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

Optimal. Leaf size=747 \[ -\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{4 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{8 a^2}-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{4 a}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a^2} \]

[Out]

-Sqrt[c + a^2*c*x^2]/(4*a^3) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(4*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^
2)/(8*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(8*a^2) + (
x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/4 + ((I/4)*c*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]) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*
Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*S
qrt[c + a^2*c*x^2]) + (((3*I)/8)*c*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]) - ((I/2)*c*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]) + ((I/2)*c*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*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) -
(3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/4)*c
*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)/4)*c*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 = 1.84627, antiderivative size = 747, normalized size of antiderivative = 1., number of steps used = 40, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4950, 4952, 4930, 4890, 4886, 4888, 4181, 2531, 6609, 2282, 6589, 261} \[ -\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{3 i c \sqrt{a^2 x^2+1} \text{PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c}}{4 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^3}{8 a^2}-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{4 a}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^3}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right ) \tan ^{-1}(a x)}{a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Sqrt[c + a^2*c*x^2]/(4*a^3) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(4*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^
2)/(8*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(4*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(8*a^2) + (
x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/4 + ((I/4)*c*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]) + (I*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/(a^3*
Sqrt[c + a^2*c*x^2]) - (((3*I)/8)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*S
qrt[c + a^2*c*x^2]) + (((3*I)/8)*c*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]) - ((I/2)*c*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]) + ((I/2)*c*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*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) -
(3*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2]) + (((3*I)/4)*c
*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)/4)*c*Sqrt[1 + a^2*x
^2]*PolyLog[4, I*E^(I*ArcTan[a*x])])/(a^3*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 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]

Rule 261

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

Rubi steps

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

Mathematica [B]  time = 12.1331, size = 1844, normalized size = 2.47 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((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)]*(-(ArcTan[a*x]*(L
og[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*(-PolyLo
g[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 - ArcTan[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 + A
rcTan[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[ArcTan[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[Arc
Tan[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]*(Cos[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[ArcTan[a*x]/
2] - Sin[ArcTan[a*x]/2])))/a^3

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Maple [A]  time = 2.878, size = 460, normalized size = 0.6 \begin{align*}{\frac{2\, \left ( \arctan \left ( ax \right ) \right ) ^{3}{a}^{3}{x}^{3}-2\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+ \left ( \arctan \left ( ax \right ) \right ) ^{3}ax+2\,\arctan \left ( ax \right ) xa+ \left ( \arctan \left ( ax \right ) \right ) ^{2}-2}{8\,{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{1}{8\,{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( \left ( \arctan \left ( ax \right ) \right ) ^{3}\ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) - \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\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}{\it polylog} \left ( 2,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +4\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,\arctan \left ( ax \right ){\it polylog} \left ( 3,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -4\,\arctan \left ( ax \right ) \ln \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,\arctan \left ( ax \right ){\it polylog} \left ( 3,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,i{\it polylog} \left ( 4,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -4\,i{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +4\,i{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,i{\it polylog} \left ( 4,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \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]

1/8/a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(2*arctan(a*x)^3*a^3*x^3-2*arctan(a*x)^2*x^2*a^2+arctan(a*x)^3*a*x+2*arctan(
a*x)*x*a+arctan(a*x)^2-2)+1/8*(c*(a*x-I)*(a*x+I))^(1/2)*(arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arc
tan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I
*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*ar
ctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*arctan
(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*I*dilog(1+I*(1+
I*a*x)/(a^2*x^2+1)^(1/2))+4*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/
2)))/a^3/(a^2*x^2+1)^(1/2)

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Maxima [F(-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 (\sqrt{a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{3}, 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(sqrt(a^2*c*x^2 + c)*x^2*arctan(a*x)^3, x)

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

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

Exception raised: TypeError