3.1172 \(\int \frac {a+b \tan ^{-1}(c x)}{x^2 (d+e x^2)^3} \, dx\)

Optimal. Leaf size=1518 \[ -\frac {7 x \left (a+b \tan ^{-1}(c x)\right ) e}{8 d^3 \left (e x^2+d\right )}-\frac {x \left (a+b \tan ^{-1}(c x)\right ) e}{4 d^2 \left (e x^2+d\right )^2}+\frac {b c \log \left (c^2 x^2+1\right ) e}{4 d^3 \left (c^2 d-e\right )}+\frac {b c \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) e}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c \log \left (e x^2+d\right ) e}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) e}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (e x^2+d\right )}-\frac {7 \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \sqrt {e}}{8 d^{7/2}}-\frac {a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \sqrt {e}}{d^{7/2}}-\frac {i b \log (i c x+1) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{c \sqrt {-d}-i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}+\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {-d}-\sqrt {e} x\right )}{\sqrt {-d} c+i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {i b \log (1-i c x) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{c \sqrt {-d}-i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}+\frac {i b \log (i c x+1) \log \left (\frac {c \left (\sqrt {e} x+\sqrt {-d}\right )}{\sqrt {-d} c+i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {7 i b c \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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 i b c \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 ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right ) \sqrt {e}}{4 (-d)^{7/2}}-\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \text {Li}_2\left (\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right ) \sqrt {e}}{32 \sqrt {-c^2} d^{7/2}}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}+\frac {b c \log (x)}{d^3}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d^3} \]

[Out]

(-a-b*arctan(c*x))/d^3/x+b*c*ln(x)/d^3-1/2*b*c*ln(c^2*x^2+1)/d^3-7/8*(a+b*arctan(c*x))*arctan(x*e^(1/2)/d^(1/2
))*e^(1/2)/d^(7/2)-a*arctan(x*e^(1/2)/d^(1/2))*e^(1/2)/d^(7/2)-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)^(7/2)-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)^(
7/2)+1/4*I*b*polylog(2,(I-c*x)*e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(7/2)+1/4*I*b*polylog(2,(I+c*x)*
e^(1/2)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(7/2)-7/32*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^(7/2)/(-c^2)^(1/2)-7/32*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^(7/2)/(-c^2)^(1/2)+7/32*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^(7/2)/(-c^2)^(1
/2)+7/32*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^(7/2)/(-c^2)^(1/2)+7/32*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^(7/2)/(-c^2)^(1/2)+7/32*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^(7/2)/(-c^2)^(1/2)+1/16*b*c*(5*c^2*d-3*e)*e*ln(c^2*x^2+1)/d^3/(c^2*d-e)^2-1/16*b*c*(5*
c^2*d-3*e)*e*ln(e*x^2+d)/d^3/(c^2*d-e)^2-7/32*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^(7/2)/(-c^2)^(1/2)-7/32*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^(7/2)/(-c^2)^(1/2)-1/4*e*x*(a+b*arctan(c*x))/d^2/(e*x^2+d)^2-7/8*e*x*(
a+b*arctan(c*x))/d^3/(e*x^2+d)+1/4*b*c*e*ln(c^2*x^2+1)/d^3/(c^2*d-e)-1/4*b*c*e*ln(e*x^2+d)/d^3/(c^2*d-e)-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)^(7/2)-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)^(7/2)+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)^(7/2)+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)^(7/2)+1/8*b*c*e/d^2/(c^2*d-e)/(e*x^2+d)

________________________________________________________________________________________

Rubi [A]  time = 2.64, antiderivative size = 1518, normalized size of antiderivative = 1.00, number of steps used = 73, number of rules used = 19, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {4980, 4852, 266, 36, 29, 31, 199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391, 444, 4910} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^3),x]

[Out]

