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

Optimal. Leaf size=638 \[ \frac {43 c^2 x \sqrt {a^2 c x^2+c}}{4032 a^2}-\frac {737 c^2 x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{10080 a}+\frac {5 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{128 a^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {a^2 c x^2+c}+\frac {17}{48} a^2 c^2 x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {83}{840} a c^2 x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {29 c^2 x^3 \sqrt {a^2 c x^2+c}}{1680}+\frac {59}{192} c^2 x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {397 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{5040 a^3}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {5 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {1373 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{20160 a^3}-\frac {1}{28} a^3 c^2 x^6 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) \]

[Out]

-397/5040*c^(5/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^3+5/64*I*c^3*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)+5/64*I*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arct
an(a*x)^2*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-5/64*I*c^3*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)+5/64*c^3*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)-5/64*c^3*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)+43/4032*c^2*x*(a^2*c*x^2+c)^(1/2)/a^2+29/1680*c^2*x^3*(a^2*c*x^2+c)^(1/2)+1/168*a^2*c^2*x^5*(a^2
*c*x^2+c)^(1/2)+1373/20160*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^3-737/10080*c^2*x^2*arctan(a*x)*(a^2*c*x^2+c)
^(1/2)/a-83/840*a*c^2*x^4*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/28*a^3*c^2*x^6*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+5/1
28*c^2*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^2+59/192*c^2*x^3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+17/48*a^2*c^2*
x^5*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+1/8*a^4*c^2*x^7*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 8.35, antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps used = 238, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4950, 4952, 4930, 217, 206, 4890, 4888, 4181, 2531, 2282, 6589, 321} \[ -\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {5 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}-\frac {5 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {1}{168} a^2 c^2 x^5 \sqrt {a^2 c x^2+c}+\frac {29 c^2 x^3 \sqrt {a^2 c x^2+c}}{1680}+\frac {43 c^2 x \sqrt {a^2 c x^2+c}}{4032 a^2}+\frac {1}{8} a^4 c^2 x^7 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {1}{28} a^3 c^2 x^6 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {17}{48} a^2 c^2 x^5 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {83}{840} a c^2 x^4 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {59}{192} c^2 x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {737 c^2 x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{10080 a}+\frac {5 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{128 a^2}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {a^2 c x^2+c}}+\frac {1373 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{20160 a^3}-\frac {397 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{5040 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(43*c^2*x*Sqrt[c + a^2*c*x^2])/(4032*a^2) + (29*c^2*x^3*Sqrt[c + a^2*c*x^2])/1680 + (a^2*c^2*x^5*Sqrt[c + a^2*
c*x^2])/168 + (1373*c^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(20160*a^3) - (737*c^2*x^2*Sqrt[c + a^2*c*x^2]*ArcTan
[a*x])/(10080*a) - (83*a*c^2*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/840 - (a^3*c^2*x^6*Sqrt[c + a^2*c*x^2]*ArcTa
n[a*x])/28 + (5*c^2*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(128*a^2) + (59*c^2*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*
x]^2)/192 + (17*a^2*c^2*x^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/48 + (a^4*c^2*x^7*Sqrt[c + a^2*c*x^2]*ArcTan[a*
x]^2)/8 + (((5*I)/64)*c^3*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])
- (397*c^(5/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(5040*a^3) - (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (((5*I)/64)*c^3*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])])/(a^3*Sqrt[c + a^2*c*x^2]) + (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)
*E^(I*ArcTan[a*x])])/(64*a^3*Sqrt[c + a^2*c*x^2]) - (5*c^3*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan[a*x])])/
(64*a^3*Sqrt[c + a^2*c*x^2])

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 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 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]

