3.5.20 \(\int x^3 (c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^3 \, dx\) [420]

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

[Out]

23/120*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^4-51/280*I*c^2*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^
2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+51/280*I*c^2*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2
*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-51/280*I*c^2*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arct
an(a*x)^2*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-51/280*c^2*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*
x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+51/280*c^2*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4
/(a^2*c*x^2+c)^(1/2)+1/420*c*x*(a^2*c*x^2+c)^(1/2)/a^3-1/140*c*x^3*(a^2*c*x^2+c)^(1/2)/a-163/840*c*arctan(a*x)
*(a^2*c*x^2+c)^(1/2)/a^4+1/60*c*x^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+1/35*c*x^4*arctan(a*x)*(a^2*c*x^2+c)^(
1/2)+9/112*c*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^3-23/280*c*x^3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a-1/14*a*c
*x^5*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)-2/35*c*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^4+1/35*c*x^2*arctan(a*x)^3*(
a^2*c*x^2+c)^(1/2)/a^2+8/35*c*x^4*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)+1/7*a^2*c*x^6*arctan(a*x)^3*(a^2*c*x^2+c)^
(1/2)

________________________________________________________________________________________

Rubi [A]
time = 4.92, antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps used = 200, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \begin {gather*} \frac {c x^2 \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}}{35 a^2}+\frac {c x^2 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{60 a^2}+\frac {1}{7} a^2 c x^6 \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}-\frac {1}{14} a c x^5 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}+\frac {8}{35} c x^4 \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}+\frac {1}{35} c x^4 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}-\frac {23 c x^3 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{280 a}-\frac {c x^3 \sqrt {a^2 c x^2+c}}{140 a}+\frac {51 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {51 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {51 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}+\frac {51 c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {51 i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{280 a^4 \sqrt {a^2 c x^2+c}}-\frac {2 c \text {ArcTan}(a x)^3 \sqrt {a^2 c x^2+c}}{35 a^4}-\frac {163 c \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{840 a^4}+\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 {9 c x \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{112 a^3}+\frac {c x \sqrt {a^2 c x^2+c}}{420 a^3} \end {gather*}

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 212

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

Rule 223

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 327

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]

Rule 2320

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 2611

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 4266

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 5008

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 5010

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 5050

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 5070

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 5072

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 6724

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]

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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \text {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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 7.72, size = 1296, normalized size = 1.99 \begin {gather*} \frac {c \left (\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (11 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+11 \pi \text {ArcTan}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (1-i e^{i \text {ArcTan}(a x)}\right )\right )-11 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-11 \text {ArcTan}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )+11 \pi \text {ArcTan}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )+11 \text {ArcTan}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcTan}(a x)}\right )\right )-11 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-20 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+11 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+20 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-11 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-11 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+22 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-22 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-22 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+22 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )\right )}{40 \sqrt {1+a^2 x^2}}-\frac {1}{960} \left (1+a^2 x^2\right )^2 \sqrt {c \left (1+a^2 x^2\right )} \left (150 \text {ArcTan}(a x)-32 \text {ArcTan}(a x)^3+8 \text {ArcTan}(a x) \left (27+20 \text {ArcTan}(a x)^2\right ) \cos (2 \text {ArcTan}(a x))+66 \text {ArcTan}(a x) \cos (4 \text {ArcTan}(a x))+12 \sin (2 \text {ArcTan}(a x))+6 \text {ArcTan}(a x)^2 \sin (2 \text {ArcTan}(a x))+6 \sin (4 \text {ArcTan}(a x))-33 \text {ArcTan}(a x)^2 \sin (4 \text {ArcTan}(a x))\right )\right )}{a^4}+\frac {c \left (-\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (309 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+309 \pi \text {ArcTan}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (1-i e^{i \text {ArcTan}(a x)}\right )\right )-309 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-309 \text {ArcTan}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )+309 \pi \text {ArcTan}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )+309 \text {ArcTan}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcTan}(a x)}\right )\right )-309 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-518 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+309 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+518 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-309 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-309 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+618 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-618 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-618 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+618 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )\right )}{1680 \sqrt {1+a^2 x^2}}-\frac {\left (1+a^2 x^2\right )^3 \sqrt {c \left (1+a^2 x^2\right )} \left (-4116 \text {ArcTan}(a x)-3648 \text {ArcTan}(a x)^3+2 \text {ArcTan}(a x) \left (-3131+896 \text {ArcTan}(a x)^2\right ) \cos (2 \text {ArcTan}(a x))-4 \text {ArcTan}(a x) \left (691+560 \text {ArcTan}(a x)^2\right ) \cos (4 \text {ArcTan}(a x))-618 \text {ArcTan}(a x) \cos (6 \text {ArcTan}(a x))-404 \sin (2 \text {ArcTan}(a x))+633 \text {ArcTan}(a x)^2 \sin (2 \text {ArcTan}(a x))-352 \sin (4 \text {ArcTan}(a x))-180 \text {ArcTan}(a x)^2 \sin (4 \text {ArcTan}(a x))-100 \sin (6 \text {ArcTan}(a x))+309 \text {ArcTan}(a x)^2 \sin (6 \text {ArcTan}(a x))\right )}{53760}\right )}{a^4} \end {gather*}