(b*c*e)/(8*d^2*(c^2*d - e)*(d + e*x^2)) - (a + b*ArcTan[c*x])/(d^3*x) - (e*x*(a + b*ArcTan[c*x]))/(4*d^2*(d +
e*x^2)^2) - (7*e*x*(a + b*ArcTan[c*x]))/(8*d^3*(d + e*x^2)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(7/2)
- (7*Sqrt[e]*(a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)) + (b*c*Log[x])/d^3 - ((I/4)*b*Sqrt[e
]*Log[1 + I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(7/2) + ((I/4)*b*Sqrt[e]*Log[1
- I*c*x]*Log[(c*(Sqrt[-d] - Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(7/2) - ((I/4)*b*Sqrt[e]*Log[1 - I*c*
x]*Log[(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] - I*Sqrt[e])])/(-d)^(7/2) + ((I/4)*b*Sqrt[e]*Log[1 + I*c*x]*Log[
(c*(Sqrt[-d] + Sqrt[e]*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(7/2) - (((7*I)/32)*b*c*Sqrt[e]*Log[(Sqrt[e]*(1 - S
qrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(7/2)) + (((7*I)
/32)*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)/S
qrt[d]])/(Sqrt[-c^2]*d^(7/2)) + (((7*I)/32)*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^(7/2)) - (((7*I)/32)*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^(7/2)) - (b*
c*Log[1 + c^2*x^2])/(2*d^3) + (b*c*(5*c^2*d - 3*e)*e*Log[1 + c^2*x^2])/(16*d^3*(c^2*d - e)^2) + (b*c*e*Log[1 +
c^2*x^2])/(4*d^3*(c^2*d - e)) - (b*c*(5*c^2*d - 3*e)*e*Log[d + e*x^2])/(16*d^3*(c^2*d - e)^2) - (b*c*e*Log[d
+ e*x^2])/(4*d^3*(c^2*d - e)) + ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(I - c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d
)^(7/2) - ((I/4)*b*Sqrt[e]*PolyLog[2, (Sqrt[e]*(1 - I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(7/2) - ((I/4)*b*S
qrt[e]*PolyLog[2, (Sqrt[e]*(1 + I*c*x))/(I*c*Sqrt[-d] + Sqrt[e])])/(-d)^(7/2) + ((I/4)*b*Sqrt[e]*PolyLog[2, (S
qrt[e]*(I + c*x))/(c*Sqrt[-d] + I*Sqrt[e])])/(-d)^(7/2) - (((7*I)/32)*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^(7/2)) + (((7*I)/32)*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^(7/2)) - (((7*I)/32)*
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^(
7/2)) + (((7*I)/32)*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^(7/2))

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 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 199

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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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 266

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 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 571

