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

Optimal. Leaf size=1027 \[ \text{result too large to display} \]

[Out]

-(a*c^2*Sqrt[c + a^2*c*x^2])/4 + (a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (21*a*c^2*Sqrt[c + a^2*c*x^2]
*ArcTan[a*x]^2)/8 - (a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x +
(7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((15*I)
/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((11*I)*a*c^3*Sqrt[
1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*
x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x
]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Poly
Log[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLo
g[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*Ar
cTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[
1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 -
I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]
- (45*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (45*a
*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (6*a*c^3*Sqrt[1
+ a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4,
(-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a
*x])])/Sqrt[c + a^2*c*x^2]

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Rubi [A]  time = 2.11121, antiderivative size = 1027, normalized size of antiderivative = 1., number of steps used = 56, number of rules used = 15, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4950, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4890, 4888, 4181, 6609, 4880, 4886, 4878} \[ -\frac{15 i a \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3 c^3}{4 \sqrt{a^2 c x^2+c}}-\frac{11 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{45 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 \sqrt{a^2 c x^2+c}}-\frac{45 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right ) c^3}{8 \sqrt{a^2 c x^2+c}}-\frac{6 i a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}+\frac{11 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{2 \sqrt{a^2 c x^2+c}}-\frac{11 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{i a x+1}}{\sqrt{1-i a x}}\right ) c^3}{2 \sqrt{a^2 c x^2+c}}-\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{45 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{45 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right ) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{6 a \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right ) c^3}{\sqrt{a^2 c x^2+c}}-\frac{45 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right ) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{45 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right ) c^3}{4 \sqrt{a^2 c x^2+c}}+\frac{7}{8} a^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3 c^2}{x}-\frac{21}{8} a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2 c^2+\frac{1}{4} a^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x) c^2-\frac{1}{4} a \sqrt{a^2 c x^2+c} c^2+\frac{1}{4} a^2 x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3 c-\frac{1}{4} a \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2 c \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*c^2*Sqrt[c + a^2*c*x^2])/4 + (a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 - (21*a*c^2*Sqrt[c + a^2*c*x^2]
*ArcTan[a*x]^2)/8 - (a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/4 - (c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x +
(7*a^2*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/8 + (a^2*c*x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/4 - (((15*I)
/4)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - ((11*I)*a*c^3*Sqrt[
1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*
x^2]*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x
]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Poly
Log[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/8)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLo
g[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*Ar
cTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[
1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (((11*I)/2)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 -
I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]
- (45*a*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (45*a
*c^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*Sqrt[c + a^2*c*x^2]) + (6*a*c^3*Sqrt[1
+ a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4,
(-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + (((45*I)/4)*a*c^3*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a
*x])])/Sqrt[c + a^2*c*x^2]

Rule 4950

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

Rule 4944

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p)/(d*f*(m + 1)), x] - Dist[(b*c*p)/(f*(m + 1)), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]

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 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 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 2531

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

Rule 2282

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

Rule 6589

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

Rule 4890

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

Rule 4888

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

Rule 4181

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

Rule 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]

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 4886

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-2*I*(a + b*ArcTan[c*x])*
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x] + (Simp[(I*b*PolyLog[2, -((I*Sqrt[1 + I*c*x])/Sqrt[1
- I*c*x])])/(c*Sqrt[d]), x] - Simp[(I*b*PolyLog[2, (I*Sqrt[1 + I*c*x])/Sqrt[1 - I*c*x]])/(c*Sqrt[d]), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]

