3.2.13 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\) [113]

Optimal. Leaf size=133 \[ -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f} \]

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3711, 3620, 3618, 65, 214} \begin {gather*} -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) - ((B - I*(A - C))*ArcT
anh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*C*Sqrt[c + d*Tan[e + f*x]])/(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 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 3711

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)+C \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx &=\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f}+\int \frac {A-C+B \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f}+\frac {1}{2} (A-i B-C) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx+\frac {1}{2} (A+i B-C) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f}+\frac {(i A+B-i C) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 f}+\frac {(i (-A-i B+C)) \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f}-\frac {(A-i B-C) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {(A+i B-C) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}\\ &=-\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d} f}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d} f}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 129, normalized size = 0.97 \begin {gather*} \frac {-\frac {i (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {i (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{d}}{f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1893\) vs. \(2(112)=224\).
time = 0.43, size = 1894, normalized size = 14.24

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1894\)
default	\(\text {Expression too large to display}\)	\(1894\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/f/d*(C*(c+d*tan(f*x+e))^(1/2)+d*(1/4/(c^2+d^2)^(3/2)/d^2*(1/2*(A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*c^2*d+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d-A*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*c*d^3+B*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*c^3-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d
^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^4-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d-C*(c^2+d^2)^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^3
)*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*A*c^2*d^3+2*A*d
^5-2*B*c^3*d^2-2*B*c*d^4-2*C*c^2*d^3-2*C*d^5-1/2*(A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d+A*(c^2
+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d-A*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*c*d^3+B*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
c^3-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^2-B*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*d^4-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*d^3+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^3)*(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/(c^2+d^2)^(3/2)/d^2*(1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*c^2*d-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d+A*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*c*d^3-B*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*c^3+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d
^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^4+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d+C*(c^2+d^2)^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^3
)*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*A*c^2*d^3+2*A*d
^5-2*B*c^3*d^2-2*B*c*d^4-2*C*c^2*d^3-2*C*d^5+1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d-A*(c^
2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d+A*(2*(c^2+d^2)^(1/2)+2*c)
^(1/2)*c*d^3-B*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*c^3+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d^2+B*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*d^4+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^2*d+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*d^3-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c^3*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*c*d^3)*(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)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)/sqrt(d*tan(f*x + e) + c), 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)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),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 {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(1/2),x)

[Out]

Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/sqrt(c + d*tan(e + f*x)), 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)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2),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 = 14.21, size = 2500, normalized size = 18.80 \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) + C*tan(e + f*x)^2)/(c + d*tan(e + f*x))^(1/2),x)

[Out]

2*atanh((32*C^2*d^2*((-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(
1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^4) - (4*C*d^3*f^2*(-16*C^4*d^2*f^4)^(1/2
))/(c^2*f^5 + d^2*f^5)) + (8*c*d^2*((-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4
+ d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*C^4*d^2*f^4)^(1/2))/((16*C^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4
) - (4*C*d^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*C^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*
C*c^2*d^3*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*C^2*c^2*d^2*f^2*((-16*C^4*d^2*f^4)^(1/2)/(16
*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^5*
f^5)/(c^2*f^4 + d^2*f^4) - (4*C*d^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*C^3*c^3*d^3*f^5)/(c
^2*f^4 + d^2*f^4) - (4*C*c^2*d^3*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*((-16*C^4*d^2*f^4)^(1/2)/(
16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((8*c*d^2*(- (-16*C^4*d^2*f^4)^(
1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*C^4
*d^2*f^4)^(1/2))/((16*C^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*d^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^
2*f^5) + (16*C^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*c^2*d^3*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f
^5)) - (32*C^2*d^2*(- (-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^
(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^4) + (4*C*d^3*f^2*(-16*C^4*d^2*f^4)^(1/
2))/(c^2*f^5 + d^2*f^5)) + (32*C^2*c^2*d^2*f^2*(- (-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^
2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*d
^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*C^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*c^2*d^3*
f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*(- (-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*
c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((32*A^2*d^2*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4))
- (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*A^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^
4) - (4*A*d^3*f^2*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (8*c*d^2*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f
^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*A^4*d^2*f^4)^(1/2)
)/((16*A^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*d^5*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*A^
3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*c^2*d^3*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*A^2*
c^2*d^2*f^2*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c
+ d*tan(e + f*x))^(1/2))/((16*A^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*d^5*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*
f^5 + d^2*f^5) + (16*A^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*c^2*d^3*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5
+ d^2*f^5)))*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) +
2*atanh((8*c*d^2*(- (-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(
1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*A^4*d^2*f^4)^(1/2))/((16*A^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*A*d^5*f^
4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*A^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) + (4*A*c^2*d^3*f^4*(
-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*A^2*d^2*(- (-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4))
- (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*A^3*c*d^3*f^3)/(c^2*f^4 + d^2*f
^4) + (4*A*d^3*f^2*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (32*A^2*c^2*d^2*f^2*(- (-16*A^4*d^2*f^4)^(1
/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*A^3
*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*A*d^5*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*A^3*c^3*d^3*
f^5)/(c^2*f^4 + d^2*f^4) + (4*A*c^2*d^3*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*(- (-16*A^4*d^2*f^4
)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((32*B^2*d^2*((B^2*c*f^
2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2
))/((16*B^3*d^2)/f - (16*B^3*c^2*d^2*f^3)/(c^2*f^4 + d^2*f^4) + (4*B*c*d^2*f^2*(-16*B^4*d^2*f^4)^(1/2))/(c^2*f
^5 + d^2*f^5)) + (8*c*d^2*((B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^
4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*B^4*d^2*f^4)^(1/2))/(16*B^3*d^4*f + 16*B^3*c^2*d^2*f - (16*B^3*c^2*
d^4*f^5)/(c^2*f^4 + d^2*f^4) - (16*B^3*c^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*B*c*d^4*f^4*(-16*B^4*d^2*f^4)^(1/
2))/(c^2*f^5 + d^2*f^5) + (4*B*c^3*d^2*f^4*(-16*B^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*B^2*c^2*d^2*f^2
*((B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*B^...

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