\(\int \frac {a+b \arctan (c x)}{x^2 (d+e x^2)^2} \, dx\) [1164]

Optimal result

Integrand size = 21, antiderivative size = 1382 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {b c \log (x)}{d^2}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \log \left (-\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \log \left (\frac {\sqrt {e} \left (1+\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{5/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {b c e \log \left (d+e x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}} \]

[Out]

(-a-b*arctan(c*x))/d^2/x-1/2*e*x*(a+b*arctan(c*x))/d^2/(e*x^2+d)+b*c*ln(x)/d^2-1/2*b*c*ln(c^2*x^2+1)/d^2+1/4*b
*c*e*ln(c^2*x^2+1)/d^2/(c^2*d-e)-1/4*b*c*e*ln(e*x^2+d)/d^2/(c^2*d-e)-a*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(5/
2)-1/2*(a+b*arctan(c*x))*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(5/2)-1/8*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)+I
*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)-I*e^(1/2)))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)-1/8*I*b*c*ln((1-x*(-c^2)^(1/2))*e^(
1/2)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1/2)))*ln(1-I*x*e^(1/2)/d^(1/2))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)+1/4*I*b*ln(1+I*c
*x)*ln(c*((-d)^(1/2)-x*e^(1/2))/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(5/2)-1/4*I*b*ln(1+I*c*x)*ln(c*((-d)^(1
/2)+x*e^(1/2))/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+1/4*I*b*polylog(2,(1+I*c*x)*e^(1/2)/(I*c*(-d)^(1/2
)+e^(1/2)))*e^(1/2)/(-d)^(5/2)-1/8*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)-I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)-I*
e^(1/2)))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)+1/4*I*b*ln(1-I*c*x)*ln(c*((-d)^(1/2)+x*e^(1/2))/(c*(-d)^(1/2)-I*e^(1/2)
))*e^(1/2)/(-d)^(5/2)+1/8*I*b*c*ln(-(1+x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)-e^(1/2)))*ln(1-I*x*e^(1
/2)/d^(1/2))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)-1/8*I*b*c*ln((1+x*(-c^2)^(1/2))*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)+e^(1
/2)))*ln(1+I*x*e^(1/2)/d^(1/2))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)+1/4*I*b*polylog(2,(1-I*c*x)*e^(1/2)/(I*c*(-d)^(1/
2)+e^(1/2)))*e^(1/2)/(-d)^(5/2)+1/8*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/2)-I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)+I
*e^(1/2)))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)-1/4*I*b*ln(1-I*c*x)*ln(c*((-d)^(1/2)-x*e^(1/2))/(c*(-d)^(1/2)+I*e^(1/2
)))*e^(1/2)/(-d)^(5/2)-1/4*I*b*polylog(2,(I-c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)-1/4*I*b*
polylog(2,(I+c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+1/8*I*b*c*polylog(2,(-c^2)^(1/2)*(d^(1/
2)+I*x*e^(1/2))/((-c^2)^(1/2)*d^(1/2)+I*e^(1/2)))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)+1/8*I*b*c*ln(-(1-x*(-c^2)^(1/2)
)*e^(1/2)/(I*(-c^2)^(1/2)*d^(1/2)-e^(1/2)))*ln(1+I*x*e^(1/2)/d^(1/2))*e^(1/2)/d^(5/2)/(-c^2)^(1/2)

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 1382, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {5100, 4946, 272, 36, 29, 31, 205, 211, 5032, 6857, 455, 5028, 2456, 2441, 2440, 2438, 5030} \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {\sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (a+b \arctan (c x))}{2 d^{5/2}}-\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (e x^2+d\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}+\frac {b c \log (x)}{d^2}+\frac {i b \sqrt {e} \log (i c x+1) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (i c x+1) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \log \left (\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \log \left (-\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \log \left (-\frac {\sqrt {e} \left (1-\sqrt {-c^2} x\right )}{i \sqrt {-c^2} \sqrt {d}-\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \log \left (\frac {\sqrt {e} \left (\sqrt {-c^2} x+1\right )}{i \sqrt {-c^2} \sqrt {d}+\sqrt {e}}\right ) \log \left (\frac {i \sqrt {e} x}{\sqrt {d}}+1\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {b c e \log \left (c^2 x^2+1\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d^2}-\frac {b c e \log \left (e x^2+d\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}}-\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}}+\frac {i b c \sqrt {e} \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} d^{5/2}} \]