Rule 4878

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

Rubi steps

\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3}{x^2} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x^2} \, dx+\left (a^2 c\right ) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^2} \, dx+\frac{1}{2} \left (a^2 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\frac{1}{4} \left (3 a^2 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+c^3 \int \frac{\tan ^{-1}(a x)^3}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{4} \left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{8} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{2} \left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx+\frac{1}{4} \left (9 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{8 \sqrt{c+a^2 c x^2}}+\frac{\left (a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}+\frac{\left (9 a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 \sqrt{c+a^2 c x^2}}+\frac{\left (a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\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}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 \sqrt{c+a^2 c x^2}}+\frac{\left (9 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}+\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (9 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (3 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{6 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{45 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{45 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (6 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \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 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \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 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (3 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{6 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{45 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{45 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (9 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{\left (9 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}-\frac{\left (3 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (3 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 i a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{\left (6 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{\left (6 a c^3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{1}{4} a c^2 \sqrt{c+a^2 c x^2}+\frac{1}{4} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{21}{8} a c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac{7}{8} a^2 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac{1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac{15 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{6 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{6 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{45 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 \sqrt{c+a^2 c x^2}}-\frac{6 i a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}+\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{11 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{6 a c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{45 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{45 a c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{6 a c^3 \sqrt{1+a^2 x^2} \text{Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt{c+a^2 c x^2}}-\frac{45 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}+\frac{45 i a c^3 \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [B]  time = 15.9356, size = 3267, normalized size = 3.18 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I)*a*c^2*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a
*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[a*x]
^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[
a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, (-I
)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (a*c^2*Sqrt[c*(1 + a^2*x^2)]*Cs
c[ArcTan[a*x]/2]*(((-7*I)*a*Pi^4*x)/Sqrt[1 + a^2*x^2] - ((8*I)*a*Pi^3*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + ((24*
I)*a*Pi^2*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 64*ArcTan[a*x]^3 - ((32*I)*a*Pi*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x
^2] + ((16*I)*a*x*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (48*a*Pi^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/Sq
rt[1 + a^2*x^2] - (96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (8*a*Pi^3*x*Log[1
+ I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (64*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*
x^2] + (192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[1 + I*E^(I*ArcTa
n[a*x])])/Sqrt[1 + a^2*x^2] - (48*a*Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (96*a
*Pi*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (64*a*x*ArcTan[a*x]^3*Log[1 + I*E^(I*Arc
Tan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (8*a*Pi
^3*x*Log[Tan[(Pi + 2*ArcTan[a*x])/4]])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*Arc
Tan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])])/Sqrt[1 +
a^2*x^2] + ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi^2*x*PolyL
og[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((192*I)*a*Pi*x*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x
])])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((
384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)
/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384
*a*x*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*Pi*x*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[
1 + a^2*x^2] - (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*a*x*PolyLog[3
, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] -
((384*I)*a*x*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2])*Sec[ArcTan[a*x]/2])/(128*Sqrt[1 + a^2*x^2
]) + a*c^2*((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(Arc
Tan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])
] - PolyLog[2, I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*(-(Pi^3*Log[Cot[(Pi/2 -
ArcTan[a*x])/2]])/8 - (3*Pi^2*((Pi/2 - ArcTan[a*x])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2
- ArcTan[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 +
(3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (
2*I)*(Pi/2 - ArcTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))]) +
2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 -
ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcT
an[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))
]))/8 - (Pi/2 + (-Pi/2 + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))] + ((3*I)/8)*(Pi
/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2)^
2 - (Pi/2 + (-Pi/2 + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))] + (I/2)*PolyLog[2, -E
^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2)^2*PolyLog[2, -E^((
2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))] - (3*(Pi/2 - ArcTan[a*x])*PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))])/4 -
(3*Pi*((I/3)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2)^3 - (Pi/2 + (-Pi/2 + ArcTan[a*x])/2)^2*Log[1 + E^((2*I)*(Pi/2 +
(-Pi/2 + ArcTan[a*x])/2))] + I*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[
a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))]/2))/2 - (3*(Pi/2 + (-Pi/2 + ArcTan[a*x])/
2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))])/2 - ((3*I)/4)*PolyLog[4, -E^(I*(Pi/2 - ArcTan[a*x]
))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-Pi/2 + ArcTan[a*x])/2))])))/(8*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1
+ a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a
^2*x^2)]*(2*ArcTan[a*x] - ArcTan[a*x]^2 - ArcTan[a*x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] - Sin[ArcT
an[a*x]/2])^2) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x
]/2] - Sin[ArcTan[a*x]/2])^3) - (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^3)/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2
] + Sin[ArcTan[a*x]/2])^4) + (Sqrt[c*(1 + a^2*x^2)]*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2]*(Co
s[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^3) + (Sqrt[c*(1 + a^2*x^2)]*(-2*ArcTan[a*x] - ArcTan[a*x]^2 + ArcTan[a*
x]^3))/(16*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2) + (Sqrt[c*(1 + a^2*x^2)]*(Sin[ArcTan
[a*x]/2] - ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]))
+ (Sqrt[c*(1 + a^2*x^2)]*(-Sin[ArcTan[a*x]/2] + ArcTan[a*x]^2*Sin[ArcTan[a*x]/2]))/(4*Sqrt[1 + a^2*x^2]*(Cos[A
rcTan[a*x]/2] - Sin[ArcTan[a*x]/2])))

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Maple [A]  time = 1.996, size = 655, normalized size = 0.6 \begin{align*} \text{result too large to display} \end{align*}

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^2,x)

[Out]

1/8*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(2*arctan(a*x)^3*x^4*a^4-2*arctan(a*x)^2*x^3*a^3+9*arctan(a*x)^3*x^2*a^2+2*a
rctan(a*x)*a^2*x^2-23*arctan(a*x)^2*x*a-8*arctan(a*x)^3-2*a*x)/x-1/8*I*a*c^2*(c*(a*x-I)*(a*x+I))^(1/2)*(-48*I*
polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+44*I*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-44*I*arctan(a*x)*
ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-90*I*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+90*I*arctan(a*x
)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-24*I*arctan(a*x)^2*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+15*I*arctan(a*
x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-15*I*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-45*arctan(a*x)
^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+45*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+48*I*po
lylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))+24*I*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+48*arctan(a*x)*polyl
og(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-48*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-44*polylog(2,-I*(1+I*
a*x)/(a^2*x^2+1)^(1/2))+90*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+44*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2
))-90*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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^{2}}, x\right ) \end{align*}

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^2,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^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )^{3}}{x^{2}}\,{d x} \end{align*}

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^2,x, algorithm="giac")

[Out]

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