3.5.25 \(\int \frac {(c+a^2 c x^2)^{3/2} \text {ArcTan}(a x)^3}{x^2} \, dx\) [425]

Optimal. Leaf size=901 \[ -\frac {3}{2} a c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {ArcTan}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \tanh ^{-1}\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {9 i a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 a c^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 i a c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a c^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

[Out]

-3*I*a*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)-9/2*I*a*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)-6*a*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)-6*I*a*c^2*arctan(a*x)*arctan(
(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+3*I*a*c^2*polylog(2,-I*(1+I*a*x)^(1/2)/
(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*I*a*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)+6*I*a*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)+9*I*a*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)-3*I*a*c^2*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*a*
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)-9*a*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)+9*a*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*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)-9*I*a*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)+9/2*I*a*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)-3/2*a*c*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)-c*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/x+1/2*a^2*c*x*arctan
(a*x)^3*(a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.88, antiderivative size = 901, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5070, 5064, 5078, 5076, 4268, 2611, 2320, 6724, 5010, 5008, 4266, 6744, 5000, 5006} \begin {gather*} -\frac {3 i a c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} a^2 c x \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3-\frac {c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3}{x}-\frac {6 a c^2 \sqrt {a^2 x^2+1} \tanh ^{-1}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{\sqrt {a^2 c x^2+c}}+\frac {9 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {9 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {3}{2} a c \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2-\frac {6 i a c^2 \sqrt {a^2 x^2+1} \text {ArcTan}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) \text {ArcTan}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {6 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)}{\sqrt {a^2 c x^2+c}}-\frac {6 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)}{\sqrt {a^2 c x^2+c}}-\frac {9 a c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {9 a c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)}{\sqrt {a^2 c x^2+c}}+\frac {3 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 a c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 a c^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {9 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {9 i a c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \text {ArcTan}(a x)}\right )}{\sqrt {a^2 c x^2+c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 - (c*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/x + (a^2*c*x*Sqrt[c + a^2
*c*x^2]*ArcTan[a*x]^3)/2 - ((3*I)*a*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] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*
x^2] - (6*a*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] + ((6*I)*a*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] + (((9*I)/2)*a*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] - (((9*I)/2)*a*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] - ((6*I)*a*c^2*Sqrt[1 + a^2*x^2]*Arc
Tan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, ((-I)*
Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - ((3*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[2, (I*Sqrt[1 +
I*a*x])/Sqrt[1 - I*a*x]])/Sqrt[c + a^2*c*x^2] - (6*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])])/Sq
rt[c + a^2*c*x^2] - (9*a*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] + (9*a*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*c^2*
Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] - ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[
4, (-I)*E^(I*ArcTan[a*x])])/Sqrt[c + a^2*c*x^2] + ((9*I)*a*c^2*Sqrt[1 + a^2*x^2]*PolyLog[4, I*E^(I*ArcTan[a*x]
)])/Sqrt[c + a^2*c*x^2]