[In]

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

[Out]

-((a + b*ArcTan[c*x])/(d^2*x)) - (e*x*(a + b*ArcTan[c*x]))/(2*d^2*(d + e*x^2)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)
/Sqrt[d]])/d^(5/2) - (Sqrt[e]*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)) + (b*c*Log[x])/d^2
+ ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(5/2) - ((I/4
)*b*Sqrt[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) + ((I/4)*b*Sqr
t[e]*Log[1 - I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(5/2) - ((I/4)*b*Sqrt[e]*Lo
g[1 + I*c*x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) - ((I/8)*b*c*Sqrt[e]*Log[(Sq
rt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2
)) + ((I/8)*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqrt[
e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*Log[-((Sqrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqr
t[d] - Sqrt[e]))]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)) - ((I/8)*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 +
Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(5/2)) - (b*c*L
og[1 + c^2*x^2])/(2*d^2) + (b*c*e*Log[1 + c^2*x^2])/(4*d^2*(c^2*d - e)) - (b*c*e*Log[d + e*x^2])/(4*d^2*(c^2*d
- e)) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(5/2) + ((I/4)*b*Sqrt
[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(5/2) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt
[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(5/2) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I + c*x))/(c*Sq
rt[-d] + I*Sqrt[e])])/(-d)^(5/2) - ((I/8)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c
^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[
e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)) - ((I/8)*b*c*Sqrt[e]*PolyLog[2, (Sqrt[-c^2]*(Sq
rt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2)) + ((I/8)*b*c*Sqrt[e]*PolyLog[2,
(Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(5/2))

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 211

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 455

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 4946

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^
n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))),
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1]

Rule 5028

Int[ArcTan[(c_.)*(x_)]/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[I/2, Int[Log[1 - I*c*x]/(d + e*x^2), x], x] -
Dist[I/2, Int[Log[1 + I*c*x]/(d + e*x^2), x], x] /; FreeQ[{c, d, e}, x]

Rule 5030

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

Rule 5032

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[u/(1 + c^2*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x]
&& (IntegerQ[q] || ILtQ[q + 1/2, 0])

Rule 5100

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> With
[{u = ExpandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x^2)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b,
c, d, e, f, m}, x] && IntegerQ[q] && IGtQ[p, 0] && ((EqQ[p, 1] && GtQ[q, 0]) || IntegerQ[m])

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \arctan (c x)}{d^2 x^2}-\frac {e (a+b \arctan (c x))}{d \left (d+e x^2\right )^2}-\frac {e (a+b \arctan (c x))}{d^2 \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {a+b \arctan (c x)}{x^2} \, dx}{d^2}-\frac {e \int \frac {a+b \arctan (c x)}{d+e x^2} \, dx}{d^2}-\frac {e \int \frac {a+b \arctan (c x)}{\left (d+e x^2\right )^2} \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac {(a e) \int \frac {1}{d+e x^2} \, dx}{d^2}-\frac {(b e) \int \frac {\arctan (c x)}{d+e x^2} \, dx}{d^2}+\frac {(b c e) \int \frac {\frac {x}{2 d \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}-\frac {(i b e) \int \frac {\log (1-i c x)}{d+e x^2} \, dx}{2 d^2}+\frac {(i b e) \int \frac {\log (1+i c x)}{d+e x^2} \, dx}{2 d^2}+\frac {(b c e) \int \left (\frac {x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{d} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {(b c) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^2}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (b c \sqrt {e}\right ) \int \frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{2 d^{5/2}}-\frac {(i b e) \int \left (\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (1-i c x)}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d^2}+\frac {(i b e) \int \left (\frac {\sqrt {-d} \log (1+i c x)}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log (1+i c x)}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{2 d^2}+\frac {(b c e) \int \frac {x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^2} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {b c \log (x)}{d^2}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 d^{5/2}}-\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{4 d^{5/2}}+\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{5/2}}+\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{5/2}}-\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{5/2}}-\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{5/2}}+\frac {(b c e) \text {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 d^2} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {b c \log (x)}{d^2}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {\left (b c \sqrt {e}\right ) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{-i c \sqrt {-d}+\sqrt {e}}\right )}{1-i c x} \, dx}{4 (-d)^{5/2}}+\frac {\left (b c \sqrt {e}\right ) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}-\sqrt {e} x\right )}{i c \sqrt {-d}+\sqrt {e}}\right )}{1+i c x} \, dx}{4 (-d)^{5/2}}-\frac {\left (b c \sqrt {e}\right ) \int \frac {\log \left (-\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{-i c \sqrt {-d}-\sqrt {e}}\right )}{1-i c x} \, dx}{4 (-d)^{5/2}}-\frac {\left (b c \sqrt {e}\right ) \int \frac {\log \left (\frac {i c \left (\sqrt {-d}+\sqrt {e} x\right )}{i c \sqrt {-d}-\sqrt {e}}\right )}{1+i c x} \, dx}{4 (-d)^{5/2}}+\frac {\left (i b c \sqrt {e}\right ) \int \left (\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{4 d^{5/2}}-\frac {\left (i b c \sqrt {e}\right ) \int \left (\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1-\sqrt {-c^2} x\right )}+\frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{2 \left (1+\sqrt {-c^2} x\right )}\right ) \, dx}{4 d^{5/2}}+\frac {\left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {\left (b c e^2\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{4 d^2 \left (c^2 d-e\right )} \\ & = -\frac {a+b \arctan (c x)}{d^2 x}-\frac {e x (a+b \arctan (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}-\frac {\sqrt {e} (a+b \arctan (c x)) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+\frac {b c \log (x)}{d^2}+\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}+\frac {i b \sqrt {e} \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {i b \sqrt {e} \log (1+i c x) \log \left (\frac {c \left (\sqrt {-d}+\sqrt {e} x\right )}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{5/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {b c e \log \left (d+e x^2\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{-i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 (-d)^{5/2}}+\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{i c \sqrt {-d}-\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 (-d)^{5/2}}+\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{-i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1-i c x\right )}{4 (-d)^{5/2}}-\frac {\left (i b \sqrt {e}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{i c \sqrt {-d}+\sqrt {e}}\right )}{x} \, dx,x,1+i c x\right )}{4 (-d)^{5/2}}+\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 d^{5/2}}+\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 d^{5/2}}-\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1-\sqrt {-c^2} x} \, dx}{8 d^{5/2}}-\frac {\left (i b c \sqrt {e}\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{1+\sqrt {-c^2} x} \, dx}{8 d^{5/2}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.75 (sec) , antiderivative size = 992, normalized size of antiderivative = 0.72 \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {a}{d^2 x}-\frac {a e x}{2 d^2 \left (d+e x^2\right )}-\frac {3 a \sqrt {e} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{5/2}}+b c^5 \left (-\frac {\arctan (c x)}{c^5 d^2 x}+\frac {\log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )}{c^4 d^2}-\frac {e \log \left (1-\frac {\left (-c^2 d+e\right ) \cos (2 \arctan (c x))}{c^2 d+e}\right )}{4 c^4 d^2 \left (c^2 d-e\right )}-\frac {3 e \left (4 \arctan (c x) \text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+2 \arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right ) \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )-\left (\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (1-\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )+\left (-\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )-2 i \text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (1-\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )+\left (\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )-2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+\left (\arccos \left (\frac {-c^2 d-e}{c^2 d-e}\right )+2 i \left (\text {arctanh}\left (\frac {c d}{\sqrt {-c^2 d e} x}\right )+\text {arctanh}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \arctan (c x)}}{\sqrt {c^2 d-e} \sqrt {c^2 d+e+\left (c^2 d-e\right ) \cos (2 \arctan (c x))}}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e-2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c^2 d+e+2 i \sqrt {-c^2 d e}\right ) \left (2 c^2 d-2 c \sqrt {-c^2 d e} x\right )}{\left (c^2 d-e\right ) \left (2 c^2 d+2 c \sqrt {-c^2 d e} x\right )}\right )\right )\right )}{8 c^4 d^2 \sqrt {-c^2 d e}}-\frac {e \arctan (c x) \sin (2 \arctan (c x))}{2 c^4 d^2 \left (c^2 d+e+c^2 d \cos (2 \arctan (c x))-e \cos (2 \arctan (c x))\right )}\right ) \]

