^2 + d^2)*Sqrt[c + d*Tan[e + f*x]])
+ (2*b^2*(a + b*Tan[e + f*x]))/Sqrt[c + d*Tan[e + f*x]])/(d*f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.31, size = 16343, normalized size = 75.66 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

Timed out

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mupad [B]  time = 13.18, size = 20864, normalized size = 96.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^3*(c + d*tan(e + f*x))^(1/2))/(d^2*f) - atan((((-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48
*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3
*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4
*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 -
24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3
*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b
^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-((
(8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 1
20*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 3
60*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*
d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2
) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2
+ 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180
*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3
*c^4*d^2*f^4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5
+ 64*c^11*d^2*f^5) - 32*b^3*d^12*f^4 + 96*a^2*b*d^12*f^4 + 64*a^3*c*d^11*f^4 + 256*a^3*c^3*d^9*f^4 + 384*a^3*c
^5*d^7*f^4 + 256*a^3*c^7*d^5*f^4 + 64*a^3*c^9*d^3*f^4 - 96*b^3*c^2*d^10*f^4 - 64*b^3*c^4*d^8*f^4 + 64*b^3*c^6*
d^6*f^4 + 96*b^3*c^8*d^4*f^4 + 32*b^3*c^10*d^2*f^4 - 192*a*b^2*c*d^11*f^4 - 768*a*b^2*c^3*d^9*f^4 - 1152*a*b^2
*c^5*d^7*f^4 - 768*a*b^2*c^7*d^5*f^4 - 192*a*b^2*c^9*d^3*f^4 + 288*a^2*b*c^2*d^10*f^4 + 192*a^2*b*c^4*d^8*f^4
- 192*a^2*b*c^6*d^6*f^4 - 288*a^2*b*c^8*d^4*f^4 - 96*a^2*b*c^10*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*b^6*
d^10*f^3 - 16*a^6*d^10*f^3 - 240*a^2*b^4*d^10*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4
*f^3 + 16*a^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a^2*b^4*c^2*d^8
*f^3 + 480*a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b^3*c^5*d^5*f^3
- 640*a^3*b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8*d^2*f^3 + 192
*a*b^5*c*d^9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*d^5*f^3 + 192*a*b^5*c^7*d^3*f^3
- 640*a^3*b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3))*(-(((8*a^6*
c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*
b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*
b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4
+ 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a
^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^
2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^
4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^
2*f^4)))^(1/2)*1i + ((-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f
^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2
+ 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f
^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^
8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^
2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 +
72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^
6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(32*b^3*d^12*f^4 + (c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^
2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^
3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*
d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*
c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3
*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*
c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^
2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)
))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2
*f^5) - 96*a^2*b*d^12*f^4 - 64*a^3*c*d^11*f^4 - 256*a^3*c^3*d^9*f^4 - 384*a^3*c^5*d^7*f^4 - 256*a^3*c^7*d^5*f^
4 - 64*a^3*c^9*d^3*f^4 + 96*b^3*c^2*d^10*f^4 + 64*b^3*c^4*d^8*f^4 - 64*b^3*c^6*d^6*f^4 - 96*b^3*c^8*d^4*f^4 -
32*b^3*c^10*d^2*f^4 + 192*a*b^2*c*d^11*f^4 + 768*a*b^2*c^3*d^9*f^4 + 1152*a*b^2*c^5*d^7*f^4 + 768*a*b^2*c^7*d^
5*f^4 + 192*a*b^2*c^9*d^3*f^4 - 288*a^2*b*c^2*d^10*f^4 - 192*a^2*b*c^4*d^8*f^4 + 192*a^2*b*c^6*d^6*f^4 + 288*a
^2*b*c^8*d^4*f^4 + 96*a^2*b*c^10*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*b^6*d^10*f^3 - 16*a^6*d^10*f^3 - 24
0*a^2*b^4*d^10*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4*f^3 + 16*a^6*c^8*d^2*f^3 + 32*
b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a^2*b^4*c^2*d^8*f^3 + 480*a^2*b^4*c^6*d^4*f^3
+ 240*a^2*b^4*c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b^3*c^5*d^5*f^3 - 640*a^3*b^3*c^7*d^3*f^3 + 48
