3.1170 \(\int \frac {x^2 (a+b \tan ^{-1}(c x))}{(d+e x^2)^3} \, dx\)

Optimal. Leaf size=966 \[ \frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {b \log \left (c^2 x^2+1\right ) c}{4 d \left (c^2 d-e\right ) e}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {b \log \left (e x^2+d\right ) c}{4 d \left (c^2 d-e\right ) e}-\frac {b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {b c}{8 \left (c^2 d-e\right ) e \left (e x^2+d\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (e x^2+d\right )}-\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (e x^2+d\right )^2}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.26, antiderivative size = 966, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 15, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4980, 199, 205, 4912, 6725, 571, 77, 4908, 2409, 2394, 2393, 2391, 444, 36, 31} \[ \frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {b \log \left (c^2 x^2+1\right ) c}{4 d \left (c^2 d-e\right ) e}+\frac {b \left (5 c^2 d-3 e\right ) \log \left (c^2 x^2+1\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {b \log \left (e x^2+d\right ) c}{4 d \left (c^2 d-e\right ) e}-\frac {b \left (5 c^2 d-3 e\right ) \log \left (e x^2+d\right ) c}{16 d \left (c^2 d-e\right )^2 e}+\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}-\frac {i b \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 ) c}{32 \sqrt {-c^2} d^{3/2} e^{3/2}}+\frac {b c}{8 \left (c^2 d-e\right ) e \left (e x^2+d\right )}+\frac {x \left (a+b \tan ^{-1}(c x)\right )}{8 d e \left (e x^2+d\right )}-\frac {x \left (a+b \tan ^{-1}(c x)\right )}{4 e \left (e x^2+d\right )^2}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c)/(8*(c^2*d - e)*e*(d + e*x^2)) - (x*(a + b*ArcTan[c*x]))/(4*e*(d + e*x^2)^2) + (x*(a + b*ArcTan[c*x]))/(8
*d*e*(d + e*x^2)) + ((a + b*ArcTan[c*x])*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + ((I/32)*b*c*Log[(S
qrt[e]*(1 - Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] + Sqrt[e])]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/
2)*e^(3/2)) - ((I/32)*b*c*Log[-((Sqrt[e]*(1 + Sqrt[-c^2]*x))/(I*Sqrt[-c^2]*Sqrt[d] - Sqrt[e]))]*Log[1 - (I*Sqr
t[e]*x)/Sqrt[d]])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*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]*d^(3/2)*e^(3/2)) + ((I/32)*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]*d^(3/2)*e^(3/
2)) + (b*c*(5*c^2*d - 3*e)*Log[1 + c^2*x^2])/(16*d*(c^2*d - e)^2*e) - (b*c*Log[1 + c^2*x^2])/(4*d*(c^2*d - e)*
e) - (b*c*(5*c^2*d - 3*e)*Log[d + e*x^2])/(16*d*(c^2*d - e)^2*e) + (b*c*Log[d + e*x^2])/(4*d*(c^2*d - e)*e) +
((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(
3/2)*e^(3/2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] - I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d] + I*Sqrt[e])])
/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) + ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(Sqrt[-c^2]*Sqrt[d
] - I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*e^(3/2)) - ((I/32)*b*c*PolyLog[2, (Sqrt[-c^2]*(Sqrt[d] + I*Sqrt[e]*x))/(S
qrt[-c^2]*Sqrt[d] + I*Sqrt[e])])/(Sqrt[-c^2]*d^(3/2)*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 77