[In]

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

[Out]

-(a/(d^2*x)) - (a*e*x)/(2*d^2*(d + e*x^2)) - (3*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(5/2)) + b*c^5*(-(
ArcTan[c*x]/(c^5*d^2*x)) + Log[(c*x)/Sqrt[1 + c^2*x^2]]/(c^4*d^2) - (e*Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*
x]])/(c^2*d + e)])/(4*c^4*d^2*(c^2*d - e)) - (3*e*(4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCo
s[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*
ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e
)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTan
h[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))
/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*(ArcTanh[(c*d
)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*
E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] + (
2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*
E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d
+ e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)
]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*
c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))])))/(8*c^4*d^2*Sqrt[-(c^2*d*e)]) - (e*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(2*c^4*
d^2*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2567 vs. \(2 (1038 ) = 2076\).

Time = 1.48 (sec) , antiderivative size = 2568, normalized size of antiderivative = 1.86

method	result	size
risch	\(\text {Expression too large to display}\)	\(2568\)
parts	\(\text {Expression too large to display}\)	\(3558\)
derivativedivides	\(\text {Expression too large to display}\)	\(3591\)
default	\(\text {Expression too large to display}\)	\(3591\)

[In]

int((a+b*arctan(c*x))/x^2/(e*x^2+d)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4*b*c^2/d*e/(c^2*d-e)/(e*d)^(1/2)*arctanh(1/2*(2*(1+I*c*x)*e-2*e)/c/(e*d)^(1/2))-1/4*b*c^3/d*e*ln(1+I*c*x)/
(c^2*d-e)/(-c^2*e*x^2-c^2*d)+1/8*b*c^4/d*e^2*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^
(1/2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*x^2-1/8*b*c^4/d*e^2*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/
2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))*x^2-1/8*b*c^2/d^2*e^3*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2
-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*x^2+1/8*b*c^2/d^2*e^3*ln(1+I*c*x)/(c^2
*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))*x^2-1/8*b*c^2/d*e^2*l
n(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))+1/8*b*
c^2/d*e^2*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)
-e))-1/4*I*b*c^4/d*e*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x+1/4*I*b*c^2/d^2*e^2*ln(1+I*c*x)/(c^2*d-e)/(-c^
2*e*x^2-c^2*d)*x+1/8*c^2*b/d*e^2*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1-I*c
*x)*e-e)/(c*(e*d)^(1/2)-e))-1/8*c^2*b/d*e^2*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(
1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))+1/4*I*c^4*b*ln(1-I*c*x)/d/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*e*x-1/4*I*c^2*b/
