3.24 \(\int \frac{A+B x}{(a+c x^2) \sqrt{d+f x^2}} \, dx\)

Optimal. Leaf size=101 \[ \frac{A \tan ^{-1}\left (\frac{x \sqrt{c d-a f}}{\sqrt{a} \sqrt{d+f x^2}}\right )}{\sqrt{a} \sqrt{c d-a f}}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+f x^2}}{\sqrt{c d-a f}}\right )}{\sqrt{c} \sqrt{c d-a f}} \]

[Out]

(A*ArcTan[(Sqrt[c*d - a*f]*x)/(Sqrt[a]*Sqrt[d + f*x^2])])/(Sqrt[a]*Sqrt[c*d - a*f]) - (B*ArcTanh[(Sqrt[c]*Sqrt
[d + f*x^2])/Sqrt[c*d - a*f]])/(Sqrt[c]*Sqrt[c*d - a*f])

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Rubi [A]  time = 0.1271, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1010, 377, 205, 444, 63, 208} \[ \frac{A \tan ^{-1}\left (\frac{x \sqrt{c d-a f}}{\sqrt{a} \sqrt{d+f x^2}}\right )}{\sqrt{a} \sqrt{c d-a f}}-\frac{B \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+f x^2}}{\sqrt{c d-a f}}\right )}{\sqrt{c} \sqrt{c d-a f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*ArcTan[(Sqrt[c*d - a*f]*x)/(Sqrt[a]*Sqrt[d + f*x^2])])/(Sqrt[a]*Sqrt[c*d - a*f]) - (B*ArcTanh[(Sqrt[c]*Sqrt
[d + f*x^2])/Sqrt[c*d - a*f]])/(Sqrt[c]*Sqrt[c*d - a*f])

Rule 1010

Int[((g_) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Dist[g, Int[(a + c
*x^2)^p*(d + f*x^2)^q, x], x] + Dist[h, Int[x*(a + c*x^2)^p*(d + f*x^2)^q, x], x] /; FreeQ[{a, c, d, f, g, h,
p, q}, x]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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]

Rubi steps

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

Mathematica [A]  time = 0.176682, size = 154, normalized size = 1.52 \[ \frac{\left (A \sqrt{c}-\sqrt{-a} B\right ) \tanh ^{-1}\left (\frac{\sqrt{c} d-\sqrt{-a} f x}{\sqrt{d+f x^2} \sqrt{c d-a f}}\right )-\left (\sqrt{-a} B+A \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{-a} f x+\sqrt{c} d}{\sqrt{d+f x^2} \sqrt{c d-a f}}\right )}{2 \sqrt{-a} \sqrt{c} \sqrt{c d-a f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-(Sqrt[-a]*B) + A*Sqrt[c])*ArcTanh[(Sqrt[c]*d - Sqrt[-a]*f*x)/(Sqrt[c*d - a*f]*Sqrt[d + f*x^2])] - (Sqrt[-a]
*B + A*Sqrt[c])*ArcTanh[(Sqrt[c]*d + Sqrt[-a]*f*x)/(Sqrt[c*d - a*f]*Sqrt[d + f*x^2])])/(2*Sqrt[-a]*Sqrt[c]*Sqr
t[c*d - a*f])

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Maple [B]  time = 0.326, size = 608, normalized size = 6. \begin{align*} -{\frac{A}{2}\ln \left ({ \left ( -2\,{\frac{af-cd}{c}}+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{af-cd}{c}}}\sqrt{ \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{af-cd}{c}}} \right ) \left ( x-{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ac}}}{\frac{1}{\sqrt{-{\frac{af-cd}{c}}}}}}-{\frac{B}{2\,c}\ln \left ({ \left ( -2\,{\frac{af-cd}{c}}+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{af-cd}{c}}}\sqrt{ \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f+2\,{\frac{f\sqrt{-ac}}{c} \left ( x-{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{af-cd}{c}}} \right ) \left ( x-{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{af-cd}{c}}}}}}+{\frac{A}{2}\ln \left ({ \left ( -2\,{\frac{af-cd}{c}}-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{af-cd}{c}}}\sqrt{ \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{af-cd}{c}}} \right ) \left ( x+{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-ac}}}{\frac{1}{\sqrt{-{\frac{af-cd}{c}}}}}}-{\frac{B}{2\,c}\ln \left ({ \left ( -2\,{\frac{af-cd}{c}}-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }+2\,\sqrt{-{\frac{af-cd}{c}}}\sqrt{ \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) ^{2}f-2\,{\frac{f\sqrt{-ac}}{c} \left ( x+{\frac{\sqrt{-ac}}{c}} \right ) }-{\frac{af-cd}{c}}} \right ) \left ( x+{\frac{1}{c}\sqrt{-ac}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{af-cd}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+a)/(f*x^2+d)^(1/2),x)

