3.13.36 \(\int (a+b \tan (e+f x))^2 (c+d \tan (e+f x))^{3/2} \, dx\) [1236]

Optimal. Leaf size=195 \[ -\frac {i (a-i b)^2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 \left (2 a b c+a^2 d-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \]

[Out]

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

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Rubi [A]
time = 0.31, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3624, 3609, 3620, 3618, 65, 214} \begin {gather*} \frac {2 \left (a^2 d+2 a b c-b^2 d\right ) \sqrt {c+d \tan (e+f x)}}{f}+\frac {4 a b (c+d \tan (e+f x))^{3/2}}{3 f}-\frac {i (a-i b)^2 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f}+\frac {i (a+i b)^2 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f}+\frac {2 b^2 (c+d \tan (e+f x))^{5/2}}{5 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(a - I*b)^2*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/f + (I*(a + I*b)^2*(c + I*d
)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/f + (2*(2*a*b*c + a^2*d - b^2*d)*Sqrt[c + d*Tan[e + f
*x]])/f + (4*a*b*(c + d*Tan[e + f*x])^(3/2))/(3*f) + (2*b^2*(c + d*Tan[e + f*x])^(5/2))/(5*d*f)

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 3609

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

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 3624

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

Rubi steps

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

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Mathematica [A]
time = 1.71, size = 202, normalized size = 1.04 \begin {gather*} \frac {\frac {6 b^2 (c+d \tan (e+f x))^{5/2}}{d}+5 i (a-i b)^2 \left (-3 (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c-3 i d+d \tan (e+f x))\right )-5 i (a+i b)^2 \left (-3 (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+\sqrt {c+d \tan (e+f x)} (4 c+3 i d+d \tan (e+f x))\right )}{15 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((6*b^2*(c + d*Tan[e + f*x])^(5/2))/d + (5*I)*(a - I*b)^2*(-3*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]
/Sqrt[c - I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c - (3*I)*d + d*Tan[e + f*x])) - (5*I)*(a + I*b)^2*(-3*(c + I*d)
^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + Sqrt[c + d*Tan[e + f*x]]*(4*c + (3*I)*d + d*Tan[e + f
*x])))/(15*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1365\) vs. \(2(165)=330\).
time = 0.45, size = 1366, normalized size = 7.01 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 21.86, size = 2500, normalized size = 12.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b^2*(c + d*tan(e + f*x))^(5/2))/(5*d*f) - atan(((((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f^2 -
2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*a*b*c^3*d^2*f^2))/f^3 - 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((8*a^4
*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*
b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 +
b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2
*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2
+ 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b
^4*c^3*f^2 - 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*
b^3*c^2*d*f^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3*f^2
- 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*c^
2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*a^
2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*
a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 + 1
8*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3*d^3*
f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a^3*b*
c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*a^2*b
^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3 + 36*
a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3*f^
2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b^3*
c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 + 4*
a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4 +
3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^4 +
18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3*d^
3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a^3*
b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2)*1i - (((16*(2*a^2*d^5*f^2 - 2*b^2*d^5*f^2 + 2*a^2*c^2*d^3*f
^2 - 2*b^2*c^2*d^3*f^2 + 4*a*b*c*d^4*f^2 + 4*a*b*c^3*d^2*f^2))/f^3 + 64*c*d^2*(c + d*tan(e + f*x))^(1/2)*(-(((
8*a^4*c^3*f^2 + 8*b^4*c^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48
*a^2*b^2*c^3*f^2 + 96*a*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*
d^6 + b^8*c^6 + b^8*d^6 + 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^
6*b^2*d^6 + 3*a^8*c^2*d^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^
4*d^2 + 18*a^4*b^4*c^2*d^4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^
2 + b^4*c^3*f^2 - 4*a*b^3*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 +
12*a*b^3*c^2*d*f^2 - 12*a^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2))*(-(((8*a^4*c^3*f^2 + 8*b^4*c^3
*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a*b
^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6 +
4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d^4
+ 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*d^2 + 12*a^2*b^6*c^2*d^4 + 12*a^2*b^6*c^4*d^2 + 18*a^4*b^4*c^2*d^
4 + 18*a^4*b^4*c^4*d^2 + 12*a^6*b^2*c^2*d^4 + 12*a^6*b^2*c^4*d^2))^(1/2) + a^4*c^3*f^2 + b^4*c^3*f^2 - 4*a*b^3
*d^3*f^2 + 4*a^3*b*d^3*f^2 - 3*a^4*c*d^2*f^2 - 3*b^4*c*d^2*f^2 - 6*a^2*b^2*c^3*f^2 + 12*a*b^3*c^2*d*f^2 - 12*a
^3*b*c^2*d*f^2 + 18*a^2*b^2*c*d^2*f^2)/(4*f^4))^(1/2) + (16*(c + d*tan(e + f*x))^(1/2)*(a^4*d^6 + b^4*d^6 - 6*
a^2*b^2*d^6 - 6*a^4*c^2*d^4 + a^4*c^4*d^2 - 6*b^4*c^2*d^4 + b^4*c^4*d^2 + 16*a*b^3*c^3*d^3 - 16*a^3*b*c^3*d^3
+ 36*a^2*b^2*c^2*d^4 - 6*a^2*b^2*c^4*d^2 - 16*a*b^3*c*d^5 + 16*a^3*b*c*d^5))/f^2)*(-(((8*a^4*c^3*f^2 + 8*b^4*c
^3*f^2 - 32*a*b^3*d^3*f^2 + 32*a^3*b*d^3*f^2 - 24*a^4*c*d^2*f^2 - 24*b^4*c*d^2*f^2 - 48*a^2*b^2*c^3*f^2 + 96*a
*b^3*c^2*d*f^2 - 96*a^3*b*c^2*d*f^2 + 144*a^2*b^2*c*d^2*f^2)^2/64 - f^4*(a^8*c^6 + a^8*d^6 + b^8*c^6 + b^8*d^6
+ 4*a^2*b^6*c^6 + 6*a^4*b^4*c^6 + 4*a^6*b^2*c^6 + 4*a^2*b^6*d^6 + 6*a^4*b^4*d^6 + 4*a^6*b^2*d^6 + 3*a^8*c^2*d
^4 + 3*a^8*c^4*d^2 + 3*b^8*c^2*d^4 + 3*b^8*c^4*...

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