0*a^4*b^2*c^2*d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8*d^2*f^3 + 192*a*b^5*c*d^9*f^3 + 192*a^5*b*c*
d^9*f^3 + 576*a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*d^5*f^3 + 192*a*b^5*c^7*d^3*f^3 - 640*a^3*b^3*c*d^9*f^3 + 576*
a^5*b*c^3*d^7*f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*
b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f
^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2
*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12
+ 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24
*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*
f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d
*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*1i)/(((-(((8*a^6
*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2
*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2
*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^
4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*
a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a
^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b
^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d
^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d
^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2
+ 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d
^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b
^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*
b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2
+ 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2
*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*
c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 32*b^3*d^12*f^4 + 96*a^2*b*d^12*f^4 + 64*
a^3*c*d^11*f^4 + 256*a^3*c^3*d^9*f^4 + 384*a^3*c^5*d^7*f^4 + 256*a^3*c^7*d^5*f^4 + 64*a^3*c^9*d^3*f^4 - 96*b^3
*c^2*d^10*f^4 - 64*b^3*c^4*d^8*f^4 + 64*b^3*c^6*d^6*f^4 + 96*b^3*c^8*d^4*f^4 + 32*b^3*c^10*d^2*f^4 - 192*a*b^2
*c*d^11*f^4 - 768*a*b^2*c^3*d^9*f^4 - 1152*a*b^2*c^5*d^7*f^4 - 768*a*b^2*c^7*d^5*f^4 - 192*a*b^2*c^9*d^3*f^4 +
288*a^2*b*c^2*d^10*f^4 + 192*a^2*b*c^4*d^8*f^4 - 192*a^2*b*c^6*d^6*f^4 - 288*a^2*b*c^8*d^4*f^4 - 96*a^2*b*c^1
0*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*b^6*d^10*f^3 - 16*a^6*d^10*f^3 - 240*a^2*b^4*d^10*f^3 + 240*a^4*b^
2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4*f^3 + 16*a^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^
4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a^2*b^4*c^2*d^8*f^3 + 480*a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*c^8*d^2*f^3 - 192
0*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b^3*c^5*d^5*f^3 - 640*a^3*b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*d^8*f^3 - 480*a^4
*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8*d^2*f^3 + 192*a*b^5*c*d^9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*a*b^5*c^3*d^7*f^3
+ 576*a*b^5*c^5*d^5*f^3 + 192*a*b^5*c^7*d^3*f^3 - 640*a^3*b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*f^3 + 576*a^5*b*c
^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2
- 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*
a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)
^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20
*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f
^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a
*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(
16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2) - ((-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^
5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2
+ 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d
*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 +
6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a
*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^
2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f
^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(32*b^3*d^12*f^4 +
(c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*
c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2
*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (1
6*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6
+ 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a
^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*
d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f
^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 64
0*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 96*a^2*b*d^12*f^4 - 64*a^3*c*d^11*f^4 - 256*a^3*c^3*d^9*f
^4 - 384*a^3*c^5*d^7*f^4 - 256*a^3*c^7*d^5*f^4 - 64*a^3*c^9*d^3*f^4 + 96*b^3*c^2*d^10*f^4 + 64*b^3*c^4*d^8*f^4
- 64*b^3*c^6*d^6*f^4 - 96*b^3*c^8*d^4*f^4 - 32*b^3*c^10*d^2*f^4 + 192*a*b^2*c*d^11*f^4 + 768*a*b^2*c^3*d^9*f^
4 + 1152*a*b^2*c^5*d^7*f^4 + 768*a*b^2*c^7*d^5*f^4 + 192*a*b^2*c^9*d^3*f^4 - 288*a^2*b*c^2*d^10*f^4 - 192*a^2*
b*c^4*d^8*f^4 + 192*a^2*b*c^6*d^6*f^4 + 288*a^2*b*c^8*d^4*f^4 + 96*a^2*b*c^10*d^2*f^4) + (c + d*tan(e + f*x))^
(1/2)*(16*b^6*d^10*f^3 - 16*a^6*d^10*f^3 - 240*a^2*b^4*d^10*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 +
32*a^6*c^6*d^4*f^3 + 16*a^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a
^2*b^4*c^2*d^8*f^3 + 480*a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b
^3*c^5*d^5*f^3 - 640*a^3*b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8
*d^2*f^3 + 192*a*b^5*c*d^9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*d^5*f^3 + 192*a*b
^5*c^7*d^3*f^3 - 640*a^3*b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3
))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*
f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*
f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 +
48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2
))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d
^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^
2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*
f^4 + 3*c^4*d^2*f^4)))^(1/2) - 16*a^9*d^9*f^2 + 48*a*b^8*d^9*f^2 - 16*b^9*c*d^8*f^2 + 128*a^3*b^6*d^9*f^2 + 96
*a^5*b^4*d^9*f^2 - 48*a^9*c^2*d^7*f^2 - 48*a^9*c^4*d^5*f^2 - 16*a^9*c^6*d^3*f^2 - 48*b^9*c^3*d^6*f^2 - 48*b^9*
c^5*d^4*f^2 - 16*b^9*c^7*d^2*f^2 + 384*a^3*b^6*c^2*d^7*f^2 + 384*a^3*b^6*c^4*d^5*f^2 + 128*a^3*b^6*c^6*d^3*f^2
+ 288*a^4*b^5*c^3*d^6*f^2 + 288*a^4*b^5*c^5*d^4*f^2 + 96*a^4*b^5*c^7*d^2*f^2 + 288*a^5*b^4*c^2*d^7*f^2 + 288*
a^5*b^4*c^4*d^5*f^2 + 96*a^5*b^4*c^6*d^3*f^2 + 384*a^6*b^3*c^3*d^6*f^2 + 384*a^6*b^3*c^5*d^4*f^2 + 128*a^6*b^3
*c^7*d^2*f^2 + 48*a^8*b*c*d^8*f^2 + 144*a*b^8*c^2*d^7*f^2 + 144*a*b^8*c^4*d^5*f^2 + 48*a*b^8*c^6*d^3*f^2 + 96*
a^4*b^5*c*d^8*f^2 + 128*a^6*b^3*c*d^8*f^2 + 144*a^8*b*c^3*d^6*f^2 + 144*a^8*b*c^5*d^4*f^2 + 48*a^8*b*c^7*d^2*f
^2))*(-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^
2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*
d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4
+ 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b
^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c
*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*
f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^
4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*2i - atan(((((((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3
*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 +
144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2
*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8
+ 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*
d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 -
72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f
^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*((((8*a^6*c^3*
f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*
c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*
c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 4
8*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c
^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^
4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*
d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^
4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d
^2*f^5) - 32*b^3*d^12*f^4 + 96*a^2*b*d^12*f^4 + 64*a^3*c*d^11*f^4 + 256*a^3*c^3*d^9*f^4 + 384*a^3*c^5*d^7*f^4
+ 256*a^3*c^7*d^5*f^4 + 64*a^3*c^9*d^3*f^4 - 96*b^3*c^2*d^10*f^4 - 64*b^3*c^4*d^8*f^4 + 64*b^3*c^6*d^6*f^4 + 9
6*b^3*c^8*d^4*f^4 + 32*b^3*c^10*d^2*f^4 - 192*a*b^2*c*d^11*f^4 - 768*a*b^2*c^3*d^9*f^4 - 1152*a*b^2*c^5*d^7*f^
4 - 768*a*b^2*c^7*d^5*f^4 - 192*a*b^2*c^9*d^3*f^4 + 288*a^2*b*c^2*d^10*f^4 + 192*a^2*b*c^4*d^8*f^4 - 192*a^2*b
*c^6*d^6*f^4 - 288*a^2*b*c^8*d^4*f^4 - 96*a^2*b*c^10*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*b^6*d^10*f^3 -
16*a^6*d^10*f^3 - 240*a^2*b^4*d^10*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4*f^3 + 16*a
^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a^2*b^4*c^2*d^8*f^3 + 480*
a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b^3*c^5*d^5*f^3 - 640*a^3*
b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8*d^2*f^3 + 192*a*b^5*c*d^
9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*d^5*f^3 + 192*a*b^5*c^7*d^3*f^3 - 640*a^3*
b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3))*((((8*a^6*c^3*f^2 - 8*
b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2
- 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^
2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^
2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 +
4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^
2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2
