3.420 \(\int x^3 (c+a^2 c x^2)^{3/2} \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=652 \[ \frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{23 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{120 a^4}-\frac{c x^3 \sqrt{a^2 c x^2+c}}{140 a}+\frac{c x \sqrt{a^2 c x^2+c}}{420 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{14} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{1}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{280 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^2}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60 a^2}+\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^4}-\frac{163 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{840 a^4} \]

[Out]

(c*x*Sqrt[c + a^2*c*x^2])/(420*a^3) - (c*x^3*Sqrt[c + a^2*c*x^2])/(140*a) - (163*c*Sqrt[c + a^2*c*x^2]*ArcTan[
a*x])/(840*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60*a^2) + (c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/3
5 + (9*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(280*a)
- (a*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/14 - (((51*I)/280)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x
])]*ArcTan[a*x]^2)/(a^4*Sqrt[c + a^2*c*x^2]) - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^4) + (c*x^2*Sqrt[
c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^2) + (8*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/35 + (a^2*c*x^6*Sqrt[c +
a^2*c*x^2]*ArcTan[a*x]^3)/7 + (23*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^4) + (((51*I)/280
)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (((51*I)/2
80)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (51*c^2*Sqr
t[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2]) + (51*c^2*Sqrt[1 + a^2*x^2]*P
olyLog[3, I*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 7.37059, antiderivative size = 652, normalized size of antiderivative = 1., number of steps used = 200, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321} \[ \frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{51 c^2 \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{a^2 c x^2+c}}-\frac{51 i c^2 \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{a^2 c x^2+c}}+\frac{23 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{120 a^4}-\frac{c x^3 \sqrt{a^2 c x^2+c}}{140 a}+\frac{c x \sqrt{a^2 c x^2+c}}{420 a^3}+\frac{1}{7} a^2 c x^6 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac{1}{14} a c x^5 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac{8}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{1}{35} c x^4 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{23 c x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{280 a}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^2}+\frac{c x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{60 a^2}+\frac{9 c x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{112 a^3}-\frac{2 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{35 a^4}-\frac{163 c \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{840 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*Sqrt[c + a^2*c*x^2])/(420*a^3) - (c*x^3*Sqrt[c + a^2*c*x^2])/(140*a) - (163*c*Sqrt[c + a^2*c*x^2]*ArcTan[
a*x])/(840*a^4) + (c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(60*a^2) + (c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/3
5 + (9*c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(112*a^3) - (23*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(280*a)
- (a*c*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/14 - (((51*I)/280)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x
])]*ArcTan[a*x]^2)/(a^4*Sqrt[c + a^2*c*x^2]) - (2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^4) + (c*x^2*Sqrt[
c + a^2*c*x^2]*ArcTan[a*x]^3)/(35*a^2) + (8*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/35 + (a^2*c*x^6*Sqrt[c +
a^2*c*x^2]*ArcTan[a*x]^3)/7 + (23*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(120*a^4) + (((51*I)/280
)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (((51*I)/2
80)*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^4*Sqrt[c + a^2*c*x^2]) - (51*c^2*Sqr
t[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2]) + (51*c^2*Sqrt[1 + a^2*x^2]*P
olyLog[3, I*E^(I*ArcTan[a*x])])/(280*a^4*Sqrt[c + a^2*c*x^2])

Rule 4950

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[
d, Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Dist[(c^2*d)/f^2, Int[(f*x)^(m + 2)*(d + e*
x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] &&
IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))

