3.13.61 \(\int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{5/2}} \, dx\) [1261]

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

[Out]

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

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Rubi [A]
time = 0.34, antiderivative size = 195, 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 = {3623, 3610, 3620, 3618, 65, 214} \begin {gather*} \frac {4 (b c-a d) (a c+b d)}{f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {2 (b c-a d)^2}{3 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {i (a-i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}+\frac {i (a+i b)^2 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(a - I*b)^2*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f) + (I*(a + I*b)^2*ArcTan
h[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) - (2*(b*c - a*d)^2)/(3*d*(c^2 + d^2)*f*(c + d*T
an[e + f*x])^(3/2)) + (4*(b*c - a*d)*(a*c + b*d))/((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 3623

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.22, size = 127, normalized size = 0.65 \begin {gather*} \frac {-\frac {2 b^2}{d}-\frac {(a-i b)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{i c+d}+\frac {(a+i b)^2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{i c-d}}{3 f (c+d \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*b^2)/d - ((a - I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(I*c + d) + ((a +
I*b)^2*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(I*c - d))/(3*f*(c + d*Tan[e + f*x])
^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(5087\) vs. \(2(169)=338\).
time = 0.51, size = 5088, normalized size = 26.09

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(5088\)
default	\(\text {Expression too large to display}\)	\(5088\)

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))^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [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))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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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))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Integral((a + b*tan(e + f*x))**2/(c + d*tan(e + f*x))**(5/2), 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))^(5/2),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 = 22.06, 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))^(5/2),x)

[Out]