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

Rule 199

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

Rule 205

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

Rule 444

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

Rule 571

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

Rule 2391

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

Rule 2393

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

Rule 2394

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

Rule 2409

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

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

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Mathematica [A]  time = 13.07, size = 1914, normalized size = 1.98 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2))
+ b*c^3*(-1/16*Log[1 - ((-(c^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)]/(c^2*d*(c^2*d - e)^2) - Log[1 - ((-(c
^2*d) + e)*Cos[2*ArcTan[c*x]])/(c^2*d + e)]/(16*(c^2*d - e)^2*e) - (4*ArcTan[c*x]*ArcTanh[(c*d)/(Sqrt[-(c^2*d*
e)]*x)] + 2*ArcCos[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] - (ArcCos[(-(c^2*d) - e)/(c^2
*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2
*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcCos[(-(c^2*d) - e)/(c^2*d - e
)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqr
t[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2
*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)])/
(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + (ArcCos[(-(c^2*d) - e)
/(c^2*d - e)] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log[(Sqrt[2]*
Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]])] + I*(P
olyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*
c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))
/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))]))/(32*c^2*d*(c^2*d - e)*Sqrt[-(c^2*d*e)]) + (4*ArcTan[c*x]*
ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + 2*ArcCos[(-(c^2*d) - e)/(c^2*d - e)]*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]] -
(ArcCos[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e - (2*I)*Sq
rt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (-ArcC
os[(-(c^2*d) - e)/(c^2*d - e)] - (2*I)*ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]])*Log[1 - ((c^2*d + e + (2*I)*Sqrt[-(c
^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] + (ArcCos[(-(c
^2*d) - e)/(c^2*d - e)] - (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[-(c^2*d*e)]]))*Log
[(Sqrt[2]*Sqrt[-(c^2*d*e)])/(Sqrt[c^2*d - e]*E^(I*ArcTan[c*x])*Sqrt[c^2*d + e + (c^2*d - e)*Cos[2*ArcTan[c*x]]
])] + (ArcCos[(-(c^2*d) - e)/(c^2*d - e)] + (2*I)*(ArcTanh[(c*d)/(Sqrt[-(c^2*d*e)]*x)] + ArcTanh[(c*e*x)/Sqrt[
-(c^2*d*e)]]))*Log[(Sqrt[2]*Sqrt[-(c^2*d*e)]*E^(I*ArcTan[c*x]))/(Sqrt[c^2*d - e]*Sqrt[c^2*d + e + (c^2*d - e)*
Cos[2*ArcTan[c*x]]])] + I*(PolyLog[2, ((c^2*d + e - (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*d - 2*c*Sqrt[-(c^2*d*e)]*x)
)/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))] - PolyLog[2, ((c^2*d + e + (2*I)*Sqrt[-(c^2*d*e)])*(2*c^2*
d - 2*c*Sqrt[-(c^2*d*e)]*x))/((c^2*d - e)*(2*c^2*d + 2*c*Sqrt[-(c^2*d*e)]*x))]))/(32*(c^2*d - e)*e*Sqrt[-(c^2*
d*e)]) + (ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(2*(c^2*d - e)*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTa
n[c*x]])^2) + (-2*c^2*d*e - c^4*d^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]] + e^2*ArcTan[c*x]*Sin[2*ArcTan[c*x]])/(8*c^
2*d*(c^2*d - e)^2*e*(c^2*d + e + c^2*d*Cos[2*ArcTan[c*x]] - e*Cos[2*ArcTan[c*x]])))

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{2} \arctan \left (c x\right ) + a x^{2}}{e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2*arctan(c*x) + a*x^2)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), 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)^3,x, algorithm="giac")

[Out]