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*(1 + a^2*x^2)]*(11*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 11*Pi*ArcTan[a*x]*Log[((-1)^(1/4)*
(1 - I*E^(I*ArcTan[a*x])))/(2*E^((I/2)*ArcTan[a*x]))] - 11*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] - 11*Arc
Tan[a*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcTan[a*x])))/E^((I/2)*ArcTan[a*x])] + 11*Pi*ArcTan[a*x]*Log[-1/2*((-1
)^(1/4)*(-I + E^(I*ArcTan[a*x])))/E^((I/2)*ArcTan[a*x])] + 11*ArcTan[a*x]^2*Log[((1 + I) + (1 - I)*E^(I*ArcTan
[a*x]))/(2*E^((I/2)*ArcTan[a*x]))] - 11*Pi*ArcTan[a*x]*Log[-Cos[(Pi + 2*ArcTan[a*x])/4]] - 20*Log[Cos[ArcTan[a
*x]/2] - Sin[ArcTan[a*x]/2]] + 11*ArcTan[a*x]^2*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] + 20*Log[Cos[ArcT
an[a*x]/2] + Sin[ArcTan[a*x]/2]] - 11*ArcTan[a*x]^2*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] - 11*Pi*ArcTa
n[a*x]*Log[Sin[(Pi + 2*ArcTan[a*x])/4]] + (22*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (22*I)*ArcTa
n[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 22*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 22*PolyLog[3, I*E^(I*ArcTan[a
*x])]))/(40*Sqrt[1 + a^2*x^2]) - ((1 + a^2*x^2)^2*Sqrt[c*(1 + a^2*x^2)]*(150*ArcTan[a*x] - 32*ArcTan[a*x]^3 +
8*ArcTan[a*x]*(27 + 20*ArcTan[a*x]^2)*Cos[2*ArcTan[a*x]] + 66*ArcTan[a*x]*Cos[4*ArcTan[a*x]] + 12*Sin[2*ArcTan
[a*x]] + 6*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] + 6*Sin[4*ArcTan[a*x]] - 33*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/960
))/a^4 + (c*(-1/1680*(Sqrt[c*(1 + a^2*x^2)]*(309*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + 309*Pi*ArcTan[a*
x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcTan[a*x])))/(2*E^((I/2)*ArcTan[a*x]))] - 309*ArcTan[a*x]^2*Log[1 + I*E^(I*Ar
cTan[a*x])] - 309*ArcTan[a*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcTan[a*x])))/E^((I/2)*ArcTan[a*x])] + 309*Pi*Arc
Tan[a*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcTan[a*x])))/E^((I/2)*ArcTan[a*x])] + 309*ArcTan[a*x]^2*Log[((1 + I
) + (1 - I)*E^(I*ArcTan[a*x]))/(2*E^((I/2)*ArcTan[a*x]))] - 309*Pi*ArcTan[a*x]*Log[-Cos[(Pi + 2*ArcTan[a*x])/4
]] - 518*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] + 309*ArcTan[a*x]^2*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[
a*x]/2]] + 518*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] - 309*ArcTan[a*x]^2*Log[Cos[ArcTan[a*x]/2] + Sin[A
rcTan[a*x]/2]] - 309*Pi*ArcTan[a*x]*Log[Sin[(Pi + 2*ArcTan[a*x])/4]] + (618*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(
I*ArcTan[a*x])] - (618*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 618*PolyLog[3, (-I)*E^(I*ArcTan[a*x])]
+ 618*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] - ((1 + a^2*x^2)^3*Sqrt[c*(1 + a^2*x^2)]*(-4116*Arc
Tan[a*x] - 3648*ArcTan[a*x]^3 + 2*ArcTan[a*x]*(-3131 + 896*ArcTan[a*x]^2)*Cos[2*ArcTan[a*x]] - 4*ArcTan[a*x]*(
691 + 560*ArcTan[a*x]^2)*Cos[4*ArcTan[a*x]] - 618*ArcTan[a*x]*Cos[6*ArcTan[a*x]] - 404*Sin[2*ArcTan[a*x]] + 63
3*ArcTan[a*x]^2*Sin[2*ArcTan[a*x]] - 352*Sin[4*ArcTan[a*x]] - 180*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]] - 100*Sin[6
*ArcTan[a*x]] + 309*ArcTan[a*x]^2*Sin[6*ArcTan[a*x]]))/53760))/a^4

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Maple [A]
time = 7.88, size = 469, normalized size = 0.72

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (240 \arctan \left (a x \right )^{3} a^{6} x^{6}-120 \arctan \left (a x \right )^{2} a^{5} x^{5}+384 \arctan \left (a x \right )^{3} a^{4} x^{4}+48 \arctan \left (a x \right ) a^{4} x^{4}-138 \arctan \left (a x \right )^{2} a^{3} x^{3}+48 \arctan \left (a x \right )^{3} a^{2} x^{2}-12 a^{3} x^{3}+28 \arctan \left (a x \right ) a^{2} x^{2}+135 \arctan \left (a x \right )^{2} a x -96 \arctan \left (a x \right )^{3}+4 a x -326 \arctan \left (a x \right )\right )}{1680 a^{4}}+\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {23 i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{60 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(469\)

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,method=_RETURNVERBOSE)

[Out]

1/1680*c/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(240*arctan(a*x)^3*a^6*x^6-120*arctan(a*x)^2*a^5*x^5+384*arctan(a*x)^3*
a^4*x^4+48*arctan(a*x)*a^4*x^4-138*arctan(a*x)^2*a^3*x^3+48*arctan(a*x)^3*a^2*x^2-12*a^3*x^3+28*arctan(a*x)*a^
2*x^2+135*arctan(a*x)^2*a*x-96*arctan(a*x)^3+4*a*x-326*arctan(a*x))+17/560*c*(c*(a*x-I)*(I+a*x))^(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)*(I+a*x))^(1/2
)*(I*arctan(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)-23/60*I*c/a^4*(c*(a*x-I)*(
I+a*x))^(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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \end {gather*}

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]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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