Rule 4952

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

Rule 4930

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 4890

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

Rule 4888

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

Rule 4181

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=c^2 \int \frac{x^3 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac{x^7 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^2}+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}-\frac{c^2 \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{a}+2 \left (\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{1}{5} \left (4 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{5} \left (3 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\right )-\frac{1}{7} \left (6 a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{7} \left (3 a^3 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{35} \left (24 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{c^2 \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^3}+\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{a^3}+\frac{c^2 \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^2}+2 \left (-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{10} \left (3 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (8 c^2\right ) \int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{15 a^2}+\frac{\left (9 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{20 a}+\frac{\left (4 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{5 a}\right )+\frac{1}{14} \left (5 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{35} \left (18 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{7} \left (a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^4}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{2 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{3 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{1}{35} \left (4 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{28} \left (5 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{1}{35} \left (9 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx-\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{a^3}-\frac{\left (16 c^2\right ) \int \frac{x \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}+2 \left (\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{\left (9 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{40 a^3}-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}-\frac{\left (8 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}-\frac{c^2 \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^2}-\frac{\left (9 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^2}-\frac{\left (4 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^2}-\frac{c^2 \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{10 a}\right )-\frac{\left (15 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{56 a}-\frac{\left (27 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{70 a}-\frac{\left (24 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}-\frac{1}{35} \left (a c^2\right ) \int \frac{x^4}{\sqrt{c+a^2 c x^2}} \, dx+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (2 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{c x^3 \sqrt{c+a^2 c x^2}}{140 a}+\frac{c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^4}-\frac{11 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{\left (15 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{112 a^3}+\frac{\left (27 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{140 a^3}+\frac{\left (12 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}-\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{a^3}+\frac{\left (48 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}+\frac{\left (8 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^2}+\frac{\left (5 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{42 a^2}+\frac{\left (6 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}+\frac{\left (15 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^2}+\frac{\left (27 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^2}+\frac{\left (24 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^2}+\frac{\left (3 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{140 a}+\frac{\left (4 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{105 a}+\frac{\left (5 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{84 a}+\frac{\left (3 c^2\right ) \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx}{35 a}+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (2 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{29 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^3}+\frac{c^2 \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}+\frac{\left (9 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{20 a^3}+\frac{\left (4 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{5 a^3}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{40 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (2 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{5 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (8 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{5 a^3 \sqrt{c+a^2 c x^2}}\right )\\ &=\frac{43 c x \sqrt{c+a^2 c x^2}}{420 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2}}{140 a}+\frac{2273 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac{11 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{5 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 \sqrt{c+a^2 c x^2}}-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^4}-\frac{\left (3 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{280 a^3}-\frac{\left (2 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^3}-\frac{\left (5 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{168 a^3}-\frac{\left (3 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^3}-\frac{\left (8 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{105 a^3}-\frac{\left (5 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{42 a^3}-\frac{\left (6 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}-\frac{\left (15 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{56 a^3}-\frac{\left (27 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{70 a^3}-\frac{\left (24 c^2\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{35 a^3}-\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{29 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{20 a^3}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{5 a^3}+\frac{\left (9 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{20 a^3}+\frac{\left (4 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{5 a^3}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{40 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (2 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (8 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}\right )-\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{112 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (27 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{140 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (12 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (48 c^2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{1+a^2 x^2}} \, dx}{35 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{43 c x \sqrt{c+a^2 c x^2}}{420 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2}}{140 a}+\frac{2273 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac{11 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{5 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 \sqrt{c+a^2 c x^2}}-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^4}+\frac{5 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}-\frac{5 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{280 a^3}-\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{105 a^3}-\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{168 a^3}-\frac{\left (3 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{70 a^3}-\frac{\left (8 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{105 a^3}-\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{42 a^3}-\frac{\left (6 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{35 a^3}-\frac{\left (15 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{56 a^3}-\frac{\left (27 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{70 a^3}-\frac{\left (24 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{35 a^3}-\frac{\left (i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{112 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (27 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{140 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (12 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (48 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{29 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (16 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (16 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}\right )\\ &=\frac{43 c x \sqrt{c+a^2 c x^2}}{420 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2}}{140 a}+\frac{2273 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac{11 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{337 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^4}+\frac{5 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}-\frac{5 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{29 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}-\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (16 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (16 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{5 a^4 \sqrt{c+a^2 c x^2}}\right )-\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (27 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{70 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (27 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{70 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (24 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (24 