3.429 \(\int x^2 (c+a^2 c x^2)^{5/2} \tan ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=1019 \[ \frac {1}{8} a^4 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^7-\frac {3}{56} a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^6+\frac {17}{48} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5+\frac {1}{56} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^5-\frac {83}{560} a c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac {59}{192} c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac {29}{560} c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac {737 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{6720 a}+\frac {5 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{128 a^2}+\frac {43 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x}{1344 a^2}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{280 a^3}+\frac {1373 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {3 c \left (a^2 c x^2+c\right )^{3/2}}{560 a^3}+\frac {397 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}-\frac {397 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {a^2 c x^2+c}}+\frac {397 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 i c^3 \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {13 c^2 \sqrt {a^2 c x^2+c}}{6720 a^3} \]

[Out]

-3/560*c*(a^2*c*x^2+c)^(3/2)/a^3-1/280*(a^2*c*x^2+c)^(5/2)/a^3+15/64*I*c^3*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^
(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+397/840*I*c^3*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1
/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-15/64*I*c^3*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1
)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+5/64*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a^2*x^2+1)^(1/2)
/a^3/(a^2*c*x^2+c)^(1/2)-397/1680*I*c^3*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a
^2*c*x^2+c)^(1/2)+15/128*I*c^3*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a
^2*c*x^2+c)^(1/2)+15/64*c^3*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c
*x^2+c)^(1/2)-15/64*c^3*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+
c)^(1/2)+397/1680*I*c^3*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)
-15/128*I*c^3*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2
)+13/6720*c^2*(a^2*c*x^2+c)^(1/2)/a^3+43/1344*c^2*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+29/560*c^2*x^3*arctan(
a*x)*(a^2*c*x^2+c)^(1/2)+1/56*a^2*c^2*x^5*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1373/13440*c^2*arctan(a*x)^2*(a^2*c*
x^2+c)^(1/2)/a^3-737/6720*c^2*x^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a-83/560*a*c^2*x^4*arctan(a*x)^2*(a^2*c*x^
2+c)^(1/2)-3/56*a^3*c^2*x^6*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+5/128*c^2*x*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^
2+59/192*c^2*x^3*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+17/48*a^2*c^2*x^5*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+1/8*a^4
*c^2*x^7*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 15.42, antiderivative size = 1019, normalized size of antiderivative = 1.00, number of steps used = 293, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4950, 4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589, 261, 266, 43} \[ \frac {1}{8} a^4 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^7-\frac {3}{56} a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^6+\frac {17}{48} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^5+\frac {1}{56} a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^5-\frac {83}{560} a c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^4+\frac {59}{192} c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x^3+\frac {29}{560} c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x^3-\frac {737 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 x^2}{6720 a}+\frac {5 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 x}{128 a^2}+\frac {43 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) x}{1344 a^2}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {\left (a^2 c x^2+c\right )^{5/2}}{280 a^3}+\frac {1373 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {3 c \left (a^2 c x^2+c\right )^{3/2}}{560 a^3}+\frac {397 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {a^2 c x^2+c}}-\frac {397 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {a^2 c x^2+c}}+\frac {397 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {15 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {15 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {13 c^2 \sqrt {a^2 c x^2+c}}{6720 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

(13*c^2*Sqrt[c + a^2*c*x^2])/(6720*a^3) - (3*c*(c + a^2*c*x^2)^(3/2))/(560*a^3) - (c + a^2*c*x^2)^(5/2)/(280*a
^3) + (43*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(1344*a^2) + (29*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/560
+ (a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/56 + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(13440*a^3)
- (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(6720*a) - (83*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)
/560 - (3*a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/56 + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(128
*a^2) + (59*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3
)/48 + (a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTa
n[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) + (((397*I)/840)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sq
rt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((15*I)/128)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*
PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((15*I)/128)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x
]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((397*I)/1680)*c^3*Sqrt[1 + a^2*x^2]*PolyLog
[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (((397*I)/1680)*c^3*Sqrt[1 + a^2*x^2]
*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan
[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (15*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]
*PolyLog[3, I*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) + (((15*I)/64)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4,
(-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (((15*I)/64)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*Arc
Tan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 261

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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 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 4886

