\(\int \frac {x^3 (a+b \arctan (c x))^2}{(d+e x^2)^2} \, dx\) [1268]

Optimal result

Integrand size = 23, antiderivative size = 943 \[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=-\frac {c^2 d (a+b \arctan (c x))^2}{2 \left (c^2 d-e\right ) e^2}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^2}+\frac {i b^2 c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b (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 e^2}-\frac {i b^2 c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b (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 e^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \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 e^2}+\frac {b^2 \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 e^2} \]

[Out]

-1/2*c^2*d*(a+b*arctan(c*x))^2/(c^2*d-e)/e^2-(a+b*arctan(c*x))^2*ln(2/(1-I*c*x))/e^2+1/2*(a+b*arctan(c*x))^2*l
n(2*c*((-d)^(1/2)-x*e^(1/2))/(1-I*c*x)/(c*(-d)^(1/2)-I*e^(1/2)))/e^2+1/2*(a+b*arctan(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^2-1/2*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^2-1/2*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^2+I*b*(a+b*arctan(c*x))*polylog(2,1-2/(1-I*c*x))/e^2-1/2*b^2*polylog(
3,1-2/(1-I*c*x))/e^2+1/4*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^2+1/
4*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^2-1/2*b*c*(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)^(1/2)/(c^2*d-e)/e^(3/2)+1/2*b*c*(a+b*a
rctan(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)^(1/2)/(c^2*d-e)/e^(3/2)-1/4
*I*b^2*c*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)^(1/2)/(c^2*d-e)/e^(3/
2)+1/4*I*b^2*c*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)^(1/2)/(c^2*d-e)
/e^(3/2)+1/4*(a+b*arctan(c*x))^2/e^2/(1-x*e^(1/2)/(-d)^(1/2))+1/4*(a+b*arctan(c*x))^2/e^2/(1+x*e^(1/2)/(-d)^(1
/2))

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 943, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {5100, 5098, 4974, 4966, 2449, 2352, 2497, 5104, 5004, 5040, 4964, 4968} \[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\frac {i c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {i c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {\operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right ) b^2}{2 e^2}+\frac {\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}{4 e^2}+\frac {\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}{4 e^2}-\frac {c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {i (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right ) b}{e^2}-\frac {i (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}{2 e^2}-\frac {i (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}{2 e^2}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (\frac {\sqrt {e} x}{\sqrt {-d}}+1\right )}-\frac {c^2 d (a+b \arctan (c x))^2}{2 \left (c^2 d-e\right ) e^2}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}+\frac {(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 e^2}+\frac {(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 )}{2 e^2} \]

[In]

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

[Out]

-1/2*(c^2*d*(a + b*ArcTan[c*x])^2)/((c^2*d - e)*e^2) + (a + b*ArcTan[c*x])^2/(4*e^2*(1 - (Sqrt[e]*x)/Sqrt[-d])
) + (a + b*ArcTan[c*x])^2/(4*e^2*(1 + (Sqrt[e]*x)/Sqrt[-d])) - ((a + b*ArcTan[c*x])^2*Log[2/(1 - I*c*x)])/e^2
- (b*c*Sqrt[-d]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/
(2*(c^2*d - e)*e^(3/2)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1
- I*c*x))])/(2*e^2) + (b*c*Sqrt[-d]*(a + b*ArcTan[c*x])*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqr
t[e])*(1 - I*c*x))])/(2*(c^2*d - e)*e^(3/2)) + ((a + b*ArcTan[c*x])^2*Log[(2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqr
t[-d] + I*Sqrt[e])*(1 - I*c*x))])/(2*e^2) + (I*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 - I*c*x)])/e^2 + ((I/
4)*b^2*c*Sqrt[-d]*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/((c^2*d
- e)*e^(3/2)) - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqr
t[e])*(1 - I*c*x))])/e^2 - ((I/4)*b^2*c*Sqrt[-d]*PolyLog[2, 1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*
Sqrt[e])*(1 - I*c*x))])/((c^2*d - e)*e^(3/2)) - ((I/2)*b*(a + b*ArcTan[c*x])*PolyLog[2, 1 - (2*c*(Sqrt[-d] + S
qrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/e^2 - (b^2*PolyLog[3, 1 - 2/(1 - I*c*x)])/(2*e^2) + (b^2*P
olyLog[3, 1 - (2*c*(Sqrt[-d] - Sqrt[e]*x))/((c*Sqrt[-d] - I*Sqrt[e])*(1 - I*c*x))])/(4*e^2) + (b^2*PolyLog[3,
1 - (2*c*(Sqrt[-d] + Sqrt[e]*x))/((c*Sqrt[-d] + I*Sqrt[e])*(1 - I*c*x))])/(4*e^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 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 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 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 5098

