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

Optimal. Leaf size=845 \[ \frac {149 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2 c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {149 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}+\frac {149 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {149 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {149 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {3}{2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right ) c^{5/2}+\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac {29}{40} a x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac {29}{20} \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac {1}{20} a x \sqrt {a^2 c x^2+c} c^2+\frac {1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3 c-\frac {3}{20} a x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2 c+\frac {1}{10} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) c+\frac {1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3 \]

[Out]

1/10*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)-3/20*a*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2+1/3*c*(a^2*c*x^2+c)^(3/2)*
arctan(a*x)^3+1/5*(a^2*c*x^2+c)^(5/2)*arctan(a*x)^3-3/2*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))-149/2
0*I*c^3*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^2*c*x^2+c)^(1/2)-2*c^3*arct
an(a*x)^3*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*I*c^3*polylog(4,(1+I*a*
x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+149/20*I*c^3*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^2*c*x^2+c)^(1/2)+3*I*c^3*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(
1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+149/20*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^
2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*c^3*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)
/(a^2*c*x^2+c)^(1/2)+149/20*c^3*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2
)-149/20*c^3*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*c^3*arctan(a*x)*
polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*I*c^3*polylog(4,-(1+I*a*x)/(a^2
*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-3*I*c^3*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/
2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-1/20*a*c^2*x*(a^2*c*x^2+c)^(1/2)+29/20*c^2*arctan(a*x)*(a^2*c*x^2+c)
^(1/2)-29/40*a*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+c^2*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.78, antiderivative size = 845, normalized size of antiderivative = 1.00, number of steps used = 54, number of rules used = 16, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4950, 4958, 4956, 4183, 2531, 6609, 2282, 6589, 4930, 4890, 4888, 4181, 4880, 217, 206, 195} \[ \frac {149 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2 c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {149 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}+\frac {149 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {149 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}-\frac {149 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) c^3}{20 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt {a^2 c x^2+c}}-\frac {3}{2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right ) c^{5/2}+\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac {29}{40} a x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac {29}{20} \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac {1}{20} a x \sqrt {a^2 c x^2+c} c^2+\frac {1}{3} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3 c-\frac {3}{20} a x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2 c+\frac {1}{10} \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) c+\frac {1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*c^2*x*Sqrt[c + a^2*c*x^2])/20 + (29*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/20 + (c*(c + a^2*c*x^2)^(3/2)*Arc
Tan[a*x])/10 - (29*a*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/40 - (3*a*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^
2)/20 + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/Sqrt[c + a^2*c*x^2] + c^2
*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3 + (c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/3 + ((c + a^2*c*x^2)^(5/2)*ArcTan
[a*x]^3)/5 - (2*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (3*c^(5/
2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/2 + ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, -E^(I
*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/Sqrt[c + a^2*c*x^2] + (((149*I)/20)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x
])])/Sqrt[c + a^2*c*x^2] - ((3*I)*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] - (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (149*c
^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) - (149*c^3*Sqrt[1 + a^2*x^2]
*PolyLog[3, I*E^(I*ArcTan[a*x])])/(20*Sqrt[c + a^2*c*x^2]) + (6*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, E
^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] + ((6*I)*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

Rule 195

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

Rule 206

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

Rule 217

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

Rule 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 4183

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

Rule 4880

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

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 4956

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[1/Sqrt[d], Sub
st[Int[(a + b*x)^p*Csc[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 4958