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 5000

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

Rule 5006

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(
ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1
- I*c*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x])]/(c*Sqrt[d])), x]) /; F
reeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 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 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*} \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3}{x^2} \, dx &=c \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x^2} \, dx+\left (a^2 c\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+c^2 \int \frac {\tan ^{-1}(a x)^3}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{2} \left (a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\left (3 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a 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,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a 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,\tan ^{-1}(a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a 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,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a 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,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {3}{2} a c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{x}+\frac {1}{2} a^2 c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {9 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 a c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a c^2 \sqrt {1+a^2 x^2} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 4.40, size = 1387, normalized size = 1.54 \begin {gather*} \frac {a c \sqrt {c+a^2 c x^2} \left (-7 i \pi ^4 \sqrt {1+a^2 x^2}-8 i \pi ^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)-384 i \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)-96 \text {ArcTan}(a x)^2-96 a^2 x^2 \text {ArcTan}(a x)^2+24 i \pi ^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2-\frac {64 \text {ArcTan}(a x)^3}{a x}-32 a x \text {ArcTan}(a x)^3+32 a^3 x^3 \text {ArcTan}(a x)^3-32 i \pi \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^3-64 i \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^3+16 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^4+48 \pi ^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \log \left (1-i e^{-i \text {ArcTan}(a x)}\right )-96 \pi \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \log \left (1-i e^{-i \text {ArcTan}(a x)}\right )-8 \pi ^3 \sqrt {1+a^2 x^2} \log \left (1+i e^{-i \text {ArcTan}(a x)}\right )+64 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^3 \log \left (1+i e^{-i \text {ArcTan}(a x)}\right )+192 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \log \left (1-e^{i \text {ArcTan}(a x)}\right )+8 \pi ^3 \sqrt {1+a^2 x^2} \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-48 \pi ^2 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \log \left (1+i e^{i \text {ArcTan}(a x)}\right )+96 \pi \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-64 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^3 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-192 \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \log \left (1+e^{i \text {ArcTan}(a x)}\right )+8 \pi ^3 \sqrt {1+a^2 x^2} \log \left (2 \sqrt {1+a^2 x^2} \sin ^2\left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+192 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{-i \text {ArcTan}(a x)}\right )+48 i \pi \sqrt {1+a^2 x^2} (\pi -4 \text {ArcTan}(a x)) \text {PolyLog}\left (2,i e^{-i \text {ArcTan}(a x)}\right )+384 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-e^{i \text {ArcTan}(a x)}\right )+192 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+48 i \pi ^2 \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-192 i \pi \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )+288 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-192 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-96 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-384 i \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,e^{i \text {ArcTan}(a x)}\right )+384 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{-i \text {ArcTan}(a x)}\right )-192 \pi \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{-i \text {ArcTan}(a x)}\right )-384 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-e^{i \text {ArcTan}(a x)}\right )+192 \pi \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )-576 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+192 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )+384 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,e^{i \text {ArcTan}(a x)}\right )-384 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,-i e^{-i \text {ArcTan}(a x)}\right )-576 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,-i e^{i \text {ArcTan}(a x)}\right )+192 i \sqrt {1+a^2 x^2} \text {PolyLog}\left (4,i e^{i \text {ArcTan}(a x)}\right )\right )}{64 \left (1+a^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*c*Sqrt[c + a^2*c*x^2]*((-7*I)*Pi^4*Sqrt[1 + a^2*x^2] - (8*I)*Pi^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x] - (384*I)*S
qrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x] - 96*ArcTan[a*x]^2 - 96*a^2*x^2*ArcTan[a*x]^2 + (24*I)*
Pi^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (64*ArcTan[a*x]^3)/(a*x) - 32*a*x*ArcTan[a*x]^3 + 32*a^3*x^3*ArcTan[a*x
]^3 - (32*I)*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3 - (64*I)*Sqrt[1 + a^2*x^2]*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*
x]^3 + (16*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^4 + 48*Pi^2*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*
x])] - 96*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])] - 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[1 + I
/E^(I*ArcTan[a*x])] + 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] + 192*Sqrt[1 + a^2*x^2]*
ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[1 + I*E^(I*ArcTan[a*x])] - 48*Pi^2*Sqr
t[1 + a^2*x^2]*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] + 96*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + I*E^(I
*ArcTan[a*x])] - 64*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])] - 192*Sqrt[1 + a^2*x^2]*ArcTa
n[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*Pi^3*Sqrt[1 + a^2*x^2]*Log[2*Sqrt[1 + a^2*x^2]*Sin[(Pi + 2*ArcTan[a*x]
)/4]^2] + (192*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] + (48*I)*Pi*Sqrt[1 + a^2*
x^2]*(Pi - 4*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] + (384*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -
E^(I*ArcTan[a*x])] + (192*I)*Sqrt[1 + a^2*x^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (48*I)*Pi^2*Sqrt[1 + a^2*x
^2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*Pi*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[
a*x])] + (288*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (192*I)*Sqrt[1 + a^2*x^2
]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (96*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*PolyLog[2, I*E^(I*ArcTan[a*x])] - (
384*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 384*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog
[3, (-I)/E^(I*ArcTan[a*x])] - 192*Pi*Sqrt[1 + a^2*x^2]*PolyLog[3, I/E^(I*ArcTan[a*x])] - 384*Sqrt[1 + a^2*x^2]
*PolyLog[3, -E^(I*ArcTan[a*x])] + 192*Pi*Sqrt[1 + a^2*x^2]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 576*Sqrt[1 + a
^2*x^2]*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 192*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[3, I*E^(I*A
rcTan[a*x])] + 384*Sqrt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] - (384*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)
/E^(I*ArcTan[a*x])] - (576*I)*Sqrt[1 + a^2*x^2]*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] + (192*I)*Sqrt[1 + a^2*x^2]
*PolyLog[4, I*E^(I*ArcTan[a*x])]))/(64*(1 + a^2*x^2))

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Maple [A]
time = 2.92, size = 602, normalized size = 0.67

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{2} \left (\arctan \left (a x \right ) a^{2} x^{2}-3 a x -2 \arctan \left (a x \right )\right )}{2 x}+\frac {3 i a c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-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 )-4 i \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 i \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 \arctan \left (a x \right ) \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (a x \right ) \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+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 )-2 \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \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}}\)	\(602\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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