[Out]

-1/2/(-a*c)^(1/2)/(-(a*f-c*d)/c)^(1/2)*ln((-2*(a*f-c*d)/c+2*f*(-a*c)^(1/2)/c*(x-1/c*(-a*c)^(1/2))+2*(-(a*f-c*d
)/c)^(1/2)*((x-1/c*(-a*c)^(1/2))^2*f+2*f*(-a*c)^(1/2)/c*(x-1/c*(-a*c)^(1/2))-(a*f-c*d)/c)^(1/2))/(x-1/c*(-a*c)
^(1/2)))*A-1/2/c/(-(a*f-c*d)/c)^(1/2)*ln((-2*(a*f-c*d)/c+2*f*(-a*c)^(1/2)/c*(x-1/c*(-a*c)^(1/2))+2*(-(a*f-c*d)
/c)^(1/2)*((x-1/c*(-a*c)^(1/2))^2*f+2*f*(-a*c)^(1/2)/c*(x-1/c*(-a*c)^(1/2))-(a*f-c*d)/c)^(1/2))/(x-1/c*(-a*c)^
(1/2)))*B+1/2/(-a*c)^(1/2)/(-(a*f-c*d)/c)^(1/2)*ln((-2*(a*f-c*d)/c-2*f*(-a*c)^(1/2)/c*(x+1/c*(-a*c)^(1/2))+2*(
-(a*f-c*d)/c)^(1/2)*((x+1/c*(-a*c)^(1/2))^2*f-2*f*(-a*c)^(1/2)/c*(x+1/c*(-a*c)^(1/2))-(a*f-c*d)/c)^(1/2))/(x+1
/c*(-a*c)^(1/2)))*A-1/2/c/(-(a*f-c*d)/c)^(1/2)*ln((-2*(a*f-c*d)/c-2*f*(-a*c)^(1/2)/c*(x+1/c*(-a*c)^(1/2))+2*(-
(a*f-c*d)/c)^(1/2)*((x+1/c*(-a*c)^(1/2))^2*f-2*f*(-a*c)^(1/2)/c*(x+1/c*(-a*c)^(1/2))-(a*f-c*d)/c)^(1/2))/(x+1/
c*(-a*c)^(1/2)))*B

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+a)/(f*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.49673, size = 2943, normalized size = 29.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+a)/(f*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

-1/4*sqrt((B^2*a - A^2*c + 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^
2*d - a^2*c*f))*log(((A*B^3*a + A^3*B*c)*f*x + (A^2*B*c^2*d - A^2*B*a*c*f + (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B
*a^3*c*f^2)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c + 2*(a
*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) + sqrt(-A^2*B^2
/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) + 1/4*
sqrt((B^2*a - A^2*c + 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d -
a^2*c*f))*log(((A*B^3*a + A^3*B*c)*f*x - (A^2*B*c^2*d - A^2*B*a*c*f + (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*
c*f^2)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c + 2*(a*c^2*
d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) + sqrt(-A^2*B^2/(a*c
^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) - 1/4*sqrt(
(B^2*a - A^2*c - 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*
c*f))*log(((A*B^3*a + A^3*B*c)*f*x + (A^2*B*c^2*d - A^2*B*a*c*f - (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*c*f^2
)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c - 2*(a*c^2*d - a
^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) - sqrt(-A^2*B^2/(a*c^3*d^
2 - 2*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x) + 1/4*sqrt((B^2*
a - A^2*c - 2*(a*c^2*d - a^2*c*f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f))
*log(((A*B^3*a + A^3*B*c)*f*x - (A^2*B*c^2*d - A^2*B*a*c*f - (B*a*c^3*d^2 - 2*B*a^2*c^2*d*f + B*a^3*c*f^2)*sqr
t(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))*sqrt(f*x^2 + d)*sqrt((B^2*a - A^2*c - 2*(a*c^2*d - a^2*c*
f)*sqrt(-A^2*B^2/(a*c^3*d^2 - 2*a^2*c^2*d*f + a^3*c*f^2)))/(a*c^2*d - a^2*c*f)) - sqrt(-A^2*B^2/(a*c^3*d^2 - 2
*a^2*c^2*d*f + a^3*c*f^2))*((B^2*a*c^2 + A^2*c^3)*d^2 - (B^2*a^2*c + A^2*a*c^2)*d*f))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{A + B x}{\left (a + c x^{2}\right ) \sqrt{d + f x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x**2+a)/(f*x**2+d)**(1/2),x)

