3.8 \(\int \frac {a+b \tan ^{-1}(c x)}{(d+e x)^4} \, dx\)

Optimal. Leaf size=206 \[ -\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}+\frac {b c^4 d \left (c^2 d^2-3 e^2\right ) \tan ^{-1}(c x)}{3 e \left (c^2 d^2+e^2\right )^3}-\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{6 \left (c^2 d^2+e^2\right )^3}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3} \]

[Out]

-1/6*b*c/(c^2*d^2+e^2)/(e*x+d)^2-2/3*b*c^3*d/(c^2*d^2+e^2)^2/(e*x+d)+1/3*b*c^4*d*(c^2*d^2-3*e^2)*arctan(c*x)/e
/(c^2*d^2+e^2)^3+1/3*(-a-b*arctan(c*x))/e/(e*x+d)^3+1/3*b*c^3*(3*c^2*d^2-e^2)*ln(e*x+d)/(c^2*d^2+e^2)^3-1/6*b*
c^3*(3*c^2*d^2-e^2)*ln(c^2*x^2+1)/(c^2*d^2+e^2)^3

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4862, 710, 801, 635, 203, 260} \[ -\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{6 \left (c^2 d^2+e^2\right )^3}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac {b c^4 d \left (c^2 d^2-3 e^2\right ) \tan ^{-1}(c x)}{3 e \left (c^2 d^2+e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/(d + e*x)^4,x]

[Out]

-(b*c)/(6*(c^2*d^2 + e^2)*(d + e*x)^2) - (2*b*c^3*d)/(3*(c^2*d^2 + e^2)^2*(d + e*x)) + (b*c^4*d*(c^2*d^2 - 3*e
^2)*ArcTan[c*x])/(3*e*(c^2*d^2 + e^2)^3) - (a + b*ArcTan[c*x])/(3*e*(d + e*x)^3) + (b*c^3*(3*c^2*d^2 - e^2)*Lo
g[d + e*x])/(3*(c^2*d^2 + e^2)^3) - (b*c^3*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2])/(6*(c^2*d^2 + e^2)^3)

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 710

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

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 4862

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a + b*
ArcTan[c*x]))/(e*(q + 1)), x] - Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{
a, b, c, d, e, q}, x] && NeQ[q, -1]

Rubi steps

\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{(d+e x)^4} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {(b c) \int \frac {1}{(d+e x)^3 \left (1+c^2 x^2\right )} \, dx}{3 e}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3\right ) \int \frac {d-e x}{(d+e x)^2 \left (1+c^2 x^2\right )} \, dx}{3 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {\left (b c^3\right ) \int \left (\frac {2 d e^2}{\left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {e^2 \left (-3 c^2 d^2+e^2\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {c^2 d \left (c^2 d^2-3 e^2\right )-c^2 e \left (3 c^2 d^2-e^2\right ) x}{\left (c^2 d^2+e^2\right )^2 \left (1+c^2 x^2\right )}\right ) \, dx}{3 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac {\left (b c^3\right ) \int \frac {c^2 d \left (c^2 d^2-3 e^2\right )-c^2 e \left (3 c^2 d^2-e^2\right ) x}{1+c^2 x^2} \, dx}{3 e \left (c^2 d^2+e^2\right )^3}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}+\frac {\left (b c^5 d \left (c^2 d^2-3 e^2\right )\right ) \int \frac {1}{1+c^2 x^2} \, dx}{3 e \left (c^2 d^2+e^2\right )^3}-\frac {\left (b c^5 \left (3 c^2 d^2-e^2\right )\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 \left (c^2 d^2+e^2\right )^3}\\ &=-\frac {b c}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {2 b c^3 d}{3 \left (c^2 d^2+e^2\right )^2 (d+e x)}+\frac {b c^4 d \left (c^2 d^2-3 e^2\right ) \tan ^{-1}(c x)}{3 e \left (c^2 d^2+e^2\right )^3}-\frac {a+b \tan ^{-1}(c x)}{3 e (d+e x)^3}+\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log (d+e x)}{3 \left (c^2 d^2+e^2\right )^3}-\frac {b c^3 \left (3 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{6 \left (c^2 d^2+e^2\right )^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.65, size = 254, normalized size = 1.23 \[ -\frac {2 \left (a+b \tan ^{-1}(c x)\right )+\frac {b c (d+e x) \left (4 c^2 d e \left (c^2 d^2+e^2\right ) (d+e x)-c^2 \left (c^2 d^2 \left (\sqrt {-c^2} d-3 e\right )+e^2 \left (e-3 \sqrt {-c^2} d\right )\right ) \log \left (1-\sqrt {-c^2} x\right ) (d+e x)^2-c^2 \left (e^2 \left (3 \sqrt {-c^2} d+e\right )-c^2 d^2 \left (\sqrt {-c^2} d+3 e\right )\right ) \log \left (\sqrt {-c^2} x+1\right ) (d+e x)^2-2 c^2 e \left (3 c^2 d^2-e^2\right ) (d+e x)^2 \log (d+e x)+e \left (c^2 d^2+e^2\right )^2\right )}{\left (c^2 d^2+e^2\right )^3}}{6 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTan[c*x])/(d + e*x)^4,x]