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 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 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 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 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 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 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 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 x^2 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+2 \left (\left (a^2 c^2\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\right )+\left (a^4 c^2\right ) \int x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^3 \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^6 c^3\right ) \int \frac {x^8 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2}+\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}-\frac {1}{2} \left (a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\frac {1}{4} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} \left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{6} \left (5 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{8} \left (7 a^4 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{4} \left (a^5 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}+\frac {1}{15} \left (4 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (-\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {3 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}+\frac {1}{15} \left (4 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx\right )+\frac {1}{48} \left (35 a^2 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{14} \left (3 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{24} \left (7 a^3 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{28} \left (a^4 c^3\right ) \int \frac {x^6}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \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 {c^2 x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {7 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{20} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{45} \left (4 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{36} \left (5 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{64} \left (35 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}+\frac {c^3 \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 (8 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{18 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {1}{35} \left (6 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{30} \left (7 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{96} \left (35 a c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{168} \left (5 a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{70} \left (3 a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{120} \left (7 a^2 c^3\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^3 \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^3 \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}}+2 \left (\frac {c^2 x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {13 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{20} c^3 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{45} \left (4 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{36} \left (5 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {\left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (3 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}-\frac {\left (8 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{18 a}-\frac {\left (5 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}+\frac {\left (3 c^3 \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}}\right )\\ &=-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}-\frac {359 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^3 \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 {c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}+\frac {1}{224} \left (5 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{280} \left (9 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{160} \left (7 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{35} \left (2 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{90} \left (7 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{288} \left (35 c^3\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{72 a^2}-\frac {c^3 \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 {\left (8 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (35 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{128 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{18 a^2}-\frac {c^3 \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 {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}-\frac {\left (3 c^3\right ) \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 (4 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{35 a}+\frac {\left (7 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}+\frac {\left (35 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{144 a}+\frac {\left (35 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{64 a}+\frac {\left (3 c^3 \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^3 \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^3 \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}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{72 a^2}-\frac {c^3 \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 {\left (8 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{18 a^2}-\frac {c^3 \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 {\left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}-\frac {\left (3 c^3\right ) \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^3 \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 (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^3 \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 {c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c^3 \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^3 \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 (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{448 a^2}-\frac {\left (9 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{560 a^2}-\frac {\left (7 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{320 a^2}+\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^2}-\frac {c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (7 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{180 a^2}+\frac {\left (2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}-\frac {\left (35 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{576 a^2}+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{72 a^2}-\frac {\left (4 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{35 a^2}-\frac {\left (7 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (8 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}-\frac {\left (35 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{144 a^2}+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{18 a^2}-\frac {\left (35 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{64 a^2}+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{8 a^2}+\frac {\left (i c^3 \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^3 \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 (5 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \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^3 \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}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 i c^3 \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 {7 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^2}+\frac {\left (2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{72 a^2}+\frac {\left (8 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{18 a^2}+\frac {\left (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{8 a^2}-\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \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^3 \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}}\right )+\frac {\left (35 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{128 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {379 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {i c^3 \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^3 \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 (5 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{448 a^2}-\frac {\left (9 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{560 a^2}-\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{320 a^2}-\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{35 a^2}-\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{180 a^2}-\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{576 a^2}-\frac {\left (4 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{35 a^2}-\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{45 a^2}-\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{144 a^2}-\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{64 a^2}-\frac {\left (3 i c^3 \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^3 \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 (35 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{128 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {3 i c^3 \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 {3 i c^3 \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^3 \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^3 \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 (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c^3 \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^3 \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 {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {7 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \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^3 \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 (5 i c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 i c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (35 c^3 \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 )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3 \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 )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \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^3 \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}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \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^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 i c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 i c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^3 \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^3 \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}}\right )\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \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^3 \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 {\left (35 i c^3 \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 )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 i c^3 \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 )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \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^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^3 \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 {3 c^3 \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 {\left (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (5 c^3 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {7 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \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^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )-\frac {\left (35 c^3 \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 )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (35 c^3 \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 )}{64 a^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {491 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}-\frac {9}{560} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1261 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{20160 a^3}-\frac {1969 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{10080 a}+\frac {29}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {21 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{128 a^2}+\frac {43}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {929 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {21 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {21 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+2 \left (-\frac {c^2 x \sqrt {c+a^2 c x^2}}{18 a^2}+\frac {1}{60} c^2 x^3 \sqrt {c+a^2 c x^2}+\frac {c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {11 c^2 x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c^2 x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}+\frac {1}{24} c^2 x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {19 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}+\frac {i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \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^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )\\ \end {align*}