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

Rule 2391

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 2393

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 2394

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 2409

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 4852

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

Rule 4908

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 4910

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 4912

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 4980

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 6725

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*} \int \frac {a+b \tan ^{-1}(c x)}{x^2 \left (d+e x^2\right )^3} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d^3 x^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d \left (d+e x^2\right )^3}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )^2}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d^3 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx}{d^3}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{d+e x^2} \, dx}{d^3}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx}{d^2}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d^3}-\frac {(a e) \int \frac {1}{d+e x^2} \, dx}{d^3}-\frac {(b e) \int \frac {\tan ^{-1}(c x)}{d+e x^2} \, dx}{d^3}+\frac {(b c e) \int \frac {\frac {x}{2 d \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{d^2}+\frac {(b c e) \int \frac {\frac {x}{4 d \left (d+e x^2\right )^2}+\frac {3 x}{8 d^2 \left (d+e x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}}{1+c^2 x^2} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^3}-\frac {(i b e) \int \frac {\log (1-i c x)}{d+e x^2} \, dx}{2 d^3}+\frac {(i b e) \int \frac {\log (1+i c x)}{d+e x^2} \, dx}{2 d^3}+\frac {(b c e) \int \left (\frac {x}{2 d \left (1+c^2 x^2\right ) \left (d+e x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}+\frac {(b c e) \int \left (\frac {x \left (5 d+3 e x^2\right )}{8 d^2 \left (1+c^2 x^2\right ) \left (d+e x^2\right )^2}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e} \left (1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d^3}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^3}+\frac {\left (3 b c \sqrt {e}\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{8 d^{7/2}}+\frac {\left (b c \sqrt {e}\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{1+c^2 x^2} \, dx}{2 d^{7/2}}-\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^3}+\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^3}+\frac {(b c e) \int \frac {x \left (5 d+3 e x^2\right )}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )^2} \, dx}{8 d^3}+\frac {(b c e) \int \frac {x}{\left (1+c^2 x^2\right ) \left (d+e x^2\right )} \, dx}{2 d^3}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {\left (3 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}{16 d^{7/2}}-\frac {\left (3 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}{16 d^{7/2}}+\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^{7/2}}-\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^{7/2}}-\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{7/2}}-\frac {(i b e) \int \frac {\log (1-i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{7/2}}+\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{4 (-d)^{7/2}}+\frac {(i b e) \int \frac {\log (1+i c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{4 (-d)^{7/2}}+\frac {(b c e) \operatorname {Subst}\left (\int \frac {5 d+3 e x}{\left (1+c^2 x\right ) (d+e x)^2} \, dx,x,x^2\right )}{16 d^3}+\frac {(b c e) \operatorname {Subst}\left (\int \frac {1}{\left (1+c^2 x\right ) (d+e x)} \, dx,x,x^2\right )}{4 d^3}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\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)^{7/2}}+\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)^{7/2}}-\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)^{7/2}}+\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)^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}-\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)^{7/2}}-\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)^{7/2}}+\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)^{7/2}}+\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)^{7/2}}+\frac {\left (3 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}{16 d^{7/2}}-\frac {\left (3 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}{16 d^{7/2}}+\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^{7/2}}-\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^{7/2}}+\frac {(b c e) \operatorname {Subst}\left (\int \left (\frac {5 c^4 d-3 c^2 e}{\left (c^2 d-e\right )^2 \left (1+c^2 x\right )}-\frac {2 d e}{\left (c^2 d-e\right ) (d+e x)^2}+\frac {e \left (-5 c^2 d+3 e\right )}{\left (-c^2 d+e\right )^2 (d+e x)}\right ) \, dx,x,x^2\right )}{16 d^3}+\frac {\left (b c^3 e\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {\left (b c e^2\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{4 d^3 \left (c^2 d-e\right )}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\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)^{7/2}}+\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)^{7/2}}-\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)^{7/2}}+\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)^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {\left (i b \sqrt {e}\right ) \operatorname {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)^{7/2}}-\frac {\left (i b \sqrt {e}\right ) \operatorname {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)^{7/2}}-\frac {\left (i b \sqrt {e}\right ) \operatorname {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)^{7/2}}+\frac {\left (i b \sqrt {e}\right ) \operatorname {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)^{7/2}}+\frac {\left (3 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}{32 d^{7/2}}+\frac {\left (3 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}{32 d^{7/2}}-\frac {\left (3 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}{32 d^{7/2}}-\frac {\left (3 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}{32 d^{7/2}}+\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^{7/2}}+\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^{7/2}}-\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^{7/2}}-\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^{7/2}}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\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)^{7/2}}+\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)^{7/2}}-\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)^{7/2}}+\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)^{7/2}}-\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {(3 b c e) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^4}+\frac {(3 b c e) \int \frac {\log \left (\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^4}-\frac {(3 b c e) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^4}-\frac {(3 b c e) \int \frac {\log \left (\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{32 \sqrt {-c^2} d^4}+\frac {(b c e) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{8 \sqrt {-c^2} d^4}+\frac {(b c e) \int \frac {\log \left (\frac {i \sqrt {e} \left (1-\sqrt {-c^2} x\right )}{\sqrt {d} \left (\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{8 \sqrt {-c^2} d^4}-\frac {(b c e) \int \frac {\log \left (-\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1-\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{8 \sqrt {-c^2} d^4}-\frac {(b c e) \int \frac {\log \left (\frac {i \sqrt {e} \left (1+\sqrt {-c^2} x\right )}{\sqrt {d} \left (-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}\right )}\right )}{1+\frac {i \sqrt {e} x}{\sqrt {d}}} \, dx}{8 \sqrt {-c^2} d^4}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\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)^{7/2}}+\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)^{7/2}}-\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)^{7/2}}+\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)^{7/2}}-\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{7/2}}+\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{7/2}}+\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{7/2}}-\frac {\left (3 i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{32 \sqrt {-c^2} d^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}-\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{7/2}}+\frac {\left (i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-c^2} x}{-\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{7/2}}-\frac {\left (i b c \sqrt {e}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-c^2} x}{\sqrt {-c^2}+\frac {i \sqrt {e}}{\sqrt {d}}}\right )}{x} \, dx,x,1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {-c^2} d^{7/2}}\\ &=\frac {b c e}{8 d^2 \left (c^2 d-e\right ) \left (d+e x^2\right )}-\frac {a+b \tan ^{-1}(c x)}{d^3 x}-\frac {e x \left (a+b \tan ^{-1}(c x)\right )}{4 d^2 \left (d+e x^2\right )^2}-\frac {7 e x \left (a+b \tan ^{-1}(c x)\right )}{8 d^3 \left (d+e x^2\right )}-\frac {a \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{7/2}}-\frac {7 \sqrt {e} \left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{7/2}}+\frac {b c \log (x)}{d^3}-\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)^{7/2}}+\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)^{7/2}}-\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)^{7/2}}+\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)^{7/2}}-\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 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 )}{32 \sqrt {-c^2} d^{7/2}}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d^3}+\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (1+c^2 x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}+\frac {b c e \log \left (1+c^2 x^2\right )}{4 d^3 \left (c^2 d-e\right )}-\frac {b c \left (5 c^2 d-3 e\right ) e \log \left (d+e x^2\right )}{16 d^3 \left (c^2 d-e\right )^2}-\frac {b c e \log \left (d+e x^2\right )}{4 d^3 \left (c^2 d-e\right )}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (1+i c x)}{i c \sqrt {-d}+\sqrt {e}}\right )}{4 (-d)^{7/2}}+\frac {i b \sqrt {e} \text {Li}_2\left (\frac {\sqrt {e} (i+c x)}{c \sqrt {-d}+i \sqrt {e}}\right )}{4 (-d)^{7/2}}-\frac {7 i b c \sqrt {e} \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \sqrt {e} \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{7/2}}-\frac {7 i b c \sqrt {e} \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{7/2}}+\frac {7 i b c \sqrt {e} \text {Li}_2\left (\frac {\sqrt {-c^2} \left (\sqrt {d}+i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{32 \sqrt {-c^2} d^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 13.43, size = 2005, normalized size = 1.32 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcTan[c*x])/(x^2*(d + e*x^2)^3),x]

