3.437 \(\int \frac{x^2 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx\)

Optimal. Leaf size=625 \[ \frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}} \]

[Out]

(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*a^2*c) + (I*Sqrt[1
+ a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcT
an[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcT
an[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcTan[
a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*
Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 +
I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*A
rcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a
^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2
]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.489111, antiderivative size = 625, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589} \[ \frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{a^2 c x^2+c}}+\frac{i \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac{3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 a^3 c}-\frac{6 i \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*a^2*c) + (I*Sqrt[1
+ a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/(a^3*Sqrt[c + a^2*c*x^2]) - ((6*I)*Sqrt[1 + a^2*x^2]*ArcT
an[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcT
an[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/2)*Sqrt[1 + a^2*x^2]*ArcTan[
a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*
Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 +
I*a*x])/Sqrt[1 - I*a*x]])/(a^3*Sqrt[c + a^2*c*x^2]) + (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*A
rcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(a
^3*Sqrt[c + a^2*c*x^2]) + ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2
]) - ((3*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2])

Rule 4952

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

Rule 4930

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

Rule 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 4886

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

Rule 4888

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

Rule 4181

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

Rule 2531

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

Rule 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 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]

Rubi steps

\begin{align*} \int \frac{x^2 \tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac{\int \frac{\tan ^{-1}(a x)^3}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^2}-\frac{3 \int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{3 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^2}-\frac{\sqrt{1+a^2 x^2} \int \frac{\tan ^{-1}(a x)^3}{\sqrt{1+a^2 x^2}} \, dx}{2 a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{a^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{c+a^2 c x^2}}-\frac{6 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{c+a^2 c x^2}}-\frac{6 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (3 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (3 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{c+a^2 c x^2}}-\frac{6 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{c+a^2 c x^2}}-\frac{6 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (3 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (3 i \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^3 c}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^3}{2 a^2 c}+\frac{i \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 \sqrt{c+a^2 c x^2}}-\frac{6 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \tan ^{-1}(a x)^2 \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{3 i \sqrt{1+a^2 x^2} \text{Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 6.00671, size = 812, normalized size = 1.3 \[ \frac{\sqrt{c \left (a^2 x^2+1\right )} \left (-\frac{1}{2} i \tan ^{-1}(a x)^4-2 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3+2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3+\frac{\tan ^{-1}(a x)^3}{\left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )^2}-\frac{\tan ^{-1}(a x)^3}{\left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )+\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )^2}+i \pi \tan ^{-1}(a x)^3+3 \pi \log \left (1-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-3 \pi \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-6 i \text{PolyLog}\left (2,-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-6 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-\frac{6 \sin \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2}{\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )}+\frac{6 \sin \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2}{\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )+\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )}-\frac{3}{4} i \pi ^2 \tan ^{-1}(a x)^2-6 \tan ^{-1}(a x)^2-\frac{3}{2} \pi ^2 \log \left (1-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+12 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac{3}{2} \pi ^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-12 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+6 i \pi \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-12 \text{PolyLog}\left (3,-i e^{-i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+12 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac{1}{4} i \pi ^3 \tan ^{-1}(a x)+\frac{1}{4} \pi ^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )-\frac{1}{4} \pi ^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-\frac{1}{4} \pi ^3 \log \left (\tan \left (\frac{1}{4} \left (2 \tan ^{-1}(a x)+\pi \right )\right )\right )-\frac{3}{2} i \pi \left (\pi -4 \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,i e^{-i \tan ^{-1}(a x)}\right )-\frac{3}{2} i \pi ^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+12 i \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-12 i \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+6 \pi \text{PolyLog}\left (3,i e^{-i \tan ^{-1}(a x)}\right )-6 \pi \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )+12 i \text{PolyLog}\left (4,-i e^{-i \tan ^{-1}(a x)}\right )+12 i \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )+\frac{7 i \pi ^4}{32}\right )}{4 a^3 c \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[c*(1 + a^2*x^2)]*(((7*I)/32)*Pi^4 + (I/4)*Pi^3*ArcTan[a*x] - 6*ArcTan[a*x]^2 - ((3*I)/4)*Pi^2*ArcTan[a*x
]^2 + I*Pi*ArcTan[a*x]^3 - (I/2)*ArcTan[a*x]^4 - (3*Pi^2*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/2 + 3*Pi*Ar
cTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] + (Pi^3*Log[1 + I/E^(I*ArcTan[a*x])])/4 - 2*ArcTan[a*x]^3*Log[1 + I/E
^(I*ArcTan[a*x])] + 12*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])] - (Pi^3*Log[1 + I*E^(I*ArcTan[a*x])])/4 - 12*A
rcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + (3*Pi^2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/2 - 3*Pi*ArcTan[a*
x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - (Pi^3*Log[Tan[(Pi + 2*ArcTa
n[a*x])/4]])/4 - (6*I)*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] - ((3*I)/2)*Pi*(Pi - 4*ArcTan[a*x])*Po
lyLog[2, I/E^(I*ArcTan[a*x])] + (12*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - ((3*I)/2)*Pi^2*PolyLog[2, (-I)*E^(
I*ArcTan[a*x])] + (6*I)*Pi*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]^2*PolyLog[2, (-I
)*E^(I*ArcTan[a*x])] - (12*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x
])] + 6*Pi*PolyLog[3, I/E^(I*ArcTan[a*x])] - 6*Pi*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 12*ArcTan[a*x]*PolyLog[
3, (-I)*E^(I*ArcTan[a*x])] + (12*I)*PolyLog[4, (-I)/E^(I*ArcTan[a*x])] + (12*I)*PolyLog[4, (-I)*E^(I*ArcTan[a*
x])] + ArcTan[a*x]^3/(Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2])^2 - (6*ArcTan[a*x]^2*Sin[ArcTan[a*x]/2])/(Cos[A
rcTan[a*x]/2] - Sin[ArcTan[a*x]/2]) - ArcTan[a*x]^3/(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])^2 + (6*ArcTan[a*
x]^2*Sin[ArcTan[a*x]/2])/(Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2])))/(4*a^3*c*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 3.78, size = 430, normalized size = 0.7 \begin{align*}{\frac{ \left ( \arctan \left ( ax \right ) xa-3 \right ) \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2\,c{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{\frac{i}{2}}}{c{a}^{3}} \left ( i \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,i\arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,i\arctan \left ( ax \right ){\it polylog} \left ( 3,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,i\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,i\arctan \left ( ax \right ){\it polylog} \left ( 3,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,{\it polylog} \left ( 4,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,{\it polylog} \left ( 4,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,{\it dilog} \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,{\it dilog} \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),x)

[Out]

1/2*(arctan(a*x)*x*a-3)*arctan(a*x)^2*(c*(a*x-I)*(a*x+I))^(1/2)/c/a^3-1/2*I*(I*arctan(a*x)^3*ln(1+I*(1+I*a*x)/
(a^2*x^2+1)^(1/2))-I*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/
(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*arctan(a*x)*ln(1+I*(1+I*a*x)/(
a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*ln(1-I*(1+I*a*x)/(
a^2*x^2+1)^(1/2))-6*I*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1
)^(1/2))+6*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*dilog(1-I*(1+I*
a*x)/(a^2*x^2+1)^(1/2)))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^(1/2)/a^3/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),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{x^{2} \arctan \left (a x\right )^{3}}{\sqrt{a^{2} c x^{2} + c}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \operatorname{atan}^{3}{\left (a x \right )}}{\sqrt{c \left (a^{2} x^{2} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^3/(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

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