- atan(-(((c + d*tan(e + f*x))^(1/2)*(96*a^2*b^2*d^18*f^3 - 16*b^4*d^18*f^3 - 16*a^4*d^18*f^3 + 320*a^4*c^4*d^
14*f^3 + 1024*a^4*c^6*d^12*f^3 + 1440*a^4*c^8*d^10*f^3 + 1024*a^4*c^10*d^8*f^3 + 320*a^4*c^12*d^6*f^3 - 16*a^4
*c^16*d^2*f^3 + 320*b^4*c^4*d^14*f^3 + 1024*b^4*c^6*d^12*f^3 + 1440*b^4*c^8*d^10*f^3 + 1024*b^4*c^10*d^8*f^3 +
320*b^4*c^12*d^6*f^3 - 16*b^4*c^16*d^2*f^3 - 1920*a^2*b^2*c^4*d^14*f^3 - 6144*a^2*b^2*c^6*d^12*f^3 - 8640*a^2
*b^2*c^8*d^10*f^3 - 6144*a^2*b^2*c^10*d^8*f^3 - 1920*a^2*b^2*c^12*d^6*f^3 + 96*a^2*b^2*c^16*d^2*f^3 - 256*a*b^
3*c*d^17*f^3 + 256*a^3*b*c*d^17*f^3 - 1280*a*b^3*c^3*d^15*f^3 - 2304*a*b^3*c^5*d^13*f^3 - 1280*a*b^3*c^7*d^11*
f^3 + 1280*a*b^3*c^9*d^9*f^3 + 2304*a*b^3*c^11*d^7*f^3 + 1280*a*b^3*c^13*d^5*f^3 + 256*a*b^3*c^15*d^3*f^3 + 12
80*a^3*b*c^3*d^15*f^3 + 2304*a^3*b*c^5*d^13*f^3 + 1280*a^3*b*c^7*d^11*f^3 - 1280*a^3*b*c^9*d^9*f^3 - 2304*a^3*
b*c^11*d^7*f^3 - 1280*a^3*b*c^13*d^5*f^3 - 256*a^3*b*c^15*d^3*f^3) + ((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 - 32*a*
b^3*d^5*f^2 + 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2
- 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 - 160*a*b^3*c^4*d*f^2 + 160*a^3*b*c^4*d*f^2 + 320*a*b^3*c^2*d^
3*f^2 - 240*a^2*b^2*c*d^4*f^2 - 320*a^3*b*c^2*d^3*f^2)^2/4 - (a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^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) - 4*a^
4*c^5*f^2 - 4*b^4*c^5*f^2 + 16*a*b^3*d^5*f^2 - 16*a^3*b*d^5*f^2 - 20*a^4*c*d^4*f^2 - 20*b^4*c*d^4*f^2 + 24*a^2
*b^2*c^5*f^2 + 40*a^4*c^3*d^2*f^2 + 40*b^4*c^3*d^2*f^2 - 240*a^2*b^2*c^3*d^2*f^2 + 80*a*b^3*c^4*d*f^2 - 80*a^3
*b*c^4*d*f^2 - 160*a*b^3*c^2*d^3*f^2 + 120*a^2*b^2*c*d^4*f^2 + 160*a^3*b*c^2*d^3*f^2)/(16*(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)*(32*b^2*d^21*f^4 - 32*a^2*d^21*f^4
- (c + d*tan(e + f*x))^(1/2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 - 32*a*b^3*d^5*f^2 + 32*a^3*b*d^5*f^2 + 40*a^4
*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3
*d^2*f^2 - 160*a*b^3*c^4*d*f^2 + 160*a^3*b*c^4*d*f^2 + 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 - 320*a^3
*b*c^2*d^3*f^2)^2/4 - (a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^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) - 4*a^4*c^5*f^2 - 4*b^4*c^5*f^2 + 16*a*b^3*d^
5*f^2 - 16*a^3*b*d^5*f^2 - 20*a^4*c*d^4*f^2 - 20*b^4*c*d^4*f^2 + 24*a^2*b^2*c^5*f^2 + 40*a^4*c^3*d^2*f^2 + 40*
b^4*c^3*d^2*f^2 - 240*a^2*b^2*c^3*d^2*f^2 + 80*a*b^3*c^4*d*f^2 - 80*a^3*b*c^4*d*f^2 - 160*a*b^3*c^2*d^3*f^2 +
120*a^2*b^2*c*d^4*f^2 + 160*a^3*b*c^2*d^3*f^2)/(16*(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)*(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 + 64
0*c^19*d^4*f^5 + 64*c^21*d^2*f^5) - 160*a^2*c^2*d^19*f^4 - 128*a^2*c^4*d^17*f^4 + 896*a^2*c^6*d^15*f^4 + 3136*
a^2*c^8*d^13*f^4 + 4928*a^2*c^10*d^11*f^4 + 4480*a^2*c^12*d^9*f^4 + 2432*a^2*c^14*d^7*f^4 + 736*a^2*c^16*d^5*f
^4 + 96*a^2*c^18*d^3*f^4 + 160*b^2*c^2*d^19*f^4 + 128*b^2*c^4*d^17*f^4 - 896*b^2*c^6*d^15*f^4 - 3136*b^2*c^8*d
^13*f^4 - 4928*b^2*c^10*d^11*f^4 - 4480*b^2*c^12*d^9*f^4 - 2432*b^2*c^14*d^7*f^4 - 736*b^2*c^16*d^5*f^4 - 96*b
^2*c^18*d^3*f^4 + 192*a*b*c*d^20*f^4 + 1472*a*b*c^3*d^18*f^4 + 4864*a*b*c^5*d^16*f^4 + 8960*a*b*c^7*d^14*f^4 +
9856*a*b*c^9*d^12*f^4 + 6272*a*b*c^11*d^10*f^4 + 1792*a*b*c^13*d^8*f^4 - 256*a*b*c^15*d^6*f^4 - 320*a*b*c^17*
d^4*f^4 - 64*a*b*c^19*d^2*f^4))*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 - 32*a*b^3*d^5*f^2 + 32*a^3*b*d^5*f^2 + 40*a
^4*c*d^4*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c
^3*d^2*f^2 - 160*a*b^3*c^4*d*f^2 + 160*a^3*b*c^4*d*f^2 + 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 - 320*a
^3*b*c^2*d^3*f^2)^2/4 - (a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^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) - 4*a^4*c^5*f^2 - 4*b^4*c^5*f^2 + 16*a*b^3*
d^5*f^2 - 16*a^3*b*d^5*f^2 - 20*a^4*c*d^4*f^2 - 20*b^4*c*d^4*f^2 + 24*a^2*b^2*c^5*f^2 + 40*a^4*c^3*d^2*f^2 + 4
0*b^4*c^3*d^2*f^2 - 240*a^2*b^2*c^3*d^2*f^2 + 80*a*b^3*c^4*d*f^2 - 80*a^3*b*c^4*d*f^2 - 160*a*b^3*c^2*d^3*f^2
+ 120*a^2*b^2*c*d^4*f^2 + 160*a^3*b*c^2*d^3*f^2)/(16*(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 1
0*c^6*d^4*f^4 + 5*c^8*d^2*f^4)))^(1/2)*1i + ((c + d*tan(e + f*x))^(1/2)*(96*a^2*b^2*d^18*f^3 - 16*b^4*d^18*f^3
- 16*a^4*d^18*f^3 + 320*a^4*c^4*d^14*f^3 + 1024*a^4*c^6*d^12*f^3 + 1440*a^4*c^8*d^10*f^3 + 1024*a^4*c^10*d^8*
f^3 + 320*a^4*c^12*d^6*f^3 - 16*a^4*c^16*d^2*f^3 + 320*b^4*c^4*d^14*f^3 + 1024*b^4*c^6*d^12*f^3 + 1440*b^4*c^8
*d^10*f^3 + 1024*b^4*c^10*d^8*f^3 + 320*b^4*c^12*d^6*f^3 - 16*b^4*c^16*d^2*f^3 - 1920*a^2*b^2*c^4*d^14*f^3 - 6
144*a^2*b^2*c^6*d^12*f^3 - 8640*a^2*b^2*c^8*d^1...

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