[Out]

-(a/(d^3*x)) - (a*e*x)/(4*d^2*(d + e*x^2)^2) - (7*a*e*x)/(8*d^3*(d + e*x^2)) - (15*a*Sqrt[e]*ArcTan[(Sqrt[e]*x
)/Sqrt[d]])/(8*d^(7/2)) + b*c^7*(-(ArcTan[c*x]/(c^7*d^3*x)) + Log[(c*x)/Sqrt[1 + c^2*x^2]]/(c^6*d^3) - (9*e*Lo
g[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(16*c^4*d^2*(c^2*d - e)^2) + (7*e^2*Log[1 - ((-(c^2*d)
+ e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)])/(16*c^6*d^3*(c^2*d - e)^2) - (15*e*(4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[
-(c^2*d*e)]*x)] + 2*ArcCos[(-(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)*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)*(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))])))/(32*c^4*d^2*(c^2*d - e)*Sqrt[-(c^2*d*e)]) + (15*
e^2*(4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCos[(-(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)*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)*(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)] + Ar
cTanh[(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))])))/(32*c^6*d
^3*(c^2*d - e)*Sqrt[-(c^2*d*e)]) + (e^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(2*c^4*d^2*(c^2*d - e)*(c^2*d + e + c^
2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])^2) + (-2*c^2*d*e^2 - 9*c^4*d^2*e*ArcTan[c*x]*Sin[2*ArcTan[c*x]]
+ 16*c^2*d*e^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]] - 7*e^3*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(8*c^6*d^3*(c^2*d - e)^
2*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])))

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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e^{3} x^{8} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{4} + d^{3} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^2/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*arctan(c*x) + a)/(e^3*x^8 + 3*d*e^2*x^6 + 3*d^2*e*x^4 + d^3*x^2), x)

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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))/x^2/(e*x^2+d)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [C]  time = 1.38, size = 6655, normalized size = 4.38 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctan(c*x))/x^2/(e*x^2+d)^3,x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, a {\left (\frac {15 \, e^{2} x^{4} + 25 \, d e x^{2} + 8 \, d^{2}}{d^{3} e^{2} x^{5} + 2 \, d^{4} e x^{3} + d^{5} x} + \frac {15 \, e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} d^{3}}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e^{3} x^{8} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{4} + d^{3} x^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctan(c*x))/x^2/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

-1/8*a*((15*e^2*x^4 + 25*d*e*x^2 + 8*d^2)/(d^3*e^2*x^5 + 2*d^4*e*x^3 + d^5*x) + 15*e*arctan(e*x/sqrt(d*e))/(sq
rt(d*e)*d^3)) + 2*b*integrate(1/2*arctan(c*x)/(e^3*x^8 + 3*d*e^2*x^6 + 3*d^2*e*x^4 + d^3*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,{\left (e\,x^2+d\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atan(c*x))/(x^2*(d + e*x^2)^3),x)

[Out]

int((a + b*atan(c*x))/(x^2*(d + e*x^2)^3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atan(c*x))/x**2/(e*x**2+d)**3,x)

[Out]

Timed out

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