+ 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/
2)*1i + (((((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c
*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c
^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f
^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^1
0*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^
6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2
*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2
*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(32*b^3*d^12*f^4 + (c + d*tan(e + f*x))^(1/2)*((((8*a^6*c^3*f^2 - 8*b^6*c^3*
f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^
4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*
a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(
a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c
^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a
^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^
3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(64*c
*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 96*a^2
*b*d^12*f^4 - 64*a^3*c*d^11*f^4 - 256*a^3*c^3*d^9*f^4 - 384*a^3*c^5*d^7*f^4 - 256*a^3*c^7*d^5*f^4 - 64*a^3*c^9
*d^3*f^4 + 96*b^3*c^2*d^10*f^4 + 64*b^3*c^4*d^8*f^4 - 64*b^3*c^6*d^6*f^4 - 96*b^3*c^8*d^4*f^4 - 32*b^3*c^10*d^
2*f^4 + 192*a*b^2*c*d^11*f^4 + 768*a*b^2*c^3*d^9*f^4 + 1152*a*b^2*c^5*d^7*f^4 + 768*a*b^2*c^7*d^5*f^4 + 192*a*
b^2*c^9*d^3*f^4 - 288*a^2*b*c^2*d^10*f^4 - 192*a^2*b*c^4*d^8*f^4 + 192*a^2*b*c^6*d^6*f^4 + 288*a^2*b*c^8*d^4*f
^4 + 96*a^2*b*c^10*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*b^6*d^10*f^3 - 16*a^6*d^10*f^3 - 240*a^2*b^4*d^10
*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4*f^3 + 16*a^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^
3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a^2*b^4*c^2*d^8*f^3 + 480*a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*
c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b^3*c^5*d^5*f^3 - 640*a^3*b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*
d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8*d^2*f^3 + 192*a*b^5*c*d^9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*
a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*d^5*f^3 + 192*a*b^5*c^7*d^3*f^3 - 640*a^3*b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*
f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3))*((((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 4
8*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^
3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^
4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2
+ 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^
3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*
b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*1i)/((((((8*a^6*c^3*f^2 - 8*b^6
*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 1
20*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 -
480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f
^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*
b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 +
60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 2
40*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*
((c + d*tan(e + f*x))^(1/2)*((((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c
*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*
d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16
*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 +
15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^
6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d
*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^
4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640
*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 32*b^3*d^12*f^4 + 96*a^2*b*d^12*f^4 + 64*a^3*c*d^11*f^4 +
256*a^3*c^3*d^9*f^4 + 384*a^3*c^5*d^7*f^4 + 256*a^3*c^7*d^5*f^4 + 64*a^3*c^9*d^3*f^4 - 96*b^3*c^2*d^10*f^4 - 6
4*b^3*c^4*d^8*f^4 + 64*b^3*c^6*d^6*f^4 + 96*b^3*c^8*d^4*f^4 + 32*b^3*c^10*d^2*f^4 - 192*a*b^2*c*d^11*f^4 - 768
*a*b^2*c^3*d^9*f^4 - 1152*a*b^2*c^5*d^7*f^4 - 768*a*b^2*c^7*d^5*f^4 - 192*a*b^2*c^9*d^3*f^4 + 288*a^2*b*c^2*d^
10*f^4 + 192*a^2*b*c^4*d^8*f^4 - 192*a^2*b*c^6*d^6*f^4 - 288*a^2*b*c^8*d^4*f^4 - 96*a^2*b*c^10*d^2*f^4) + (c +
d*tan(e + f*x))^(1/2)*(16*b^6*d^10*f^3 - 16*a^6*d^10*f^3 - 240*a^2*b^4*d^10*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a
^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4*f^3 + 16*a^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^
8*d^2*f^3 - 480*a^2*b^4*c^2*d^8*f^3 + 480*a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7
*f^3 - 1920*a^3*b^3*c^5*d^5*f^3 - 640*a^3*b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3