[Out]

-1/6*(2*(a + b*ArcTan[c*x]) + (b*c*(d + e*x)*(e*(c^2*d^2 + e^2)^2 + 4*c^2*d*e*(c^2*d^2 + e^2)*(d + e*x) - c^2*
(c^2*d^2*(Sqrt[-c^2]*d - 3*e) + e^2*(-3*Sqrt[-c^2]*d + e))*(d + e*x)^2*Log[1 - Sqrt[-c^2]*x] - c^2*(e^2*(3*Sqr
t[-c^2]*d + e) - c^2*d^2*(Sqrt[-c^2]*d + 3*e))*(d + e*x)^2*Log[1 + Sqrt[-c^2]*x] - 2*c^2*e*(3*c^2*d^2 - e^2)*(
d + e*x)^2*Log[d + e*x]))/(c^2*d^2 + e^2)^3)/(e*(d + e*x)^3)

________________________________________________________________________________________

fricas [B]  time = 1.87, size = 642, normalized size = 3.12 \[ -\frac {2 \, a c^{6} d^{6} + 5 \, b c^{5} d^{5} e + 6 \, a c^{4} d^{4} e^{2} + 6 \, b c^{3} d^{3} e^{3} + 6 \, a c^{2} d^{2} e^{4} + b c d e^{5} + 2 \, a e^{6} + 4 \, {\left (b c^{5} d^{3} e^{3} + b c^{3} d e^{5}\right )} x^{2} + {\left (9 \, b c^{5} d^{4} e^{2} + 10 \, b c^{3} d^{2} e^{4} + b c e^{6}\right )} x + 2 \, {\left (6 \, b c^{4} d^{4} e^{2} + 3 \, b c^{2} d^{2} e^{4} + b e^{6} - {\left (b c^{6} d^{3} e^{3} - 3 \, b c^{4} d e^{5}\right )} x^{3} - 3 \, {\left (b c^{6} d^{4} e^{2} - 3 \, b c^{4} d^{2} e^{4}\right )} x^{2} - 3 \, {\left (b c^{6} d^{5} e - 3 \, b c^{4} d^{3} e^{3}\right )} x\right )} \arctan \left (c x\right ) + {\left (3 \, b c^{5} d^{5} e - b c^{3} d^{3} e^{3} + {\left (3 \, b c^{5} d^{2} e^{4} - b c^{3} e^{6}\right )} x^{3} + 3 \, {\left (3 \, b c^{5} d^{3} e^{3} - b c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (3 \, b c^{5} d^{4} e^{2} - b c^{3} d^{2} e^{4}\right )} x\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (3 \, b c^{5} d^{5} e - b c^{3} d^{3} e^{3} + {\left (3 \, b c^{5} d^{2} e^{4} - b c^{3} e^{6}\right )} x^{3} + 3 \, {\left (3 \, b c^{5} d^{3} e^{3} - b c^{3} d e^{5}\right )} x^{2} + 3 \, {\left (3 \, b c^{5} d^{4} e^{2} - b c^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (c^{6} d^{9} e + 3 \, c^{4} d^{7} e^{3} + 3 \, c^{2} d^{5} e^{5} + d^{3} e^{7} + {\left (c^{6} d^{6} e^{4} + 3 \, c^{4} d^{4} e^{6} + 3 \, c^{2} d^{2} e^{8} + e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{7} e^{3} + 3 \, c^{4} d^{5} e^{5} + 3 \, c^{2} d^{3} e^{7} + d e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{8} e^{2} + 3 \, c^{4} d^{6} e^{4} + 3 \, c^{2} d^{4} e^{6} + d^{2} e^{8}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x+d)^4,x, algorithm="fricas")

[Out]

-1/6*(2*a*c^6*d^6 + 5*b*c^5*d^5*e + 6*a*c^4*d^4*e^2 + 6*b*c^3*d^3*e^3 + 6*a*c^2*d^2*e^4 + b*c*d*e^5 + 2*a*e^6
+ 4*(b*c^5*d^3*e^3 + b*c^3*d*e^5)*x^2 + (9*b*c^5*d^4*e^2 + 10*b*c^3*d^2*e^4 + b*c*e^6)*x + 2*(6*b*c^4*d^4*e^2
+ 3*b*c^2*d^2*e^4 + b*e^6 - (b*c^6*d^3*e^3 - 3*b*c^4*d*e^5)*x^3 - 3*(b*c^6*d^4*e^2 - 3*b*c^4*d^2*e^4)*x^2 - 3*
(b*c^6*d^5*e - 3*b*c^4*d^3*e^3)*x)*arctan(c*x) + (3*b*c^5*d^5*e - b*c^3*d^3*e^3 + (3*b*c^5*d^2*e^4 - b*c^3*e^6
)*x^3 + 3*(3*b*c^5*d^3*e^3 - b*c^3*d*e^5)*x^2 + 3*(3*b*c^5*d^4*e^2 - b*c^3*d^2*e^4)*x)*log(c^2*x^2 + 1) - 2*(3
*b*c^5*d^5*e - b*c^3*d^3*e^3 + (3*b*c^5*d^2*e^4 - b*c^3*e^6)*x^3 + 3*(3*b*c^5*d^3*e^3 - b*c^3*d*e^5)*x^2 + 3*(
3*b*c^5*d^4*e^2 - b*c^3*d^2*e^4)*x)*log(e*x + d))/(c^6*d^9*e + 3*c^4*d^7*e^3 + 3*c^2*d^5*e^5 + d^3*e^7 + (c^6*
d^6*e^4 + 3*c^4*d^4*e^6 + 3*c^2*d^2*e^8 + e^10)*x^3 + 3*(c^6*d^7*e^3 + 3*c^4*d^5*e^5 + 3*c^2*d^3*e^7 + d*e^9)*
x^2 + 3*(c^6*d^8*e^2 + 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 + d^2*e^8)*x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x+d)^4,x, algorithm="giac")

[Out]

sage0*x

________________________________________________________________________________________

maple [A]  time = 0.04, size = 282, normalized size = 1.37 \[ -\frac {c^{3} a}{3 \left (c e x +d c \right )^{3} e}-\frac {c^{3} b \arctan \left (c x \right )}{3 \left (c e x +d c \right )^{3} e}-\frac {c^{3} b}{6 \left (c^{2} d^{2}+e^{2}\right ) \left (c e x +d c \right )^{2}}+\frac {c^{5} b \ln \left (c e x +d c \right ) d^{2}}{\left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {c^{3} b \,e^{2} \ln \left (c e x +d c \right )}{3 \left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {2 c^{4} b d}{3 \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c e x +d c \right )}-\frac {c^{5} b \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2 \left (c^{2} d^{2}+e^{2}\right )^{3}}+\frac {c^{3} b \,e^{2} \ln \left (c^{2} x^{2}+1\right )}{6 \left (c^{2} d^{2}+e^{2}\right )^{3}}+\frac {c^{6} b \arctan \left (c x \right ) d^{3}}{3 e \left (c^{2} d^{2}+e^{2}\right )^{3}}-\frac {c^{4} b e \arctan \left (c x \right ) d}{\left (c^{2} d^{2}+e^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x))/(e*x+d)^4,x)

[Out]

-1/3*c^3*a/(c*e*x+c*d)^3/e-1/3*c^3*b/(c*e*x+c*d)^3/e*arctan(c*x)-1/6*c^3*b/(c^2*d^2+e^2)/(c*e*x+c*d)^2+c^5*b/(
c^2*d^2+e^2)^3*ln(c*e*x+c*d)*d^2-1/3*c^3*b*e^2/(c^2*d^2+e^2)^3*ln(c*e*x+c*d)-2/3*c^4*b*d/(c^2*d^2+e^2)^2/(c*e*
x+c*d)-1/2*c^5*b/(c^2*d^2+e^2)^3*ln(c^2*x^2+1)*d^2+1/6*c^3*b*e^2/(c^2*d^2+e^2)^3*ln(c^2*x^2+1)+1/3*c^6*b/e/(c^
2*d^2+e^2)^3*arctan(c*x)*d^3-c^4*b*e/(c^2*d^2+e^2)^3*arctan(c*x)*d