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + c^2*
x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*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 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 \frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{x} \, dx &=c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {1}{5} (3 a c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx+c^2 \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {1}{10} \left (a c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx-\frac {1}{20} \left (9 a c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx-\left (a c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+c^3 \int \frac {\tan ^{-1}(a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {1}{20} \left (a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{40} \left (9 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{20} \left (9 a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\left (a c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx-\left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\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}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {1}{20} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )-\frac {1}{20} \left (9 a c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )-\left (a c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (9 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{40 \sqrt {c+a^2 c x^2}}-\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{40 \sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \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-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\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+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\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 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\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 (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \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 (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {\left (i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \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 (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{20} a c^2 x \sqrt {c+a^2 c x^2}+\frac {29}{20} c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{10} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac {29}{40} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3}{20} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{20 \sqrt {c+a^2 c x^2}}+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3}{2} c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {3 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \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 (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {149 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt {c+a^2 c x^2}}-\frac {149 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{20 \sqrt {c+a^2 c x^2}}+\frac {6 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c^3 \sqrt {1+a^2 x^2} \text {Li}_4\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [A]  time = 7.17, size = 1066, normalized size = 1.26 \[ \frac {1}{8} \sqrt {c \left (a^2 x^2+1\right )} \left (\frac {2 i \tan ^{-1}(a x)^4}{\sqrt {a^2 x^2+1}}+\frac {8 \log \left (1-e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {a^2 x^2+1}}-\frac {8 \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {a^2 x^2+1}}+8 \tan ^{-1}(a x)^3-\frac {24 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}+\frac {24 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}+\frac {24 i \text {Li}_2\left (e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}+\frac {24 i \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}-\frac {48 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+\frac {48 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+\frac {48 \text {Li}_3\left (e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-\frac {48 \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+\frac {48 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-\frac {48 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-\frac {48 i \text {Li}_4\left (e^{-i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-\frac {48 i \text {Li}_4\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 x^2+1}}-\frac {i \pi ^4}{\sqrt {a^2 x^2+1}}\right ) c^2+2 \left (\frac {\sqrt {c \left (a^2 x^2+1\right )} \left (i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+\text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-\text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )\right )}{\sqrt {a^2 x^2+1}}+\frac {1}{12} \left (a^2 x^2+1\right ) \sqrt {c \left (a^2 x^2+1\right )} \tan ^{-1}(a x) \left (4 \tan ^{-1}(a x)^2-3 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+6 \cos \left (2 \tan ^{-1}(a x)\right )+6\right )\right ) c^2+\left (\frac {\sqrt {c \left (a^2 x^2+1\right )} \left (-11 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2+11 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-11 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+10 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-11 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+11 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )\right )}{20 \sqrt {a^2 x^2+1}}-\frac {1}{960} \left (a^2 x^2+1\right )^2 \sqrt {c \left (a^2 x^2+1\right )} \left (-32 \tan ^{-1}(a x)^3+6 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2-33 \sin \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2+8 \left (20 \tan ^{-1}(a x)^2+27\right ) \cos \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+66 \cos \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+150 \tan ^{-1}(a x)+12 \sin \left (2 \tan ^{-1}(a x)\right )+6 \sin \left (4 \tan ^{-1}(a x)\right )\right )\right ) c^2 \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c*(1 + a^2*x^2)]*(((-I)*Pi^4)/Sqrt[1 + a^2*x^2] + 8*ArcTan[a*x]^3 + ((2*I)*ArcTan[a*x]^4)/Sqrt[1 + a
^2*x^2] + (8*ArcTan[a*x]^3*Log[1 - E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (24*ArcTan[a*x]^2*Log[1 - I*E^(I
*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (24*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*Arc
Tan[a*x]^3*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((24*I)*ArcTan[a*x]^2*PolyLog[2, E^((-I)*ArcTan[a*x
])])/Sqrt[1 + a^2*x^2] + ((24*I)*ArcTan[a*x]^2*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*Arc
Tan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[
a*x])])/Sqrt[1 + a^2*x^2] + (48*ArcTan[a*x]*PolyLog[3, E^((-I)*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*ArcTan[a
*x]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (48*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x
^2] - (48*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((48*I)*PolyLog[4, E^((-I)*ArcTan[a*x])])/Sqrt[
1 + a^2*x^2] - ((48*I)*PolyLog[4, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2]))/8 + 2*c^2*((Sqrt[c*(1 + a^2*x^2)]*(
I*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - I*ArcTan[a*x]*PolyLog[2, (-I)*E
^(I*ArcTan[a*x])] + I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - PolyL
og[3, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + ((1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]*(6 + 4*ArcTa
n[a*x]^2 + 6*Cos[2*ArcTan[a*x]] - 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/12) + c^2*((Sqrt[c*(1 + a^2*x^2)]*((-11*I
)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2
, (-I)*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTa
n[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]))/(20*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]] - 3
3*ArcTan[a*x]^2*Sin[4*ArcTan[a*x]]))/960)

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 1.27, size = 562, normalized size = 0.67 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 \arctan \left (a x \right )^{3} x^{4} a^{4}-18 \arctan \left (a x \right )^{2} x^{3} a^{3}+88 \arctan \left (a x \right )^{3} x^{2} a^{2}+12 \arctan \left (a x \right ) a^{2} x^{2}-105 \arctan \left (a x \right )^{2} x a +184 \arctan \left (a x \right )^{3}-6 a x +186 \arctan \left (a x \right )\right )}{120}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (40 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-40 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 i \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+120 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+149 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-149 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-298 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+298 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+240 \arctan \left (a x \right ) \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-240 \arctan \left (a x \right ) \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+240 i \polylog \left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-240 i \polylog \left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+120 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+298 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-298 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) c^{2}}{40 \sqrt {a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/120*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(24*arctan(a*x)^3*x^4*a^4-18*arctan(a*x)^2*x^3*a^3+88*arctan(a*x)^3*x^2*a^
2+12*arctan(a*x)*a^2*x^2-105*arctan(a*x)^2*x*a+184*arctan(a*x)^3-6*a*x+186*arctan(a*x))+1/40*(c*(a*x-I)*(I+a*x
))^(1/2)/(a^2*x^2+1)^(1/2)*(40*arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-40*arctan(a*x)^3*ln(1+(1+I*a*x)
/(a^2*x^2+1)^(1/2))-120*I*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+120*I*arctan(a*x)^2*polylog(2,-
(1+I*a*x)/(a^2*x^2+1)^(1/2))+149*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-149*arctan(a*x)^2*ln(1-I*(1
+I*a*x)/(a^2*x^2+1)^(1/2))-298*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+298*I*arctan(a*x)*polyl
og(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+240*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-240*arctan(a*x)*pol
ylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+240*I*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))-240*I*polylog(4,-(1+I*a*x)/(
a^2*x^2+1)^(1/2))+120*I*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))+298*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-298*
polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))*c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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