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-2*I*(a + b*ArcTan[c*x])*
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x] + (Simp[(I*b*PolyLog[2, -((I*Sqrt[1 + I*c*x])/Sqrt[1
- I*c*x])])/(c*Sqrt[d]), x] - Simp[(I*b*PolyLog[2, (I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 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 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 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 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 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 6609

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

Rubi steps

\begin {align*} \int x^2 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ &=c^2 \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx+2 \left (\left (a^2 c^2\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\right )+\left (a^4 c^2\right ) \int x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=c^3 \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^8 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2}+\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{2} \left (a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (7 a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (3 a^5 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3}-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{2} c^3 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}+\frac {\left (9 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}+\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{4 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{2} c^3 \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}+\frac {\left (9 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}+\frac {1}{5} \left (2 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{5} \left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{48} \left (35 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{28} \left (9 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{16} \left (7 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{28} \left (3 a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {7 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{20} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{15} \left (4 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{12} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{64} \left (35 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {c^3 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}-\frac {\left (4 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{12 a}-\frac {\left (15 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a}-\frac {1}{20} \left (a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{35} \left (9 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{20} \left (7 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{64} \left (35 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{56} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{70} \left (9 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{40} \left (7 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{56} \left (a^3 c^3\right ) \int \frac {x^5}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{4 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {1}{20} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{15} \left (4 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{12} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {c^3 \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}-\frac {\left (4 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{12 a}-\frac {\left (15 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a}-\frac {1}{20} \left (a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {c^2 \sqrt {c+a^2 c x^2}}{4 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {359 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt {c+a^2 c x^2}}-\frac {6 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {1}{224} \left (15 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{280} \left (27 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{160} \left (21 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (6 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{30} \left (7 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{96} \left (35 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{24 a^2}+\frac {\left (35 c^3\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{128 a^2}+\frac {\left (8 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{6 a^2}+\frac {\left (15 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {\left (3 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{40 a}+\frac {\left (2 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {\left (6 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}+\frac {\left (5 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{24 a}+\frac {\left (7 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{30 a}+\frac {\left (35 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{96 a}+\frac {\left (105 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{128 a}+\frac {1}{224} \left (5 a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{40} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{280} \left (9 a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{160} \left (7 a c^3\right ) \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{112} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 \sqrt {c+a^2 c x^2}}{4 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{24 a^2}+\frac {\left (8 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{6 a^2}+\frac {\left (15 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {\left (3 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{40 a}+\frac {\left (2 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {\left (5 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{24 a}-\frac {1}{40} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {c^2 \sqrt {c+a^2 c x^2}}{6 a^3}+\frac {491 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{448 a^2}-\frac {\left (27 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{560 a^2}-\frac {\left (21 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{320 a^2}-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (7 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{60 a^2}-\frac {\left (35 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{192 a^2}-\frac {\left (12 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (7 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}-\frac {\left (35 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{48 a^2}-\frac {\left (105 c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{64 a^2}-\frac {\left (15 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{448 a}-\frac {\left (27 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{560 a}-\frac {\left (21 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{320 a}-\frac {\left (3 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}-\frac {\left (7 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{60 a}-\frac {\left (35 c^3\right ) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{192 a}+\frac {1}{448} \left (5 a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{560} \left (9 a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )+\frac {1}{320} \left (7 a c^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {1}{40} \left (a c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {1}{112} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{a^4 \sqrt {c+a^2 c x}}-\frac {2 \sqrt {c+a^2 c x}}{a^4 c}+\frac {\left (c+a^2 c x\right )^{3/2}}{a^4 c^2}\right ) \, dx,x,x^2\right )+\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{40 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{24 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{128 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{6 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c^2 \sqrt {c+a^2 c x^2}}{6 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {7 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {7 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}-\frac {1}{40} \left (a c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{40 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{24 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (8 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{6 a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {2239 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}-\frac {c \left (c+a^2 c x^2\right )^{3/2}}{210 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {491 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {379 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {379 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {379 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {1}{448} \left (5 a c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac {1}{560} \left (9 a c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )+\frac {1}{320} \left (7 a c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {13 c^2 \sqrt {c+a^2 c x^2}}{60 a^3}-\frac {c \left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}+\frac {9 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {9 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{448 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (27 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{560 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (21 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{320 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (7 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{60 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{192 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (12 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{35 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (7 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{48 a^2 \sqrt {c+a^2 c x^2}}-\frac {\left (105 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{64 a^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2899 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}+\frac {47 c \left (c+a^2 c x^2\right )^{3/2}}{1680 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {491 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {c+a^2 c x^2}}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (105 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (105 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {13 c^2 \sqrt {c+a^2 c x^2}}{60 a^3}-\frac {c \left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {9 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {9 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2899 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}+\frac {47 c \left (c+a^2 c x^2\right )^{3/2}}{1680 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {491 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {21 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (105 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (105 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {13 c^2 \sqrt {c+a^2 c x^2}}{60 a^3}-\frac {c \left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {2899 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}+\frac {47 c \left (c+a^2 c x^2\right )^{3/2}}{1680 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {491 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {13 c^2 \sqrt {c+a^2 c x^2}}{60 a^3}-\frac {c \left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {9 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {9 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (15 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (105 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (105 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2899 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}+\frac {47 c \left (c+a^2 c x^2\right )^{3/2}}{1680 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {491 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {13 c^2 \sqrt {c+a^2 c x^2}}{60 a^3}-\frac {c \left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (105 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (105 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2899 c^2 \sqrt {c+a^2 c x^2}}{6720 a^3}+\frac {47 c \left (c+a^2 c x^2\right )^{3/2}}{1680 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{280 a^3}+\frac {491 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{1344 a^2}-\frac {27}{560} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{56} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{13440 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{6720 a}+\frac {29}{560} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{56} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{840 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{128 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {929 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{1680 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {63 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {63 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {13 c^2 \sqrt {c+a^2 c x^2}}{60 a^3}-\frac {c \left (c+a^2 c x^2\right )^{3/2}}{60 a^3}-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a^2}+\frac {1}{20} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{240 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{120 a}-\frac {1}{10} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {19 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{120 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}