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 374, normalized size = 1.82 \[ -\frac {1}{6} \, {\left (c {\left (\frac {{\left (3 \, c^{4} d^{2} - c^{2} e^{2}\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}} - \frac {2 \, {\left (3 \, c^{4} d^{2} - c^{2} e^{2}\right )} \log \left (e x + d\right )}{c^{6} d^{6} + 3 \, c^{4} d^{4} e^{2} + 3 \, c^{2} d^{2} e^{4} + e^{6}} + \frac {4 \, c^{2} d e x + 5 \, c^{2} d^{2} + e^{2}}{c^{4} d^{6} + 2 \, c^{2} d^{4} e^{2} + d^{2} e^{4} + {\left (c^{4} d^{4} e^{2} + 2 \, c^{2} d^{2} e^{4} + e^{6}\right )} x^{2} + 2 \, {\left (c^{4} d^{5} e + 2 \, c^{2} d^{3} e^{3} + d e^{5}\right )} x} - \frac {2 \, {\left (c^{6} d^{3} - 3 \, c^{4} d e^{2}\right )} \arctan \left (c x\right )}{{\left (c^{6} d^{6} e + 3 \, c^{4} d^{4} e^{3} + 3 \, c^{2} d^{2} e^{5} + e^{7}\right )} c}\right )} + \frac {2 \, \arctan \left (c x\right )}{e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e}\right )} b - \frac {a}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/(e*x+d)^4,x, algorithm="maxima")

[Out]

-1/6*(c*((3*c^4*d^2 - c^2*e^2)*log(c^2*x^2 + 1)/(c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6) - 2*(3*c^4*d^2
- c^2*e^2)*log(e*x + d)/(c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6) + (4*c^2*d*e*x + 5*c^2*d^2 + e^2)/(c^
4*d^6 + 2*c^2*d^4*e^2 + d^2*e^4 + (c^4*d^4*e^2 + 2*c^2*d^2*e^4 + e^6)*x^2 + 2*(c^4*d^5*e + 2*c^2*d^3*e^3 + d*e
^5)*x) - 2*(c^6*d^3 - 3*c^4*d*e^2)*arctan(c*x)/((c^6*d^6*e + 3*c^4*d^4*e^3 + 3*c^2*d^2*e^5 + e^7)*c)) + 2*arct
an(c*x)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e))*b - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))/(d + e*x)^4,x)

[Out]

int((a + b*atan(c*x))/(d + e*x)^4, x)

________________________________________________________________________________________

sympy [A]  time = 20.04, size = 9229, normalized size = 44.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x))/(e*x+d)**4,x)

[Out]

