\(\int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx\) [308]

Optimal result

Integrand size = 24, antiderivative size = 436 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \]

[Out]

-1/6*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))*c^(1/2)/a^3+1/4*I*c*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a
*x)^2*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-1/4*I*c*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*
(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+1/4*I*c*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^
2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+1/4*c*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*
c*x^2+c)^(1/2)-1/4*c*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+1/12*x
*(a^2*c*x^2+c)^(1/2)/a^2+1/12*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^3-1/6*x^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a+1/
8*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^2+1/4*x^3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{6 a}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{8 a^2}+\frac {1}{4} x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}+\frac {x \sqrt {a^2 c x^2+c}}{12 a^2}-\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{12 a^3}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a^3} \]

[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 212

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

Rule 223

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 327

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]

Rule 2320

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 2611

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 4266

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 5008

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 5010

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 5050

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 5070

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 5072

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 6724

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*} \text {integral}& = c \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} (3 c) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}-\frac {1}{2} (a c) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} c \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {(3 c) \int \frac {\arctan (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 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {(3 c) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (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} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (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 \text {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 ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (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 {\arctan (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} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}-\frac {c \text {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 \text {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) \text {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 ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (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} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\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} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (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} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\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} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (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 ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (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} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\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} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (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} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\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} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {\sqrt {c+a^2 c x^2} \left (8 \left (3 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-3 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+3 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-3 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+\left (1+a^2 x^2\right )^{3/2} \left (\arctan (a x) \left (2+6 \sqrt {1+a^2 x^2} \cos (3 \arctan (a x))\right )-3 \arctan (a x)^2 \left (-7 a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )+2 \left (a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )\right )\right )}{96 a^3 \sqrt {1+a^2 x^2}} \]

[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])

Maple [A] (verified)

Time = 1.40 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 a^{3} \arctan \left (a x \right )^{2} x^{3}-4 a^{2} \arctan \left (a x \right ) x^{2}+3 a \arctan \left (a x \right )^{2} x +2 a x +2 \arctan \left (a x \right )\right )}{24 a^{3}}-\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-8 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 a^{3} \sqrt {a^{2} x^{2}+1}}\)	\(302\)

[In]

int(x^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/24/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(6*a^3*arctan(a*x)^2*x^3-4*a^2*arctan(a*x)*x^2+3*a*arctan(a*x)^2*x+2*a*x+2*
arctan(a*x))-1/24*I*(c*(a*x-I)*(I+a*x))^(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)

Fricas [F]

\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[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)

Sympy [F]

\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]

[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)

Maxima [F]

\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{2} \,d x } \]

[In]

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

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \]

[In]

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

[Out]

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