3.4.16 \(\int x^2 (c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^2 \, dx\) [316]

Optimal. Leaf size=531 \[ \frac {c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}+\frac {31 c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{360 a^3}-\frac {19 c x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)+\frac {c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2+\frac {i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {41 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}} \]

[Out]

-41/360*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^3+1/8*I*c^2*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arc
tan(a*x)^2*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)-1/8*I*c^2*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)+1/8*I*c^2*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)+1/8*c^2*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)-1/8*c^2*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)+1/36*c*x*(a^2*c*x^2+c)^(1/2)/a^2+1/60*c*x^3*(a^2*c*x^2+c)^(1/2)+31/360*c*arctan(a*x)*(a^2*c*x^2+c)^(1/2
)/a^3-19/180*c*x^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a-1/15*a*c*x^4*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1/16*c*x*arc
tan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^2+7/24*c*x^3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)+1/6*a^2*c*x^5*arctan(a*x)^2*(a
^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 2.22, antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 92, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \begin {gather*} -\frac {19 c x^2 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{180 a}+\frac {c x \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}{16 a^2}+\frac {1}{6} a^2 c x^5 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}-\frac {1}{15} a c x^4 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}+\frac {7}{24} c x^3 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}+\frac {c x \sqrt {a^2 c x^2+c}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {a^2 c x^2+c}-\frac {i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {31 c \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}{360 a^3}-\frac {41 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{360 a^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*Sqrt[c + a^2*c*x^2])/(36*a^2) + (c*x^3*Sqrt[c + a^2*c*x^2])/60 + (31*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(
360*a^3) - (19*c*x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(180*a) - (a*c*x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/15 +
(c*x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(16*a^2) + (7*c*x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/24 + (a^2*c*x^
5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/6 + ((I/8)*c^2*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]) - (41*c^(3/2)*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(360*a^3) - ((I/8)*c^2*Sq
rt[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]) + ((I/8)*c^2*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^2*Sqrt[1 + a^2*x^2]*Pol
yLog[3, (-I)*E^(I*ArcTan[a*x])])/(8*a^3*Sqrt[c + a^2*c*x^2]) - (c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, I*E^(I*ArcTan
[a*x])])/(8*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*} \int x^2 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=c^2 \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2}+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}+2 \left (\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{4} \left (3 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{6} \left (5 a^2 c^2\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^2\right ) \int \frac {x^5 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{8} \left (5 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+2 \left (-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^2 \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}\right )+\frac {1}{15} \left (4 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^2\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^2 \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 {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {1}{20} c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{45} \left (4 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{36} \left (5 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (5 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}+\frac {c^2 \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 (8 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}-\frac {\left (5 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{18 a}-\frac {\left (5 c^2\right ) \int \frac {x \tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}+\frac {\left (3 c^2 \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 {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^2 \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^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{72 a^2}+\frac {\left (8 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{18 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c^2 \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^2 \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 {\left (3 c^2\right ) \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^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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 (5 c^2 \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 {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {i c^2 \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^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}-\frac {i c^2 \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^2 \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 {c^2 \text {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^2\right ) \text {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^2\right ) \text {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^2\right ) \text {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^2\right ) \text {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^2\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \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^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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 {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 i c^2 \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 {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {i c^2 \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^2 \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}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \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^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \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^2 \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^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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}}\right )+\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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 {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 i c^2 \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 {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \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 {13 i c^2 \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^2 \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^2 \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^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \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^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \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^2 \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 c^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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 {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 i c^2 \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 {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \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 {13 i c^2 \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^2 \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^2 \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}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \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^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \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^2 \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 c^2 \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^2 \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}}\right )+\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \text {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^2 \sqrt {1+a^2 x^2}\right ) \text {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 {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {13 i c^2 \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 {799 c^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \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 {13 i c^2 \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 {13 c^2 \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 {13 c^2 \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 x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {3 i c^2 \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^{3/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \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^2 \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 c^2 \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^2 \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}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1101\) vs. \(2(531)=1062\).
time = 2.28, size = 1101, normalized size = 2.07 \begin {gather*} \frac {c \sqrt {c+a^2 c x^2} \left (184 a x \sqrt {1+a^2 x^2}+128 a^3 x^3 \sqrt {1+a^2 x^2}-56 a^5 x^5 \sqrt {1+a^2 x^2}+252 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)+264 a^2 x^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)+12 a^4 x^4 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)+3690 a x \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+4860 a^3 x^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+1170 a^5 x^5 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2+830 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+1770 a^2 x^2 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+1050 a^4 x^4 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))+110 a^6 x^6 \text {ArcTan}(a x) \cos (3 \text {ArcTan}(a x))-90 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-270 a^2 x^2 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-270 a^4 x^4 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-90 a^6 x^6 \text {ArcTan}(a x) \cos (5 \text {ArcTan}(a x))-720 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )-720 \pi \text {ArcTan}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (1-i e^{i \text {ArcTan}(a x)}\right )\right )+720 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+720 \text {ArcTan}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-720 \pi \text {ArcTan}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )-720 \text {ArcTan}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcTan}(a x)}\right )\right )+720 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+1312 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-720 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-1312 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+720 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+720 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-1440 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+1440 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )+1440 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )-1440 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )+132 \sin (3 \text {ArcTan}(a x))+156 a^2 x^2 \sin (3 \text {ArcTan}(a x))-84 a^4 x^4 \sin (3 \text {ArcTan}(a x))-108 a^6 x^6 \sin (3 \text {ArcTan}(a x))-1065 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-2835 a^2 x^2 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-2475 a^4 x^4 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-705 a^6 x^6 \text {ArcTan}(a x)^2 \sin (3 \text {ArcTan}(a x))-52 \sin (5 \text {ArcTan}(a x))-156 a^2 x^2 \sin (5 \text {ArcTan}(a x))-156 a^4 x^4 \sin (5 \text {ArcTan}(a x))-52 a^6 x^6 \sin (5 \text {ArcTan}(a x))+45 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+135 a^2 x^2 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+135 a^4 x^4 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))+45 a^6 x^6 \text {ArcTan}(a x)^2 \sin (5 \text {ArcTan}(a x))\right )}{11520 a^3 \sqrt {1+a^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c*Sqrt[c + a^2*c*x^2]*(184*a*x*Sqrt[1 + a^2*x^2] + 128*a^3*x^3*Sqrt[1 + a^2*x^2] - 56*a^5*x^5*Sqrt[1 + a^2*x^
2] + 252*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 264*a^2*x^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 12*a^4*x^4*Sqrt[1 + a^2*x
^2]*ArcTan[a*x] + 3690*a*x*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 4860*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 11
70*a^5*x^5*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 830*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 1770*a^2*x^2*ArcTan[a*x]*Cos
[3*ArcTan[a*x]] + 1050*a^4*x^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 110*a^6*x^6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 9
0*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 270*a^2*x^2*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 270*a^4*x^4*ArcTan[a*x]*Cos[5*
ArcTan[a*x]] - 90*a^6*x^6*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 720*ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] - 72
0*Pi*ArcTan[a*x]*Log[((-1)^(1/4)*(1 - I*E^(I*ArcTan[a*x])))/(2*E^((I/2)*ArcTan[a*x]))] + 720*ArcTan[a*x]^2*Log
[1 + I*E^(I*ArcTan[a*x])] + 720*ArcTan[a*x]^2*Log[((1/2 + I/2)*(-I + E^(I*ArcTan[a*x])))/E^((I/2)*ArcTan[a*x])
] - 720*Pi*ArcTan[a*x]*Log[-1/2*((-1)^(1/4)*(-I + E^(I*ArcTan[a*x])))/E^((I/2)*ArcTan[a*x])] - 720*ArcTan[a*x]
^2*Log[((1 + I) + (1 - I)*E^(I*ArcTan[a*x]))/(2*E^((I/2)*ArcTan[a*x]))] + 720*Pi*ArcTan[a*x]*Log[-Cos[(Pi + 2*
ArcTan[a*x])/4]] + 1312*Log[Cos[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 720*ArcTan[a*x]^2*Log[Cos[ArcTan[a*x]/2
] - Sin[ArcTan[a*x]/2]] - 1312*Log[Cos[ArcTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + 720*ArcTan[a*x]^2*Log[Cos[ArcTan
[a*x]/2] + Sin[ArcTan[a*x]/2]] + 720*Pi*ArcTan[a*x]*Log[Sin[(Pi + 2*ArcTan[a*x])/4]] - (1440*I)*ArcTan[a*x]*Po
lyLog[2, (-I)*E^(I*ArcTan[a*x])] + (1440*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 1440*PolyLog[3, (-I)
*E^(I*ArcTan[a*x])] - 1440*PolyLog[3, I*E^(I*ArcTan[a*x])] + 132*Sin[3*ArcTan[a*x]] + 156*a^2*x^2*Sin[3*ArcTan
[a*x]] - 84*a^4*x^4*Sin[3*ArcTan[a*x]] - 108*a^6*x^6*Sin[3*ArcTan[a*x]] - 1065*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]
] - 2835*a^2*x^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 2475*a^4*x^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 705*a^6*x^
6*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 52*Sin[5*ArcTan[a*x]] - 156*a^2*x^2*Sin[5*ArcTan[a*x]] - 156*a^4*x^4*Sin[
5*ArcTan[a*x]] - 52*a^6*x^6*Sin[5*ArcTan[a*x]] + 45*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 135*a^2*x^2*ArcTan[a*x]
^2*Sin[5*ArcTan[a*x]] + 135*a^4*x^4*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 45*a^6*x^6*ArcTan[a*x]^2*Sin[5*ArcTan[a
*x]]))/(11520*a^3*Sqrt[1 + a^2*x^2])

________________________________________________________________________________________

Maple [A]
time = 0.70, size = 338, normalized size = 0.64

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 \arctan \left (a x \right )^{2} a^{5} x^{5}-48 \arctan \left (a x \right ) a^{4} x^{4}+210 \arctan \left (a x \right )^{2} a^{3} x^{3}+12 a^{3} x^{3}-76 \arctan \left (a x \right ) a^{2} x^{2}+45 \arctan \left (a x \right )^{2} a x +20 a x +62 \arctan \left (a x \right )\right )}{720 a^{3}}-\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (45 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-45 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-164 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{720 a^{3} \sqrt {a^{2} x^{2}+1}}\)	\(338\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/720*c/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(120*arctan(a*x)^2*a^5*x^5-48*arctan(a*x)*a^4*x^4+210*arctan(a*x)^2*a^3*
x^3+12*a^3*x^3-76*arctan(a*x)*a^2*x^2+45*arctan(a*x)^2*a*x+20*a*x+62*arctan(a*x))-1/720*I*c*(c*(a*x-I)*(I+a*x)
)^(1/2)*(45*I*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-45*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1
)^(1/2))+90*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-90*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+90*arcta
n(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-90*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-164*a
rctan((1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2+1)^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________