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Mathematica [A]  time = 4.84, size = 759, normalized size = 1.19 \[ \frac {c^2 \sqrt {a^2 c x^2+c} \left (7006 a x \left (a^2 x^2+1\right )^{7/2}-25088 a x \left (a^2 x^2+1\right )^{5/2}+53760 a x \left (a^2 x^2+1\right )^{3/2}+185325 a x \left (a^2 x^2+1\right )^{7/2} \tan ^{-1}(a x)^2-38134 \left (a^2 x^2+1\right )^{7/2} \tan ^{-1}(a x)+524160 a x \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x)^2+5376 \left (a^2 x^2+1\right )^{5/2} \tan ^{-1}(a x)+564480 a x \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2+53760 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)-203264 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-93975 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+12246 \left (a^2 x^2+1\right )^4 \sin \left (3 \tan ^{-1}(a x)\right )+41685 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )+7678 \left (a^2 x^2+1\right )^4 \sin \left (5 \tan ^{-1}(a x)\right )-1575 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^2 \sin \left (7 \tan ^{-1}(a x)\right )+2438 \left (a^2 x^2+1\right )^4 \sin \left (7 \tan ^{-1}(a x)\right )-315840 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-48384 \left (a^2 x^2+1\right )^3 \sin \left (3 \tan ^{-1}(a x)\right )+20160 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )-23296 \left (a^2 x^2+1\right )^3 \sin \left (5 \tan ^{-1}(a x)\right )-80640 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+53760 \left (a^2 x^2+1\right )^2 \sin \left (3 \tan ^{-1}(a x)\right )-7658 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-10990 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )+3150 \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x) \cos \left (7 \tan ^{-1}(a x)\right )+49280 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-40320 \left (a^2 x^2+1\right )^3 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )+161280 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-201600 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+201600 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+201600 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-201600 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )+201600 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2\right )}{2580480 a^3 \sqrt {a^2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c + a^2*c*x^2]*(53760*a*x*(1 + a^2*x^2)^(3/2) - 25088*a*x*(1 + a^2*x^2)^(5/2) + 7006*a*x*(1 + a^2*x^
2)^(7/2) + 53760*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] + 5376*(1 + a^2*x^2)^(5/2)*ArcTan[a*x] - 38134*(1 + a^2*x^2)^
(7/2)*ArcTan[a*x] + 564480*a*x*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2 + 524160*a*x*(1 + a^2*x^2)^(5/2)*ArcTan[a*x]^
2 + 185325*a*x*(1 + a^2*x^2)^(7/2)*ArcTan[a*x]^2 + (201600*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - 203264
*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + 161280*(1 + a^2*x^2)^2*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 49280*(1 + a^2*x^2
)^3*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 7658*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 40320*(1 + a^2*x^2)
^3*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 10990*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[5*ArcTan[a*x]] + 3150*(1 + a^2*x^2)^
4*ArcTan[a*x]*Cos[7*ArcTan[a*x]] - (201600*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (201600*I)*ArcT
an[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 201600*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 201600*PolyLog[3, I*E^(I
*ArcTan[a*x])] + 53760*(1 + a^2*x^2)^2*Sin[3*ArcTan[a*x]] - 48384*(1 + a^2*x^2)^3*Sin[3*ArcTan[a*x]] + 12246*(
1 + a^2*x^2)^4*Sin[3*ArcTan[a*x]] - 80640*(1 + a^2*x^2)^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 315840*(1 + a^2*x
^2)^3*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 93975*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 23296*(1 + a
^2*x^2)^3*Sin[5*ArcTan[a*x]] + 7678*(1 + a^2*x^2)^4*Sin[5*ArcTan[a*x]] + 20160*(1 + a^2*x^2)^3*ArcTan[a*x]^2*S
in[5*ArcTan[a*x]] + 41685*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 2438*(1 + a^2*x^2)^4*Sin[7*ArcTan
[a*x]] - 1575*(1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[7*ArcTan[a*x]]))/(2580480*a^3*Sqrt[1 + a^2*x^2])

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fricas [F]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 1.76, size = 376, normalized size = 0.59 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (5040 \arctan \left (a x \right )^{2} x^{7} a^{7}-1440 \arctan \left (a x \right ) x^{6} a^{6}+14280 \arctan \left (a x \right )^{2} x^{5} a^{5}+240 x^{5} a^{5}-3984 \arctan \left (a x \right ) x^{4} a^{4}+12390 \arctan \left (a x \right )^{2} x^{3} a^{3}+696 a^{3} x^{3}-2948 \arctan \left (a x \right ) a^{2} x^{2}+1575 \arctan \left (a x \right )^{2} x a +430 a x +2746 \arctan \left (a x \right )\right )}{40320 a^{3}}+\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1575 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-1575 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6352 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40320 a^{3} \sqrt {a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40320*c^2/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(5040*arctan(a*x)^2*x^7*a^7-1440*arctan(a*x)*x^6*a^6+14280*arctan(a*
x)^2*x^5*a^5+240*x^5*a^5-3984*arctan(a*x)*x^4*a^4+12390*arctan(a*x)^2*x^3*a^3+696*a^3*x^3-2948*arctan(a*x)*a^2
*x^2+1575*arctan(a*x)^2*x*a+430*a*x+2746*arctan(a*x))+1/40320*I*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(1575*I*arctan(a
*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1575*I*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3150*arctan
(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3150*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*arctan(a*x
)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6352*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]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{2} \arctan \left (a x\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*atan(a*x)^2*(c + a^2*c*x^2)^(5/2),x)

[Out]

int(x^2*atan(a*x)^2*(c + a^2*c*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2, x)

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