[Out]

Integral((A + B*x)/((a + c*x**2)*sqrt(d + f*x**2)), x)

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Giac [C]  time = 3.82123, size = 11297, normalized size = 111.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x^2+a)/(f*x^2+d)^(1/2),x, algorithm="giac")

[Out]

-1/2*(3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d
*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d)
- 2*a*f/(c*abs(d)))))^3*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) - (2*a*c^2*sqrt(-d)*d^2*f - 2*
a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cosh(1/2*imag_part(
arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 - 9*(2*a*c^2
*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B
*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d
)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*
abs(d))))) + 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt
(-a*c*d*f + a^2*f^2))*B*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d/
abs(d) - 2*a*f/(c*abs(d)))))^3*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) + 9*(2*a*c^2*sqrt(-d)*
d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*r
eal_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(
1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2
- 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f
+ a^2*f^2))*B*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*a
*f/(c*abs(d)))))^3*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*
a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(a
rccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_p
art(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 + (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)
*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))
^3*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 - 4*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (
c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*ab
s(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/
(c*abs(d))))) + 8*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f
+ a^2*f^2))*A*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*
a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) -
2*a*f/(c*abs(d))))) - 4*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-
a*c*d*f + a^2*f^2))*A*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d
) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 + (2*a*c^2*sqrt(-d)*d^2*f -
2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cosh(1/2*imag_pa
rt(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) - (2*a*c^2*sq
rt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*si
n(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))
)*arctan(-(sqrt(f)*x + (d^2)^(1/4)*cos(1/2*arccos((c*d - 2*a*f)/(c*abs(d)))) - sqrt(f*x^2 + d))/((d^2)^(1/4)*s
in(1/2*arccos((c*d - 2*a*f)/(c*abs(d))))))/(a*c^3*d^3*f - a^2*c^2*d^2*f^2) - 1/2*(3*(2*a*c^2*sqrt(-d)*d^2*f -
2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part
(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sin(1/2*re
al_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) - (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-
d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))
)))^3*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 - 9*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)
*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d)
- 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d
/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) + 3*(2*a*c^2*sqrt(-d)*d
^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cosh(1/2*i
mag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sin
h(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) + 9*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (
c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(
c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*
a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a
^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cosh(1/2*imag_part(a
rccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sinh(1/2*imag_p
art(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-
d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))
))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs
(d)))))^3 + (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a
*c*d*f + a^2*f^2))*B*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sinh(1/2*imag_part(arccos(d/abs
(d) - 2*a*f/(c*abs(d)))))^3 + 4*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))
*sqrt(-a*c*d*f + a^2*f^2))*A*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos
(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) - 8*(2*a*c^2*d^2*f^(
3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cos(1/2*real_part(a
rccos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_par
t(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) + 4*(2*a*c^2*
d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cos(1/2*real
_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*i
mag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 + (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqr
t(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(
d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) - (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*
f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*sin(1/2*real_part(arccos(d/abs(d) -
2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))))*arctan(-(sqrt(f)*x - (d^2)^(1/4)
*cos(1/2*arccos((c*d - 2*a*f)/(c*abs(d)))) - sqrt(f*x^2 + d))/((d^2)^(1/4)*sin(1/2*arccos((c*d - 2*a*f)/(c*abs
(d))))))/(a*c^3*d^3*f - a^2*c^2*d^2*f^2) + 1/4*((2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-
d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))
))^3*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 - 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)
*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d)
- 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sin(1/2*real_part(arccos(d/a