d^2*e^2*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x-a/x/d^2-1/4*c^2*b/d*e/(c^2*d-e)/(e*d)^(1/2)*arctanh(1/2*(2*
(1-I*c*x)*e-2*e)/c/(e*d)^(1/2))-1/4*c^3*b/d*e*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)-1/8*c^2*b/d^2*e^3*ln(1-
I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*x^2+1/8*c^
4*b*ln(1-I*c*x)/d/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))
*e^2*x^2-1/8*c^4*b*ln(1-I*c*x)/d/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1-I*c*x)*e-e)/(c*
(e*d)^(1/2)-e))*e^2*x^2+1/8*c^2*b/d^2*e^3*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/
2)+(1-I*c*x)*e-e)/(c*(e*d)^(1/2)-e))*x^2+1/2*b*c/d^2*ln(I*c*x)-1/2*b*c/d^2*ln(1+I*c*x)+1/4*b/d^2*e*ln(1-I*c*x)
/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1-I*c*x)*e-e)/(c*(e*d)^(1/2)-e))+1/2*c^2*a/d^2*e*x/(-c^2*e*x^2-c^2*d)-1/4*b/d^
2*e*ln(1-I*c*x)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))-1/8*c*b/d^2*e/(c^2*d-e)*ln((1-
I*c*x)^2*e-c^2*d-2*(1-I*c*x)*e+e)-3/2*I*a/d^2*e/(e*d)^(1/2)*arctanh(1/2*(2*(1-I*c*x)*e-2*e)/c/(e*d)^(1/2))+1/2
*c*b/d^2*ln(-I*c*x)-1/2*c*b/d^2*ln(1-I*c*x)+1/8*c^4*b*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln(
(c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))*e-1/4*c^3*b/d^2*e^2*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*
x^2-1/8*c^4*b*ln(1-I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1-I*c*x)*e-e)/(c*(e*d)^(
1/2)-e))*e-1/4*b*c^3/d^2*e^2*ln(1+I*c*x)/(c^2*d-e)/(-c^2*e*x^2-c^2*d)*x^2+1/8*b*c^4*e*ln(1+I*c*x)/(c^2*d-e)/(-
c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))-1/8*b*c^4*e*ln(1+I*c*x)/(c^2*
d-e)/(-c^2*e*x^2-c^2*d)/(e*d)^(1/2)*ln((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))-3/8*b/d^2*e/(e*d)^(1/2
)*dilog((c*(e*d)^(1/2)-(1-I*c*x)*e+e)/(c*(e*d)^(1/2)+e))+3/8*b/d^2*e/(e*d)^(1/2)*dilog((c*(e*d)^(1/2)+(1-I*c*x
)*e-e)/(c*(e*d)^(1/2)-e))-1/2*I*b/d^2*ln(1-I*c*x)/x-3/8*b/d^2*e/(e*d)^(1/2)*dilog((c*(e*d)^(1/2)-(1+I*c*x)*e+e
)/(c*(e*d)^(1/2)+e))+3/8*b/d^2*e/(e*d)^(1/2)*dilog((c*(e*d)^(1/2)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))+1/2*I*b/d^
2*ln(1+I*c*x)/x-1/8*b*c/d^2*e/(c^2*d-e)*ln((1+I*c*x)^2*e-c^2*d-2*(1+I*c*x)*e+e)-1/4*b/d^2*e*ln(1+I*c*x)/(e*d)^
(1/2)*ln((c*(e*d)^(1/2)-(1+I*c*x)*e+e)/(c*(e*d)^(1/2)+e))+1/4*b/d^2*e*ln(1+I*c*x)/(e*d)^(1/2)*ln((c*(e*d)^(1/2
)+(1+I*c*x)*e-e)/(c*(e*d)^(1/2)-e))

Fricas [F]

\[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))/x^2/(e*x^2+d)^2,x, algorithm="fricas")

[Out]

integral((b*arctan(c*x) + a)/(e^2*x^6 + 2*d*e*x^4 + d^2*x^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*atan(c*x))/x**2/(e*x**2+d)**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*arctan(c*x))/x^2/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arctan \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{2}} \,d x } \]

[In]

integrate((a+b*arctan(c*x))/x^2/(e*x^2+d)^2,x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arctan (c x)}{x^2 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

int((a + b*atan(c*x))/(x^2*(d + e*x^2)^2),x)

[Out]

int((a + b*atan(c*x))/(x^2*(d + e*x^2)^2), x)