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2)^2, x_Symbol] :> Dist[1/(4*d^2*Rt[-e/
d, 2]), Int[(a + b*ArcTan[c*x])^p/(1 - Rt[-e/d, 2]*x)^2, x], x] - Dist[1/(4*d^2*Rt[-e/d, 2]), Int[(a + b*ArcTa
n[c*x])^p/(1 + Rt[-e/d, 2]*x)^2, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[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 {d x (a+b \arctan (c x))^2}{e \left (d+e x^2\right )^2}+\frac {x (a+b \arctan (c x))^2}{e \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x (a+b \arctan (c x))^2}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx}{e} \\ & = \frac {\int \frac {(a+b \arctan (c x))^2}{\left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 \sqrt {-d} e^{3/2}}-\frac {\int \frac {(a+b \arctan (c x))^2}{\left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )^2} \, dx}{4 \sqrt {-d} e^{3/2}}+\frac {\int \left (-\frac {(a+b \arctan (c x))^2}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {(a+b \arctan (c x))^2}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e} \\ & = \frac {(a+b \arctan (c x))^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(b c) \int \left (\frac {\sqrt {-d} e (a+b \arctan (c x))}{\left (c^2 d-e\right ) \left (-\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 d \left (\sqrt {-d}+\sqrt {e} x\right ) (a+b \arctan (c x))}{\sqrt {-d} \left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e^2}-\frac {(b c) \int \left (\frac {\sqrt {-d} e (a+b \arctan (c x))}{\left (-c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {c^2 \left (d+\sqrt {-d} \sqrt {e} x\right ) (a+b \arctan (c x))}{\left (c^2 d-e\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e^2}-\frac {\int \frac {(a+b \arctan (c x))^2}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{3/2}}+\frac {\int \frac {(a+b \arctan (c x))^2}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{3/2}} \\ & = \frac {(a+b \arctan (c x))^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}+\frac {(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 e^2}+\frac {(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 e^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^2}-\frac {i b (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 e^2}-\frac {i b (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 e^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \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 e^2}+\frac {b^2 \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 e^2}-\frac {\left (b c^3\right ) \int \frac {\left (d+\sqrt {-d} \sqrt {e} x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac {\left (b c^3 \sqrt {-d}\right ) \int \frac {\left (\sqrt {-d}+\sqrt {e} x\right ) (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2}-\frac {\left (b c \sqrt {-d}\right ) \int \frac {a+b \arctan (c x)}{-\sqrt {-d}+\sqrt {e} x} \, dx}{2 \left (c^2 d-e\right ) e}+\frac {\left (b c \sqrt {-d}\right ) \int \frac {a+b \arctan (c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \left (c^2 d-e\right ) e} \\ & = \frac {(a+b \arctan (c x))^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^2}-\frac {i b (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 e^2}-\frac {i b (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 e^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \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 e^2}+\frac {b^2 \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 e^2}-\frac {\left (b c^3\right ) \int \left (\frac {d (a+b \arctan (c x))}{1+c^2 x^2}+\frac {\sqrt {-d} \sqrt {e} x (a+b \arctan (c x))}{1+c^2 x^2}\right ) \, dx}{2 \left (c^2 d-e\right ) e^2}+\frac {\left (b c^3 \sqrt {-d}\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 \left (c^2 d-e\right ) e^2}+\frac {\left (b^2 c^2 \sqrt {-d}\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 \left (c^2 d-e\right ) e^{3/2}}-\frac {\left (b^2 c^2 \sqrt {-d}\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 \left (c^2 d-e\right ) e^{3/2}} \\ & = \frac {(a+b \arctan (c x))^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^2}+\frac {i b^2 c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b (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 e^2}-\frac {i b^2 c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b (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 e^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \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 e^2}+\frac {b^2 \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 e^2}-2 \frac {\left (b c^3 d\right ) \int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx}{2 \left (c^2 d-e\right ) e^2} \\ & = -\frac {c^2 d (a+b \arctan (c x))^2}{2 \left (c^2 d-e\right ) e^2}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1-\frac {\sqrt {e} x}{\sqrt {-d}}\right )}+\frac {(a+b \arctan (c x))^2}{4 e^2 \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}-\frac {(a+b \arctan (c x))^2 \log \left (\frac {2}{1-i c x}\right )}{e^2}-\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {b c \sqrt {-d} (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 \left (c^2 d-e\right ) e^{3/2}}+\frac {(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 e^2}+\frac {i b (a+b \arctan (c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{e^2}+\frac {i b^2 c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b (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 e^2}-\frac {i b^2 c \sqrt {-d} \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 \left (c^2 d-e\right ) e^{3/2}}-\frac {i b (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 e^2}-\frac {b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-i c x}\right )}{2 e^2}+\frac {b^2 \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 e^2}+\frac {b^2 \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 e^2} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{3} \left (a +b \arctan \left (c x \right )\right )^{2}}{\left (e \,x^{2}+d \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

1/2*a^2*(d/(e^3*x^2 + d*e^2) + log(e*x^2 + d)/e^2) + integrate((b^2*x^3*arctan(c*x)^2 + 2*a*b*x^3*arctan(c*x))
/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

Giac [F]

\[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )}^{2} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b \arctan (c x))^2}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

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

[Out]

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