Piecewise((a*x/d**4, Eq(c, 0) & Eq(e, 0)), (-a/(3*d**3*e + 9*d**2*e**2*x + 9*d*e**3*x**2 + 3*e**4*x**3), Eq(c,
0)), (24*a*d**3/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) - 21*I*b*d**3*atanh(e
*x/d)/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) - 10*I*b*d**3/(-72*d**6*e - 216*
d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) + 9*I*b*d**2*e*x*atanh(e*x/d)/(-72*d**6*e - 216*d**5*e**
2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) - 9*I*b*d**2*e*x/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x
**2 - 72*d**3*e**4*x**3) + 9*I*b*d*e**2*x**2*atanh(e*x/d)/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 -
72*d**3*e**4*x**3) - 3*I*b*d*e**2*x**2/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3
) + 3*I*b*e**3*x**3*atanh(e*x/d)/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3), Eq(c
, -I*e/d)), (24*a*d**3/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) + 21*I*b*d**3*a
tanh(e*x/d)/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) + 10*I*b*d**3/(-72*d**6*e
- 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) - 9*I*b*d**2*e*x*atanh(e*x/d)/(-72*d**6*e - 216*d*
*5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3) + 9*I*b*d**2*e*x/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*
e**3*x**2 - 72*d**3*e**4*x**3) - 9*I*b*d*e**2*x**2*atanh(e*x/d)/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*
x**2 - 72*d**3*e**4*x**3) + 3*I*b*d*e**2*x**2/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**
4*x**3) - 3*I*b*e**3*x**3*atanh(e*x/d)/(-72*d**6*e - 216*d**5*e**2*x - 216*d**4*e**3*x**2 - 72*d**3*e**4*x**3)
, Eq(c, I*e/d)), ((a*x + b*x*atan(c*x) - b*log(x**2 + c**(-2))/(2*c))/d**4, Eq(e, 0)), (-2*a*c**6*d**6/(6*c**6
*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d
**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c*
*2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 6
*a*c**4*d**4*e**2/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c
**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54
*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9
*x**2 + 6*e**10*x**3) - 6*a*c**2*d**2*e**4/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c
**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3
+ 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 1
8*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 2*a*e**6/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*
e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4
*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 +
6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 6*b*c**6*d**5*e*x*atan(c*x)/(6*c**6*d**9*e + 1
8*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x
+ 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**
7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 6*b*c**6*d**
4*e**2*x**2*atan(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 +
18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5
+ 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*
e**9*x**2 + 6*e**10*x**3) + 2*b*c**6*d**3*e**3*x**3*atan(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d
**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*
c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**
3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 3*b*c**5*d**5*e*log(x**2 + c**(-2))/(6*c**
6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*
d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c
**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) +
6*b*c**5*d**5*e*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*
x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**
5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x
+ 18*d*e**9*x**2 + 6*e**10*x**3) - 5*b*c**5*d**5*e/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x*
*2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e
**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*
e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 9*b*c**5*d**4*e**2*x*log(x**2 + c**(-2))/(6*c**6*d**9
*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e
**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d*
*3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 18*b*c
**5*d**4*e**2*x*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*
x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**
5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x
+ 18*d*e**9*x**2 + 6*e**10*x**3) - 9*b*c**5*d**4*e**2*x/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e*
*3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d
**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*
d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 9*b*c**5*d**3*e**3*x**2*log(x**2 + c**(-2))/(6*c
**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**
4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54
*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3)
+ 18*b*c**5*d**3*e**3*x**2*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6
*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 +
18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d
**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 4*b*c**5*d**3*e**3*x**2/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 1
8*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x*
*2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*
e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 3*b*c**5*d**2*e**4*x**3*log(x**2 +
c**(-2))/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7
*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d*
*4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 +
6*e**10*x**3) + 6*b*c**5*d**2*e**4*x**3*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*
x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4
*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**
3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 12*b*c**4*d**4*e**2*atan(c*x)/(6*c**6*d**9*e + 18*c
**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 5
4*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x
**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 18*b*c**4*d**3*
e**3*x*atan(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c*
*4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*
c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*
x**2 + 6*e**10*x**3) - 18*b*c**4*d**2*e**4*x**2*atan(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*
e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4
*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 +
6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 6*b*c**4*d*e**5*x**3*atan(c*x)/(6*c**6*d**9*e
+ 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4
*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*
e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + b*c**3*d*
*3*e**3*log(x**2 + c**(-2))/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x
**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5
*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x +
18*d*e**9*x**2 + 6*e**10*x**3) - 2*b*c**3*d**3*e**3*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c*
*6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 +
18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8
*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 6*b*c**3*d**3*e**3/(6*c**6*d**9*e + 18
*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x +
54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7
*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 3*b*c**3*d**2
*e**4*x*log(x**2 + c**(-2))/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x
**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5
*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x +
18*d*e**9*x**2 + 6*e**10*x**3) - 6*b*c**3*d**2*e**4*x*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*
c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2
+ 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e*
*8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 10*b*c**3*d**2*e**4*x/(6*c**6*d**9*e
+ 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**
4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3
*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 3*b*c**3
*d*e**5*x**2*log(x**2 + c**(-2))/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e
**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2
*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**
8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 6*b*c**3*d*e**5*x**2*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x
+ 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5
*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d*
*2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 4*b*c**3*d*e**5*x**2/(6*c**6*d*
*9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6
*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*
d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + b*c*
*3*e**6*x**3*log(x**2 + c**(-2))/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e
**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2
*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**
8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 2*b*c**3*e**6*x**3*log(d/e + x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x +
18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x
**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2
*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 6*b*c**2*d**2*e**4*atan(c*x)/(6*c
**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**
4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54
*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3)
- b*c*d*e**5/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d
**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2
*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2
+ 6*e**10*x**3) - b*c*e**6*x/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**3*x**2 + 6*c**6*d**6*e**4
*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d**4*e**6*x**3 + 18*c**2*d*
*5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d**3*e**7 + 18*d**2*e**8*x
+ 18*d*e**9*x**2 + 6*e**10*x**3) - 2*b*e**6*atan(c*x)/(6*c**6*d**9*e + 18*c**6*d**8*e**2*x + 18*c**6*d**7*e**
3*x**2 + 6*c**6*d**6*e**4*x**3 + 18*c**4*d**7*e**3 + 54*c**4*d**6*e**4*x + 54*c**4*d**5*e**5*x**2 + 18*c**4*d*
*4*e**6*x**3 + 18*c**2*d**5*e**5 + 54*c**2*d**4*e**6*x + 54*c**2*d**3*e**7*x**2 + 18*c**2*d**2*e**8*x**3 + 6*d
**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3), True))

________________________________________________________________________________________