bs(d) - 2*a*f/(c*abs(d)))))^2 - 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c
*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*cosh(1/2*
imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) + 9
*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^
2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(
c*abs(d)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) -
2*a*f/(c*abs(d))))) + 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*
d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*cosh(1/2*imag_part(
arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - 9*(2*a*c^2*
sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*
cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))
))*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs
(d)))))^2 - (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a
*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sinh(1/2*imag_part(arccos(d/abs
(d) - 2*a*f/(c*abs(d)))))^3 + 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*s
qrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_
part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 - 2*(2
*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cos(1
/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^
2 + 2*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))
*A*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs
(d)))))^2 + 4*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a
^2*f^2))*A*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*
f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) - 4*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*
f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cosh(1/2*imag_part(arccos(d/abs(d) -
2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs
(d) - 2*a*f/(c*abs(d))))) - 2*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*s
qrt(-a*c*d*f + a^2*f^2))*A*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos
(d/abs(d) - 2*a*f/(c*abs(d)))))^2 + 2*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^
(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_par
t(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 + (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d
^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*c
osh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) - (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (
c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(
c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))))*log(2*(d^2)^(1/4)*(sqrt(f)*x - sqrt(f*x
^2 + d))*cos(1/2*arccos((c*d - 2*a*f)/(c*abs(d)))) + (sqrt(f)*x - sqrt(f*x^2 + d))^2 + sqrt(d^2))/(a*c^3*d^3*f
- a^2*c^2*d^2*f^2) - 1/4*((2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-
d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*cosh(1/2*imag_pa
rt(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 - 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d
)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))
)*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(
d)))))^2 - 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-
a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*cosh(1/2*imag_part(arccos(d/ab
s(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) + 9*(2*a*c^2*sqrt(-d)*d^
2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*rea
l_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/
2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) +
3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f +
a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a
*f/(c*abs(d)))))*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - 9*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^
2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arc
cos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(
arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - (2*a*c^2*
sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*
cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)
))))^3 + 3*(2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d*f)*sqrt(-a*
c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sin(1/2*real_part(arccos(d/abs(d)
- 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^3 + 2*(2*a*c^2*d^2*f^(3/2) -
2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cos(1/2*real_part(arccos(d
/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - 2*(2*a*c^2*d^2*f^
(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cosh(1/2*imag_part
(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2 - 4*(2*a*c^
2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cos(1/2*re
al_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(
1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))) + 4*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sq
rt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^2))*A*cosh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*
sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)
)))) + 2*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f + a^2*f^
2))*A*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2*a*f/(c*
abs(d)))))^2 - 2*(2*a*c^2*d^2*f^(3/2) - 2*a^2*c*d*f^(5/2) + (c^2*d^2*sqrt(f) - 2*a*c*d*f^(3/2))*sqrt(-a*c*d*f
+ a^2*f^2))*A*sin(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))^2*sinh(1/2*imag_part(arccos(d/abs(d) - 2
*a*f/(c*abs(d)))))^2 + (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*a*c*sqrt(-d)*d
*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*cosh(1/2*imag_part(arc
cos(d/abs(d) - 2*a*f/(c*abs(d))))) - (2*a*c^2*sqrt(-d)*d^2*f - 2*a^2*c*sqrt(-d)*d*f^2 + (c^2*sqrt(-d)*d^2 - 2*
a*c*sqrt(-d)*d*f)*sqrt(-a*c*d*f + a^2*f^2))*B*cos(1/2*real_part(arccos(d/abs(d) - 2*a*f/(c*abs(d)))))*sinh(1/2
*imag_part(arccos(d/abs(d) - 2*a*f/(c*abs(d))))))*log(-2*(d^2)^(1/4)*(sqrt(f)*x - sqrt(f*x^2 + d))*cos(1/2*arc
cos((c*d - 2*a*f)/(c*abs(d)))) + (sqrt(f)*x - sqrt(f*x^2 + d))^2 + sqrt(d^2))/(a*c^3*d^3*f - a^2*c^2*d^2*f^2)