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (96 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (96 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}\\ &=\frac{43 c x \sqrt{c+a^2 c x^2}}{420 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2}}{140 a}+\frac{2273 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac{11 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{337 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^4}+\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{5 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{5 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (15 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (15 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{56 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (27 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{70 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (27 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{70 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (24 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (24 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (96 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (96 i c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{29 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}-\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (4 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{5 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (16 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{5 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (16 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{5 a^4 \sqrt{c+a^2 c x^2}}\right )\\ &=\frac{43 c x \sqrt{c+a^2 c x^2}}{420 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2}}{140 a}+\frac{2273 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac{11 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{337 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^4}+\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{5 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}+\frac{5 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{29 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}-\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{89 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}\right )-\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{56 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (15 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{56 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (27 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{70 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (27 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{70 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (24 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (24 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{35 a^4 \sqrt{c+a^2 c x^2}}-\frac{\left (96 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{35 a^4 \sqrt{c+a^2 c x^2}}+\frac{\left (96 c^2 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{35 a^4 \sqrt{c+a^2 c x^2}}\\ &=\frac{43 c x \sqrt{c+a^2 c x^2}}{420 a^3}-\frac{c x^3 \sqrt{c+a^2 c x^2}}{140 a}+\frac{2273 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{840 a^4}-\frac{11 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{60 a^2}+\frac{1}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{131 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{112 a^3}+\frac{61 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{280 a}-\frac{1}{14} a c x^5 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{118 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^4}+\frac{59 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{105 a^2}-\frac{6}{35} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{7} a^2 c x^6 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac{337 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{120 a^4}+\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{2543 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}-\frac{2543 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}+\frac{2543 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{280 a^4 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{c x \sqrt{c+a^2 c x^2}}{20 a^3}-\frac{29 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{20 a^4}+\frac{c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{10 a^2}+\frac{5 c x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}-\frac{3 c x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{20 a}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{8 c \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^4}-\frac{4 c x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{15 a^2}+\frac{1}{5} c x^4 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{2 a^4}-\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 i c^2 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}+\frac{89 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}-\frac{89 c^2 \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{20 a^4 \sqrt{c+a^2 c x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 3.26548, size = 538, normalized size = 0.83 \[ \frac{c \sqrt{a^2 c x^2+c} \left (64 \left (-309 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+309 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+309 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-309 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-259 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+309 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+2688 \left (11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-11 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-11 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+11 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )+10 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )-11 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )+\left (a^2 x^2+1\right )^{7/2} \left (\frac{8 \tan ^{-1}(a x) \left (764 \cos \left (2 \tan ^{-1}(a x)\right )+309 \cos \left (4 \tan ^{-1}(a x)\right )+647\right )}{a^2 x^2+1}-3 \tan ^{-1}(a x)^2 \left (211 \sin \left (2 \tan ^{-1}(a x)\right )-60 \sin \left (4 \tan ^{-1}(a x)\right )+103 \sin \left (6 \tan ^{-1}(a x)\right )\right )+4 \left (101 \sin \left (2 \tan ^{-1}(a x)\right )+88 \sin \left (4 \tan ^{-1}(a x)\right )+25 \sin \left (6 \tan ^{-1}(a x)\right )\right )+64 \tan ^{-1}(a x)^3 \left (-28 \cos \left (2 \tan ^{-1}(a x)\right )+35 \cos \left (4 \tan ^{-1}(a x)\right )+57\right )\right )-56 \left (a^2 x^2+1\right )^{5/2} \left (\frac{48 a x}{\left (a^2 x^2+1\right )^2}+\tan ^{-1}(a x)^2 \left (6 \sin \left (2 \tan ^{-1}(a x)\right )-33 \sin \left (4 \tan ^{-1}(a x)\right )\right )+32 \tan ^{-1}(a x)^3 \left (5 \cos \left (2 \tan ^{-1}(a x)\right )-1\right )+6 \tan ^{-1}(a x) \left (36 \cos \left (2 \tan ^{-1}(a x)\right )+11 \cos \left (4 \tan ^{-1}(a x)\right )+25\right )\right )\right )}{53760 a^4 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c + a^2*c*x^2]*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x
^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a
*x])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyLog[3, I*E^(I*ArcTan[a*x])]) + 2688*((-11*I)*ArcTan[E
^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(
I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] +
11*PolyLog[3, I*E^(I*ArcTan[a*x])]) - 56*(1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1
+ 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*
(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]])) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[
a*x]] + 35*Cos[4*ArcTan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 309*Cos[4*ArcTan[a*x]]))/(1 +
a^2*x^2) + 4*(101*Sin[2*ArcTan[a*x]] + 88*Sin[4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*S
in[2*ArcTan[a*x]] - 60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))))/(53760*a^4*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 4.163, size = 469, normalized size = 0.7 \begin{align*}{\frac{c \left ( 240\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{6}{a}^{6}-120\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{5}{a}^{5}+384\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{4}{a}^{4}+48\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-138\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}+48\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{x}^{2}{a}^{2}-12\,{a}^{3}{x}^{3}+28\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+135\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa-96\, \left ( \arctan \left ( ax \right ) \right ) ^{3}+4\,ax-326\,\arctan \left ( ax \right ) \right ) }{1680\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{17\,c}{560\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}-3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{17\,c}{560\,{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}+6\,i\arctan \left ( ax \right ){\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}\ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{{\frac{23\,i}{60}}c}{{a}^{4}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*c/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*(240*arctan(a*x)^3*x^6*a^6-120*arctan(a*x)^2*x^5*a^5+384*arctan(a*x)^3*
x^4*a^4+48*arctan(a*x)*x^4*a^4-138*arctan(a*x)^2*x^3*a^3+48*arctan(a*x)^3*x^2*a^2-12*a^3*x^3+28*arctan(a*x)*a^
2*x^2+135*arctan(a*x)^2*x*a-96*arctan(a*x)^3+4*a*x-326*arctan(a*x))+17/560*c*(c*(a*x-I)*(a*x+I))^(1/2)*(I*arct
an(a*x)^3-3*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+
1)^(1/2))-6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-17/560*c*(c*(a*x-I)*(a*x+I))^(1/2
)*(I*arctan(a*x)^3+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(
a^2*x^2+1)^(1/2))-6*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-23/60*I*c/a^4*(c*(a*x-I)*(
a*x+I))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{5} + c x^{3}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c*x^5 + c*x^3)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError