3.1162 \(\int \frac {x^2 (a+b \tan ^{-1}(c x))}{(d+e x^2)^2} \, dx\)

Optimal. Leaf size=1335 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (e x^2+d\right )}+\frac {a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (c^2 x^2+1\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (e x^2+d\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \text {Li}_2\left (\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.96, antiderivative size = 1335, normalized size of antiderivative = 1.00, number of steps used = 45, number of rules used = 14, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4980, 199, 205, 4912, 6725, 444, 36, 31, 4908, 2409, 2394, 2393, 2391, 4910} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 \sqrt {d} e^{3/2}}-\frac {x \left (a+b \tan ^{-1}(c x)\right )}{2 e \left (e x^2+d\right )}+\frac {a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\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 )}{4 \sqrt {-d} e^{3/2}}+\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 )}{4 \sqrt {-d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {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 )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {b c \log \left (c^2 x^2+1\right )}{4 \left (c^2 d-e\right ) e}-\frac {b c \log \left (e x^2+d\right )}{4 \left (c^2 d-e\right ) e}+\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} (i-c x)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} (1-i c x)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} (i c x+1)}{i \sqrt {-d} c+\sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \text {PolyLog}\left (2,\frac {\sqrt {e} (c x+i)}{\sqrt {-d} c+i \sqrt {e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b c \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {i b c \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (\sqrt {d}-i \sqrt {e} x\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}-\frac {i b c \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}-i \sqrt {e}}\right )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}}+\frac {i b c \text {PolyLog}\left (2,\frac {\sqrt {-c^2} \left (i \sqrt {e} x+\sqrt {d}\right )}{\sqrt {-c^2} \sqrt {d}+i \sqrt {e}}\right )}{8 \sqrt {-c^2} \sqrt {d} e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 10.54, size = 877, normalized size = 0.66 \[ -\frac {a x}{2 e \left (e x^2+d\right )}+\frac {a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 \sqrt {d} e^{3/2}}+\frac {b c \left (-\frac {2 \log \left (\frac {d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}{d c^2+e}\right )}{c^2 d-e}+\frac {-4 \tan ^{-1}(c x) \tanh ^{-1}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )+2 \cos ^{-1}\left (\frac {d c^2+e}{e-c^2 d}\right ) \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )+\left (\cos ^{-1}\left (\frac {d c^2+e}{e-c^2 d}\right )-2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (-\frac {2 c^2 d \left (i e+\sqrt {-c^2 d e}\right ) (c x-i)}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}\right )+\left (\cos ^{-1}\left (\frac {d c^2+e}{e-c^2 d}\right )+2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {2 i c^2 d \left (e+i \sqrt {-c^2 d e}\right ) (c x+i)}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}\right )-\left (\cos ^{-1}\left (\frac {d c^2+e}{e-c^2 d}\right )-2 i \tanh ^{-1}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )+2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{-i \tan ^{-1}(c x)}}{\sqrt {e-c^2 d} \sqrt {-d c^2-e+\left (e-c^2 d\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )-\left (\cos ^{-1}\left (\frac {d c^2+e}{e-c^2 d}\right )+2 i \tanh ^{-1}\left (\frac {\sqrt {-c^2 d e}}{c e x}\right )-2 i \tanh ^{-1}\left (\frac {c e x}{\sqrt {-c^2 d e}}\right )\right ) \log \left (\frac {\sqrt {2} \sqrt {-c^2 d e} e^{i \tan ^{-1}(c x)}}{\sqrt {e-c^2 d} \sqrt {-d c^2-e+\left (e-c^2 d\right ) \cos \left (2 \tan ^{-1}(c x)\right )}}\right )+i \left (\text {Li}_2\left (\frac {\left (d c^2+e-2 i \sqrt {-c^2 d e}\right ) \left (d c^2+\sqrt {-c^2 d e} x c\right )}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}\right )-\text {Li}_2\left (\frac {\left (d c^2+e+2 i \sqrt {-c^2 d e}\right ) \left (d c^2+\sqrt {-c^2 d e} x c\right )}{\left (c^2 d-e\right ) \left (c^2 d-c \sqrt {-c^2 d e} x\right )}\right )\right )}{\sqrt {-c^2 d e}}-\frac {4 \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )}{d c^2+e+\left (c^2 d-e\right ) \cos \left (2 \tan ^{-1}(c x)\right )}\right )}{8 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \arctan \left (c x\right ) + a x^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2*arctan(c*x) + a*x^2)/(e^2*x^4 + 2*d*e*x^2 + d^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(x^2*(a+b*arctan(c*x))/(e*x^2+d)^2,x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 2.37, size = 2315, normalized size = 1.73 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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