\(\int \frac {(a+b \arctan (c x))^2}{x^2 (d+e x^2)^2} \, dx\) [1273]

Optimal result

Integrand size = 23, antiderivative size = 1141 \[ \int \frac {(a+b \arctan (c x))^2}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {i c (a+b \arctan (c x))^2}{d^2}-\frac {i c e (a+b \arctan (c x))^2}{2 d^2 \left (c^2 d-e\right )}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac {b c e (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{d^2 \left (c^2 d-e\right )}-\frac {b c e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2}}-\frac {b c e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{2 d^2 \left (c^2 d-e\right )}+\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2}}+\frac {2 b c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d^2 \left (c^2 d-e\right )}-\frac {i b^2 c \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )}{d^2}-\frac {i b^2 c e \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^2 \left (c^2 d-e\right )}+\frac {i b^2 c e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2 \left (c^2 d-e\right )}+\frac {3 i b \sqrt {e} (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2}}+\frac {i b^2 c e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 d^2 \left (c^2 d-e\right )}-\frac {3 i b \sqrt {e} (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2}}-\frac {3 b^2 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{5/2}}+\frac {3 b^2 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{\left (c \sqrt {-d}+i \sqrt {e}\right ) (1-i c x)}\right )}{8 (-d)^{5/2}} \]

[Out]

-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d^2-1/2*I*b^2*c*e*polylog(2,1-2/(1-I*c*x))/d^2/(c^2*d-e)-(a+b*arctan(c*x))^
2/d^2/x+b*c*e*(a+b*arctan(c*x))*ln(2/(1-I*c*x))/d^2/(c^2*d-e)-b*c*e*(a+b*arctan(c*x))*ln(2/(1+I*c*x))/d^2/(c^2
*d-e)+2*b*c*(a+b*arctan(c*x))*ln(2-2/(1-I*c*x))/d^2-1/2*b*c*e*(a+b*arctan(c*x))*ln(2*c*((-d)^(1/2)-x*e^(1/2))/
(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/d^2/(c^2*d-e)-1/2*b*c*e*(a+b*arctan(c*x))*ln(2*c*((-d)^(1/2)+x*e^(1/2))/(1
-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))/d^2/(c^2*d-e)+1/4*I*b^2*c*e*polylog(2,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)
/(c*(-d)^(1/2)+I*e^(1/2)))/d^2/(c^2*d-e)+1/4*I*b^2*c*e*polylog(2,1-2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d
)^(1/2)-I*e^(1/2)))/d^2/(c^2*d-e)-3/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(
c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*((-d)^(1/2)-x*e^(1/2))/(
1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(5/2)-1/2*I*c*e*(a+b*arctan(c*x))^2/d^2/(c^2*d-e)-3/4*(a+b*arc
tan(c*x))^2*ln(2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+3/4*(a+b*arct
an(c*x))^2*ln(2*c*((-d)^(1/2)+x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)-1/2*I*b^2*c*e*
polylog(2,1-2/(1+I*c*x))/d^2/(c^2*d-e)-I*c*(a+b*arctan(c*x))^2/d^2-3/8*b^2*polylog(3,1-2*c*((-d)^(1/2)-x*e^(1/
2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+3/8*b^2*polylog(3,1-2*c*((-d)^(1/2)+x*e^(1/2))/(1-I
*c*x)/(c*(-d)^(1/2)+I*e^(1/2)))*e^(1/2)/(-d)^(5/2)+1/4*(a+b*arctan(c*x))^2*e^(1/2)/d^2/((-d)^(1/2)-x*e^(1/2))-
1/4*(a+b*arctan(c*x))^2*e^(1/2)/d^2/((-d)^(1/2)+x*e^(1/2))

Rubi [A] (verified)

Time = 1.44 (sec) , antiderivative size = 1141, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {5100, 4946, 5044, 4988, 2497, 5034, 4974, 4966, 2449, 2352, 5104, 5004, 5040, 4964, 4968} \[ \int \frac {(a+b \arctan (c x))^2}{x^2 \left (d+e x^2\right )^2} \, dx=-\frac {i c e \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b^2}{2 d^2 \left (c^2 d-e\right )}-\frac {i c \operatorname {PolyLog}\left (2,\frac {2}{1-i c x}-1\right ) b^2}{d^2}-\frac {i c e \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right ) b^2}{2 d^2 \left (c^2 d-e\right )}+\frac {i c e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 d^2 \left (c^2 d-e\right )}+\frac {i c e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{4 d^2 \left (c^2 d-e\right )}-\frac {3 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{5/2}}+\frac {3 \sqrt {e} \operatorname {PolyLog}\left (3,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b^2}{8 (-d)^{5/2}}+\frac {c e (a+b \arctan (c x)) \log \left (\frac {2}{1-i c x}\right ) b}{d^2 \left (c^2 d-e\right )}-\frac {c e (a+b \arctan (c x)) \log \left (\frac {2}{i c x+1}\right ) b}{d^2 \left (c^2 d-e\right )}-\frac {c e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 d^2 \left (c^2 d-e\right )}-\frac {c e (a+b \arctan (c x)) \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{2 d^2 \left (c^2 d-e\right )}+\frac {2 c (a+b \arctan (c x)) \log \left (2-\frac {2}{1-i c x}\right ) b}{d^2}+\frac {3 i \sqrt {e} (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 (-d)^{5/2}}-\frac {3 i \sqrt {e} (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right ) b}{4 (-d)^{5/2}}-\frac {i c e (a+b \arctan (c x))^2}{2 d^2 \left (c^2 d-e\right )}-\frac {(a+b \arctan (c x))^2}{d^2 x}+\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\sqrt {e} (a+b \arctan (c x))^2}{4 d^2 \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {i c (a+b \arctan (c x))^2}{d^2}-\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{\left (c \sqrt {-d}-i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2}}+\frac {3 \sqrt {e} (a+b \arctan (c x))^2 \log \left (\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{\left (\sqrt {-d} c+i \sqrt {e}\right ) (1-i c x)}\right )}{4 (-d)^{5/2}} \]

[In]

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

[Out]

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

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 4964

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

Rule 4966

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

Rule 4968

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^2/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^2)*(Log[
2/(1 - I*c*x)]/e), x] + (Simp[(a + b*ArcTan[c*x])^2*(Log[2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] + S
imp[I*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1 - 2/(1 - I*c*x)]/e), x] - Simp[I*b*(a + b*ArcTan[c*x])*(PolyLog[2, 1
- 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/e), x] - Simp[b^2*(PolyLog[3, 1 - 2/(1 - I*c*x)]/(2*e)), x] + Si
mp[b^2*(PolyLog[3, 1 - 2*c*((d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(2*e)), x]) /; FreeQ[{a, b, c, d, e}, x] &&
NeQ[c^2*d^2 + e^2, 0]

Rule 4974

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

Rule 4988

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

Rule 5004

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

Rule 5034

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcTan[c*x])^p, (d + e*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[q] && IGtQ[p, 0]

Rule 5040

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 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 5104

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

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

Mathematica [F(-1)]

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

[In]

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

[Out]

$Aborted

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((a+b*arctan(c*x))^2/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]

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

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

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

[In]

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

[Out]

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