sage0*x

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maple [B]  time = 1.08, size = 3801, normalized size = 3.93 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*I*c^3*b*(c^2*e*d)^(1/2)/(c^4*d^2-2*c^2*d*e+e^2)/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))+1/16*I/c*b*(c^2*e*d)^(1/2)/(c^4*d^2-2*c^2*d*e+e^2)/d^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))+1/8*a/d/e/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))+1/8*c^4*a/
(c^2*e*x^2+c^2*d)^2/d*x^3-1/16*b*(d*e)^(1/2)*e/d^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^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)-1/4*c*b/(c^4*d^2-2*c^2*d*e+e^2)*e/d/(c^2*d-e)*ln((1+I*c*x)
/(c^2*x^2+1)^(1/2))-1/16*c^7*b*d^2*arctan(c*x)^2/e^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/32/c*b*e^2*po
lylog(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))/d^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d
)^(1/2)-1/32*c^7*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))/e^2/(c^4*d^2-
2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)+1/32*c^3*b*(c^2*e*d)^(1/2)/(c^4*d^2-2*c^2*d*e+e^2)/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))+1/32/c*b*(c^2*e*d)^(1/2)/(c^4*d^2-2*c^2*d*e+e^2)/d^2*polyl
og(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))+1/16/c*b*(c^2*e*d)^(1/2)/(c^4*d^2-2*c^2*d
*e+e^2)/d^2*arctan(c*x)^2-1/8*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*x^2-1/8*c^5*b/(c^4*d^2-2*c^2
*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*x^2-1/8*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*x^4+1/16*c^3*b*(c^
2*e*d)^(1/2)/(c^4*d^2-2*c^2*d*e+e^2)/e^2*arctan(c*x)^2-1/16*c^4*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^4*d^2-2*c^2*d*e+e^2)+1/4*I*c^5*b*d*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)/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/8*I*c^
5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e^2/d*arctan(c*x)*x^4-1/8*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e
*x^2+c^2*d)^2*d-3/16*c^3*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^4*d^2-
2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-3/8*c^3*b*arctan(c*x)^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/16*b*(d*e
)^(1/2)/d^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^4*d^2-2*c^2*d*e+e^
2)-1/8*c*b*(c^2*e*d)^(1/2)/d/e/(c^4*d^2-2*c^2*d*e+e^2)*arctan(c*x)^2+1/4*c*b*e*arctan(c*x)^2/d/(c^4*d^2-2*c^2*
d*e+e^2)^2*(c^2*e*d)^(1/2)-1/16/c*b*e^2*arctan(c*x)^2/d^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/16*c^2*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^4*d^2-2*c^2*d*e
+e^2)/(c^2*d-e)-1/16*c*b*(c^2*e*d)^(1/2)/d/e/(c^4*d^2-2*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^5*b*d*arctan(c*x)^2/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/8*c^2
*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^4*d^2-2*c^2
*d*e+e^2)+1/4*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/e*d/(c^2*d-e)*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))+1/8*c^5*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))/e/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)*d-
1/16*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/e*d/(c^2*d-e)*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)-1/8*c^4*a/(c^2*e*x^2+c^2*d)^2/e*x+1/4*c
^6*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*arctan(c*x)*x-1/8*c^4*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+
c^2*d)^2*e*arctan(c*x)*x+1/8*c^8*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*arctan(c*x)*x^3-1/4*c^6*b/(c^
4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x^3+1/16*c^4*b*(d*e)^(1/2)/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^4*d^2-2*c^2*d*e+e^2)/(c^2*d-e)+1/16*c*b/(c^4*d^2-2*c^2*d
*e+e^2)*e/d/(c^2*d-e)*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)+1/8*c*b*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))/d/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-3/8*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)/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/8*I*c^5*b/(c^4*d^2-2*
c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*arctan(c*x)-1/8*I*c*b*(c^2*e*d)^(1/2)/d/e/(c^4*d^2-2*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))+1/4*I*c*b*e*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^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/16*I/c*b*
e^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)/d^2/(c^4*d^2-2*c^2*d*e+e^
2)^2*(c^2*e*d)^(1/2)-1/16*I*c^7*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))*arc
tan(c*x)/e^2/(c^4*d^2-2*c^2*d*e+e^2)^2*(c^2*e*d)^(1/2)-1/8*c^8*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/e
*d^2*arctan(c*x)*x+1/8*c^4*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e^2/d*arctan(c*x)*x^3+1/16*c^6*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^4*d^2-2*c^2*d*e+
e^2)/(c^2*d-e)-1/8*I*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e*arctan(c*x)*x^4-1/4*I*c^7*b/(c^4*d^2-
2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*d*arctan(c*x)*x^2-1/4*I*c^5*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2*e
*arctan(c*x)*x^2-1/8*I*c^7*b/(c^4*d^2-2*c^2*d*e+e^2)/(c^2*e*x^2+c^2*d)^2/e*d^2*arctan(c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*((e*x^3 - d*x)/(d*e^3*x^4 + 2*d^2*e^2*x^2 + d^3*e) + arctan(e*x/sqrt(d*e))/(sqrt(d*e)*d*e)) + 2*b*integr
ate(1/2*x^2*arctan(c*x)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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