3.2.24 \(\int \frac {(a+b \tan (e+f x)) (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(c+d \tan (e+f x))^{5/2}} \, dx\) [124]

Optimal. Leaf size=273 \[ -\frac {(a-i b) (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}+\frac {(i a-b) (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}+\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]

[Out]

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

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Rubi [A]
time = 0.54, antiderivative size = 271, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3716, 3709, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (b c-a d) \left (A d^2-B c d+c^2 C\right )}{3 d^2 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (a d^2 \left (2 c d (A-C)-B \left (c^2-d^2\right )\right )+b \left (-c^2 d^2 (A-3 C)+A d^4-2 B c d^3+c^4 C\right )\right )}{d^2 f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {(b+i a) (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}+\frac {(-b+i a) (A+i B-C) \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])*(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)*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) + ((I*a - b)*
(A + I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) + (2*(b*c - a*d)*(c^2*C - B
*c*d + A*d^2))/(3*d^2*(c^2 + d^2)*f*(c + d*Tan[e + f*x])^(3/2)) - (2*(b*(c^4*C - c^2*(A - 3*C)*d^2 - 2*B*c*d^3
+ A*d^4) + a*d^2*(2*c*(A - C)*d - B*(c^2 - d^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 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]

Rule 3716

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

Rubi steps

\begin {align*} \int \frac {(a+b \tan (e+f x)) \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(c+d \tan (e+f x))^{5/2}} \, dx &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {\int \frac {a d (A c-c C+B d)+b \left (c^2 C-B c d+A d^2\right )+d (A b c+a B c-b c C-a A d+b B d+a C d) \tan (e+f x)+b C \left (c^2+d^2\right ) \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx}{d \left (c^2+d^2\right )}\\ &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {-d \left (a \left (c^2 C-2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )-d \left (2 a A c d-2 a c C d-A b \left (c^2-d^2\right )-a B \left (c^2-d^2\right )+b \left (c^2 C-2 B c d-C d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{d \left (c^2+d^2\right )^2}\\ &=\frac {2 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {((a-i b) (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) (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 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {((i a+b) (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 {((i a-b) (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 (b c-a d) \left (c^2 C-B c d+A d^2\right )}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {((a+i b) (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 )}{(i c-d)^2 d f}+\frac {((a-i b) (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 {(i a+b) (A-i B-C) \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-b) (A+i B-C) \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) \left (c^2 C-B c d+A d^2\right )}{3 d^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (b \left (c^4 C-c^2 (A-3 C) d^2-2 B c d^3+A d^4\right )+a d^2 \left (2 c (A-C) d-B \left (c^2-d^2\right )\right )\right )}{d^2 \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 = 1.96, size = 300, normalized size = 1.10 \begin {gather*} -\frac {2 (c-i d) (c+i d) (2 b c C+b B d-2 a C d)+d (A b c+a B c-b c C-a A d+b B d+a C d) \left (i (c+i d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )+6 C (c-i d) (c+i d) d (a+b \tan (e+f x))-3 (A b+a B-b C) d \left (i (c+i d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )-(i c+d) \, _2F_1\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^2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Tan[e + f*x])*(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 - I*d)*(c + I*d)*(2*b*c*C + b*B*d - 2*a*C*d) + d*(A*b*c + a*B*c - b*c*C - a*A*d + b*B*d + 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)]) + 6*C*(c - I*d)*(c + I*d)*d*(a + b*Tan[e + f*x]) - 3*(A*b + a*B -
b*C)*d*(I*(c + I*d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] - (I*c + d)*Hypergeometric
2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])*(c + d*Tan[e + f*x]))/(d^2*(c^2 + d^2)*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. \(9671\) vs. \(2(246)=492\).
time = 0.55, size = 9672, normalized size = 35.43

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(9672\)
default	\(\text {Expression too large to display}\)	\(9672\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))*(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

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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))*(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

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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))*(A+B*tan(f*x+e)+C*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 ) \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\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))*(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))*(A + B*tan(e + f*x) + C*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))*(A+B*tan(f*x+e)+C*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 = 88.47, size = 2500, normalized size = 9.16 \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))*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(c + d*tan(e + f*x))^(5/2),x)

[Out]