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Mathematica [B]  time = 24.43, size = 6517, normalized size = 6.40 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3,x]

[Out]

Result too large to show

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="fricas")

[Out]

integral((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3, x)

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giac [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^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 2.05, size = 566, normalized size = 0.56 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1680 \arctan \left (a x \right )^{3} x^{7} a^{7}-720 \arctan \left (a x \right )^{2} x^{6} a^{6}+4760 \arctan \left (a x \right )^{3} x^{5} a^{5}+240 \arctan \left (a x \right ) x^{5} a^{5}-1992 \arctan \left (a x \right )^{2} x^{4} a^{4}+4130 \arctan \left (a x \right )^{3} a^{3} x^{3}-48 a^{4} x^{4}+696 \arctan \left (a x \right ) x^{3} a^{3}-1474 \arctan \left (a x \right )^{2} x^{2} a^{2}+525 \arctan \left (a x \right )^{3} x a -168 a^{2} x^{2}+430 \arctan \left (a x \right ) x a +1373 \arctan \left (a x \right )^{2}-94\right )}{13440 a^{3}}-\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (525 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-1575 i \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-525 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+1575 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \polylog \left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{13440 a^{3} \sqrt {a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x)

[Out]

1/13440*c^2/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(1680*arctan(a*x)^3*x^7*a^7-720*arctan(a*x)^2*x^6*a^6+4760*arctan(a*
x)^3*x^5*a^5+240*arctan(a*x)*x^5*a^5-1992*arctan(a*x)^2*x^4*a^4+4130*arctan(a*x)^3*a^3*x^3-48*a^4*x^4+696*arct
an(a*x)*x^3*a^3-1474*arctan(a*x)^2*x^2*a^2+525*arctan(a*x)^3*x*a-168*a^2*x^2+430*arctan(a*x)*x*a+1373*arctan(a
*x)^2-94)-1/13440*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(525*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1575*I*
arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-525*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+1
575*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3176*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1
/2))+3150*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3150*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))
-3176*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2
))-3150*I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3176*I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3176*I*dilog
(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2+1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(5/2)*x^2*arctan(a*x)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*atan(a*x)^3*(c + a^2*c*x^2)^(5/2),x)

[Out]

int(x^2*atan(a*x)^3*(c + a^2*c*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a**2*c*x**2+c)**(5/2)*atan(a*x)**3,x)

[Out]

Integral(x**2*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3, x)

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