- 240*a^4*b^2*c^8*d^2*f^3 + 192*a*b^5*c*d^9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*
d^5*f^3 + 192*a*b^5*c^7*d^3*f^3 - 640*a^3*b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*
a^5*b*c^7*d^3*f^3))*((((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2
+ 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 +
144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4
+ 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*
b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*
f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 7
2*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*
f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2) - (((((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*
b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*
f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*
c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^
4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a
^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*
f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*
d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(32*b^3*d^12*f^4 + (c + d*tan(e + f*x
))^(1/2)*((((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c
*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c
^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f
^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^1
0*b^2))^(1/2) - 4*a^6*c^3*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^
6*c*d^2*f^2 - 60*a^2*b^4*c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2
*d*f^2 + 180*a^2*b^4*c*d^2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2
*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*
c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 96*a^2*b*d^12*f^4 - 64*a^3*c*d^11*f^4 - 256*a^3*c^3*d^9*f^4 - 384*a^3*c^5*d^7
*f^4 - 256*a^3*c^7*d^5*f^4 - 64*a^3*c^9*d^3*f^4 + 96*b^3*c^2*d^10*f^4 + 64*b^3*c^4*d^8*f^4 - 64*b^3*c^6*d^6*f^
4 - 96*b^3*c^8*d^4*f^4 - 32*b^3*c^10*d^2*f^4 + 192*a*b^2*c*d^11*f^4 + 768*a*b^2*c^3*d^9*f^4 + 1152*a*b^2*c^5*d
^7*f^4 + 768*a*b^2*c^7*d^5*f^4 + 192*a*b^2*c^9*d^3*f^4 - 288*a^2*b*c^2*d^10*f^4 - 192*a^2*b*c^4*d^8*f^4 + 192*
a^2*b*c^6*d^6*f^4 + 288*a^2*b*c^8*d^4*f^4 + 96*a^2*b*c^10*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*b^6*d^10*f
^3 - 16*a^6*d^10*f^3 - 240*a^2*b^4*d^10*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4*f^3 +
16*a^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a^2*b^4*c^2*d^8*f^3 +
480*a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b^3*c^5*d^5*f^3 - 640
*a^3*b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8*d^2*f^3 + 192*a*b^5
*c*d^9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*d^5*f^3 + 192*a*b^5*c^7*d^3*f^3 - 640
*a^3*b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3))*((((8*a^6*c^3*f^2
- 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3
*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d
^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c
^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3*
f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*c
^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^2
*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))
)^(1/2) - 16*a^9*d^9*f^2 + 48*a*b^8*d^9*f^2 - 16*b^9*c*d^8*f^2 + 128*a^3*b^6*d^9*f^2 + 96*a^5*b^4*d^9*f^2 - 48
*a^9*c^2*d^7*f^2 - 48*a^9*c^4*d^5*f^2 - 16*a^9*c^6*d^3*f^2 - 48*b^9*c^3*d^6*f^2 - 48*b^9*c^5*d^4*f^2 - 16*b^9*
c^7*d^2*f^2 + 384*a^3*b^6*c^2*d^7*f^2 + 384*a^3*b^6*c^4*d^5*f^2 + 128*a^3*b^6*c^6*d^3*f^2 + 288*a^4*b^5*c^3*d^
6*f^2 + 288*a^4*b^5*c^5*d^4*f^2 + 96*a^4*b^5*c^7*d^2*f^2 + 288*a^5*b^4*c^2*d^7*f^2 + 288*a^5*b^4*c^4*d^5*f^2 +
96*a^5*b^4*c^6*d^3*f^2 + 384*a^6*b^3*c^3*d^6*f^2 + 384*a^6*b^3*c^5*d^4*f^2 + 128*a^6*b^3*c^7*d^2*f^2 + 48*a^8
*b*c*d^8*f^2 + 144*a*b^8*c^2*d^7*f^2 + 144*a*b^8*c^4*d^5*f^2 + 48*a*b^8*c^6*d^3*f^2 + 96*a^4*b^5*c*d^8*f^2 + 1
28*a^6*b^3*c*d^8*f^2 + 144*a^8*b*c^3*d^6*f^2 + 144*a^8*b*c^5*d^4*f^2 + 48*a^8*b*c^7*d^2*f^2))*((((8*a^6*c^3*f^
2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^
3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*
d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*
c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) - 4*a^6*c^3
*f^2 + 4*b^6*c^3*f^2 + 24*a*b^5*d^3*f^2 + 24*a^5*b*d^3*f^2 + 12*a^6*c*d^2*f^2 - 12*b^6*c*d^2*f^2 - 60*a^2*b^4*
c^3*f^2 + 60*a^4*b^2*c^3*f^2 - 80*a^3*b^3*d^3*f^2 - 72*a*b^5*c^2*d*f^2 - 72*a^5*b*c^2*d*f^2 + 180*a^2*b^4*c*d^
2*f^2 + 240*a^3*b^3*c^2*d*f^2 - 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)
))^(1/2)*2i - (2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(d^2*f*(c^2 + d^2)*(c + d*tan(e + f*x))^
(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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