((2*(C*b*c^3 + A*b*c*d^2 - B*b*c^2*d))/(3*(c^2 + d^2)) - (2*(c + d*tan(e + f*x))*(A*b*d^4 + C*b*c^4 - 2*B*b*c*
d^3 - A*b*c^2*d^2 + 3*C*b*c^2*d^2))/(c^2 + d^2)^2)/(d^2*f*(c + d*tan(e + f*x))^(3/2)) - atan(-(((c + d*tan(e +
f*x))^(1/2)*(16*A^2*b^2*d^18*f^3 - 16*B^2*b^2*d^18*f^3 + 16*C^2*b^2*d^18*f^3 - 320*A^2*b^2*c^4*d^14*f^3 - 102
4*A^2*b^2*c^6*d^12*f^3 - 1440*A^2*b^2*c^8*d^10*f^3 - 1024*A^2*b^2*c^10*d^8*f^3 - 320*A^2*b^2*c^12*d^6*f^3 + 16
*A^2*b^2*c^16*d^2*f^3 + 320*B^2*b^2*c^4*d^14*f^3 + 1024*B^2*b^2*c^6*d^12*f^3 + 1440*B^2*b^2*c^8*d^10*f^3 + 102
4*B^2*b^2*c^10*d^8*f^3 + 320*B^2*b^2*c^12*d^6*f^3 - 16*B^2*b^2*c^16*d^2*f^3 - 320*C^2*b^2*c^4*d^14*f^3 - 1024*
C^2*b^2*c^6*d^12*f^3 - 1440*C^2*b^2*c^8*d^10*f^3 - 1024*C^2*b^2*c^10*d^8*f^3 - 320*C^2*b^2*c^12*d^6*f^3 + 16*C
^2*b^2*c^16*d^2*f^3 - 32*A*C*b^2*d^18*f^3 - 128*A*B*b^2*c*d^17*f^3 + 128*B*C*b^2*c*d^17*f^3 - 640*A*B*b^2*c^3*
d^15*f^3 - 1152*A*B*b^2*c^5*d^13*f^3 - 640*A*B*b^2*c^7*d^11*f^3 + 640*A*B*b^2*c^9*d^9*f^3 + 1152*A*B*b^2*c^11*
d^7*f^3 + 640*A*B*b^2*c^13*d^5*f^3 + 128*A*B*b^2*c^15*d^3*f^3 + 640*A*C*b^2*c^4*d^14*f^3 + 2048*A*C*b^2*c^6*d^
12*f^3 + 2880*A*C*b^2*c^8*d^10*f^3 + 2048*A*C*b^2*c^10*d^8*f^3 + 640*A*C*b^2*c^12*d^6*f^3 - 32*A*C*b^2*c^16*d^
2*f^3 + 640*B*C*b^2*c^3*d^15*f^3 + 1152*B*C*b^2*c^5*d^13*f^3 + 640*B*C*b^2*c^7*d^11*f^3 - 640*B*C*b^2*c^9*d^9*
f^3 - 1152*B*C*b^2*c^11*d^7*f^3 - 640*B*C*b^2*c^13*d^5*f^3 - 128*B*C*b^2*c^15*d^3*f^3) + ((((8*A^2*b^2*c^5*f^2
- 8*B^2*b^2*c^5*f^2 + 8*C^2*b^2*c^5*f^2 - 80*A^2*b^2*c^3*d^2*f^2 + 80*B^2*b^2*c^3*d^2*f^2 - 80*C^2*b^2*c^3*d^
2*f^2 + 16*A*B*b^2*d^5*f^2 - 16*A*C*b^2*c^5*f^2 - 16*B*C*b^2*d^5*f^2 + 40*A^2*b^2*c*d^4*f^2 - 40*B^2*b^2*c*d^4
*f^2 + 40*C^2*b^2*c*d^4*f^2 + 80*A*B*b^2*c^4*d*f^2 - 80*A*C*b^2*c*d^4*f^2 - 80*B*C*b^2*c^4*d*f^2 - 160*A*B*b^2
*c^2*d^3*f^2 + 160*A*C*b^2*c^3*d^2*f^2 + 160*B*C*b^2*c^2*d^3*f^2)^2/4 - (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)*(A^4*b^4 + B^4*b^4 + C^4*b^4 - 4*A*C^3*b^4 - 4*A^3
*C*b^4 + 2*A^2*B^2*b^4 + 6*A^2*C^2*b^4 + 2*B^2*C^2*b^4 - 4*A*B^2*C*b^4))^(1/2) + 4*A^2*b^2*c^5*f^2 - 4*B^2*b^2
*c^5*f^2 + 4*C^2*b^2*c^5*f^2 - 40*A^2*b^2*c^3*d^2*f^2 + 40*B^2*b^2*c^3*d^2*f^2 - 40*C^2*b^2*c^3*d^2*f^2 + 8*A*
B*b^2*d^5*f^2 - 8*A*C*b^2*c^5*f^2 - 8*B*C*b^2*d^5*f^2 + 20*A^2*b^2*c*d^4*f^2 - 20*B^2*b^2*c*d^4*f^2 + 20*C^2*b
^2*c*d^4*f^2 + 40*A*B*b^2*c^4*d*f^2 - 40*A*C*b^2*c*d^4*f^2 - 40*B*C*b^2*c^4*d*f^2 - 80*A*B*b^2*c^2*d^3*f^2 + 8
0*A*C*b^2*c^3*d^2*f^2 + 80*B*C*b^2*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)*(128*A*b*c^15*d^6*f^4 - 32*B*b*d^21*f^4 - 736*A*b*c^3*d^18*f^4 - 2432*A*
b*c^5*d^16*f^4 - 4480*A*b*c^7*d^14*f^4 - 4928*A*b*c^9*d^12*f^4 - 3136*A*b*c^11*d^10*f^4 - 896*A*b*c^13*d^8*f^4
- (c + d*tan(e + f*x))^(1/2)*((((8*A^2*b^2*c^5*f^2 - 8*B^2*b^2*c^5*f^2 + 8*C^2*b^2*c^5*f^2 - 80*A^2*b^2*c^3*d
^2*f^2 + 80*B^2*b^2*c^3*d^2*f^2 - 80*C^2*b^2*c^3*d^2*f^2 + 16*A*B*b^2*d^5*f^2 - 16*A*C*b^2*c^5*f^2 - 16*B*C*b^
2*d^5*f^2 + 40*A^2*b^2*c*d^4*f^2 - 40*B^2*b^2*c*d^4*f^2 + 40*C^2*b^2*c*d^4*f^2 + 80*A*B*b^2*c^4*d*f^2 - 80*A*C
*b^2*c*d^4*f^2 - 80*B*C*b^2*c^4*d*f^2 - 160*A*B*b^2*c^2*d^3*f^2 + 160*A*C*b^2*c^3*d^2*f^2 + 160*B*C*b^2*c^2*d^
3*f^2)^2/4 - (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)
*(A^4*b^4 + B^4*b^4 + C^4*b^4 - 4*A*C^3*b^4 - 4*A^3*C*b^4 + 2*A^2*B^2*b^4 + 6*A^2*C^2*b^4 + 2*B^2*C^2*b^4 - 4*
A*B^2*C*b^4))^(1/2) + 4*A^2*b^2*c^5*f^2 - 4*B^2*b^2*c^5*f^2 + 4*C^2*b^2*c^5*f^2 - 40*A^2*b^2*c^3*d^2*f^2 + 40*
B^2*b^2*c^3*d^2*f^2 - 40*C^2*b^2*c^3*d^2*f^2 + 8*A*B*b^2*d^5*f^2 - 8*A*C*b^2*c^5*f^2 - 8*B*C*b^2*d^5*f^2 + 20*
A^2*b^2*c*d^4*f^2 - 20*B^2*b^2*c*d^4*f^2 + 20*C^2*b^2*c*d^4*f^2 + 40*A*B*b^2*c^4*d*f^2 - 40*A*C*b^2*c*d^4*f^2
- 40*B*C*b^2*c^4*d*f^2 - 80*A*B*b^2*c^2*d^3*f^2 + 80*A*C*b^2*c^3*d^2*f^2 + 80*B*C*b^2*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 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) + 160*A*b*c^17*d^4*f^4 +
32*A*b*c^19*d^2*f^4 - 160*B*b*c^2*d^19*f^4 - 128*B*b*c^4*d^17*f^4 + 896*B*b*c^6*d^15*f^4 + 3136*B*b*c^8*d^13*
f^4 + 4928*B*b*c^10*d^11*f^4 + 4480*B*b*c^12*d^9*f^4 + 2432*B*b*c^14*d^7*f^4 + 736*B*b*c^16*d^5*f^4 + 96*B*b*c
^18*d^3*f^4 + 736*C*b*c^3*d^18*f^4 + 2432*C*b*c^5*d^16*f^4 + 4480*C*b*c^7*d^14*f^4 + 4928*C*b*c^9*d^12*f^4 + 3
136*C*b*c^11*d^10*f^4 + 896*C*b*c^13*d^8*f^4 - 128*C*b*c^15*d^6*f^4 - 160*C*b*c^17*d^4*f^4 - 32*C*b*c^19*d^2*f
^4 - 96*A*b*c*d^20*f^4 + 96*C*b*c*d^20*f^4))*((((8*A^2*b^2*c^5*f^2 - 8*B^2*b^2*c^5*f^2 + 8*C^2*b^2*c^5*f^2 - 8
0*A^2*b^2*c^3*d^2*f^2 + 80*B^2*b^2*c^3*d^2*f^2 - 80*C^2*b^2*c^3*d^2*f^2 + 16*A*B*b^2*d^5*f^2 - 16*A*C*b^2*c^5*
f^2 - 16*B*C*b^2*d^5*f^2 + 40*A^2*b^2*c*d^4*f^2 - 40*B^2*b^2*c*d^4*f^2 + 40*C^2*b^2*c*d^4*f^2 + 80*A*B*b^2*c^4
*d*f^2 - 80*A*C*b^2*c*d^4*f^2 - 80*B*C*b^2*c^4*...

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