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

Optimal. Leaf size=436 \[ -\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c}}{12 a^2}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^2}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{12 a^3}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{6 a^3} \]

[Out]

(x*Sqrt[c + a^2*c*x^2])/(12*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTa
n[a*x])/(6*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^2) + (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/4 + ((
I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2]) - (Sqrt[c]*ArcTanh
[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^3) - ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])]
)/(a^3*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x
^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.13451, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321} \[ -\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}-\frac{c \sqrt{a^2 x^2+1} \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{a^2 c x^2+c}}{12 a^2}+\frac{1}{4} x^3 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac{x^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a^2}+\frac{i c \sqrt{a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{12 a^3}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{6 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[c + a^2*c*x^2])/(12*a^2) + (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(12*a^3) - (x^2*Sqrt[c + a^2*c*x^2]*ArcTa
n[a*x])/(6*a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(8*a^2) + (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/4 + ((
I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2)/(a^3*Sqrt[c + a^2*c*x^2]) - (Sqrt[c]*ArcTanh
[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(6*a^3) - ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Arc
Tan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + ((I/4)*c*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])]
)/(a^3*Sqrt[c + a^2*c*x^2]) + (c*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/(4*a^3*Sqrt[c + a^2*c*x
^2]) - (c*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/(4*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 217

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

Rule 206

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

Rule 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 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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin{align*} \int x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx &=c \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{1}{4} (3 c) \int \frac{x^2 \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx-\frac{c \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{2 a^2}-\frac{c \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a}-\frac{1}{2} (a c) \int \frac{x^3 \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}-\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{1}{6} c \int \frac{x^2}{\sqrt{c+a^2 c x^2}} \, dx+\frac{(3 c) \int \frac{\tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{8 a^2}+\frac{c \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{a^2}+\frac{c \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{3 a}+\frac{(3 c) \int \frac{x \tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{4 a}-\frac{\left (c \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)^2}{\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}}{12 a^2}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac{c \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{12 a^2}-\frac{c \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{3 a^2}-\frac{(3 c) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx}{4 a^2}+\frac{c \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{a^2}-\frac{\left (c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \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)^2}{\sqrt{1+a^2 x^2}} \, dx}{8 a^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a^3}-\frac{c \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{12 a^2}-\frac{c \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{3 a^2}-\frac{(3 c) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )}{4 a^2}+\frac{\left (3 c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}-\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (i c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (i c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{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 \log \left (1-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 x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}-\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\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 \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 i c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{\left (c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{\left (c \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}-\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{c \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 \sqrt{c+a^2 c x^2}}-\frac{c \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \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 \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{i \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 \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}\\ &=\frac{x \sqrt{c+a^2 c x^2}}{12 a^2}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac{x^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}+\frac{x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac{1}{4} x^3 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{6 a^3}-\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{i c \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \text{Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}+\frac{c \sqrt{1+a^2 x^2} \text{Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}-\frac{c \sqrt{1+a^2 x^2} \text{Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a^3 \sqrt{c+a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 1.21586, size = 267, normalized size = 0.61 \[ \frac{\sqrt{a^2 c x^2+c} \left (\left (a^2 x^2+1\right )^{3/2} \left (-3 \tan ^{-1}(a x)^2 \left (\sqrt{a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )-7 a x\right )+2 \left (\sqrt{a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )+a x\right )+\tan ^{-1}(a x) \left (6 \sqrt{a^2 x^2+1} \cos \left (3 \tan ^{-1}(a x)\right )+2\right )\right )+8 \left (-3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+3 i \tan ^{-1}(a x) \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )+3 \text{PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )-3 \text{PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )-2 \tanh ^{-1}\left (\frac{a x}{\sqrt{a^2 x^2+1}}\right )+3 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )\right )}{96 a^3 \sqrt{a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[c + a^2*c*x^2]*(8*((3*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 2*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] -
(3*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 3*P
olyLog[3, (-I)*E^(I*ArcTan[a*x])] - 3*PolyLog[3, I*E^(I*ArcTan[a*x])]) + (1 + a^2*x^2)^(3/2)*(ArcTan[a*x]*(2 +
6*Sqrt[1 + a^2*x^2]*Cos[3*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(-7*a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]) + 2
*(a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]))))/(96*a^3*Sqrt[1 + a^2*x^2])

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Maple [A]  time = 0.59, size = 302, normalized size = 0.7 \begin{align*}{\frac{6\, \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{3}{a}^{3}-4\,\arctan \left ( ax \right ){a}^{2}{x}^{2}+3\, \left ( \arctan \left ( ax \right ) \right ) ^{2}xa+2\,ax+2\,\arctan \left ( ax \right ) }{24\,{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{\frac{i}{24}}}{{a}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 3\,i \left ( \arctan \left ( ax \right ) \right ) ^{2}\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}\ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +6\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{-i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -6\,\arctan \left ( ax \right ){\it polylog} \left ( 2,{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +6\,i{\it polylog} \left ( 3,{-i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -6\,i{\it polylog} \left ( 3,{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -8\,\arctan \left ({\frac{1+iax}{\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)^2*(a^2*c*x^2+c)^(1/2),x)

[Out]

1/24/a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(6*arctan(a*x)^2*x^3*a^3-4*arctan(a*x)*a^2*x^2+3*arctan(a*x)^2*x*a+2*a*x+2*
arctan(a*x))-1/24*I*(c*(a*x-I)*(a*x+I))^(1/2)*(3*I*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*arcta
n(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*arctan(
a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog(3,I*(1
+I*a*x)/(a^2*x^2+1)^(1/2))-8*arctan((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)^2*(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 )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arctan(a*x)^2*(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)^2, 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}^{2}{\left (a x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*sqrt(c*(a**2*x**2 + 1))*atan(a*x)**2, 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)^2*(a^2*c*x^2+c)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError