\(\int \frac {(c+a^2 c x^2)^{3/2} \arctan (a x)^3}{x^4} \, dx\) [427]

Optimal result

Integrand size = 24, antiderivative size = 788 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

-1/3*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^3-a^3*c^(3/2)*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))+3*I*a^3*c^2*arctan
(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-7*a^3*c^2*arctan(a*x)^
2*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-7*I*a^3*c^2*arctan(a*x)*polylog(2
,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+7*I*a^3*c^2*arctan(a*x)*polylog(2,-(1+I*a*
x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-3*I*a^3*c^2*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a
^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-2*I*a^3*c^2*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(
a*x)^3*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-7*a^3*c^2*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/
2)/(a^2*c*x^2+c)^(1/2)-6*a^3*c^2*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*
c*x^2+c)^(1/2)+6*a^3*c^2*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^
(1/2)+7*a^3*c^2*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*I*a^3*c^2*polyl
og(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+6*I*a^3*c^2*polylog(4,I*(1+I*a*x)/(
a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-a^2*c*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/x-1/2*a*c*arctan
(a*x)^2*(a^2*c*x^2+c)^(1/2)/x^2-a^2*c*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5070, 5064, 5082, 272, 65, 214, 5078, 5076, 4268, 2611, 2320, 6724, 5010, 5008, 4266, 6744} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{x}-\frac {a^2 c \arctan (a x) \sqrt {a^2 c x^2+c}}{x}-\frac {a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 x^2}-\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 x^3}-\frac {7 a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {7 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {7 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {7 a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {7 a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 i a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 i a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right ) \]

[In]

Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^4,x]

[Out]

-((a^2*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x) - (a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(2*x^2) - (a^2*c*Sqrt[c
+ a^2*c*x^2]*ArcTan[a*x]^3)/x - ((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/(3*x^3) - ((2*I)*a^3*c^2*Sqrt[1 + a^2*x
^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3)/Sqrt[c + a^2*c*x^2] - (7*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*
ArcTanh[E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - a^3*c^(3/2)*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]] + ((7*I)*a^
3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a^3*c^2*Sqrt[
1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((3*I)*a^3*c^2*Sqrt[1 + a
^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((7*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*
ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (7*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I
*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - (6*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x
])])/Sqrt[c + a^2*c*x^2] + (6*a^3*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])])/Sqrt[c +
a^2*c*x^2] + (7*a^3*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((6*I)*a^3*c^2*
Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((6*I)*a^3*c^2*Sqrt[1 + a^2*x^2]*P
olyLog[4, I*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 272

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

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 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && 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 5064

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Dist[b*c*(p/(f*(m + 1))), Int[(
f*x)^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e,
c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -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 5076

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[1/Sqrt[d], Sub
st[Int[(a + b*x)^p*Csc[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 5078

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 + c^2*
x^2]/Sqrt[d + e*x^2], Int[(a + b*ArcTan[c*x])^p/(x*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 5082

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] + (-Dist[b*c*(p/(f*(m + 1))), Int[(f*x
)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Dist[c^2*((m + 2)/(f^2*(m + 1))), 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] && G
tQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]

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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = c \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^4} \, dx+\left (a^2 c\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx \\ & = -\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}+(a c) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^3} \, dx+\left (a^2 c^2\right ) \int \frac {\arctan (a x)^3}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^2\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}+\left (a c^2\right ) \int \frac {\arctan (a x)^2}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\left (a^3 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\left (3 a^3 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^4 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}+\left (a^2 c^2\right ) \int \frac {\arctan (a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^3 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}+\left (a^3 c^2\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {8 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {1}{2} \left (a^3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {8 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {8 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+(a c) \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )-\frac {\left (2 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {8 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {8 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 9.61 (sec) , antiderivative size = 1508, normalized size of antiderivative = 1.91 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\frac {a^3 c \sqrt {c \left (1+a^2 x^2\right )} \csc \left (\frac {1}{2} \arctan (a x)\right ) \left (-\frac {7 i a \pi ^4 x}{\sqrt {1+a^2 x^2}}-\frac {8 i a \pi ^3 x \arctan (a x)}{\sqrt {1+a^2 x^2}}+\frac {24 i a \pi ^2 x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}-64 \arctan (a x)^3-\frac {32 i a \pi x \arctan (a x)^3}{\sqrt {1+a^2 x^2}}+\frac {16 i a x \arctan (a x)^4}{\sqrt {1+a^2 x^2}}+\frac {48 a \pi ^2 x \arctan (a x) \log \left (1-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {96 a \pi x \arctan (a x)^2 \log \left (1-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {8 a \pi ^3 x \log \left (1+i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {64 a x \arctan (a x)^3 \log \left (1+i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {192 a x \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {8 a \pi ^3 x \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {48 a \pi ^2 x \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {96 a \pi x \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {64 a x \arctan (a x)^3 \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {192 a x \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {8 a \pi ^3 x \log \left (\tan \left (\frac {1}{4} (\pi +2 \arctan (a x))\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {192 i a x \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {48 i a \pi x (\pi -4 \arctan (a x)) \operatorname {PolyLog}\left (2,i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {384 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {48 i a \pi ^2 x \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {192 i a \pi x \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {192 i a x \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {384 a x \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {192 a \pi x \operatorname {PolyLog}\left (3,i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 a x \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {192 a \pi x \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 a x \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {384 a x \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 i a x \operatorname {PolyLog}\left (4,-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 i a x \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )}{128 \sqrt {1+a^2 x^2}}+\frac {a^3 c^2 \sqrt {1+a^2 x^2} \left (-12 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right )-2 \arctan (a x)^3 \cot \left (\frac {1}{2} \arctan (a x)\right )-3 \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\frac {a x \arctan (a x)^3 \csc ^4\left (\frac {1}{2} \arctan (a x)\right )}{2 \sqrt {1+a^2 x^2}}+12 \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )-12 \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+24 \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-24 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-24 \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+24 \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )+3 \arctan (a x)^2 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-\frac {8 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^3 \sin ^4\left (\frac {1}{2} \arctan (a x)\right )}{a^3 x^3}-12 \arctan (a x) \tan \left (\frac {1}{2} \arctan (a x)\right )-2 \arctan (a x)^3 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{24 \sqrt {c \left (1+a^2 x^2\right )}} \]

[In]

Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^4,x]

[Out]

(a^3*c*Sqrt[c*(1 + a^2*x^2)]*Csc[ArcTan[a*x]/2]*(((-7*I)*a*Pi^4*x)/Sqrt[1 + a^2*x^2] - ((8*I)*a*Pi^3*x*ArcTan[
a*x])/Sqrt[1 + a^2*x^2] + ((24*I)*a*Pi^2*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 64*ArcTan[a*x]^3 - ((32*I)*a*Pi*
x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + ((16*I)*a*x*ArcTan[a*x]^4)/Sqrt[1 + a^2*x^2] + (48*a*Pi^2*x*ArcTan[a*x]*L
og[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])])/Sqrt[1
+ a^2*x^2] - (8*a*Pi^3*x*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (64*a*x*ArcTan[a*x]^3*Log[1 + I/E^
(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (
8*a*Pi^3*x*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*a*Pi^2*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*
x])])/Sqrt[1 + a^2*x^2] + (96*a*Pi*x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (64*a*x*A
rcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*x*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x]
)])/Sqrt[1 + a^2*x^2] + (8*a*Pi^3*x*Log[Tan[(Pi + 2*ArcTan[a*x])/4]])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[
a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((48*I)*a*Pi*x*(Pi - 4*ArcTan[a*x])*PolyLog[2,
I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^
2*x^2] + ((48*I)*a*Pi^2*x*PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((192*I)*a*Pi*x*ArcTan[a*x]*
PolyLog[2, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ((192*I)*a*x*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan
[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (384*
a*x*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*Pi*x*PolyLog[3, I/E^(I*ArcTan[a
*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*PolyLog[3, -E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (192*a*Pi*x*PolyLog[3,
(-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (384*a*x*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1
+ a^2*x^2] + (384*a*x*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)/E^(I*Arc
Tan[a*x])])/Sqrt[1 + a^2*x^2] - ((384*I)*a*x*PolyLog[4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2])*Sec[ArcTan
[a*x]/2])/(128*Sqrt[1 + a^2*x^2]) + (a^3*c^2*Sqrt[1 + a^2*x^2]*(-12*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - 2*ArcTan[
a*x]^3*Cot[ArcTan[a*x]/2] - 3*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - (a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^4)/(2
*Sqrt[1 + a^2*x^2]) + 12*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] - 12*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])
] + 24*Log[Tan[ArcTan[a*x]/2]] + (24*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (24*I)*ArcTan[a*x]*PolyLo
g[2, E^(I*ArcTan[a*x])] - 24*PolyLog[3, -E^(I*ArcTan[a*x])] + 24*PolyLog[3, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x]
^2*Sec[ArcTan[a*x]/2]^2 - (8*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^3*Sin[ArcTan[a*x]/2]^4)/(a^3*x^3) - 12*ArcTan[a*x
]*Tan[ArcTan[a*x]/2] - 2*ArcTan[a*x]^3*Tan[ArcTan[a*x]/2]))/(24*Sqrt[c*(1 + a^2*x^2)])

Maple [A] (verified)

Time = 4.73 (sec) , antiderivative size = 557, normalized size of antiderivative = 0.71

method	result	size
default	\(-\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (8 x^{2} \arctan \left (a x \right )^{2} a^{2}+6 a^{2} x^{2}+3 x \arctan \left (a x \right ) a +2 \arctan \left (a x \right )^{2}\right )}{6 x^{3}}-\frac {i c \,a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-12 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-7 i \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-2 i \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+6 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+12 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-14 i \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-14 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+14 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+7 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+14 i \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-12 \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+12 \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\)	\(557\)

[In]

int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/6*c*(c*(a*x-I)*(I+a*x))^(1/2)*arctan(a*x)*(8*x^2*arctan(a*x)^2*a^2+6*a^2*x^2+3*x*arctan(a*x)*a+2*arctan(a*x
)^2)/x^3-1/2*I*c*a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(2*I*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*I*arc
tan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-7*I*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-2*I*ln(
(1+I*a*x)/(a^2*x^2+1)^(1/2)+1)+6*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)^2*polylo
g(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+12*I*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-14*I*polylog(3,-
(1+I*a*x)/(a^2*x^2+1)^(1/2))-14*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+14*arctan(a*x)*polylog(2,(
1+I*a*x)/(a^2*x^2+1)^(1/2))+7*I*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*I*ln((1+I*a*x)/(a^2*x^2+1)^(
1/2)-1)+14*I*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*arctan(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-12*p
olylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+12*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)

Fricas [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^4,x, algorithm="fricas")

[Out]

integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x^4, x)

Sympy [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x^{4}}\, dx \]

[In]

integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**3/x**4,x)

[Out]

Integral((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)**3/x**4, x)

Maxima [F]

\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^4,x, algorithm="maxima")

[Out]

integrate((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x^4, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x^4} \,d x \]

[In]

int((atan(a*x)^3*(c + a^2*c*x^2)^(3/2))/x^4,x)

[Out]

int((atan(a*x)^3*(c + a^2*c*x^2)^(3/2))/x^4, x)